Combinatorial Solution to the Problem of Optimal Routing in
Progressive “tradings V. E. BENES 865

Integral Equation for Simultaneous Diagonalization of Two
Covariance Kernels T.T. KADOTA 883

Principles of Design of Magnetic Devices for Attitude Control
of Satellites M.S. GLASS 893

A High-Capacity Digital Light Deflector Using Wollaston
Prisms W. J. TABOR 957

Transistor Distortion Analysis Using Volterra Series Repre-
sentation S. NARAYANAN 991

B.S.T.J. Briefs: Realizability Conditions for the Impedance
Function of the Lossless Tapered Transmission Line—A Critique
E. N. PROTONOTARIOS 1047

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A simple analysis shows that the unit-cube conductance is a figure of
merit in semiconductor device design theory. The unit-cube conductance,
G, ws gwen by 2Kv, where K is the permittivity of the semiconductor and
vq 18 the limiting drift velocity.

The space-charge resistance, h,, , due to carrier generated under ava-
lanche condition is derived for p-n junctions. It 1s found that for parallel-
plane structure, R,. = 1/GN, where N is the number of unit cubes in the
depletion region with cube edge equal to the depletion width or N = A/W*
where W is the depletion width and A the junction area. The disturbance
in voltage caused by the space-charge effect is given by I/GN = JW’*/G
where I and J are the current and current density, respectively. Similar
results are obtained for p-n junctions with coaxial-cylinder and concentric-
sphere structures.

For silicon, the value of G is approximately 40 umhos. The transconduct-
ance of a silicon surface-controlled avalanche transistor in terms of the
unit-cube expression 1s about 12.5 N wmhos.

A simple analysis of ‘“‘avalanche resistance’ can be given for the
limiting case in which carriers are generated at one boundary surface
of the depletion region of a p-n junction and travel across the depletion
region with a limiting drift velocity v,. Structures satisfying these
conditions can be of the n° pp” form. It will be shown that the quantity

2Kv,z (where K is the permittivity of the semiconductor) is a figure of
merit in semiconductor device design theory which limits the perform-
ance of space-charge-limited devices. This quantity is a combination
of ‘material constants” similar to F',v, (where F’; is ‘‘breakdown field’’)
which limits the frequency-power performance of transistors’’’ and
K/c (o is the conductivity) which limits the gain-bandwidth product®
of solid-state devices. |

Tor a structure in which the space-charge layer is bounded by parallel
planes of area A and spacing W, it will be shown that the effective space-
charge resistance can be interpreted as due to N unit-cube conductances
in parallel, where the unit-cube conductance G is given by

and N is number of unit cubes in the depletion layer with cube edge
equal to the depletion width, or

For coaxial-cylinder and concentric-sphere structures, similar results
are obtained for the R,,. The number of unit cubes (or curvilinear
cubes), however, depends on the radius of the surface upon which ava-
lanche occurs and the length of the cylinder (for the coaxial-cylinder
structure). These functional dependences are derived below.

An interesting application of the space-charge resistance and the
unit-cube expression is given for a surface-controlled avalanche tran-
sistor (SCAT).°

As represented in Fig. 1(a) the depletion layer of an n*pp” structure
extends through the p layer with a doping of NV,, and is bounded by
the planes at x = 0 and x = W. When the applied voltage V is equal
to the breakdown voltage V, the electric field E(x) has its maximum
absolute value fF’, at « = 0 and decreases to Fz — (qN.W/K) atx = W.
This insures breakdown at x = 0. Furthermore, if gV,W/K < 0.9 Fz,
then the field is everywhere = F',,/10, so that holes have their limiting
drift velocity vz all across W.

Fig. 1— p-n junction geometry of (a) parallel plane, (b) coaxial cylinder,
and (ce) concentric sphere structures.

where p is the carrier-charge density and A the area. Since &# at z = 0
is assumed to be equal to F,, the disturbance A(x) in the electric
field due to p is

so that the disturbance in voltage caused by the carriers (1.e., the average
field times W) is obtained by integrating A(z)

Consider first that the maximum field occurs at the mner surface.
As shown in Fig. 1(b) the depletion layer extends through the intrinsic
region of an nip” coaxial-cylinder structure and is bounded by the
cylinders of radii r = a and r = b. When V = V,, the electric field
E(r) has its maximum absolute value F', at r = a, and decreases to
Fza/b atr = b.

so that p varies as 1/r. Integrating Poisson’s equation leads to a dis-
turbance AK(r) in the electric field and AV, in the voltage due to p
given by

A, is the area on the outer cylinder surface that corresponds to one
unit-cube conductance G.

Equation (10) may be interpreted as the resistance of N unit curvilinear
cubes in parallel. These cubes are formed by intersection of equipotential
surfaces with the orthogonal family of electric field lines. Each cube has
a conductance (2Kv,), and the number of cubes N is given by (11).
The area A, approaches (b — a)” when a — b, and approaches 2(b — a)’
when a — 0, and consequently remains finite even as the inner cylinder
approaches a line.

The maximum field may be caused to occur on the outer surface
r = b by adjusting the chemical charges in the depletion layer appro-
priately, such as a p pn” structure with the pn” junction at r = 8,
the p*p boundary at r = a. In this case, the area A, approaches (b — a)’
when a — b, and approaches 20° In (b/a) when a — 0. Hence, the space-
charge resistance has the same value as given by (10) and (11) when
a — b, but approaches infinity as a — 0.

As shown in Fig. 1(c), the depletion layer extends through the in-
trinsic region of an n“ip~ concentric sphere structure and is bounded by
the spheres of radii r = a andr = b. When V = Vz, the electric field
E(r) has its maximum value F’, at the inner surface r = a, and decreases
to Fsa’/b’ at r = b.

and A, is quantity of area on the outer sphere surface that corresponds
to one curvilinear unit-cube conductance G. When a is finite or a — Bb,
(15) may be interpreted as the resistance of N unit curvilinear cubes
in parallel, where each cube has a conductance (2Kv,) and the number
of cubes N is given by (16). Unlike the n*ip”* coaxial-cylinder case,
R,, of the concentric-sphere structure approaches infinite resistance as
the inner sphere approaches a point.

Fs can occur on the outer sphere for a p*pn” structure, for example.
The results of A, are the same for both limiting cases as given in (17).

An interesting application of the unit-cube expression is that for a
surface-controlled avalanche transistor (SCAT)* with a total junction
perimeter of P and a space-charge layer W. Because the space-charge
resistance is finite for an n“ip* coaxial cylinder structure as the inner
cylinder approaches a line [see (12)], it is feasible to make calculations
for SCAT on the basis of an avalanche line source. There are N = P/W
such unit cubes around the edge, and the total transconductance, g,, ,

TABLE I— Unit-CusE Expressions (W = b — a)
Structures Parallel Coaxial Concentric
plane cylinders spheres
N A a—b a—0 a—>b
(No. of unit cubes) Ws or Lb Lh And?
W2 b Ww?
Cube edge W W Vb W
(cm)
R
8c 1 1
conn) GN ~ (2Kva)N
or
Gm = (Lex) = : : (19)
0 TR oe

For a silicon SCAT with a device geometry of P = 1000 » and W =
0.5 uw, there are 2000 unit cubes with cube edge 0.5 nw. The space-charge
resistance is 12.5 ohms, and the transconductance is 25,400 wmhos.

Another application is to calculate the voltage disturbance AV, in
a Read diode.’ For a silicon Read diode with drift region of 10 um and
an operating current density of 1000 amp/cm’, the value of AV;, as
obtained from (6), is approximately 25 volts.

A summary of the number of unit cubes and other pertinent quantities
is presented in Table I. It has been shown that the unit-cube conductance
(2Kv,) is a figure of merit in semiconductor device design theory. The
unit-cube expressions are shown to be useful for calculation of the space-
charge resistance.

1. Johnson, E. O., Physical Limitation on Frequency and Power Parameters of
Transistors, IEEE International Convention Record, Part 5, March, 1965,
pp. 27-34. Also appearing in RCA Review, June, 1965, pp. 163-177.

2. De Loach, B. C., Recent Advances in Solid State Microwave Generators, a
chapter of Advances in Microwaves, to be published by Academic Press
and edited by Leo Young.

3. Rose, A., An Analysis of the Gain-Bandwidth Limitations of Solid State
Triodes, RCA Review, December, 1963, pp. 627-640.

4. Shockley, W. and Hooper, W. W., The Surface Controlled Avalanche Transis-
tors, Wescon Meeting, Los Angeles, August, 1964.

5. Read, W. T., A Proposed High-Frequency, Negative-Resistance Diode,
BS.T.J., 37, March, 1958, pp. 401.

Consideration of large alphabet digital communication systems ts of both
theoretical and practical interest. Although performance bounds on optimum
systems for the Gaussian channel are available, constructive methods for
approaching these bounds are unknown, except in a few very special cases.
Specific systems have been proposed and evaluated relative to these bounds,
but exact evaluation of error probability is generally a difficult numerical
task. It 1s of interest to consider simpler performance criteria which permit:
comparison of various systems without extensive computation.

An easily evaluated criterion (based on the alphabet size and minimum
distance between signal vectors) 1s shown to yield a simple sufficient condi-
tion for one system to be better than another (smaller error probability
for the same energy-per-bit). The criterion is applied to orthogonal, bi-
orthogonal, simplex, and more general permutation modulation systems.
In addition to comparing the various systems, we consider ways of obtain-
ing good special cases of permutation modulation. Finally, we assess a
recently proposed system (‘‘N-orthogonal phase modulation’’) and show
that zt 1s generally inferior to more conventional techniques.

The choice of waveforms for communicating over the Gaussian
additive noise channel is a classic problem in communication theory.
Orthogonal modulation systems (i.e., digital communications in which
the alphabet consists of orthogonal waveforms) are known to result
in good power efficiency at the expense of poor bandwidth utilization.’
As the alphabet size M is increased, the energy-per-bit # required to
achieve a given error probability P, diminishes, but the information
rate to bandwidth ratio (#/W) diminishes even more rapidly. Bi-
orthogonal and simplex modulation afford somewhat improved per-
formance, but are likewise restricted to low values of R/W.

There is considerable interest in finding large alphabet systems
which have both good power efficiency and good bandwidth utilization.

Slepian® has given bounds on what can be achieved, but constructive
techniques for approaching these bounds are generally unknown.

Although computer evaluation is ultimately required for precise
knowledge of error probability, it is of interest to consider simpler
performance criteria which permit at least a qualitative comparison
of various systems without extensive computation. It is the purpose
of this paper to demonstrate the utility of the latter approach.

After defining the problem more precisely in Section IJ, some well-
known bounds on the error probability are employed in Section III
to obtain a simple analytic criterion for comparing systems in the limit
of low P,. In Section IV this criterion is applied to systems (PSK,
FSK, biorthogonal, and simplex) for which extensive exact computa-
tions are available and for which the conclusions drawn are already
well-known. After these illustrative examples, permutation modula-
tion* is considered in Section V and N-orthogonal phase modulation’’®
in Section VI. It is shown that the former can yield better performance
than conventional techniques, but that the latter is generally inferior.
Iinally, in Section VII limits on our performance criterion, obtained
from sphere-packing arguments, are presented.

We consider an M/-ary modulation system of equienergy waveforms
S,(t),2 = 1, --- , M, on (0,7), having the correlation matrix

E, is the energy of each waveform so that p;; = 1, and —1 S p,, $ 1.
It is conventional®’’ to define a normalized information rate,
(2 log. M7)/n, where n S M is the rank of the correlation matrix (di-
mensionality of the signal space). We choose to call this normalized
rate the ‘information to bandwidth ratio, R/W”’ motivated by the

where the second equality follows if we set n = 27W, which is at
least partially justified for large n by the work of Pollak and Landau.*
_ For our purposes, the right-hand side of (2) may be considered as the
definition of R/W.

correlation matrix can be written as a permutation of the first row.
Considering the waveforms as vectors in an n-dimensional linear vector
space, this means that each waveform sees an identical environment
of neighboring waveforms. This restriction is a desirable one if it is
desired to transmit each waveform with equal a prior? probability.
The restriction is satisfied by the various modulation systems mentioned
in the introduction.* Slepian’ has termed such systems ‘‘group codes
for the Gaussian channel.”’

where n(¢) is a sample function from a white Gaussian noise process
of spectral density N, ; 1.e.,

On the basis of this observation we wish to decide with minimum prob-
ability of error (P,) which of the 7 waveforms was transmitted. The
optimum (minimum P,) receiver is known’’ to consist of 14 matched
filters which give

and decision that the kth waveform was transmitted is made if 2, > 2;
for allj # k;i.e., the decision is made on the basis of the largest matched
filter output.

* The only commonly employed M-ary system (known to the author) which does
not satisfy this restriction is M-level amplitude modulation.

846 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1967
The error probability of this system is given by*
-1-| ef dey ++ dew plas, -** 24), (8)

where p(a,, °*- , Yar) is the multi-variate zero-mean Gaussian dis-
tribution with covariance given by (7), and the region of integration
Q,; 1s defined by the condition

Landau and Slepian’* have proved the long-conjectured result that
P, is minimized for a given M (but n unrestricted) by the simplex
configuration in which the correlation matrix has the formy

For this case the expression for P, may be reduced to a single integral’®
and numerical results are readily obtained.”

Weber’* has derived locally optimum configurations when M/2 <
n <= M — 1. Forn = M/2 a local optimum is the biorthogonal con-
figuration in which the signal vectors are located along the coordinate
axes (+ and —) of the n-dimensional vector space such that

* Actually this is the error probability assuming the 7th signal is transmitted.
However, under the assumption of equal a priori transmission of all signals, and the
permutation property assumed for the correlation matrix, this probability is inde-
pendent of z and is equal to the system probability of error.

+ The “local optimality” of the simplex configuration (viz., that P, has a local
minimum) had been proved previously by Balakrishnan.”

In this case P, may also be expressed as a single integral which is
readily evaluated by machine techniques. Although for a given value
of M, biorthogonal modulation requires slightly more energy-per-bit
to achieve a given P, than simplex,* it is noted that (for large M)
R/W for biorthogonal is essentially twice that of simplex. Further-
more, for biorthogonal half of the waveforms are the negatives of the
remaining half; consequently, M/2 rather than M matched filters are
required. For these reasons biorthogonal is generally preferred to sim-
plex, and indeed has been employed for deep-space communications.”

The disadvantage of both simplex and biothogonal modulation is
that good power efficiency is associated with large values of M (as it
must be for any modulation system) which from (10) and (12) imply
small values of R/W. Weber’s** results indicate locally optimum sys-
tems with R/W between simplex and biorthogonal, but these are then
also restricted to relatively small R/W.

Optimum systems (in the sense of minimum P,) are not known for
n < M/2. However, bounds on the error probability of optimum sys-
tems have been obtained’ and evaluated.* (The upper bound is ob-
tained by random coding arguments, and the lower bound by sphere
packing arguments.) These bounds are extremely useful in assessing
the performance of specific systems; however, to do so involves ex-
plicit evaluation of P, for the specific systems of interest. This is at
best a difficult numerical task. Furthermore, we may find in comparing
two systems that one is better if we are interested in P, } 10%,
whereas the reverse is true when P, = 10-°. Also, in comparing sys-
tems with different values of M it may be unrealistic to compare P.,,
since P, is the word error probability, and the systems contain a dif-
ferent number of bits per word. Comparison on the basis of bit error
probability involves a difficult conversion from word to bit error
probability which involve coding arguments separate from the modula-
tion system performance.** For all of these reasons it 1s desirable to
find a simpler criterion than P, which permits at least a gross com-
parison of modulation systems. :

requires less energy per bit. The simple unqualified statement that simplex is
the optimum modulation system is misleading.

One approach to comparing modulation systems is to obtain lower
and upper bounds on the true error probability*

If two systems are close in performance, the above procedure may
not enable us to determine which is better unless the bounds are close.
On the other hand, close bounds may be difficult to evaluate and may
not lead to a simple performance criterion. We adopt the viewpoint
here that it is desirable to have bounds, which although quite loose,
lead to a simple sufficient condition for determining when one system
is better than another.

That is, p is the largest non-diagonal entry of the correlation matrix.
It is readily established thati

The lower bound is obtained by observing that P, for an M-ary system
can be no less than that of the binary system containing nearest
neighbor waveforms. The upper bound follows from

where the first inequality in (17) 1s a consequence of the symmetry
property of the system and the fact that the probability of a union of
events is less than the sum of the probabilities of the events. The

* We consider here bounds on word error probability, which, however, may be
easily converted to bounds on bit error probability. For example, if a word is in
error at least one bit is in error, and at most all bits are in error. Hence,
P./log:M and P. are lower and upper bounds on the bit error probability.

‘~ These bounds are generally well known and appear widely in the literature;
eg., Refs. 7, 17, 18, 19. Also, as noted in the previous footnote, these bounds on
word error probability are readily converted into bounds on bit error probability.

second inequality in (17) follows simply by observing that a sum of
(MZ — 1) terms is no greater than (M — 1) times the largest term.

In comparing modulation systems with different alphabet size it is
more appropriate to consider the energy per bit # rather than the
signal energy E,, where

Indeed, the parameter #/N, is an appropriate measure of the power
efficiency of a modulation system. The Shannon channel capacity
formula requires that E/N, > log.2 to achieve arbitrarily small P.,
conventional systems generally require values of E/N, at least 4 times
the Shannon minimum.? |

In terms of the parameter L/N, the error probability bounds may
be rewritten

(2) In the limit of large H/N,, Ay > Ke is sufficient to ensure that
system 1 is better than system 2. (If K, > Ke we will say that sys-
tem 1 is “asymptotically better” than system 2.)

(22) If system I is asymptotically better than system 2, then there
exists a valuc of H/N, above which system 1 is better than system 2.
Below this value of H/N, our formalism is generally inadequate to
determine which system is better. (The critical value of E/N, may
be obtained by replacing the inequality in (21) by an equality.)

(22) A binary system that is asymptotically better than an M-ary
system is always better than the M-ary system.

Thus, we can always determine quite simply which of two systems is
asymptotically better, and may, in many special cases, be able to make
comparisons at specific H/N, of interest.

of the P, obtained with the two systems when operated at the same
average power and information rate. To complete the comparison, the
bandwidth requirements of the two systems should also be considered.
Thus, the parameter R/W, as well as K should be used in comparing
systems.

In the following sections of this paper specific systems will be con-
sidered and represented by points on a K, R/W plot. This will enable
an immediate comparison of the asymptotic performance of systems
having the same R/W. It should be noted that Gilbert*” used a simi-
lar plot in his 1952 paper which addressed the same subject con-
sidered here. Gilbert employed a (SNR, R/W) plot in which the effec-
tive signal-to-noise ratio (SNR) was obtained for a given P, by using
the upper bound in (19). Since the SNR is related to our 1/K, better
systems correspond to smaller SNR. Our purpose in writing this paper
is not to argue that our plot is a better way to present the results
than Gilbert’s. (Indeed, since in general, P, is much closer to the upper
than to the lower bound, his method of comparison is somewhat bet-
ter, although somewhat less convenient to use.) Our purpose rather is
to resurrect these old methods which have been largely discarded
since the advent of high-speed computation, and to illustrate their
applicability to recently proposed modulation systems.

Note that K = 2 for both M = 2 and M = 4* and falls off thereafter.
Since the dimensionality of the signal space is n = 1 for M = 2 and
n= 2 for M > 2, it follows that h/W is given by

R 2 for M = 2,
vo : (23)
log, M for M > 2.
* K is maximized (for integer M) when M = 3. In practice, it is generally

desirable to consider only those values of M which are integer powers of 2 (ie.,
each symbol conveys an integer number of bits). We shall restrict our numerical
examples to such cases.

Table I lists the K and R/W values for phase-shift modulation, and
these are denoted by dots in Fig. 1. It is apparent that M' = 2 and
M = 4 are asymptotically better than the higher-order systems, and
from our previous results this implies that the binary system is always
better than the general M-ary case with M > 4.* Recall that we are

Fig. 1— (K,R/IW) plot for phase-shift, orthogonal, biorthogonal, and simplex
modulations.

consistently using the term “better” to mean smaller P, for a given
E/N,. Large alphabet phase modulation may still be desirable because
of the larger R/W.

4.2 Frequency Shift (Orthogonal) Modulation
For M orthogonal signals (e.g., frequency-shifted signals with es-
sentially non-overlapping spectra), p = 0 and
K = log. M. (24)
The dimensionality of the signal space is the number of orthogonal
vectors, nm = M, so that

Table II lists the K and R/W values for orthogonal modulation, and
these are denoted by circles in Fig. 1. Larger values of 1/7 correspond to

systems which are asymptotically better, at the expense, however, of
smaller values of R/W. It is clear that binary orthogonal is inferior
to binary and quarternary PSK both in terms of a smaller K and
smaller h/W.7

A biorthogonal system consists of 11/2 orthogonal waveforms and
their negatives. The maximum correlation coefficient 1s p = 0 for M 2
4, but p = —1 for M = 2. Therefore,

* This conclusion is confirmed by the exact calculations of P. for M-ary PSK
by C. R. Cahn.?°

+ Binary FSK may still be employed, of course, for simplicity reasons or
because the channel phase coherence may not be consistent with phase-shift
modulation.

Vv mM | on
Table III lists the K and R/W values for biorthogonal modulation,
and these are denoted by []’s in Fig. 1. Note that M = 2 and M = 4
biorthogonal are equivalent, respectively, to binary and quarternary
PSK.

Clearly, for fixed R/W, biorthogonal is asymptotically better than
orthogonal. For example, consider M = 4 orthogonal and M = 16
biorthogonal, both of which have R/W = 1. From (21), the biortho-
gonal system is better than the orthogonal system for all E/N, > 2.8,
which corresponds to all P, of practical interest. (P, < 3(10)-*).

In simplex modulation, the M code vectors form a regular simplex
in M — 1 dimensions. (All vectors are equally spaced from all other
vectors. This corresponds to an equilateral triangle in two dimensions,
and a regular tetrahedron in three dimensions.) All correlation coef-
ficients are equal and are given by?%??3 » = —1/(M — 1). Therefore,

WwW M—1 29)
Comparison of (28), (29) with (24), (25) indicates that for large M
simplex modulation is essentially identical to orthogonal modulation.
Table IV lists the K and R/W values for simplex modulation, and
these are denoted by /\’s in Fig. 1. A quick glance at Fig. 1 indicates

that depending on the R/W of interest, biorthogonal or PSK modula-
tion offers the best asymptotic performance of the systems considered
so far. (The dashed line in Fig. 1 is drawn through these “best”
points.) Note that although simplex provides the largest K for a fixed
value of M, it does not do so for fixed R/W.*

Slepian* has recently described an exceedingly general modulation
system (permutation modulation) for which all of the systems con-
sidered in the previous section are special cases. The optimum de-
modulation algorithm is particularly simple, but the actual evaluation
of P,, and the finding of good special cases is somewhat more complex.
We restrict ourselves here to a special subclass of permutation modula-
tion. This subclass is suggested both as the simplest generalization of
biorthogonal systems, and because perusal of Slepian’s results indicate
that systems taken from this subclass are amongst the better of the
moderate-sized alphabet examples which he considers.

Following Slepian we define an (n,m) permutation modulation sys-
tem as follows. The time interval T is divided into n subintervals (n =
2TW). The first waveform of the alphabet consists of a signal with
amplitude unity in the first m subintervals (m < n), and zero ampli-
tude in the remaining subintervals. The remainder of the waveforms
consist of all possible permutations of the subintervals, allowing also
all combinations of plus and minus amplitudes. For example, the
(3,2) system contains twelve waveforms which we may represent as

It is also noted that the special case (n,1) corresponds to biorthogonal
modulation.*

This (n,m) modulation clearly satisfies the symmetry requirements
of our theory. All members of the alphabet have equal energy? and the
correlation matrix has the desired permutation property. It is readily
seen that the maximum correlation coefficient is given by

m— 1
pre (31)
Thus,
K = (log, M)(1 — p) (32)
=[+ Z logs te )
m m
so that (n,m) modulation always achieves K > 1. Also
fk _ 2 log, M
W n (33)
Seo
n

Since m = 1 leads to biorthogonal modulation which has many de-
sirable properties, it is natural to look next at the special case m = 2.
From (32) and (33) it follows that for (n,2),

+ With the normalization employed above, the signal energy is m. However,
all code words may be multiplied by a constant to achieve any desired £E,.

values of the K and R/W are given in Table V and are plotted as M’s
in Fig. 2. (For reference, Fig. 2 also contains the biorthogonal and
PSK results from Fig. 1.) Thus, similar to biorthogonal, as n becomes
large KC increases but R/W decreases. It is seen from Fig. 2 that (n,2)
modulation gives better performance (larger K for a given R/W) than
biorthogonal or PSK.*

(2,1) corresponds to M = 4 biorthogonal, which from our earlier
results gives K = 2, R/W = 2. It is seen from Table V that (4,2)
gives K = 2.30, R/W = 2.30 which corresponds to both better
asymptotic performance and better bandwidth utilization. It is ap-

parent from (33) that whenever n = 2m, R/W = K, and an im-
mediate question 1s how large can we make these two quantities.
With n = 2m, it follows from (32) that

2m\ 1 re
so that for large m, K — 3. It is easily shown that K increases mono-
tonically towards this asymptotic value as m is increased. Thus,
(2m, m) modulation does not permit attainment of arbitrarily large
values of K, and hence cannot attain arbitrarily low P, with finite
H/N,. This is consistent with Slepian’s statement* that permutation
modulation cannot approach channel capacity arbitrarily closely at
non-zero h/W.

* This is, of course, achieved only at the expense of a larger alphabet size. It
may also be noted from Table V that the alphabet size is not generally, a power
of 2 which may also be a practical disadvantage.

As an immediate generalization of the above, consider the more
general case n = km where k > 1.* Then, from (83)

For large k, the right-hand-side of (41) increases as log k; however,
as seen from (88) R/W diminishes as k-. The locus of K, R/W values
obtained with different values of k (but large m so that the approxima-
tion (41) applies) is shown by the solid curve in Fig. 2. In the limit
as k > 1 (but m always sufficiently large such that (kK — 1) m > 1),
K > 1 and R/W — 2. As k increases, both R/W and K increase until
k =~ 1.5 at which point R/W ~% 3.2 and K = 2.8. Further increases in
k result in a reduction in R/W but continued increase in K.

The above results indicate that (n,m) codes can be found with
F/W as large as octary PSK (R/W = 8) and with a better
asymptotic performance.

In the previous examples we have compared by approximate meth-
ods modulation systems which have already been analyzed exactly.
Although perhaps additional insight into the relative performance of
these systems has been obtained, many of our conclusions may be
inferred from existing exact calculations. We now wish to consider a
new system, recently proposed by Reed and Scholtz,°:® which (to our
knowledge) has not yet been evaluated numerically.

Consider an alphabet M divided into M; groups, each group con-
taining M, members. Thus,

The different groups may be considered to be sufficiently separated in
frequency so that waveforms from different groups are orthogonal.
Within a group the waveforms have the correlation properties as-
sociated with phase-shift modulation. Thus for M, 2 4 the maximum
correlation coefficient is p = cos (27/M,), and

M, M,
In the special case of M; = 1 it is apparent that this system reduces to
simple phase-shift modulation (Section 4.1). In the special case of
M, = 4 it reduces to the biorthogonal case (Section 4.38). A question
of interest then is whether choices of M, > 4, M; > 1 lead to better
performance than either phase-shift or biorthogonal modulation.*

shown for the combined phase orthogonal modulation. The solid curves
are for constant values of M; (noted on the curve); the uppermost
point on each such curve corresponds to M, = 4, and each lower point
corresponds to M, increased by a factor of two. The dashed curve
goes through the M, = 4 points (biorthogonal). It is apparent from
this figure that in this class of systems, for R/W = 2, the M, = 4
biorthogonal systems give the largest value of K. For R/W = 2, the
M; = 1 phase-shift systems give the largest value of K. Thus, in terms

generating waveforms with the above correlation properties, rather than in a
comparative evaluation of performance.

of asymptotic performance, choice of M, > 4, M; > 1 always gives
poorer performance than systems which achieve the same R/W with
either M, = 4 (biorthogonal) or M,; = 1 (simple phase shift).

For example, consider M; = 2, M, = 8. This yields R/W = 2 and
K = 1.17. However, R/W = 2 is also achieved with M; = 1, M, = 4
(quaternary PSK), and for this case K = 2. From (21) we can con-
clude that the latter system is better than the former for all #/N, >
2.5, which includes all P, of interest. The significance of these results
is that we can make this comparison with only a simple slide-rule
calculation.

In the above comparison we considered only M, = 4. The case of
M, = 1, M; > 1 is the orthogonal modulation previously considered.
The case of M, = 2, M; > 1 gives the same performance as biorthog-
onal but achieves only 4 the R/W and consequently is of little in-
terest. The case of M, = 3, M; > 1 consists of orthogonal combina-
tions of two-dimensional simplexes (equilateral triangles). Reed and
Scholtz® conjecture that for M = 3M,;, the three-phase orthogonal
system gives a smaller P, than any other collection of 3M; signal func-
tions in a space of dimensionality 247;. Although this conjecture may
well be true, we wish to point out that if the comparison is made on
the basis of fixed R/W (rather than fixed M) then biorthogonal is
asymptotically better than three-phase orthogonal. One way of seeing
this is by noting that three-phase orthogonal has the same K but
smaller R/W than the four-phase (biorthogonal) system of the same
dimensionality. To increase the R/W of the three-phase system re-
quires a reduction in K which makes it asymptotically poorer than
the corresponding biorthogonal system.

It has been shown that the (K,R/W) plot provides a useful tech-
nique for comparing the performance of various modulation systems.
Although our main concern here is in the comparison of specific sys-
tems, it is still natural to ask whether there are bounds on what may
be achieved in the (K,R/W) plane.

that if no constraint is placed on alphabet size or signal space
dimensionality, K can, in principle, be made arbitrarily large for any

R/W. This corresponds to the fact that the Shannon channel capacity
formula implies that arbitrarily small P, may be achieved at all
(finite) #/W with finite H/N>.

If M is held fixed but n is unconstrained, then the maximum J is
achieved by the simplex modulation’? (Section 4.4) for which case
Kk = [M/(M — 1)] logeM and R/W = (2 loge’T)/M — 1.

Perhaps of more practical interest is the opposite case where the
signal space dimensionality n is fixed, but M is unconstrained. Here,
sphere-packing arguments may be used to show that?

M$ Tonal s+, 4), (47)
where I,,(p,q) is the incomplete beta-function which is extensively
tabulated.” Thus, for a given p-and n, an upper bound to M may be
calculated from (47). Since I,(p,q) is monotonic increasing in 2, this
also gives a lower bound on p for fixed M and n. Considered in this
latter context we can then determine an upper bound on K with which
is associated a given value of R/W = (2/n) logsM. This upper bound,
K,, is plotted in Fig. 4 as a function of R/W for n = 5 and n = 10.
Both curves indicate that K, achieves a maximum value. This is un-
derstandable since for large R/W, 1 — p decreases more rapidly than
log,M increases. On the other hand, as R/W decreases, logeM keeps
decreasing, whereas 1 — p is of course always less than 2. Thus, it is
not surprising that there exists an R/W at which K, is a maximum.

It should be noted, however, that K, is an upper bound which likely
cannot be achieved. For example, when R/W = (2/n) loge2n, cor-
responding to M = 2n, the optimum configuration is widely con-
jectured to be the biorthogonal case.2* The corresponding K and R/W
values for biorthogonal with n = 10 and n = 5 are shown by the
points marked (10,1) and (5,1) on the dashed curves of Fig. 4. These
points lie well below the upper bounds.

Biorthogonal is a special case (m = 1) of the (n,m) permutation
modulation considered in Section V. Fig. 4 (dashed curves) shows the
K and &/W values for the (10,m) and (5,m) cases. As must be,
these curves lie below the upper bounds given by the solid curves.

Finally, we note from Fig. 4 that (n,m) permutation modulation
possesses the interesting feature that as m is increased (for a fixed n)
a maximum f/W is achieved. Both the properties of the maxima of
K, and the maxima of the R/W of (n,m) modulation are probably
worthy of further study.

Tig. 4— Bounds on (K,R/W) for fixed n and comparison with permutation
modulation.

The main conclusion to be drawn is that the K, R/W plot provides
an exceedingly useful technique for comparing modulation systems.
We have restricted ourselves to modulation systems in which the
signal alphabet consists of equienergy waveforms for which all rows
of the correlation matrix are permutations of a given row. (Geo-
metrically, the alphabet consists of M points on the surface of an n-
dimensional sphere such that all points see exactly the same environ-
ment.) This class of systems, although somewhat limited, is sufficiently
broad to cover most systems of theoretical and practical interest.
Given two systems in this class such that K, > Ko; then in the limit
of large E/N, (low P,) Per < Peo for the same H/N,. Furthermore,
we have obtained a simple sufficient condition on the E/N, above
which this inequality is valid. These results are in reality not new.

They are implicit in the results of Shannon’ and in many other works.*®
What is perhaps new is that many interesting results and comparisons
can be obtained by such simple techniques.

Considerably more precise comparisons can of course be made by
exact computation of P, rather than by comparison of K. The latter
procedure however is considerably quicker and allows ready con-
sideration of entire classes of systems (e.g., the (n,m) permutation
modulation and the combined phase-shift orthogonal modulations
considered in the previous sections). The comparisons discussed here
are not meant to supplant exact evaluation, but rather as a coarse sieve
for delineating systems worthy of more extensive calculation.

1. Viterbi, A. J., On Coded Phase-Coherent Communications, IRE Trans. Space
Elec. Tel., SE7’-7, 1, March, 1961, pp. 3-14 (see also Chapter 7 of Digztal
Commumecations with Space Applications, by S. W. Colomb, et al.,
Prentice Hall, 1964).

2. Jacobs, I., Theoretical and Practical Limitations of Weak-Signal Processing
eo Space Research II, North-Holland Publishing Co., 1961, pp.

5. Reed, I. 8. and Scholtz, R. A., N-Orthogonal Phase-Modulated Codes, Univer-
sity of Southern California USCEE Report 134, May, 1965, (Presented at
the First IEEE Annual Communications Convention, Boulder, Colorado,
June 7-9, 1965.)

6. Reed, I. 8. and Scholtz, R. A., N-Orthogonal Phase-Modulated Codes, IEEE
Trans. Inform. Theor., 7-12, July, 1966, pp. 388-395.

7. Shannon, C. E., Probability of Error for Optimal Codes in a Gaussian Chan-
nel, B.8.T.J., 38, May, 1959, pp. 611-656.

8. Landau, H. J. and Pollak, H. O., Prolate Spheroidal Wave Functions,
Fourier Analysis and Uncertainty-III: The Dimension of the Space of
Kssentially Time- and Band Limited Signals, B.S.T.J., 47, July, 1962, pp.
1295-1336.

9. Slepian, D., Group Codes for the Gaussian Channel, Notes prepared for use
at Summer School on Coding, Royan, France, August-September 1965.

10. Kotelnikov, V. A., The Theory of Optimum Noise Immunity, Translated by
R. A. Silverman, McGraw-Hill Book Co., Ine., New York, 1959.

11. Landau, H. J. and Slepian, D., On the Optimality of the Regular Simplex
Code, B.S.T.J. 45, October, 1966, pp. 1247-1272.

12. Balakrishnan, A. V., Signal Selection Theory for Space Communication Chan-
nels, Advances in Communication Systems: Theory and Application, Vol.
1, Chapter 1, Academic Press, 1965.

13. Nuttall, A. H., Error Probabilities for Equi-Correlated M-ary Signals Under
Phase-Coherent and Phase-Incoherent Reception, IRE Trans. Inform.
Theor., JT-8, July, 1962, pp. 305-314.

14. Weber, C. L., Signal Design for Space Communication Channels, Part I, Uni-
versity of Southern California USCEE Report 129, February, 1965.

15. Sanders, R. W., The Digilock Orthogonal Modulation System, Advances in
Communication Systems: Theory and Application, ed. by A. V. Balakrish-
nan, Vol. 1, Chapter 3, Academic Press, 1965.

18. Arthurs, E. and Dym, H., On the Optimum Detection of Digital Signals in
the Presence of White Gaussian Noise, IRE Trans. Commun. Syst., CS-10,
December, 1962, pp. 336-372.

19. Wozencraft, J. M. and Jacobs, I. M., Principles of Communication Engineer-
ang, John Wiley & Sons, Inc., New York, 1965, Chapters 4 and 5.

20. Cahn, C. R., Performance of Digital Phase-Modulation Communication Sys-
tems, IRE Trans. Commun. Syst., CS-7, May, 1959, pp. 3-6.

21. Pearson, K., Tables of the Incomplete Beta-Function, University Press,
Cambridge, England, 1934.

Combinatorial Solution to the Problem of
Optimal Routing in Progressive Gradings

The grading or graded multiple proposed by E. A. Gray ts a certain
kind of one-stage, two-sided, partial access telephone connecting network
for switching customers’ lines to trunks all having the same destination.
Its essential feature is that traffic from lines not having identical access
patterns can be offered to a common trunk, and so pooled. In a progressive
grading the trunk groups are partially ordered in a hierarchy, 7.e., some
provide primary routes, others function as secondary routes which handle
traffic overflowing from primary routes, as well as originating traffic, etc.,
up to final routes.

A call which zs using an overflow or “‘later’”’ trunk when it could be using
a primary or ‘earlier’ group 1s said to make a “hole in the multiple’.
It was recognized early in the development of gradings that such holes were
undesirable.

The problem of optimal routing in telephone networks, considered in
general in the author’s earlier work, 1s here specialized to progressive grad-
ings. It had been shown that for networks with certain combinatorial
properties the optimal choices of routes for accepted calls (so as to minimize
the loss under perfect information) could be described in a simple and in-
tuitive way in terms of these properties. The present paper gives a proof
that all progressive gradings have such a combinatorial property, associated
with the hierarchical nature of the grading. The optimal policy for routing
accepted calls 1s related to the phenomenon of “‘holes in the multiple’,
and can be paraphrased in the traditional telephone terminology thus:
filling a hole in the multiple is preferable to using a final route, and filling
an earlier hole is preferable to filling a later one.

In this paper, we consider some ways in which the concept of a
hierarchy of routes is relevant to the problem of optimal routing as
formulated in previous work.? Naturally, such a hierarchy can be
relevant to routing only if it is in a suitable way related to those
combinatorial properties of the network which distinguish the ‘good’
from the ‘poor’ ways of completing calls. (Examples of such properties
were given in Ref. 2.) It shall be shown that natural hierarchies as-
sociated with certain gradings hold the key to the routing problem in
these one-stage networks.

It is now known? that if a network possesses one of certain com-
binatorial properties, then this property can be used to describe in a
simple way the optimal choices of routes for accepted calls so as to
minimize the loss under perfect information. The next natural ques-
tion is, then, what networks possess some of these properties? We
shall prove that the members of an important subclass of connecting
networks, that of progressive gradings, all have a combinatorial prop-
erty similar to the strongest of those of Ref. 2; this property is as-
sociated with a natural hierarchy of routes, and leads to a solution of
the routing problem for accepted calls.

We first discuss and clarify some of the usage and terminology as-
sociated with gradings. Since about* 1905 the noun ‘grading’ and the
adjective ‘graded’ have been used in telephony to describe a certain
kind of one-stage two-sided network for connecting customers’ lines
to trunks all having the same destination. Roughly speaking, a grading
has this property: some trunk is such that two lines have access to
it which do not have access to the same trunks. The essential feature
is that traffic from distinguishable lines (i.e., ones not having identical
access patterns) can be offered to a common trunk.

*. A. Gray proposed the “graded multiple” in 1905, and was granted a
patent for it (No. 1002388) in 1911.

It appears, though, that the word ‘grading’ has been used in a wider
sense in Europe than in the United States. In particular, the American
usage® implies a certain order in the pattern of access that the lines
have to the trunks, whereas in the European meaning this implication
is absent. The order implicit in the American usage amounts to this:
the trunks are partitioned into groups which are so partially ordered
that no group has more than one successor in the ordering; a line that
has access to one group has access to all groups that follow it in the
ordering. (This ordering usually determines the order in which the
lines hunt over the trunks.) Thus, e.g., a trunk group with no predeces-
sors in the ordering can be used by exactly one group of lines, for
which it is the “primary” route. In one European sense of “grading,”
however, a trunk group which is the first one hunted over by one line
group may be the nth one (n > 1) hunted over by some other line
group.* The distinction drawn here is of some importance, inasmuch
as the order structure implicit in the American usage gives rise to a
natural hierarchy of routes that is directly relevant to routing, whereas
in the more general case this hierarchy is not necessarily present.

Recently, in an effort to establish a uniform terminology, the
nomenclature committee of the International Teletraffic Congress de-
cided® that the terms ‘grading’ and ‘graded multiple’ should be in-
terchangeable, and the structures described in R. I. Wilkinson’s paper?
as graded multiples be called, more specifically, progressive graded
multiples or progressive gradings, the word ‘progressive’ here re-
ferrmg to the order structure we have described as characteristic of
the American usage. The usage recommended by this committee is
adopted herein.

Since the present work can be viewed as a continuation of Ref. 2,
we take the liberty of assuming familiarity with the notations and
concepts used there, and we include only occasional reminders of the
meanings of important notions.

It will be convenient to have a notation for routes. A route r for a
eall c is just a way in which c can be put up or realized in a network »,
and so it can be identified with the state in which the only call in pro-
egress is c using route r. Thus, a route for c is any element of y~*(c).*
We use the variables g and r (over the set L, of states with one call in
progress) to denote routes.

* We recall that if xis a state, y(z)is the assignment of inlets to outlets realized by z.

By a hierarchy of routes we mean a partial ordering ~ contained in
Cae ao le

It is apparent that > can hold only between alternative routes for
the same call. (Of course, not every hierarchy of routes is relevant to
routing; only those that have a suitable relation to the ways in which
calls in progress block new calls will be of interest. The problem is
to clarify the meaning of ‘suitable’.)

A hierarchy of routes, being a partial ordering of the states with one
call in progress, can be extended to, or can induce, a partial ordering
of the whole set S of states in several natural ways. Since > can hold
only between alternative routes for the same call, it is reasonable to
confine attention to extensions which hold only between states that are
equivalent in the sense of ~ in Ref. 2, i.e., are (possibly) different ways
of realizing the same assignment. An obvious first candidate for such
an extension is given by the condition

x~y and rSuaqsyr~q imply rDgq. (1)
However, we eschew this definition in favor of a stronger one: let us set

x Dy = « is reachable from y by sequentially moving calls in pro-
gress from routes that are lower (later) (in the sense of
— on L,) to routes that are higher (earlier) .*

It is intended here not merely that, as in (1), each call have a higher
route in x than in y, but that it should be possible to pass from y to z by
a sequence of equivalent states each differing from the previous one
in that one call has been rerouted on a higher route. This stronger
condition is rendered formally by first defining

* In an attitude prejudiced and justified by the principal results (Theorems 1 and
2) we are working toward, we use the words ‘lower’, ‘earlier’, and their antonyms so
as to suggest consistently that lower routes are less desirable than higher, earlier ones
are preferable to later, etc.

In a one-stage connecting network vy = (G,J,Q,S), with J the set of
customers’ lines (inlets) and © that of trunks (outlets), the graph G
giving network structure is determined entirely by the access relation
A such that

The set S of states of »y can be represented by the set of all subsets of
A which are one-to-one correspondences. The range of x, rng (2), is
the set of trunks which are busy in 2.

The access relation A can be used to give a simple definition of a
progressive grading. We use X X Y for the Cartesian product of X
and Y, i.e., the set of pairs (x,y) with « e X and y ¢ Y. If X is a set,
| X | denotes the number of elements of X.

Definition: v is a progressive grading if and only if it is a one-stage
network for which there exist partitions II and Z of Q and J, respectively,
and a partial ordering = of II, such that for 7, U, V eM andleaz

@) (2 XT) an A # @ implies (L X T) CA,
(ii) (L X U) CA, V = U imply (LX V) CA,
(iii) U= T,V = Timply U S$ VorV s$ U
@) |L|2z| U TI.

The first condition simply says that if a line has access to some trunk
from a group 7’, then all lines in its line group have access to every trunk
in 7. The second condition says (roughly) that a line with access to
a trunk group 7’ has access to all groups that are later than 7' in the
partial ordering. The third condition says that a trunk group 1s followed
(in the partial ordering) by at most one other group; if the ‘“‘later’’
groups are thought of as overflow groups, this means that each group
has at most one group to which to overflow traffic. Finally, the fourth
condition rules out the relatively uninteresting cases in which some line
group has access to more trunks zn toto than there are lines in the group.

It is apparent that if a trunk group 7’, is later than one T’, , then every
line with access to 7’, has access to 7’, . This is the “progressive” prop-
erty. In analogy with the intuitions expressed in Ref. 2, it should be
better to use an earlier trunk group than a later one, if both are available.
Thus, the structure of a progressive grading at once suggests the con-
jecture that optimal routing will consist of using the early routes in

preference to the later or (to anticipate a bit) overflow groups. This
conjecture is true and follows from Theorem 2. In traditional telephone
terminology (see E. C. Molina’s appendix in Ref. 3) it states that filling
a hole in the multiple is preferable to using a final route, and that
filling an earlier hole is preferable to filling a later one.

A line group L is said to be a bye if it has access only to ‘‘overflow”’
trunk groups, i.e., if

It is easily seen that in a progressive grading a hierarchy of routes
can be defined by this rule: r D gq if and only if r ~ q and g(q) 2 g(r),
where g(r) is the trunk growp used by route r.* This is the natural
hierarchy of routes associated with a progressive grading; here r _) q
if and only if 7 ~ q and r is on an “earlier” trunk group than g. In this
instance, = is also a simple ordering on each g(y~*(y(r))). These simple
orderings forming the hierarchy of course correspond exactly to the
preference relation among routes suggested by the natural intuition
(already mentioned) that there is no point in using a later or “‘overflow”’
trunk when an earlier one is available, because possibly fewer lines have
access to the latter. The relation > defined above on L, extends by
(2) to all of S.

In a proof to be given later we shall use the fact that the set of pro-
gressive gradings can be partially ordered by a relation D according
to the following definition of covering: », covers v, if and only if v2 is
obtained from », by removing, for some line group L, either (case 1) a
trunk from the first (in S,) trunk group to which L has access together
with one line of L if L has access to more than one trunk, or (case 2)
the trunk to which L has access together with L itself if L has access
to exactly one trunk. That is, if », is defined by partitions TI, , 4,, a
partial ordering =, of I], , and an access relation A, , then v, covers v2
provided that there exist t ¢ II, andl ¢ L ¢ Z. with

For practical purposes a network in which some line group has access
to no trunks is in all respects equivalent to the same network with those
lines omitted. For this reason the definition of covering was divided into
cases 1 and 2, so as to build this equivalence right into the definition.

As we have said, », covers v, if and only if », results from », by ripping
out (7) some trunk from a “‘primary”’ group, (27) a line with access to
it, and (222) all crosspoints associated with these terminals, with the
proviso that if this leaves some lines with access to no trunks, then these
lines are also to be removed. Because of this, there exists a natural or
canonical map yu of the states S(v,) of », into those S(v.) of »., defined
roughly by the condition that ux is what is left of x after the line and
trunk that define the covering of v2. by », have been ripped out. The
canonical map can be defined formally very simply, as follows: A state
x of v, is representable as a subset of A, which is also a one-to-one cor-
respondence; similarly, a state of v. is just a one-to-one map contained
in A; what is left of x after the ripping-out process is Just

Le = 7 fr A» .
Thus, if » corresponds to ripping out line / and trunk ¢, and « = {(1,t)},
then uz = 6 = zero state. If z = {(1,t,)} ora = {(1,, H} with, #1
and t, ~ t, then again wx = zero state. Ifa = {(,,4)} v y, withl, = 1
ort, = t, then wx = py. It is easy to see that if u rips out J and ¢, then
uwS 18 isomorphic with the ‘‘cone”’
{ve Six = {(l,D}},

because it does not make any difference whether / and ¢ are present in
the system and connected to each other, or are just absent. That is,

uwS is essentially the set of states of S that remain available if | is con-
nected to ¢ with a holding-time of +o.
This notion of a canonical map provides many useful notations.
It is convenient to extend the p-notation as follows: For 7 ¢ II

Clearly, u7' is what is left of the trunk group T after the line J and the
trunk ¢ associated with » have been ripped out. Also, we set

{(ux uy): D y, pe ¥ Oorxe = O, and py + Gory = 4}.
The relation » 2 can be seen to be identical with 2, ; it is a useful
mnemonic; it defines the hierarchy of routes in the “reduced” system
v2; the partial ordering induced in S(»,)[= wS(r,)] by this hierarchy is
precisely yp 2.

VI. PRELIMINARY RESULTS
In Ref. 2, for a general partial ordering A, the notation
sup A,,
R

whenever this set was nonempty. The notation was chosen to denote
a set of R-maximal elements of A,,, rather than an actual R-maximal
element itself, so as not to prejudge the question as to how many there
were. It will be shown that if the network » under study is a progressive
grading, and R = 2 = natural hierarchy, then unless c is blocked in
x (and A,, is empty) A., always has a C-maximal element which is
unique to within equivalence under permutations of lines within their
line groups and trunks within their trunk groups. |

Let now x be a state and let ¢ ¢ x be a call which is not blocked in z.
It is apparent that for y, 2 ¢ A., we have either g(y — x) 2 g(z — 2)
or g(y — x) S&S g(e — x). Hence, there is a yp € A,, such that

for all w ¢ A,, , and yo is unique to within equivalence. (Recall the con-
struction of Q in Section III, and the fact that D is J u Q.) Hence,

exists, and equals 7(yo), 7(-) being the natural homomorphism of S
into the quotient S/(2 n C). (See Ref. 2.)
We now consider policies g(-,-) such that
ea x—h if eisa hangup h,
be sup A,, if eis a new call c not blocked in z.

Such a policy expresses the routing rule of always choosing the earliest
available trunk in the natural hierarchy characteristic of a progressive
grading.

The relation B (for ‘‘better’’) was defined in Ref. 2 by the condition
xBy ifandonlyif «2 ~ y and every call blocked in x 1s also blocked
in y.

By Theorem 1, to be proved shortly, it will follow that x D y implies
x B y, which in turn implies s(x) 2 s(y). Thus, the policies ¢(-,-) coin-
cide with the ‘‘maximum s(-)’ policies suggested in Ref. 2. (See Ref.
2 for notations.)

Let / and t be the line and trunk, respectively, associated with yu. There
exists a trunk ¢* such that

Let T* denote the set (trunk group) achieving the infimum on the right.
Since t is busy in x and ¢* is not, t ¥ ¢*. Thus, 7* ¥ {t}, and w(7*) + 8.

We next show that »7’ € rng (ux) implies 7 € rng (x). If not, then
there exists t, ¢ uJ’, hence ¢ T such that ¢, ¢ rng (ux) and ¢, € rng (2).
But

Now 7* S T for every T such that 7 € rng (x) and [*A7. From the
two previous paragraphs, it follows that
(uT*)(u S)pT
for every 7 such that u7 € rng (ux), *AuT, uT ¥ 0. That is,

wy ESUP Agius
pS
and since y was arbitrary within sup A,, , the lemma is proved.
>

Now let ¢ be a call from line / which is blocked in x but not in y. Then
c is not blocked in z either. The only trunk which is busy in x and not
in z is that used by the call y(a — z). Thus, since c is blocked in x and
not in z, g(x — 2) is a trunk group usable for the call c. However, by
property (22) of progressive gradings, {l} & g(y — z) C A, Le, J has
access to the group g(y — z) as well. Hence, some trunk of g(y — 2)
is idle in x, since the call y(2 — z) has a choice of routes in state z, one
of these being on g(y — z). Thus, c is not blocked in x, and zx B y.

Theorem 1: In a progressive grading, the partial ordering > induced by
the natural hierarchy of routes 1s contained in B.

Proof: Immediate from Lemma 2 and the facts that > is the transitive
closure of J u Q, and that B is transitive.

Lemma 8: If x D y, then x 1s obtainable from y by moving calls to earlier
routes in such a way that each call is moved at most once.

Proof: The result is true if only one move is made. Suppose it to hold
if nm moves 2n toto are made. Let x be obtainable from y by sequence of
(n + 1) moves. The trunk groups available for a given call c form a
set simply ordered by S, and so can be indexed 1, 2, --- , the S-earlier
receiving the lower integer. For c S y(z), let n(c,x) be the index of the
group used by ¢c in x. Some call c that is moved in obtaining x from y
achieves

Starting in state y it is possible to move such a call (once) directly to
its route in xz, to get a state 2 in which it is still possible to carry out
exactly each of the moves that take y into x except those involving c.
These are at most 7 in number, so each call involved need be moved at
most once.

for every event e that is either a hangup or a new call not blocked in
either x or y. It has been shown in Ref. 3 for a general network that if
gy preserves B then it embodies the optimal routing policy for accepted
calls.

The main theorem we prove (Section VII) states that a sup A.,
policy, i.e., one satisfying (3), preserves >. The method to be used in
the proof of this result is illustrated in part by the following remarks:
consider linear arrays 2, y, 2, --- each of m urns, n = 2, each urn con-
taining at most one ball, with fewer than n nonempty urns per array.
Let x D y mean that x is obtainable from y by moving balls to the left.
Let yx denote the result of adding a ball in the leftmost empty urn.

Proof: The result is obviously true for n = 2 by enumeration. Let it
hold for a given value n = 2, and consider arrays x, y of n urns satisfying
the hypotheses. Let yz denote the result of removing the leftmost urn
from z, and bz that of adding an urn containing one ball at the left of
z. There are two cases: (2) the leftmost urns are empty in both x and

Case (iz): In obtaining x from y some ball moved into the leftmost
urn. Obtain z from y by moving just this one ball to the leftmost urn.
Then + D zD y, ot D v2, gy = blz, oz = bez. Since gz is obtained
from wy by removing some ball, and replacing it in the leftmost empty
urn of the resulting array, we have gyz D Wy, and so gz D gy.

In cases 3 and 4 of the proof of the next theorem, the analog of the
inductive index n will be the partial ordering of the set of progressive
eradings.

Theorem 2: In a progressive grading v let D be the partial ordering in-
duced by the natural hierarchy of routes in v, and let ¢ be a policy with the
property that

Proof: The proof is by induction over the partial ordering > of the
set of progressive gradings which is defined by the definition of covering
given earlier. A grading v that is minimal in D has no “‘overflow groups’’,
1e.. 2 = identity relation, so that no trunk group has a successor in
the order 2 characteristic of »v. Thus, v consists entirely of trunk groups
serving line groups on a one-to-one basis, so that for some n

In this minimal case > is the identity relation, and ¢ obviously preserves
it.

As a hypotheses of induction, we now suppose that every progressive
grading covered by v has the property that any sup A,, -policy preserves
—. Let now x D y inv and let e ¢ x. The induction argument will have
four cases, the last two of which are analogous to the observation made
earlier.

e987 jeder yn— 1.
This sequence indicates how one would get y from x by moving calls
to “preferred” routes. By Lemma 3 it is no restriction to assume that
no call is rerouted more than once. Let the route of h be rin x and q in y.

If h is one of the calls whose route is changed in the above sequence, say
to take z, into 2,,, by changing the route of h from r to q, then

is a sequence which shows that (2 — r) D (y — q). If the route of h is
not changed, then r = g and the same conclusion follows.

Case 2: x D y, and e € x is a new call c blocked in x. By Theorem 1,
x B y, so c is also blocked in y. Then,

Case 3: x D y, eis a new call c not blocked in either x or y, and the line
group L of c is not a bye. Let

Subcase 3.1: T is full in neither x nor y. Then there exist routes r, q¢
such that g(r) = g(q) = T,

21,22, °° » 2, = Y IS a Sequence with
2; O 2341 g=1,°-°: ,n—I,
showing that x > y, then
LUr=2, UU yk UrT=YuUrY Y gis a sequence which
shows that

This is because we can assume without loss of generality that the trans-
formations which change y into x reroute a call at most once, and thus
move no calls onto 7’. (Lemma 3.)

Subcase 3.2: T is full in both x and y. Since L is not a bye, there exist
l_.melLandt, ue T with

Because 1, m and é, wu are respectively interchangeable, 1.e., since lines
and trunks are permutable within their respective groups, no loss of
generality is incurred if it is supposed that 1 = m andt = n. Let pu be
the canonical map corresponding to ripping out / and Z.

in —< in case 1
I — {7} in case 2,
ume ‘— in case 1
m— {D} in case 2,
> with 7 — {t} replacing 7’ throughout, in case 1
w= = 2,
2 Sy Xt) in case 2,
th A cane in case 1
A— (UX {t}) - ZX Q) in case 2,
with

po(c,x) Sd A eur)
we(c,y) Esup Agu) .
re)
Since x D y, and either both wx = 0, py = 0, or neither, we have
(ux my) € wD. (4)
Let ~ be a policy for », with
E(d uz) € hes A atue) ? Vd E pe. (5)

But ¢(c,x) differs from ye(c,x) and ¢g(c,y) from py(c,y), only in having
an additional line 1 and an additional trunk ¢ connected to each other.
Hence,

The argument of subcase 3.2, basic to Theorem 2, can be appreciated
by looking at it thus: x D y means 4z,, °°: , & with 2; Q 241,7 =
lee, n—14=2,2 = y. Since

using r is not moved as y is transformed into zx by moving calls. Thus,
(2; —7r) Q @is1 — 1), ~=1,°---,n-1
@—rnQdy >).

But the ‘‘cone” {z:z = r} is isomorphic to the states of a grading (», of
the proof) covered by v and the isomorphism, 22z., uw restricted to the
cone, has the basic property, for x, y in the cone

Subcase 3.3: T is full in x, but not in y. Since ZL is not a bye it is S-
minimal, and hence there exists a call d with d S y(x) nm y(y) such that
dis on T in x is not on T in y, and can be moved to T in state y to give
rise to a new state z without rendering impossible any the remaining
moves which transform y into x. Thus, 7 D z D y. Since x n z ¥ 8,
subcase 3.1 gives v(c,x) D (cz). Further, the route of c in ¢(c,z) is
no higher (later) in $ than the one in y left by d as it was moved to
T to give rise to state z. Hence, to within equivalence

Case 4: x D y, € is a new call c not blocked in either of x or y, and the
line group of c is a bye. There is at least one other line group LZ which

Subcase 4.1: L XT nxw4#0LXTanyxroboorLxTnz= 8,
LXToay = 6. Since L is not a bye, there exist 1,m eLandt,u eT
with

(In the second instance, property (2) of the definition of a progressive
grading has been used to conclude that there must be idle lines on L
if there are idle trunks on 7.)

As in subcase 3.2, no loss of generality is incurred if it is supposed
that 1 = m and ¢ = wu. Let uw be the canonical map corresponding to
ripping out J and ¢. The argument now continues as in subcase 3.2.

Subcase 4.2: (L XT) nx4#90,(LXT) ny = 6.Since Lis S-minimal,
there exists a call d with d = y(x) n y(y) such that d is on T in @, is
not on 7 in y, and can be moved to 7 in state y to give rise to a new
state z without rendering impossible any of the remaining moves which
transform y into x. Thus, x D z D y. Since x n z ¥ 6, subcase 3.1 gives

T,. = gQnf {S:*AS,S € rng (y)}).
Here 7’, is the earliest group c could be put on in y. Let also y denote
the operation of moving d from T, to T, and for any call f
A; = tor): = v(r)}
= {S:the line of f has access to S}.

Case (1): Tz € A, N Ag, Ta S T,. Then moving d from T,; to T means
that c can use 7', in Zz, so o(c,z) > o(c,y), because ¢(c,z) results from
y(c,y) by moving first d to J, and then ¢ to T, , so actually

1. Weber, J. H., Some Traffic Characteristics of Communication Networks
with Automatic Alternate Routing, B.S.TJ., 41, July, 1962, pp. 1201-1247.

2. Benes, V. E., Programming and Control Problems Arising from Optimal
Routing in Telephone Networks, B.S.T.J., 45, November, 1966, pp. 1373-
1438; Abstract in SIAM J. Control, 4, 1966, pp. 6-18.

3. Wilkinson, R. J., The Interconnection of Telephone Systems—Graded Multi-
ples, B.S.T.J., 10, October, 1931, pp. 531-564.

4. Syski, R., Introduction to Congestion Theory in Telephone Systems, Oliver
and Boyd, Edinburgh and London, 1960.

Integral Equation for Simultaneous
Diagonalization of ‘Two
Covariance Kernels

Let K,(s,t) and K,(s,t), —T S s,t S T, be real, symmetric, continuous
and strictly positiwe-definite kernels, and denote by K, and Kz the cor-
responding integral operators. Let x(t) be a sample function of erther of
two zero-mean processes with covariances K,(s,t) and K.(s,t). We prove
a generalized version of the following: If the integral equation

has formal solutions \; and W;(t) which may contain 6-functions, and
af {K,;} forms a complete set in &.[—T,T], then (2) the two kernels have
the following sumultaneous diagonalization:

in the stochastic mean, uniformly in t, and the coefficients are simul-
taneously orthogonal, 7.e.,

Et (w,wi)(@v;)} = 6s , Eni (x,vi)(a,W;)} = rz bi; ,
where (x,;) ts obtained by formally integrating y(t) against x(t).

Kk, the integral operators with kernels K,(s,t) and K.(s,t). We have
previously’ established that, if Ky7?K.K7? is a densely defined and
bounded operator on £, (the space of all square-integrable functions
on [—T7,T]) and if its extension to the whole of £, has eigenvalues
\; and complete orthonormal eigenfunctions ¢,(t), 7 = 0, 1, --+ , then
the two kernels have the following simultaneous diagonalization:

uniformly on [—T7,7] X [—T7,T]. In addition, if x() is a sample func-
tion of either of two (separable and measurable) zero-mean processes

a(t) = » ni(2)(Kig,) (t) | (2)
in the stochastic mean, uniformly in ¢t. Moreover,*
Ey {ni(a)nj(a)} = 43; , Lat n(x) ni(ax)} = rx be
wheret
n(x) = lim (2,K7*y;,) (3)

in the stochastic mean, and {¢;,} 1s any sequence of functions in the
domain of Kj? such that lim || 9; — ¢;, || = 0.°°? Furthermore, if the
two kernels have continuous 27th derivatives (0°’/ds’dt’)K,(s,t), p =
1, 2, then (1) and (2) can be differentiated term-by-term r times while
retaining the same senses of convergence.’

We remarked in Ref. 1 that, if y; is in the domain of K;?, ¥; = Ky»;
satisfies the integral equation

Slepian (private communication) has long conjectured that, if (4)
admits formal solutions \; and y; , 7 = 0,1, --- , where y; may contain

* Ey, p = 1, 2, denotes the expectation with respect to P
+ For any f, g eL2 , (f,g) denotes the inner product of f and g, and || f || the norm
of f.

6-functions and their derivatives, then the expansion coefficients and
functions of (2) are given by formally substituting such y, into (5).*
This conjecture, proved here, is significant since it provides a concrete
means of obtaining the expansions (1) and (2). To illustrate the point,

(iid)n; = c,(z,b;) as.,* Kip; = c;Ky);, which verifies Slepian’s
conjecture for this example. Here c; is a normalization constant given by

Cox+1 = Cox lon—dx ;
(that is, Co.4; is obtained by replacing @, with 6, in co,), up; and fp; ,
p = 1,2,7 =0,1,--- , are the eigenvalues and orthonormal eigenfunc-
tions of K,, and (y; , f:;) 1s defined analogously to (9).
In this paper we prove the generalization of (2), (27), and (222), starting
with abstract kernels K,(s,t) and K,(s,t) and a generalized version of
the integral equation (4).

II. MAIN RESULT
Theorem: Let K,(s,t), p = 1, 2, -—T S s,t S T, be real, symmetric,
strictly positive-definite kernels with continuous 2 rth derivatives
(0°"/dsdt’)K,(s,t). If there exist sequences of real numbers {Qitm},

for some constants b, and bz, , and sequences of square-integrable functions
{W.1}, which satisfy the equation

such that the right-hand side of (12) forms a complete set in £2 , then
(() K7?K.K;? is a densely defined and bounded operator on £2 ,
(22) ats extension to the whole of &2 has eigenvalues and complete ortho-
normal eigenfunctions, which are the \; and

and by the right-hand side of (12) without \;. Here, K?,,, p = 1, 2,
denotes an integral operator whose kernel is defined as

2 Mil pi (fri Q) = a
uniformly in (s,t).” It follows from this that (15) converges in the
mean in (s,f) as well. Hence, from Fubini’s theorem, K?,,(s,t) is a
square-integrable function of t for almost every s. Thus, ¢;(s) of (18)
is well defined. We assume without loss of generality that 9;, 7 =
0,1, --- , are normalized.

the right-hand side of (12) without \; is given by (10), and completeness
of {cos 6,t, sin 6,¢} follows from (18) and a gap-and-density theorem.°

For notational simplicity, we write K,.1, p = 1, 2, for the integral
operator whose kernel is

where the second equality follows from the mean convergence of (15)
and the third from the uniform convergence of (16). To prove (18),
consider

which vanishes as 7 — © since (16) converges uniformly in (s,t).
(7) K;?Ki and K;?K? are densely defined and bounded on &, .
To prove this, apply Kz? on both sides of (12) and use (18) to obtain

where the second equality follows from (17) and (18), the third from
k time differentiation of (12) and from (17) and (18), and the last
from (13). Hence, with 9; being normalized,

Now {y;} is complete since the right-hand side of (12) without ), ,
which forms a complete set by hypothesis, is equal to Kiy;, and K?
is strictly positive-definite. Hence, from (11), Ky?K? is densely defined
and bounded.

To prove that K;?K3 is also densely defined and bounded, define
¢,; as the normalized right-hand side of (18) with the subscript 1 re-
placed by 2. Completeness of {¢;} is similarly deduced via (12). Now,
by following the same procedure with the roles of K, and K, inter-
changed, we obtain

|| KriK3¢,
Then, the assertion follows immediately from (11).
(vit) The ranges of K? and K? are equal, namely,
Ki(£2) = Rie):

To prove this, denote by ZL and M the extensions to the whole of
&, of K>?K? and Ky'K? respectively, which exist as a result of (it).
Since the domains of K?Z and KiM are &, , which is also the domains
of K? and K?, we have

Ki = KiL, K} = Kim.
Then, from the first equality, Ki(£.) C Ki(£,), while, from the second,
K3(£2) C Ki(£2). Hence, the assertion holds.

To prove (19), note first that fi; , 7 = 0,1, --- , are in the domain
of Kz? as a result of (i772) and also that (Ky*f1;, K3f:;) = 6;; from
orthonormality of {f,;}. Thus, {K7#/,;} and {K3f,;} form a pair of
mutually reciprocal bases of £, . Hence,

But, according to (22), the right-hand side converges uniformly. Hence,
the above partial sum must converge uniformly to f,; . Suppose for
some k,0 Sk <1,

APO = > (hi fedf PO)
uniformly in ¢.” Hence, by induction, (23) holds for every k,O Sk Sr
Therefore, from (22),
(Ke*hi; , Kin()) = fP?@, b=0,1, +--+ ,7.
Then, (19) follows from (21) and the above.
To prove (20), we expand K3,, g relative to {K?},,}:

(v) To prove (2) of the theorem, we note from (727) and (72) that
Kk7?K? is everywhere-defined and bounded on £2. Hence, its adjoint
(K;?K3)* is also everywhere-defined and bounded. Now, for any

f « & and g « D(K;'), the domain of K;?, we have (Kj?K3f,g) =
(f,K3Ky*g). Thus, KiKy*g = (Ky*K3)*g, g « D(Ky*). Hence, KiK;y}
is bounded. Since (Kj?) is dense in £,, we conclude that Ky7?K.K;?
is densely defined and bounded.

where the second equality follows from (19), (20), (15) and (18), and
third from (12) and (13). Now denote by Q the extension of K7?K,Ky7?
to the whole of £, . Then,

which vanishes as nm — ©. Therefore, Qy; = X,9,. Lastly, since {g;}
is complete in £,, {d;} constitutes the entire spectrum of Q.
(viz) To prove (272) of the theorem, note from (8), (24) and (vz) that

both of which vanish as n — © by virtue of (19) and (20). Therefore,
upon combination of the above results, (14) is proved.

1. Kadota, T. T., Simultaneous Diagonalization of Two Covariance Kernels
and Application to Second-order Stochastic Processes, submitted for publi-
cation in SIAM J. Appl. Math.

2. Root, W. L., Singular Gaussian Measures in Detection Theory, Proc. Symp.
on Time Series Analysis, edited by M. Rosenblatt, John Wiley & Sons,
New York, 1963, pp. 292-315.

3. Yaglom, A. M., On the Equivalence and Perpendicularity of Two Gaussian
Measures in Function Space, Proc. Symp. on Time Series Analysis, edited
by M. Rosenblatt, John Wiley & Sons, New York, 1968, pp. 327-348.

4. Huang, R. Y. and Johnson, R. A., Information Capacity of Time Continuous
Channels, IRE Trans. Inform. Theory, 77-8, September, 1962, pp. 191-205.

5. Kadota, T. T., Simultaneously Orthogonal Expansion of Two Stationary
Gaussian Processes-Examples, B.S.T.J., 45, September, 1966, pp. 1071-1096.

6. Kadota, T. T. and Shepp, L. A., On the Best Finite Set of Linear Observables
for Discriminating Two Gaussian Signals, IEEE Trans. Inform. Theory,
April, 1967.

7. Kadota, T. T., Term-by-term Differentiability of Mercer’s Expansion, Proc.
Am. Math. Soc., 78, February, 1967, pp. 69-72.

8. Levinson, N., Gap and Density Theorems, Am. Math. Soc. Colloquium Pub-
lications, 26, Am. Math. Soc., Providence, Rhode Island, 1940, p. 3,
Theorem II.

9. Apostol, T. M., Mathematical Analysis, Addison-Wesley, Reading, Massachu-
setts, 1957, p. 403.

Principles of Design of Magnetic Devices for
Attitude Control of Satellites

Magnetic devices mounted within an orbiting satellite interact with
the earth’s magnetic field and produce torque to modify the attitude or
angular adjustment of the satellite axis of spin. The satellite environment
dictates that these devices be designed for minimum weight or minimum
power consumption, or a suitable compromise between these two minima.
Principles of design of magnetic devices to satisfy these requirements are
developed in this paper. The resulting design equations and charts enable
the ready optimization of design and selection of preferred materials.
While most of this work was directed initially at the Telstar” satellite
project, the design charts and formulas are found useful in other areas
of magnet design. Methods of magnetic measurement devised for the satellite
are discussed.

Satellites with directional instrumentation, such as the antenna sys-
tem of the communications satellites, require attitude control to keep
this instrumentation properly on target. For example, a spin imparted
to the satellite at time of launch gives it a sort of gyroscopic stability.
However, complete attitude control requires some available torque to
correct the direction of the spin axis.

In the orbiting satellite the earth’s gravitational field is balanced by
centrifugal force, leaving the earth’s magnetic field as a convenient
means for interaction torque. Suitable interaction with the earth’s
magnetic field can be set up by electromagnets, or by air-core coils of
large area, either of which can be turned on or off at will to provide
attitude correction as needed. Small permanent magnets can be de-
signed and installed to cancel out residual magnetic moment in the
satellite, which if permitted to interact with the earth’s field could
cause precession of the spin axis. Other miscellaneous torque applica-

Limitations of payload and of available power in the satellite gen-
erally make it necessary to design with quantitative accuracy and to
optimize the factors which control weight and power consumption. To
this end, the magnet designer may select from various geometries of
magnet and coil and from various available materials. This selection
and optimization is facilitated by the use of suitable design formulas
and charts. In this paper, we review the derivation and illustrative
use of such formulas and charts. While the work reported here has
been aimed specifically at certain problems of the Telstar® satellite,
it is evident that the technique of magnet design presented here is ap-
plicable to any similar set of problems.

For the convenience of the magnet designer who buys magnets and
magnet wire by the pound, measures them in feet, inches, or mils,
and measures torque in pound-inches, all of the derived design
formulae and graphs are built around the practical units (inches,
pounds, oersteds, gauss, etc.). This avoids the necessity of converting
units, which is time consuming and can lead to costly errors. There
is included for convenience a table of the most frequently used con-
version factors (Table I).

The torque characteristic of the air-core coil is derived from the
galvanometer formula which, in some textbooks, is written in MKS
units:

Copper Aluminum
Rom | aaa ace
E(volts) 0.75 X SERED 1.21 x Wa)
W(watts) 2.18 x 1.21 X as (NI)?
Wet.(Ibs.) 0.321 AP 0.0983 AP

NI: Required ampere turns
N: Number of turns used
A: Cross section area of winding (inch?)
(N times the section area of a single turn)
P: Average length of turn in winding (inch)

Here 7’, is the maximum torque exerted on the coil when its axis 1s
perpendicular to the field H,, and NIA is the required product of am-
pere turns and area enclosed by the coil to deliver that amount of
torque.

It is convenient to set up a table of formulas from which one may
translate the geometry and ampere-turn characteristics of the coil into
power and weight requirements. The power and weight will depend
upon the winding material used, but practical considerations usually
limit this to copper or aluminum. So one may take the weight and
resistivity characteristics of copper and aluminum from handbook
tables and with the aid of Ohm’s Law derive the formulas of Table II.
Using (2) and Table II, one may estimate readily the power and
weight of an air-core coil to satisfy specified torque requirements. It is
evident that copper has the advantage in lower power consumption,
but that aluminum offers a greater advantage in weight reduction. If
power and weight are of about equal importance, then the power-
weight product should be minimized. It is evident that aluminum has a
factor-of-two advantage over copper in this characteristic.

A magnetized bar, either a permanent magnet or the core of an
electromagnet, displays a moment, or normalized torque, proportional
to the product of the volume of the bar by the intrinsic induction
within the bar. The magnetic moment, M,,, is identified as normalized
torque in the familar equation

and the relation between magnetic moment, intrinsic induction in the
bar, and the geometry of the bar is given by another familiar equation

in which B — AH is the intrinsic induction, A is the cross-section area
of the bar at the median plane, and S is the effective distance between
poles. For a magnet of length / and diameter d, one may define a
shortening factor, Rg = S/l, which evaluates the effect of recession of
the poles, and rewrite (4) as

The shortening factor, Rs, has been evaluated by Okoshi.t Okoshi’s
values are plotted as a function of lJ/d of the bar in Fig. 1, with a

broken line extrapolation guided by experimental data. If one com-
bines (3) and (5) after conversion to practical units, the result is

and the required volume of bar to produce a specified magnetic mo-
ment is given by rearrangement of (6)

Here H, is the applied magnetizing field, and B and H describe the
condition of magnetization within the bar. The demagnetizing factor,
Dz», which is partially defined by (8) is used to express the dependence
of the intrinsic induction within the bar upon the aspect ratio (l/d) of
the bar. It has been tabulated and charted as a function of l/d by
Okoshi,? Bozorth and Chapin,? and others. These sources agree upon
the value of Dz for long slender magnets. For shorter magnets, where
there is some disagreement, we find the Okoshi data to be in agreement
with experiment.

In plotting the characteristics of magnetic materials we normally
plot the intrinsic induction (B — H) or the flux density (6) as the
dependent variable, and the field strength (H, or AH,) as the in-
dependent variable. Hence, the ratio

becomes the slope of the generalized load line of the magnetized bar.
This reciprocal of the demagnetizing factor is found useful in numerous
magnetic calculations and possibly deserves a name and symbol of its
own. We have elected to call it the loading factor, with the symbol U,

and have plotted it as a function of //d in Fig. 2. In the electromagnet
core operating below saturation, H is generally negligibly small compared
with either B or H, , and the expression for the loading factor reduces
to U = B/H,. In the permanent magnet, H, disappears and the ex-
pression for loading factor becomes U = (B — H)/(—4H). For long
magnets, (I/d > 5) H is negligibly small compared with B, and the
loading factor is further simplified to U = B/(—4H). In this restricted
form the loading factor is identified with the “‘permeance coefficient’
and similar terms used in the literature of permanent magnets.

In Fig. 3 we illustrate the application of the loading factor to the
analysis of permanent magnets and electromagnets. For this illustra-
tion each is assumed to have I/d = 33 so that the loading factor, U &
400. In the permanent magnet (Remendur) the magnetomotive force

Fig. 3— Application of load factor to magnet design: (a) permanent magnet
(Remendur 38), (b) electromagnet (Permalloy 45 core).

is generated within the magnet and varies with the loading of the
magnet as indicated by the B, H curve. (Here 4H is sufficiently small
so that B is indistinguishable from B — #H). The operating point is
determined by the intersection of the B, H curve with the load line
of slope U. This is a fixed point for a particular magnet with a par-
ticular condition of magnetization. The conditions of Fig. 3 were chosen
to match the characteristics of Remendur. In the electromagnet when
operated below saturation, H is negligibly small so the load line rep-
resents the relation between flux B in the bar, and applied field, H, ,
up to the region (around B = 10,000 for Permalloy 45) where the core
material starts saturation, and the characteristic starts deviating from
the straight load line.

3.2 Design of Permanent Magnets for High Torque-W eight Ratio
The weight of the magnet in pounds is

in which p is density of magnet material in Ibs/in°. One may combine
(10) with (6) to obtain

Here the left-hand side of (11) is the normalized torque-weight ratio.
The design objective is to maximize this ratio.

The dependence of the operating point of the permanent magnet
upon the load factor, U, has been illustrated in Fig. 3. On similar charts
one may plot intrinsic induction (B — H) as a function of field (A) .
for various magnet materials as in Fig. 4. Values of B and # for these
plots may be derived readily from the regular demagnetization curves
supplied by magnet manufacturers. Then, for a particular value of

Fig. 4—JIntrinsic induction of permanent magnets; (a) directional grain
ceramic, (b) Alnico 9, (c) Alnico 5.

Fig. 5— Variation of torque-weight ratio with aspect ratio; (a) directional
grain ceramic, (b) Alnico 9, (c) Alnico 5, (d) Remendur 38.

l/d one may pick off the corresponding value of U from ITig. 2, and
using this as the slope of the load line may find the value of (B — H)
for a particular magnet material at the point of intersection. This
value of (B — H) and the value of p appropriate to the material may
be inserted in (11) to give the normalized torque-weight ratio. For
example, at l/d = 4, U & 17.5. The load line of that slope intersects
the intrinsic induction curve for Alnico 5 at (B — H) = 10,000. In-
serting this value and the value of Rs in (11) and using p = 0.26 for
Alnico, one obtains

WH,
Repeating this procedure for various values of l/d and for various
materials, one can assemble the necessary data to plot the curves of
Fig. 5.

It is evident that for each magnet material there is a value of l/d
above which the torque-weight ratio is essentially constant, and below
which the torque-weight ratio falls off rapidly with decreasing [/d.
This follows the shape of the demagnetization curves of Fig. 4. This
value of I/d is large for magnets having low coercivity, and small for
magnets having high coercivity. One would normally design the mag-
net to operate on the flat part of the characteristic to obtain high
torque-weight ratio.

The electromagnet is assumed to consist of a cylindrical core of
ferromagnetic material with a solenoid wound around it. The design
formula for the core is the same as that for the permanent magnet and

is given in (7). This gives the required volume to produce a specified
moment, operating the core at a specified value of flux density. The
ampere-turn requirements are derived as follows.

NI
2.021
If, as is usually the case in the electromagnet, H is negligibly small
compared with H,, then one may combine (8) and (138) to obtain

Since Dy, and Rg are functions of ee ratio (lJ/d) one may define
an aspect ratio factor,

For any proposed geometry of an Seeeauane with specified value
of magnetic moment (7',/H,) the ampere-turn requirement may be
determined from (19) and then translated into power and weight re-
quirements by reference to Table II.

A permanent magnet material with low coercivity and high rema-
nence, as exhibited by Remendur in Fig. 3, offers the inviting possi-

bility of easy magnetization and reversal of field by means of short
pulses of current through a winding. Between pulses it acts as a perma-
nent magnet, with polarity determined by the direction of the preced-
ing pulse. Thus, it provides a switchable field with very low expendi-
ture of power. It requires, however, for complete demagnetization, or
“knock-down,” much more sophisticated circuitry. Also, the certainty
of complete demagnetization from an applied pulse or series of pulses,
depends to a considerable extent upon the preceding history of the
magnet. For this reason, it is not likely to replace readily the simple
air-core coil or electromagnet unless the available power is so
severely limited as to justify the added circuit development effort.

In the preceding sections we have developed design formulas and
design graphs which enable us to estimate with fair quantitative ac-
curacy the size and weight of various magnetic structures to satisfy
torque requirements as specified. Intercomparison of the air-core coil
and electromagnet offers an interesting illustration of the use of these
techniques. We consider a typical example which assumes a spherical
satellite of 45 inches effective diameter in which there is required an
available magnetic moment of 0.2 lb-in per oersted which can be

turned on or off at will. It is further assumed that an upper limit of
nine pounds weight and twelve watts power consumption is to be
imposed upon the magnetic circuitry.

First, we assume that an air-core coil is laid out around the equator
of the satellite to enclose maximum area, and that this area is

formula for aluminum from Table II we can show that the power X
weight product is

power X weight = 0.119 * 10°°(141)°(210)” = 105.
If we use the total weight allowance of nine pounds for the winding
then the required power is

This is within the permitted 12 watts, so we have shown that it is
feasible to use an equatorial coil.

Turning now to the design of the electromagnet, one inserts T,/H, =
0.2 and (B — H) = 10,000 in (7) to show that the required volume of
core material is

The characteristics of the winding, however, are closely dependent
upon the aspect ratio of the core. To illustrate this point we consider
two shapes, one to be 10 inches long, the other to be 45 inches long
to just fit in along the spin axis of the satellite. For the 10-inch core,

If we assume the average diameter of the winding is 1.6 inches then
the average length of turn, P = 5.05 inches. We insert these numbers
into the power X weight formula for aluminum in Table IT and obtain

If we assume that the average diameter of the winding is 0.85 inch,
then the average length of turn, P = 2.67 inches, and

So we may use 0.5 pound of winding to bring the total weight only to
five pounds, and the required power will be only 0.78 watt. This
illustrates the advantage of the long slender electromagnet over the
short one for purposes of producing torque.

It is evident that the specified conditions of the example can be
satisfied by the equatorial coil or by the long slender electromagnet.
On the basis of the calculated results one might well prefer the
electromagnet, which satisfies the requirements with a substantial
saving of power and weight. However, other factors must be con-
sidered. It is not likely to be convenient to mount the magnet full
length along the spin axis because of interference with other equipment.
In a core of this length, a very small amount of residual magnetization
after removal of current will result in a considerable magnetic mo-
ment, rather than the desired zero magnetic moment which is charac-
teristic of the de-energized air-core coil. The weight distribution of
the electromagnet along the spin axis decreases the spin stability,
while the weight distribution of the equatorial coil enhances the spin

stability of the satellite. For these and other reasons the equatorial
coil remains a favored method of attitude control in the communica-
tions satellite.

There have been various proposals to provide friction damping of
roll or precession of the spin axis by mounting a small magnet within
a hollow spherical enclosure attached to the satellite. The magnet
would tend to maintain its alignment in the earth’s field and to pro-
vide damping through friction contact with the interior of the sphere.
For this application, if it exists, or for any similar application, one
would wish to design for maximum normalized torque and minimum
normalized period of oscillation in the field and within the confines
of the sphere.

5.1 Design for Maximum Moment within Limiting Spheres
Referring to (6) and dividing through by D* where D is diameter
of sphere in inches, and D? = (d?+/?)%/?,
i
H,D°
The relation expressed by (20) is displayed in Fig. 7 for the same

g. 7— Normalized moment within limiting sphere; (a) ceramic, (b) Alnico
9, co “Alnico 5. :

This normalized period of oscillation is displayed graphically as a
function of l/d in Fig. 8. In designing a magnet for friction damping

Fig. 8 — Period of oscillation within limiting sphere; (a) ceramic, (b) Alnico 9,
(c) Alnico 5. ,

one would probably select the best compromise between maximum
torque displayed in Fig. 7 and minimum period of oscillation as shown
in Fig. 8. This would suggest the use of Alnico 9 and design for J/d & 1.5.

The spheroids are a family of solids the surfaces of which are
generated by ellipses revolving around an axis. Revolution around a
major axis generates a prolate spheroid for which //d > 1. Revolution
around a minor axis generates an oblate spheroid for which l/d < 1.
Revolution of a circle around a diameter generates a sphere for
which l/d = 1.

Values of load factor, U, for spheroids are plotted in Fig. 1. The
volume of the spheroid is only two thirds that of a cylinder having
the same I and d, so (11) becomes, for spheroids,

From solutions of (26) one may plot curves for normalized torque-
weight ratio. In Fig. 9 we show a curve for spheroids of Alnico to-
gether with a curve for cylinders of Alnico borrowed from Fig. 5.
While the spheroids show a somewhat better torque-weight ratio
than the cylinders, it is doubtful whether the advantage is sufficient
to offset the added cost of shaping and mounting.

The sphere might have unique advantages mounted in a spherical
enclosure for friction damping. For the sphere of diameter D, one may
rewrite (6)

This is an expression for the total normalized torque that can be
packed into a specified spherical enclosure. Solutions of (28) for
various magnet materials are collected in Table III. The moment of
inertia of the sphere is

Equations (26), (28), and (31) define for the sphere the same
normalized quantities which are plotted for the cylinder in Figs. 5, 7,
and 8. Solutions of these equations for various magnet materials are
listed in Table III. The combination of the table and the three figures
gives all the information required to select the preferred material and
geometry for a specified application and to arrive at a quantitative
design of the magnet.

Satellites with spin stabilization introduce two magnetic measuring
problems—measurement of “drag” and measurement of residual mo-
ment. The “drag” results from eddy currents induced in the rotating
metal shell of the satellite by the earth’s magnetic field. The energy
dissipated in these eddy currents must be derived from the rotational
energy of the satellite, and there results a decay of the spin rate.
One wishes to evaluate the rate of this decay to forecast when the
spin rate will fall below the minimum required for stability. The
moment measurement is to detect any residual magnetic moment
perpendicular to the axis of spin which will interact with the earth’s
magnetic field to induce precession of the spin axis. After an accurate
measurement this moment is canceled out by mounting in the satellite
a small permanent magnet of equal moment and opposite polarity.
Both measurements—drag and moment—can be made conveniently
with a specially designed coil array.

The drag test requires a reasonably uniform field over the volume
of the satellite. A paper analysis reveals that this can be provided
by an array of coils of reasonable size with a particular distribution
of ampere turns. Two coils, each of radius r, and of N turns are spaced
-+r/4 from an assumed zero point on a common axis. Two other coils,
each of radius r, and 7N/3 turns, are spaced +r from the assumed

zero along the common axis. The coils are connected in series to run
at the same current so that the outer coils have effectively 7N/3 times
as many ampere turns as the inner pair. The arrangement of coils and
the resulting distribution of field along the axis are shown in Fig. 10.

It was established by measurements that the region of uniform
field extended out radially from the axis to include a spherical volume,
the radius of which is roughly two thirds the radius of the coils.
Hence, an array of coils five feet in diameter easily provided uniform
field over the volume of the satellite. (If a conventional Helmholtz
array were used the coils would have to be about 10 feet in diameter
to achieve reasonably uniform field over the same volume.) This
array was mounted on a turntable and rotated around the satellite
which was supported by a calibrated torque suspension. From the
result of this drag measurement it was possible to calculate the rate
of decay of satellite spin in the earth’s magnetic field.

The magnetic moment perpendicular to the spin axis of the satellite
was measured by rotating 1t within a coil array similar to the one
used for drag tests except that the windings were connected to an inte-
grating fluxmeter. One reasons intuitively that if a magnetic object is
aligned parallel to the axis of the coils and rotated 180°, it will give a
deflection of the integrating fluxmeter proportional to the moment,
and that the proportionality constant will be unaffected by the position

of the magnet in the array as long as it is within the volume in which
the array produces uniform field. This intuitive reasoning has been
confirmed by various measurements. The proportionality constant
for the array is established by calibration with a small air-coil, of
known NIA for which the moment can be calculated from (2). A
two-to-one scale down of the array has proved to be convenient for
bench measurements of magnetic moment of small magnetic objects.

VIII. ACKNOWLEDGMENT
The author is indebted to Mr. L. Rongved for stimulating discus-
sions of the dynamics of the orbiting satellite.

APPENDIX
Definition of Symbols
The following letter symbols have been adopted for use in this paper.

Rs = shortening ratio, from recession of poles.
LT = torque between magnet and perpendicular field.
T = period of mechanical oscillation.

1. Okoshi, Takanori, Demagnetizing Factors of Rods and Tubes Computed from
Analog Measurements, J.A.P., 36, August, 1965, pp. 2382-2387.

We consider, from a number of different viewpoints, the tensor coefficients
which describe second harmonic generation, optical rectification, and the
Pockels or linear electro-optic effect in acentric crystals. Stationary per-
turbation theory 1s used to calculate the low-frequency limit of the intrinsic
electronic nonlinearity neglecting all effects due to local fields or lattice
polarization. Solid methane is used as an example and the result used to
estimate the coefficient in hexamethylene tetramine. The calculated result
is within a factor of 2 of the expervmental figure. The method 1s susceptible
lo further refinement and, since rt requires only a knowledge of ground
state wave functions, and is essentially very simple, it appears to offer a
useful approach to the calculation of the coefficients.

The classical anharmonic oscillator model 1s briefly covered and the model
is related to a quantal treatment. We find that the anharmonic potential
used in the model is directly related to the actual crystalline potential. It
can also be related to the charge distribution in the electronic ground state.

Local field corrections and the effects of lattice polarization are presented.
These alter the nonlinear properties in a sumple and obvious way, but one
which has been misunderstood in some of the literature.

Our results form a theoretical background to Muller's empirical rule
relating the nonlinear coefficients to the linear susceptibilities. An extensive
table of Muller-reduced tensor coefficients collated from the published litera-
ture is presented.

Finally, we draw together some of the threads of the previous sections.
An appendix deals with the vexing question of definitions.

Second harmonic generation, optical rectification, and the linear
electro-optic effect are particular aspects of a process in which two
fields, H’e’* and E7ve'™', generate a polarization

Our concern is with the tensor coefficients d%"." which (Nye’) necessarily
vanish in centric (centrosymmetric) crystals and which, in acentric
crystals, are subject to symmetry restrictions, (Kleinman’) which often
leave only one or two independent components.

Iixperimentally, the values of the allowed components of d in different
materials and at different frequencies range from about 2.10°*° esu
(cm/stat volt) to about 6.10°° esu. This range may be contrasted with
the linear optical susceptibility ~ which is between 0.1 and 0.3 for the
vast majority of materials and only quite exceptionally exceeds unity.
There 1s, however, a connection between the tensor d and x which is
expressed by an important empirical rule due to Miller.* If we write
dh” as

ty = XeexGXeAsa ’ (2)
where x,° 1s the 227 component of x at a frequency a, and if we have chosen
& principal axis system for x, then the allowed components of A;;,
for all effects and all materials are similar in magnitude. We shall see
in a later section that for very many materials in both the visible and
10 w region of the spectrum (Patel*), A;;, is near 3 X 10°° esu. No
materials with A above 20 X 10°° esu have yet been found and very
few are known to have a value below 0.2 X 10°° esu. In the case of
NH,H,PO, where the best measurements of s.h.g., optical rectification
and the electro-optic effect are available (Francois,’ Ward,’ Carpenter’)
the value of A,.; from all three effects is 3 X 10°° esu within the experi-
mental error of 15 percent. The fact that s.h.g., a purely optical effect,
leads to the same value of A as rectification and the electro-optic effect
indicates quite clearly that the basic mechanism of the nonlinearities
is common to all three effects and must therefore reside in the electronic
motion of the system. In the next section, we shall concentrate on this
aspect of the problem and neglect the effects of local fields and lattice
polarization.

A number of authors (see Section IV for references) have given quantal
treatments of the optical nonlinearities whose end result 1s an expression
for the coefficients d*?,” in terms of sums of rather inaccessible matrix
elements. Useful as these expressions are, in establishing some of the
general properties of the coefficients, they are not a practical step on the
road to calculating the coefficients from other empirical quantities. At
the other extreme, the classical anharmonic oscillator model has been
used to demonstrate some of the qualitative features of nonlinear be-
havior (see Section III). This treatment, though simple and appealing,
suffers from the defect that the relation between the parameters of

the model and those of the real system is obscure. In Section IV, we
shall remedy this defect and show that the two approaches are closely
related.

First, however, we give an approximate method of calculating the
low-frequency limit of the coefficients from stationary perturbation
theory in a form in which it has been successfully applied to the linear
properties of n electron atoms (see, e.g., Dalgarno’).

At low frequencies, 1.e., well below any electronic resonance we can
use stationary perturbation theory to calculate, to arbitrary order in
the applied field # the energy W of the ground state. The polarization
is then given by

ow
P,;=- al, (3)
Weshall assume that we are dealing with a crystal containing N identical
atoms or molecules in unit volume whose individual ground state energies
are w, so that W = Nw.

m=1
is the dipole moment operator of the molecule, and the sum extends over
all n valence electrons. We can neglect the core electrons because of
their high binding energies. If we expand w in increasing order in # as

the term w, gives the permanent dipole moment of the molecule, w,
gives the linear susceptibility and w; gives a polarization quadratic in
FE which leads to the desired nonlinear coefficients. The electric field
will perturb the state function y and we shall write the perturbed func-
tion as either

Knowledge of y or | ) to first order in # is sufficient to determine w, , wz ,
and ws; for

Moreover, the correct value of y, or | 1) is determined by the require-
ment that it minimize w. Thus, we can obtain | 1) by a variational
procedure and the only element of choice left to us is that of the trial
wave function.

Minimizing w is equivalent (see Dalgarno and Lewis’) to the simpler
problem of minimizing

| ABA |0) ~~~ (0| AAA {0)
The unperturbed Hamiltonian of the system is of the form
__l Sy:
A, 2 Im pe Wa, + V. (13)
and so, in the denominator of (12),
Hh = hl, + eB ie a (14)
Thus,
“ eh’
(0| bith 0) = ——= fy, DO Btw B-DVadedr, — (15)

where dr is an element of configuration space and we have used H | 0) =
0. Equation (15) can be written as

where the discarded first integration part vanishes at the limits, if
these are infinite, or if they are the boundary of a cell in periodic lattice,
provided only that H does not vary appreciably within a cell (dipole
approximation).

(0| AHA |0) = I
a somewhat unfamiliar form of the sum rule. If, on the other hand,
we are dealing with overlapping molecules in a periodic lattice, the
variational problem is to minimize the contribution to w from a single
cell of the lattice. Thus, in (11) and all succeeding equations, the inte-
grals implied by the expectation values are to be taken only over the
interior of a cell. This will also apply to all integrals involved in evalua-
ting w, = (1 |h| 0) and w, = (1 | # | 1). In this case (18) remains un-
changed. This can be shown to be a general consequence of time reversal
invariance and the commutation rule

4 (r2\2
ame (As (24)
For the H atom, this gives a = 4a} instead of the correct value 4.5a;,
while for the helium atom, taking an effective nuclear charge Z =
27/16 gives 1.8 X 10°*° ces. The experimental value is 2.1 X 10~*° ces.
In general, (24) is a lower limit to a, if we evaluate (X’) correctly as the
expectation value of the mean square moment of all the electrons. If
the electrons are uncorrelated

where (£’) refers to one electron. We used this procedure in helium since
the two electrons are in orthogonal spin states and are automatically
uncorrelated. In more complicated atoms correlation exists and almost
always results in

(X*) < n(é’) (26)
since electrons repel each other. Thus, while (24) is a lower limit we
cannot say anything about the sign of the error in

We note, in passing, that, in a solid with overlapping molecules, a the
polarizability is large. This leads to an element of instability in the situa-
tion for as a increases the screening of the coulomb potential becomes
more effective and the electrons less localized leading to a further in-
crease in a and eventually metallic behavior. For this reason, most
materials, which are not regular insulators, are metals. Those rare
materials which have values of Na appreciably greater than 0.3 (n > 2.2)
owe their existence to a rather delicate balance of forces.
The third-order energy is

NONLINEAR OPTICAL COEFFICIENTS 919
With N molecules in unit volume this gives a nonlinear coefficient
2S SS ey (31)

Equation (31) is the central result of this section. It expresses d;;, in
terms of the linear (corrected for local fields) susceptibility x and a
cubic moment (third-order semi-invariant) of the electronic distribution
in the ground state.

If we neglect overlap and, for simplicity, also assume that the electrons
are uncorrelated so that 7';;, = nt,;;, where ¢;;, refers to a single electron
we have

and 7';;, 1s now apart from numerical factors the octupole moment of
the charge distribution. If the electron density in the molecule is p(r)

If we account for local fields through a Lorentz correction the correct
value of x to insert in (33) is obtained from

At first sight (83) seems to imply that d is proportional to x in conflict
with Miller’s rule. However, x depends on N/n(R?Y + nN’ and N is
inversely proportional to (r’)? so that x = nr, while d = r*. Thus, d is
in fact more nearly proportional to x* than x.

We now consider as an example, the tetrahedral molecule methane
CH, , which crystallizes in the tetrahedral space group F43m with a
lattice parameter ~6 A and a molar volume of 32 ces. If we take Car-
tesian axes along the sides of the cubic cell, the bonds point in the 111,

and tetrahedrally related, 111, 111, 111, directions. From symmetry,
there is only one independent component of d;;, , in which all the sub-
scripts are unequal.

The shortest c-c distance is 4.2 A and from the size of the free molecule
we conclude that overlap is unimportant.

Turner, Saturno, Hauk and Parr’’ haveused one center wave functions
to calculate the electronic density p in the molecule. F'rom this we can
obtain f,.3 using (34).

and this is not very sensitive to the limits of integration. The experi-
mental molar susceptibility of CH, is 1.6 ces and so x = 0.05. Thus, if
we neglect correlations between the eight valence electrons we have
from (33)

In a similar way, neglecting correlations, we can calculate the molar
susceptibility from (27). Turner et al’s charge density leads to

and so, with cight valence electrons, we obtain a = 6.5 X 10°” and
x = a molar susceptibility of 3.9 ccs, rather over twice the experimental
value.

This is a clear indication that the electrons are correlated. However,
the correlation enters twice in x but only once in d (since we have ex-
pressed d in terms of the experimental x). Thus, d problaby lies between
2x 10° and 3 X 107° esu.

To see whether 3 X 107° esu is a reasonable value for d,23 we compute
the Miller reduced tensor d/x* = Ayo3 = 24 X 107° esu.

This is quite exceptionally high. Most materials have allowed com-
ponents of A near 3 X 107° esu and only one coefficient in LiNbO;
(9 X 10°) and Ajo; in hexamethylene tetramine (15 X 107°) approach
this value.

However, we believe it is in fact not far wrong. In most materials
geometric factors conspire to reduce d by various factors of cos @ and
the atomic groups are in the first instance less aspherical than CH, .
In CH, the effects of every electron are directly additive.

Hexamethylene tetramine (HMT), the other exception to Muiller’s
rule, is, like methane, a tetrahedral molecule N,(CH.), in a tetrahedral
143m crystal. The 4 nitrogen atoms form the 111 1II, 111, 111, corners

of a regular tetrahedron and the CH, groups occupy the edges but the
N-C-N bonds are bent outwards in such a way that all the angles are
very nearly tetrahedral. The carbon atoms occupy the six sites 2, 0, 0
etc. (see Kitaigorodskii’’).

The refractivity of the molecule as a whole can be very satisfactorily
accounted for by a system of additive bond refractions. (See Lelevre’”
for a review of bond refractions.) The three basic units are 12 C-—H
bonds pointing in the 1, 1, 1, and related directions, 4 nonbonding orbitals
on the nitrogen atoms pointing in the 111 and related directions, and
12 N-C bonds in the 111 directions.

Since the refractivities are additive, these units appear to act inde-
pendently in determining the molar refractivity. Le Fevre (loc. cit.) gives
values of R = (47/3)La (where L is Avagadro’s number) of 2.8 ces for
each unbonded nitrogen pair, 1.65 ces for each C—H bond and 0.62 ces
for each N-C bond. Thus, the N—C bonds make a rather small con-
tribution to x, and probably even less to d since they have an approxi-
mate inversion centre at the centre of the bond (C and N are similar
atoms as compared with C and H). We therefore neglect them.

The 12 CH bonds in the 111 direction are roughly equivalent to 3
methane molecules in the molecular volume 105 ccs, and further the
electrons will be less correlated than in methane. Thus, their contribution
to dy23 18

To calculate the effect of the nonbonding nitrogen electrons we assume
that they occupy SP* hybrid orbitals directed along 111 etc. with Slater
radial wave functions Ar exp (—2.5r/2a,). It is then straightforward
to show that for one electron

Thus, the total value of dis; is —4.5 X 10°-° esu. This could be slightly
increased by the effects of atomic overlap, and possibly by contributions
from the N-C bonds. It could be either increased or decreased by elec-
tron correlations on individual CH, groups. The experimental values

for the electro-optic effect Heilmeyer’® and second harmonic generation,
Heilmeyer, Ockman, Braunstein, and Kramer,’* when corrected for local
field effects using a Lorentz factor, both give d = +8.2 X 10°” esu.
Thus, our calculation is within a factor 2 of the observed value.

This method is therefore capable, in simple cases, of predicting the
magnitude of d rather successfully. Moreover, the experimental value
of d for HMT suggests that we were correct in assuming that CH,
will have an anomalously large reduced tensor Aj); .

The fact that the division of a complex molecule such as HMT into
simple components leads to a reasonable value for d leads us to hope
that a similar procedure will be possible in other cases. It might then
be possible to assign empirical values of d to basic components such as
the C-H bond or the N: nonbonding pair, and to combine these ad-
ditively (with a proper attention to geometry) to predict the values of
d for even more complex molecules. This would not be surprising since
a similar procedure (see LeFevre loc. cit.) works very satisfactorily for
the linear susceptibilities.

It is then obvious that large nonlinear effects will only result, if the
molecule contains polarizable groups disposed in an arrangement which
results in a ground state of, far from inversion, symmetry. The large
value of A in HMT results from the fortunate coincidence that the
most polarizable components are themselves strongly asymmetric and
so oriented that their effects are additive. The much smaller values of
A commonly observed can then be explained as partly due to no group
in the crystal being quite so asymmetric as N: or CH. in HMT and
partly due to unfavorable geometric relations between the groups. I’or
example, if our approach is correct we should expect the analogous
compound adamantane (CH),(CH:), in which the nitrogens are re-
placed by CH groups with the CH bond along 111, etc. to have a djo3
appropriate to 2 (=3 — 1) CH, molecules in 105 ccs, ie., di23 2 2 X 107°
esu or about half the value for HMT.

Exceptionally small values of A will occur in materials where most
of the molecule possesses local inversion symmetry, so that only a
fraction of the molecule contributes to d, while the whole molecule
contributes to x. We shall consider an example of this in a later section.

Overlap between adjacent molecules is neccessarily bound to lend
further uncertainty to the calculation in materials with a pronounced
band structure, but it seems possible that rough approximations should
be obtainable from, for example, the relation between bandgap and
the corresponding separation in the isolated atoms. In fact, since what
we actually require is 7',;;,/n, which, if the electrons are uncorrelated,

Finally, we may remark that very much better approximations to
d;;, can obviously be made if we know the ground state wave function
explicitly and also use more sophisticated trial wave functions in the
variational calculation. It is at first sight surprising that a knowledge
of the ground state wave function alone is sufficient to determine x and
d which, in the more usual treatments involve the properties of excited
states. However, we should remember that a knowledge of the exact
ground state wave function is, except in pathological circumstances,
sufficient to determine the unperturbed Hamiltonian; thus, the whole
spectrum of states.

Although the considerations of the preceding section are sufficient
to determine the magnitude of d at low frequencies, they offer little
guide to the variation of d with frequency and, if recast in terms of time
dependent perturbation theory they tend to lose their attractive sim-
plicity. In the next section we shall show that a more familiar form of
time dependent theory leads to results which can be represented in
terms of a classical anharmonic oscillator model. Here, we discuss the
properties of the model itself.

We assume that unit volume of the material contains N optical elec-
trons which move in a potential

V = 3mQja; + V i jnUsXjXy, , (43)
where a sum over repeated subscripts is implied. The potential V;;,

There will be a similar response to a field H%e'”’ and, if we introduce
these responses back into the nonlinear term in (44) we obtain a re-
sponse at the sum frequency a = 6 + ¥y given by

3V 55,Ne" 1 1 1
oe faeearaae |
Thus, the symmetry of d;;, mimics that of V;;, if we neglect the res-
onance denominators. |

The linear susceptibility obtained from (45) is the familiar expression
» Ne 1

If we assume that V;;, is electrostatic in origin its order of magnitude
will be e’/d* where d is an atomic spacing and we shall also have Nd’ ~ 1.
Thus,

With d equal to 2 A this is 2.5 X 107° esu, about the mean value of A
for most materials. In a later section, we shall give another estimate of A.

The potential V;;,v;7;2, distorts the shape of the ground state of
the harmonic oscillator and as a result the system acquires a cubic
moment ¢;;, [defined in (82)] which we now calculate.

Let | 0) represent the unperturbed ground state wave function in
the absence of the anharmonic term and | 7p) be an excited state, then
the perturbed wave function is

The expectation values of even operators such as (27), (v;a7;) are un-
altered by V, while the expectation value of an odd operator such as
XU; OY ©;2;X, 18 given by

Dp hw,
It will suffice if we calculate t,;;, with ¢ # 7 # k. Since (;7;) = 0 if
t ~ i we only require (7;%;x7,) and contributions to this come only from
the 6 = 3! terms in V with2 + 7 ¥ k. The only state which contributes
to the sum is |p) = | 1, 1, 1) with an energy A(Q, + Q, + Q;). The matrix

holds for all the components of ¢;,; .
If we substitute this relation in (47) and take the limit as afy — 0
we obtain

This is twice the value obtained in (83) because there we treated {,;,
as a fixed property of the ground state which was then perturbed by E;
whereas here we have considered an even ground state perturbed by #
and a fixed potential.

Thus, if the real system has a cubic moment #¢;;, in the ground state,
the equivalent anharmonic oscillator model requires an anharmonic
potential

3 a4
tik = ee bs in (56)
and this will result in a cubic moment ¢/,, = 3¢;;, in the oscillator ground
state.

in molecular crystals of strongly covalent compounds such as CH,.
But, in ionic crystals it may be more sensible to consider the ions as
spheres perturbed by a crystal potential V‘,;,. In a later section we
shall see that there is a simple relation between the model potential
and V¢;, -

The classical anharmonic oscillator model has previously been used
by Bloembergen,’’ Garrett and Robinson”® and Kurtz’’ to give a qualita-
tive account of nonlinear phenomena. The latter authors also discuss
in some detail its relation to Miller’s rule.

Obviously, the model is the nonlinear analogue of the classical har-
monic oscillator model used with such success for the last 60 years in
the discussion of linear behavior such as dispersion, and, just as the
harmonic model is directly related to the results of a quantum mechan-
ical treatment, we may expect the anharmonic oscillator to have a
similar basis. In the next section we explore this relation.

Ducuing and Pershan,” Butcher and McLean,’® Kelley,’® Cheng and
Miller”’ and Ward” have given rigorous quantal treatments of optical
nonlinearities in solids. We select an expression due to Armstrong,
et al (loc. cit.) which expresses the nonlinear coefficients in terms of the
energies fw, of excited states and the matrix elements (0 | 2; | 7),
(p | x; | q), ete. of the dipole operator between states. The ground state
is (0 |.

This expression is valid, cither for an assembly of N isolated atoms
in unit volume or, in the dipole approximation, for a real solid where
the wave functions overlap. In the latter case, the solid must be divided
into cells of the periodic lattice, and N is then the density of cells, while
the matrix elements are to be evaluated only over the interior of a cell.
The periodicity of the lattice ensures that contributions from parts of
the wave function outside a cell cancel in the crystal as a whole.

To avoid a plethora of subscripts we let each of x, y, and z serve to
represent any one of the components and we can then write the expres-
sion for d as

This expression vanishes if the states | 0), | p), etc. have adefinite parity,
its value therefore depends on the existence of matrix elements whose
presence is contingent on the absence of inversion symmetry. For this
reason, it 1s almost impossible to make an informed guess about its
magnitude or behavior.

A familiar approximation to x is obtained if we note that in (58) the
variation of the summand with | p) is almost exclusively due to the
matrix elements. These not only obey selection rules, but also decrease
rapidly in magnitude as the state | ») increases in energy, and therefore
overlaps the ground state less and less. For example, in the H atom with
a 1S ground state the matrix element x,, vanishes unless p is one of
the states 2P, 3P, etc. Moreover, as we go from the 2P state to the 8P
state x,,7,, decreases by over a hundredfold. At the same time, w,
changes by less than 30 percent. Thus, except near a resonance, we can
treat w, as a constant 2, somewhere near the first allowed transition and
write (3.2) as

where the primed sum excludes p = 0. Now
De’ ZosYv0 = Dy Lose — LocYoo = (LY)oo — LooYoo
= ((« — (x))(y — (y)) (60)
where a ( ) denotes a ground state expectation value. Thus,

We shall not pursue the further manipulations of (61) using the sum
rule which lead back to (27) but we remark that in many cases a form
such as (61) for x, involving a single Sellmeier or classical oscillator
term, gives an excellent account of optical dispersion, and that when

applied to the hydrogen atom, with iQ set equal to 2e’/a, the 1S — 2P
energy, it leads to a value of x at low frequencies

Before we can adopt a similar procedure with the nonlinear coefficient
we must first satisfy ourselves that there is no essential difference be-
tween a sum with three matrix clements and one with two. In x all
matrix elements terminate on | 0) but in d it is quite possible in a term
such a8 %o,Ype2ao » With p gq corresponding to highly excited states, of
great spatial extent, that the term y,, may be large enough to compensate
for the smallness of x,,2,. . If this were the case it would be possible for
the exact value of the sum to depend critically on cancellations between
large terms involving highly excited states. The removal of the frequen-
cles w, , etc. as a single average would then have disastrous results on
the sum.

We will advance three arguments why this is unlikely. Consider first
an even higher-order calculation, that of the fourth-order Stark shift
of the ground state of atomic hydrogen due to a field F. In atomic
units this is given by an exact calculation (Dalgarno’’) as

W? = —33359* ~ —56F". (62)
We can also express W“ (Dalgarno, loc. cit.) as
?
Ww? er — > ye 4 L opt pnglar& ro a by Lon po a come Pe.
p q r WWW> p Wp Wy

This is close to the correct result (36), but despite the fact that we have
taken the lowest possible value of © it is too small. This is a clear indica-
tion that some cancellation of higher terms, which we have aggravated
by our cavalier treatment of w, , etc., is occurring. This is not surprising
for, if in the triple sum we consider the lowest possible sequence of levels
1S 2P 2S 2P 18 for which wo, = o, = w, = Q the product of the matrix
elements is 5 a.u. while for the sequence 1S 8P 8S 8P 1S the product

is 10 a.u. made up of a contribution of 0.0033 from the two 1S 8P
elements and 3000 from the 8S 8P elements.

However, this is not quite so serious as it appears, for in a real solid
no matrix element can exceed the linear dimensions of a cell say 5 a.u.
so that the product in the low transition would remain at 5 a.u. while
the product for the upper transition would be reduced to 0.08.

Our final argument is empirical. If cancellations between large terms
are critically important, the relevant feature of our procedure is the
change in the ratio w,/w, it causes for highly excited neighboring states.
In hydrogen the ratio of the 1S — 8P energy to the 1S — 7P energy
is 1.005 and we replace this by unity. In a time dependent theory res-
onance denominators appear, and if the sum is really so critically
balanced, we expect the observed quantity, in this case the hyper-
polarizability, to vary rapidly with frequency when » ~ 0.005 0”,
i.e., at a frequency 10 times lower than the first absorption edge. In
nonlinear optics, no such variation is observed until one of the frequen-
cles approaches much more closely (about 70 percent) to the absorption
edge (Chang, Ducuing, and Bloembergen”*).

Taken together these arguments give us reasonable grounds for
hoping that the sums will not bite us if we remove w, , etc. from under
the summation sign.

In the sum in (57) there is no restriction on 7 or q, in particular terms
with either p = 0 or g = 0 occur. These will lead to trouble if we attempt
to approximate the sums as they stand. We therefore first segregate
all such terms. Let { } denote the entire summand in (57), then

W, —~ VY Yor Wr — Q@
a Zortro _ B oe ATi ra
Yoo B pie: ws __ eine vidos 5 2 — Bp
8 Ure ye cseoeeee :
Zoo 7 00 i eee 65
a yea pt ued Ei fet (65)

Three single sums remain unprimed, but because a = 8 + vy the terms
with r = 0 cancel and so we may regard all the sums as primed.

We can now remove w, , w, and w, as a single average Q, and this leads
to an expression containing terms such as

Each of the sums on the right is now a ground state expectation value.
When all the terms are collected together we obtain

in terms of the, by now, familiar cubic moment 7’,,,. This expression
bears an obvious resemblance to (61) for x.

Our expression (66) or (67) would be very nearly exact if all the optical
levels had very nearly the same energy. It would then correspond to
the fictitious two level system (see Refs. 15, 16, 18) often used to oblit-
erate some of the intractable features of (57). Unlike this model, how-
ever, our expression retains the geometry of the system implicit in the
selection and sum rules.

Equation (66) is possibly valid up to a frequency where one of a, 8,
or y approaches the first allowed transition frequency. At somewhat
lower frequencies, it is legitimate to drop the term By — a’ in the numer-
ator. This then allows us to make a further generalization at no increase
in complexity.

By removing w, and w, from (57) as a single average we have tacitly
neglected the possibility that the system might be birefringent. We
can remedy this by noting that in (57) each frequency , or a,
is uniquely associated with a matrix element such as x,, or 2,. which
terminates on the ground state | 0) and therefore also appears in x.
Thus, we can consistently introduce three averages 2, , 2, , and Q, as-
sociated with correspondingly polarized transitions. If we follow this
process through all its tedious ramifications, we find that, except in
the term By — a’ which we are omitting, it leads to the surprisingly
simple result that D(Q,a,8,7) is replaced by

If we compare this with the result for the classical anharmonic oscillator
obtained by combining (54) with (47) we see that they are identical
except for a factor 2 which once again arises because in one case we
assumed that T,,, was a fixed parameter while in the other it was V,,, .

We now see that the classical model is equivalent to the quantal
treatment, except near a resonance, in the following sense.

If we construct the model, by choosing Q, , 2, , and 2, to give the
correct linear properties then we must choose the anharmonic term in
the potential to produce a cubic moment in the ground state of the model
equal to $ the corresponding moment in the real system. The dynamical
properties of the two systems are then equivalent and the model can
be used to treat more complicated systems where the quantal treatment
is too difficult.

We now consider the relation of V/,, to the actual potential respon-
sible for the existence of 7',,,. Obviously, the relation is obtained by
requiring that both potentials yield the same cubic moment, one in the
model, the other in the real system. In this case, there will be no factor
of 2.

For simplicity, we consider only a system which is isotropic before
the application of the anharmonic potential. Further, we restrict our-
selves to atoms in which there is only one valence electron. Our results
will, however, be directly applicable to atoms with more electrons if
we can neglect electron correlations.

We already have an expression for the oscillator (54) which we can
write as
Vou

V = Vi;.0;2;2,, the crystal potential, satisfies Laplace’s equation.
We will consider a more general potential of the form

where P’?} is an associated Legendre polynomial normalized to unity.
This potential satisfies Laplace’s equation only if n = lL.

If the unperturbed ground state wave function is Y the first-order
correction y, due to V satisfies

Since 7 is an odd moment we need only consider odd terms in V (in
fact only / = 1 andl = 3) and for these /, vanishes since yp has definite

parity.
We let
vi = flo (73)
and then
(Hy — Eo)fvo = —Vo
but, since |
Ho -* Wa
this leads to
Vif + 2VP-V log vo = V. (7-4)

and in this new ground state we can easily evaluate expectation values
such as

In evaluating 7;;, we shall need (z;), (x7), and (x;7,;2,). The even
moments are unchanged by V and we obtain the odd moments by ex-
panding x; and x;2;x, in terms of Legendre polynomials and powers
of r.

We omit most of the gruesome details of the calculation, and further
restrict V to contain only terms of the type

The term in V;} which does not satisfy Laplace’s equation is necessary
to obtain the most general form of the cubic part of the potential
V ;;42%,2;2, . This contains 10 independent parameters while P’} has only
7. The missing 3 are supplied by P’.

and it is easy to check that the coefficients of S; and X,; vanish, so that
we recover (70).

If V satisfies Laplace’s equation S; = O and in the absence of an
internal field X; every component is given by

Thus, in this case 7';;, and the reduced Miller tensor A;;, have the same
symmetry as V;,;,. Therefore, since S; = 0 we have

Ass3 = —2Ag11 . (94)
This relation is rather well obeyed by the coefficients for the 6-mm

crystals listed in the Table I. Signs are available only for the electro-optic

coefficients and so the s.h.g. results represent moduli only. References
to the experimental data are given in conjunction with later tables.
In the case of the electro-optic data, the experimental figure is for Aj13
and we have assumed that Kleinman’s rule (Kleinman’) holds and that
this 1s equal to A;,,. Except for s.h.g. in ZnO the ratio is —2 within
the experimental error.

and the expected ratio is +3. In crystals where both terms occur in
V with arbitrary strength, any value of the ratio is possible. This is
observed in the ferro-electric crystals BaTiO, ratio +4 and LiNbO,
where it is +1.7 for the electro-optic effect and +11 for s.h.g. It is
perhaps somewhat surprising that the ratio is so exactly —2 in the
6-mm crystals since this is a polar point group and an internal field
X3 1s not forbidden by symmetry.

If V does not satisfy Laplace’s equation, (it need only satisfy Poisson’s
equation) there is no direct relation between the components of T,;,
and those of V;;, even in the absence of a field X;, although, since
xyz 1s a spherical harmonic, we still have

T 123 = 358333 Vins : (96)
However, since the coefficients of S; vanish for the harmonic oscillator

we may expect them to be small in other cases. We gain some support
for this view by considering the hydrogen atom for which

The coefficient of V;;, in each term of 7';;, is then —315/16(a/ha,)
while the coefficient of S; in 7';;; is a factor 23/200 smaller. In 7;;; it
is 23/600 smaller. Thus, in the absence of X; the non-Laplacean terms
in V cause no more than an 11 percent departure from the relation

Since we expect the dominant terms in V to satisfy Laplace’s equation
it appears that 7';;,, A.;, and the model potential V/,, will be very
nearly proportional to the corresponding terms in V.

The potential V’ required in the model is related to the crystal
potential by

Tor a hydrogen atom this gives V/,, ~ 5V;;, thus, insofar as real atoms
behave like hydrogen atoms, a model with the same spatial extent
a = a, and the same first allowed transition w,  w, will require a
potential roughly five times as strong as the actual potential. This
reflects the obvious fact that a harmonic oscillator is a stiffer system
with more sharply localized (Y ~ e7"’) wave functions than an atom
(y ye").

We have now shown that, with an appropriate choice of parameters
a classical anharmonic oscillator model is a very good approximation
to the intrinsic electronic nonlinearities of real systems.

In the next section, we use the model to consider the effect of lattice
polarizability which we have so far neglected.

We have already remarked in the introduction that the seat of the
nonlinearities resides in the electronic motion. It is, however, consider-
ably modified by local field corrections and in the case of optical rectifica-
tion and the linear electro-optic effect by lattice polarization.

Miller’s rule states that d?{" is proportional to the product of the
observed linear susceptibilities X 4 , etc. at the appropriate frequencies.
If one of these is de we are to take the actual de susceptibility and not
the extrapolated long wavelength limit of the optical susceptibility.

At first sight, 1t seems plausible that this is simply the effect of in-
ternal fields, which cause the local field experienced by an atom to be
greater than the applied field. We now examine this hypothesis and
show that it is inadequate.

Microscopic calculations yield the polarization of single atoms due to
local fields. In the linear case, if we have N atoms per unit volume of
polarizability a

In some cases the Lorentz value of ! = 47/3 is applicable and we then
obtain the well-known relation between the refractive index n, or the
dielectric constant « = n’ and a.

In nonlinear optics the two driving fields Ef and E% are obviously
modified according to (105) but, as Armstrong, Bloembergen, Ducuing
and Pershan ** have shown, there is a further factor in P. This arises
because the nonlinear polarization

produced directly on the atoms, further polarizes the surrounding
medium.
We have
PY = py + IT NoP?, (107)
so that
a Dp; a a
P= 1 — TNa = (1 + Ixii)pe . (108)

‘Therefore, even if d does not vary with x, D will do so. ‘This is, however,
not enough to explain the observed variation of D with x. For example,
in semiconductors it is very likely that IT’ is small if not zero and yet

the measured values of D appear to obey Miller’s rule and be propor-
tional to x°. Thus, the intrinsic coefficient d itself must have a similar
dependence on_X,

in terms of the measured susceptibilities (i.e., n° — 1), which is the con-
tent of Miller’s rule, and then use (104) to express D in terms of the
atomic polarizabilities we obtain

Thus, the reduced tensor is the same whether or not we apply local
field corrections as long as we do it consistently. To obtain a more or
less constant value of A we must have d varying as a’.

We repeat that optical nonlinearities have an electronic origin.
Electrons in atoms do not move in a harmonic potential. Second har-
monic generation, which can only involve electronic motion, is much
the same in covalent organic materials, ionic crystals and ferro-electrics.
Large values of d’* are associated exclusively with large refractive in-
dices. Thus, nonlinearities in the ionic motion play a secondary role
in nonlinear optics; however important they may be in determining
the ferro-electric properties.

We shall attempt to construct a model, just sufficiently general to
exhibit the gross features of ferro-electric behavior, and show that it
modifies the nonlinear optical behavior exactly as predicted by Miller’s
rule. The model is not put forward as an explanation of ferro-electricity
although it has a venerable past in that connection, but as a demonstra-
tion that a simple system with singular dielectric properties behaves
in a way consistent with Miller’s rule.

In Fig. 1, we illustrate a moderately realistic one-dimensional model
in which electrons of mass m are coupled to ions of mass JM in a lattice.
Forces act between like and unlike particles and of these by far the

strongest 1s K,,,,, which is responsible for the electronic optical spectrum.
The remaining forces determine the lattice spectrum. The important
nonlinearities are associated with K,,,,. The linear behavior of this
model is formidably complicated and we therefore assume that its
salient features are already evident in the much simpler model of Fig. 2.

The electron of mass m, = m is coupled to the ion of mass mz = M
by the force constant k,. which replaces K,,. . It is anharmonic. The
electron and the ion are also coupled to rigid supports representing the
rest of the crystal by forces k, and k, . It is as though we had gone di-
rectly from the Born-Von Karman theory of specific heats to the Kin-
stein theory without mentioning Debye.

Let x, be the displacement of the electron of charge e, and z, that of
the ion of charge e, . We shall assume that the potential energy is

so that the anharmonic term is exclusively associated with the ‘“‘atomic”’
binding of the electron to its parent ion. It will be convenient to define
Vo, = —Vi2. The equation of motion in a field H’e*”’ is then

To obtain the sum frequency polarization due to two fields E’e
and E7e*’’’ we substitute the linear responses back into the nonlinear
term in (114). The result is a nonlinear coefficient

We note first that if, as seems most reasonable, e, = —e, then g(a) =
g(8) = g(y) = 1. In any case at optical frequencies g(w) ~ 1 for all
reasonable values of e./e, and at de

ke “Se = ky
(0) = 2 (123)
ke -++ (2) ky -+F kaa ae )
€y C1
which is also near unity if e. © —e,. Thus, to all intents
Am — Sos (124)

Thus, A is an intrinsic electronic property and the effect of ionic
motion is entirely contained in its effect on x. We note, however, that
in some ferro-electrics, where the departure from inversion symmetry
is both small and temperature dependent, A will also be temperature
dependent.

If k,, ki2, and k, are all positive, the de susceptibility is greater
than the low frequency limit of x° but not dramatically so. There is,
however, no reason why one of these constants should not be negative.
Negative compliances are familiar in classical mechanics, a well-known
example is the common automatic door stop which exhibits a positive
compliance as the door is first opened but a negative compliance when
the door is almost fully open. The force between atoms as a whole in a
lattice exhibits a positive compliance but if we separate this force into
nuclear-nuclear and electron-electron repulsion and nuclear-electron
attraction, it is quite reasonable to assume that at the equilibrium
distance the latter component has a negative compliance.

It is immaterial which term in (113) we take as negative although
on physical grounds it seems most suitable to take k, and this is also
a convenient choice.

the system remains in stable equilibrium at x, = a = 0.
The natural resonance w, and we, of the system satisfy
mmwiws = kiko + kiki + kokis (127)

and so as 7 — 1 one of these frequencies —0. At the same time the de
susceptibility (for simplicity we take —e, = e, = e)

becomes infinite, while the low-frequency limit of the optical suscepti-
bility remains finite.

If » exceeds unity there is a spontaneous polarization limited only
by terms such as wx} which we have failed to include in ¢.

All this is reminiscent of ferro-electric behavior if » is temperature
dependent and the Curie point corresponds to y = 1.

The inclusion of a term wx; in ¢ will, in fact, make 7 temperature
dependent, for the effect of this term is to replace k. by an effective value
for low-frequency displacements

where 2? is the mean square thermal displacement. As a result if 7, is
the value at 7 = 0 we have

if we define 7’, as the temperature at which y = 1. This is of course a
crude approximation to a Curie-Weiss Law.

By ascribing all the temperature dependence to changes in k,, it
is obvious from (117) that X° is temperature independent. For 7 to
be equal to unity, it is not necessary for —k, to be of the same magnitude
as ky. , all we require [see (126)] is that —k, be near k, . Thus, from (117),

we do not expect any very anomalous values of x° in ferro-electrics,
except in so far as materials with a high electronic polarizability are
more likely to be ferro-electric.

We have now shown that it is possible to incorporate in our model
features which lead to quite different behavior for the optical and de
dielectric constants without either invalidating Miller’s rule or even
changing the value of A which is essentially a purely electronic property.

We should, therefore, expect the temperature variation of D%A” to
correspond to that of x,<x,x,, and this is well borne out by the measure-
ments of Zwicker and Scherrer’ of the electro-optic coefficients and
Bass, Franken, and Ward’ of the optical rectification coefficients in
the dihydrogen phosphates. Both coefficients are directly proportional
to the de dielectric constant which obeys a Curie Weiss Law.

In KDP there is almost no change of the s.h.g. coefficient (Van de
Ziel and Bloembergen*’) with temperature above or below the Curie
point, in accord with our expectations, but at the Curie point there is
a small discontinuous change. In an orthorhombic coordinate system
d2* and dZ% are equal above 7’, but below T,, d,4 increases and d;,3
decreases while at the same time x,; — x33 decreases and x22 — X33
increases. With a constant A this is not compatible with Miller’s rule.

However, at the transition there is a change in crystal class in which
a, increases and a, decreases, Jona and Shirane.”” It is not unreasonable
to assume that this increases 73,, and decreases 7'32.. by more than
enough to compensate for the changes in x,, and x22.

The classical anharmonic oscillator model, which we have shown to
be a good approximation to the behavior of a real system, leads directly
to that part of Miller’s rule which refers to the geometric properties
and frequency dependence of the nonlinear coefficients in a single
material. We have also in (51) advanced a crude argument to show that
A will not vary much from material to material.

When we examine the experimental data we shall see that the allowed
components of A are between 1 X 107° and 6 X 107° esu for most mate-
rials but that there are a few materials with significantly higher values
and a number with values as low as 0.1 X 10° esu.

In most cases, these exceptional values have a rather simple explana-
tion and we have therefore to explain a constancy of A to within a
factor of about 10.

Neglecting the effects of lattice polarization and local field corrections,

which we have shown are irrelevant, the results for the classical an-
harmonic oscillator model are, from (50) and (56),

This is also the result from the static perturbation treatment of Section
IT.

If we use
= wy an ©)
x = 4N aes gN ar, (133)
we arrive at
_ 243 do Vin.
Ai, = 16 e N27?) (134)
Now the volume occupied by the oscillator is both 1/N and 8 ¢r’)' and
SO
4/ 23 T 5%
Aijr wd, 10 (r » (?)3 esu (135)

where we have inserted numerical values for a, and e. This expresses
A;;, as the product of a scale factor (r’)? and a dimensionless shape
factor T’/r’. Whether we assign to each oscillator the volume per valence
electron, per atom or per group of atoms, (r’)? is likely to be between
0.75 and 3 A; so that A will be sensibly constant near 3 X 10°° esu, if
the shape factor is of the order of 0.01 to 0.05. We have from Turner,
Saturno, Hank and Parr’s’’ results for CH, a shape factor of 0.05,
and so this range of shape factors is not unreasonable. It corresponds
to a linear distortion 0.02? ~ 25 percent. It is also not unreasonable
that the distortion should be of this general order, wherever it is al-
lowed by symmetry. We may speculate that much smaller values of 7'/r°
would imply very weak interatomic forces and much larger values would
lead to a structure unstable relative to a more symmetric arrangement.

Thus, qualitatively, the relative constancy of A reflects relatively
constant shape factors, although we can hardly claim that this is more
than a sophisticated form of dimensional analysis. It does, however,
suggest that A is determined primarily by the geometric properties of
the molecular and crystal structure.

Large values of A will occur only when the molecules themselves
depart considerably from inversion symmetry and are arranged in the
crystal in such a way that the effects of individual parts of the molecule

are additive. Small values of A will occur when sections of the molecule
have local near inversion symmetry or when their disposition in the
crystal favors the cancellation of effects from different atomic groupings.
However the molecules are arranged in the lattice, A will be small if
the molecules themselves have near inversion symmetry, or consist
of uncoupled parts with the same property.

published s.h.g. data, 7 values from optical rectification data and 50
from electro-optic data. Definitions and conventions are discussed in
the appendix and a separate list of references is given for the data in
the appendix. Probable errors vary from measurement to measurement.
It is probably safe to say that no measurement has an accuracy better
than +10 percent and in many cases the probable error is greater. That
for the s.h.g. data at 10.6 u is 30 percent and except for ADP the rectifica-
tion data is only good to a factor of 3. In addition, a few materials have
discordant results reported by different groups and this suggests that,
especially in the case of crystals which are difficult to grow, the data
should be regarded rather critically. In the case of CuCl, Sterzer,
Blattner and Miniter’? have constructed a modulator whose behavior
is consistent with the higher value of the electro-optic coefficient. This
casts some doubt on the low value for CuBr reported in conjunction
with CuCl. In the case of the linear e-o coefficient in HMT, Heilmeyer’s™
value dios = 32 X 10°° esu is the most recent and reliable.

The average of all the s.h.g. data is A = 3.3 X 10°° esu, and only
two coefficients d,.. in HMT and dz33 in LiNbO; exceed 6 X 107° esu
by more than the probable error. One coefficient dj... in LiNbO is
unambiguously less than 1 X 107° esu. The sole accurate rectification
coefficient, Ajo; in NH,HPO,, is 3 X 107° esu which is remarkably
close to the values 3.15 and 3.2 X 10°° esu for s.h.g. and the electro-
optic effect.

Whereas the s.h.g. data have a rather compact distribution about
3 X 10°° esu the linear e-o data are more straggled. The mean value
is 2.3 * 10°° esu but there are a considerable number of materials
with A < 1 X 10°° esu. The difference between the averages A,,, and
A,., i8 not due to the different materials in the two lists, it persists if we

Material Class | A 123 A | aie) A Ref.
KH2POs “Jam |0.55| +65 4.0 | —50 |1.7 constant stress 29
IWH2PO. 42m 60 aad constant strain 30
KD2POs 42m 160 4.0 constant stress 31
KHeAsOa 42m 77 3.7 84 1.7 constant stress 32
RbHeAsO, 42m 92 3.5 constant stress 32
NH4HePO, 42m +55 4.4 | ~—146 |3.4 constant stress 32,29
NHaHe2PO, 42m 36 3.9 constant strain

eliminate all materials not common to both lists and may therefore, be
either a real effect or a systematic error.

A few materials [e.g., SrS.0,-H.0, Cs;H,,0;NaBr-H,O and K.Mg,
(SO,)3] have very low values of A. The latter is especially interesting
since the isomorphous (NH,).Cd.(SO.)3 and (NH,).Mn,.(SOx,)3 salts
have somewhat larger values. The ammonium cadmium salt 1s known
to be ferro-electric at very low temperatures and the ammonium man-
ganese salt is also suspected of ferro-electricity (Jona and Shirane’*).
More significant, perhaps, is the fact that the divalent ions have very
nearly regular octahedral coordination (Zemann and Zemann’*’) and
so form a unit with near inversion symmetry and contribute little to d.
The main contribution comes from the monovalent ions and their
irregularly placed neighbors. The difference between the potassium and
ammonium salts would then be due to the difference in the polariza-
bility of the two ions. For K” the refractivity is 2.45 ccs and for NH4
it is 4.05 ces (see Le Fevre’’). If this enters d as a cube the expected
ratio of the d coefficients would be 4.5. The observed value is greater
than about 5. Note also that NH{ itself lacks inversion symmetry.

The tabulated values of A show that Miller’s rule is an excellent rough
guide to the probable value of d. If the component is allowed by sym-
metry

However, the rule by itself is not infallible. Occasionally, a value of
d much higher than that predicted by (136) will occur. In some cases,
(e.g., d333 in LiNbO3) this is accompanied by a very low value of another
coefficient and it is then plausible to assume that this is due to a partic-
ularly critical geometric configuration. In other cases, (e.g., HMT) it
is quite clearly due to the coincidence of a number of favorable factors.
The atoms, the molecule and the crystal all have the same symmetry
and moreover, as we saw in Section 1, all the separate contributions to
d have the same sign. Thus, it is likely that the value of A ~ 15 X 10°
esu for HMT represents something of an upper limit to what is possible.

More often (186) will overestimate d. This is especially hkely to
occur if the molecules themselves, or large sections of the molecule have
near inversion symmetry, but it may also occur if the crystal structure
itself departs only very slightly from a centro-symmetric structure.

If reasonably good ground state wave functions are available, the
direct perturbation method of Section II seems most suitable as a

basis for calculating the magnitudes of the coefficients. It gives the
intrinsic nonlinear coefficient
38x" T esx
Sep eee EE 13
in = (137)
in terms of the intrinsic low-frequency limit of the optical suscepti-
bility x* and a cubic moment in the ground state. If the electrons are
uncorrelated, this can be replaced by
3x”
eo 7 best (188)
and t,;;, can be obtained from the charge distribution. From (138) we
obtain the reduced tensor
3 beck
ace (x°)”

Asin, = (139)
and we can then incorporate this directly in Miller’s rule using the
observed susceptibilities x,4, etc. to obtain dfh7 .

This continuation of the basic perturbation calculation with Miller’s
rule appears to be the most straight-forward approach to the coefficients.
Apart from the cubic moment ¢,;;, it involves only experimental
quantities.

The analogy with the classical anharmonic oscillator established in
Section IV seems most likely to be fruitful in qualitative discussions
of the general behavior of the coefficients. It appears to have both
empirical and theoretical justification.

A calculation based on fourth-order perturbation theory and a lavish
use of sum rules leads to

where (r’) is the mean square radius of the charge distribution and
Q;.. 18 the semi-invariant

950 THE BELL SYSTEM TECHNICAL JOURNAL, MAY-JUNE 1967
° e °
If we take (r’)? as 1 A this gives
Ave 3 xX 10° ase esu. (143)

We have seen that in the lower-order processes 7'/r* is of the order of
2 X 10°. This does not imply that Q/r* is of the order (2 X 107*)*? ~
5 X 10°° for whereas T is nonzero only because of asymmetric molecular
and intermolecular forces, Q is nonzero even for free atoms or ions. For
example, in the hydrogen atom Q,;;;/(r°)’ = 5/18 and Q;,;;/(r°)’ =
1/9 so that we expect A to be of the order of 3 X 10°"’ to 10°” esu.
For calcite with x ptiecay = 0.1 and xa, = 0.55 this gives a value of d
between 3 X 10°" and 10°” esu. Bjorkholm and Siegman*’ have meas-
ured 3 X 10°“ esu.

We have seen that the reduced tensor A,;, is proportional to the cubic
moment,

and it is therefore clearly symmetric in all its indices. This is in agree-
ment with Kleinman’s’ hypothesis and follows from the origin of the
nonlinear behavior in the electronic motion.

Finally, we remark that nothing increases d like large values of the
linear susceptibilities yet, although, the values of most allowed reduced
tensor components A,;;, are near 3 X 10~° esu they can vary by a factor
100:1. The molecular geometry will often indicate which end of the
range is likely to apply to a particular material.

We have throughout adopted a notation, originally introduced by
Bloembergen*” and his colleagues, in which two fields with complex
time dependence H%e*** and E%e'’' produce a polarization P%e'*' at
the algebraic sum frequency a = 8 + y according to

If the actual fields vary as cos w,¢ and cos wet there will be terms in
P at w, + w. and w, — w, obtained from (144) by letting 8 = a, , a, ,
etc. where @, = —,.

This notation has several advantages in theoretical calculations for
much the same reason that the use of complex numbers simplifies ac
circuit theory, and for much the same reason it has a number of dis-
advantages in calculating numerical values. For this reason, it has not
gained general acceptance by experimentalists who tend to use a number
of different notations, some of which, especially in electro-optics, are
of respectable antiquity. The difference between the two notations
introduces various factors of 2. These are independent of the general
reluctance of physicists to state unequivocally whether they are using
peak or rms fields. In particularly fertile ground, these various factors
can luxuriate and blossom as factors of 8 in the final answer. We use
peak fields in all our definitions.

where we have used d%23 = d%? and suppressed the first superscript,
which 1s always the algebraic sum of the second and third superscripts.
We can also write (146) as

P,(t) = 4{dtos + diss} FoF cos e + QO)t
+ 4fdt3 + dis}F2Fs cos (m — Ot. (147)
If w = Q, this gives
Pit) = 3fdies*® + diss’ } Fol’; cos Qué + g{diss’ + dige’ } Fos . (148)
Now the usual experimental definition would be
Pi) = (dig, + dizz)FoF's cos 2wt + (dtog + d)g2) FoF’; (149)
and so we see that
sh.g. dif, = 4d79°° (150)
rectification d{,, = 4473. (151)

P,(t) = iss’ + diss’) FFs cos wt. (152)
The experimental definition reads
P(t) = dyo3F2F'; cos wt (153)
1.€.,
6x12 = Aish’, , (154)

so that it is possible to contract the last two suffices according to the
scheme

(di23 + d332)F'oFs cos Qut
2d?3,F.F cos 2uwt = d°.4-2F.F cos 2wt. (156)
It is therefore common to define the “vector” §

6
P; = 2 dipSp ° (158)
p=l
With this notation d°4 = dij = df ~ djs + d28 but also ds = dj.
In the electro-optic case, the subscripts referring to optical fields
can be contracted

The alternative ordering (155) leads to df, ..
Note that in this case, since d operates on two distinct fields, one
optical the other dc, there is no possibility of constructing a ‘‘vector”’

Electro-optic data are often presented as coefficients r,, in the sus-
ceptibility ellipsoid. If n is the refractive index (assumed isotropic),

In the MKS system, the units are meters per volt, in the cgs system
they are centimeters per stat-volt. One MKS unit is 3 X 10° esu and
so numerical values of d in esu are the larger numbers.

We have not discussed the influence of a mixed use of rms and peak
fields but we note that if rms fields are used throughout the values of
the coefficients will all be v2 times larger than if peak fields are used
throughout. No one is, however, likely to use an rms de field.

Experimental values of the electro-optic coefficients are usually
expressed in absolute units and the only ambiguity that can occur is
associated with whether the measurements were made at constant stress
(unclamped) or constant strain (clamped). It is safe to assume that
constant stress is implied by the absence of any definite statement to
the contrary.

Second harmonic coefficients are sometimes given in absolute units
but more often relative to the coefficient d;., in KH,PO,. An absolute
measurement of this by Ashkin, Boyd, and Diedzic®” gave

but this is now believed to be too large. The most recent measurements,
Francois’, Bjorkholm,”’ give

for the coefficient in NH,H,PO, . Relative measurements show that it is
identical in KDP and ADP. We have used a rounded off, compromise
value

in compiling the tables. It affects all values of d°* at optical frequencies
but not at 10.6 pu.

It will be apparent that in comparing theory or experiment with
experiment, considerable care is needed to be sure that like definitions
are being compared with like.

2. Kleinman, D. A., Nonlinear Dielectric Polarization in Optical Media, Phys.
Rev., 126, 1962, p. 1977.

3. Miller, R. C., Optical Second Harmonic Generation in Piezo-Electric Crystals,
Appl. Phys. Lett., 5, 1964, p. 17.

. Dalgarno, A., Atomic Polarisabilities and Shielding Factors, Adv. in Physics,

Kitaigorodskii, A. I., Organic Chemical Crystallography, Consultants Bureau,
New York, 1957.

Le Fevre, R. J. W., Molecular Refractivity and Polarizability, Adv. in Phys.
Organic Chem., 2. 1964, p. 1.

Heilmeyer, G. H., Dieleetne and Electro-optic Properties of Hexamine, Appl.
Optics, 3, 1964, D. 1281.

Heilmeyer, G. H, Ockman, N., Braunstein, R., and Kramer, D. A., Second
Harmonie Generation and the Electro-optic Effect in Hexamine, Appl.
Phys. Lett., 5, 1964, p. 229.

Garrett, C. G. B. and Robinson, F. N. J., Miller’s Phenomenological Rule
for Computing Nonlinear Susceptibilities, J. Quantum Electronics, QE-2,
1966, p. 328.

Armstrong, J. A. , Bloembergen, N.,, Ducuing, J., and Pershan, P., Light Waves
in a Nonlinear Dielectric, Phys. Rev., 127, 1962, p. 1918.

Butcher, P. N. and McLean, Reel Ss The Nonlinear Constitutive Relation in
Solids at Optical Frequencies, Proc. Phys. Soc., 81, 1963, p. 219.

Ward, J. F., Calculation of Nonlinear Optical Susceptabilities, Rev. Mod.
Phys., 37, 1965, p. 1.

Chang, R. K., Ducuing, J., and Bloembergen, N., Dispersion of the Optical
Nonlinearitv in Semiconductors, Phys. Rev. Lett., 14, 1965, p. 415.

Van der Ziel, J. P. and Bloembergen, N., Temperature Dependence of Optical
Harmonic Generation in Ferro-Electrics, Phys. Rev., 135, 1964, p. A1662.

Sterzer, I’., Blattner, D., and Miniter, S. J., Coprous Chloride Light Modula-
tors, Opt. Soc. Am., 44, 1964, p. 62.

Second-Harmonic Generation in Ammonium Dihvdrogen Phosphate and
Calcite, Phys. Rev., 154, February, 1967, p. 851.

. Ashkin, A., Boyd, G. D., and Diedzic, J. M., Phys. Rev. Lett., 17, 1963. p. 14.
. Bjorkholm, J. E., Optical Second Harmonic Generation Using a Focused

Heilmeyer, G. H., Ockman, N., Braunstein, R., and Kramer, D. A., Second
Harmonic Generation and the Electro-optic Effect in Hexamine, Appl. Phys.
Lett., 5, 1964, p. 229.

. Miller, R. C., Optical Second Harmonic Generation in Piezo-electric Crystals,

—An Efficient Phase Matchable Nonlinear Material: Appl. Phys. Lett., 9,
1964, p. 234; Miller, R. C. and Savage, A., Temperature Dependence of
the Optical ‘Properties of Ferro-electric LiNbO, and LiTaO:, Appl. Phys.
Lett., 9, 1966, p. 169.

. Shiker, T. R., Linear Electro-optic Effects, J. Opt. Soc. Am., 54, 1964, p. 1281.
: McQuaid, R. W., Pockels Effect of Hexamethylene-tetramine, Appl. Opt, 2 Z,

: Nitsche, R., Crystal Growth and Electro-optic Effect of Bismuth Germanate,

k McQuaid, R. W., Proc. IRE 50, 1962, p. 2484; ibed. 61, 1963, p. 470.
. Sliker, T. R. and Jost, J. M., Electro-optic Effect and Refractive Index of

. Thornber, K. K., Kurtzig, A. Je and Turner, E. H., unpublished work.
. Turner, E. H. and Kaminow, I. P, J. Opt. Soc. Am., 63, 1963, p. 528.
Landolt-Bornstein, Zahlenwerte und Funktionen II Band 8 Teil, Optische

. Thornber, K. K. and Turner, E. H., unpublished work.
. Buhrer, C. F. and Ho, L., Electro-optic Effect in (NH:)2 Cde(SO.)s and

. Turner, E. H., unpublished work.
. Carpenter, R. O’B., Electro-optic Sound on Film Modulator, J. Opt. Soc. Am.,

Sliker, T. R. and Burlage, 8S. R., Dielectric and Optical Properties of K Dz
PQ,, J. Appl. Phys., 34, 1963, p. 1837.

Ott, J. H. and Sliker, T. R., Electro-optic Effects in KH-sPO,; and its
Isomorphs, J. Opt. Soc. Am., 54, 1964, p. 1442.

Kaminow, I. P., Barium Titanate Light Phase Modulator, Appl. Phys. Lett.,
7, 1965, p. 123; zbid., 8, 1966, p. 54.

Johnston, A. R. and Weingart, J. M., Low-Frequency Electro-optie Effect in
Ba Ti Os, J. Opt. Soc. Am., 55, 1965, p. 828.

Turner, E. H., High-Frequency Electro-optic Coefficients of Lithium Niobate,
Appl. Phys. Lett., 8, 1966, p. 303.

Bloch, O. G., Zheludev, I. S., and Shamburov, U. A., Electro-optic Effect in
Penta-erythritol, Sov. Phys. Crystallog., 8, 1963, p. 37.

The problems associated with «imperfect modulators are discussed and
the characteristics of three different schemes of operation are calculated.
Results from experiments with one such scheme, the reflection mode of
operation, are given. They compare favorably with the calculations.

A digital light deflector (DLD) is a device that can switch a light
beam to a number of distinguishable positions and has been previously
described by a number of authors.+-* Such a device can be made from
a number of modulators and passive deflectors. The modulator, for
this application, 1s one that is capable of switching the sense of
polarization, and the deflector unit is a passive element which has
different optical paths corresponding to the two senses of polarization.
A basic unit of a DLD is shown in Fig. 1. It has been previously

shown’ that such units in series can generate 2” distinguishable
positions.

A modulator can be made from any material which can become
birefringent with the application of an external signal. A minimum of
7 retardation is needed in order to switch the sense of polarization.
Examples of modulators that have been considered for this applica-
tion are Kerr cells,® stressed plate shutters,° and crystals such as
KDP? and KTN7® which exhibit an electro-optic effect. The Kerr
cells, although very fast, cannot be used at high repetition rates be-
cause of heating difficulties. Stressed plate shutters, since they depend
on mechanical strain, are limited to lower frequencies. The most at-
tractive modulator materials are the electro-optic crystals and are the
ones that are being seriously considered for the DLD.

The passive deflectors that have been considered for this application
are uniformly thick sections of properly oriented uniaxial crystals
such as calcite,2.? prisms of the same materials) and Wollaston
prisms. The uniformly thick pieces of calcite are used with con-
verging light for the maximum number of resolvable positions® *? but
with such use suffer from aberrations that are caused by the variation
of angle in the converging beam. A converging beam of light passing
through a thick piece of calcite oriented for a displaced beam shows
aberrations which for the most part appear like astigmatism. Prisms,
when used with plane waves, can deviate the angle of the plane wave
without distortion, and therefore, a DLD using prisms can give results
that are limited only by diffraction theory. A DLD using prisms
also uses much less birefringent material than one based on uniformly
thick pieces of the same material. A disadvantage of simple prisms is
that the difference in angle between the two oppositely polarized
beams is only a small variation superimposed on the much larger
normal type prism deflection. This difficulty can be minimized if the
prisms are immersed in an oil whose index of refraction is near that of
the prism. Wollaston prisms do not have the deflection associated with
simple prisms—instead, the only deviation is that between the two
oppositely polarized beams—and are therefore well suited for DLD
use.

In this paper, the design of a high-capacity DLD using Wollaston
prisms will be discussed along with experimental results which will
show that this system does lead to densities limited primarily by
diffraction effects. The problem of an imperfect modulator is also
discussed, and calculations on several systems are given which relate
the signal to background ratio to the modulation efficiency.

Since a Wollaston prism is used to deflect a beam in angle, it is
convenient to think of the operation of a DLD in terms of angular
space. Later it will be convenient to place a lens after the DLD
which will focus the beams of light, each with a different angle with
respect to the axis of the lens, into corresponding points on the image
plane.

The capacity of a DLD is determined by the values of two angles.
One is the largest angle that is allowed in the system and the other
is the minimum angular separation between adjacent positions. The
capacity of the DLD is then just the square of this ratio. The
minimum value is determined by either diffraction effects or imperfec-
tions in the optical system, and the largest value is determined by the
maximum angular aperture of the system. First, let us consider the
lower limit set only by diffraction theory. The light emerging from a
circular aperture illuminated by a uniformly intense plane wave will
have a spread of angles that is caused by diffraction. The intensity of
the light as a function of angle is given by the well-known Airy func-
tion (Fig. 2).° The smallest deflection angle in a DLD must be suffi-
ciently large so that the deflected beam must be resolved from the
undeflected one. If we set the criterion that the two beams should be
separated in angle such that in the far field the first dark ring of
each beam overlap, then the minimum angle is given by 2.44 »/D
where A is the wavelength of light and D is the diameter of the circular
aperture (Fig. 2).

It is possible to estimate the crosstalk, e.g., the ratio of light within
the first dark ring to the light within a circle of equal size as the first
dark ring but displaced, by examining the Airy function. The light
within the first dark ring contains 84 percent of the total energy,® and
a ring displaced by one diameter (corresponding to a separation of the
two directions of 2.44 A/D) falls within an annulus of 1.22 A/D to 3.66
A/D which contains 10.6 percent of the total energy.® Since a circle
can be surrounded by six circles of the same diameter, this displaced
ring contains somewhat less than 3 (10.6 percent) = 1.77 percent.
The crosstalk ratio 1s then 84/1.77 = 47 or 16.7 dB. For a separation
between the beams of twice the above, i.e., 4.88 A/D, the crosstalk
ratio can be estimated to be 210 or 23.2 dB.

If, in addition to diffraction effects, the wavefront is distorted further
by some aberration, then the focused spot size will increase and thereby
decrease the capacity of the DLD. The amount of wavefront distortion
depends somewhat on the type of aberration, but for wavefront distor-

tion of \/4 or less the increase in the focal spot size is not very significant.”°
The aberrations in the DLD will result from inhomogeneities in the
material and from poorly worked surfaces. It should be emphasized
that the value of \/4 is the maximum variation allowed after passing
through the entire DLD, and therefore, the maximum variation for
any individual unit is much less. For a DLD with 10° resolvable posi-
tions, the total number of units would be 20, i.e., 2°° is approximately
10°; and therefore the maximum wavefront error in any unit should
be less than \/4+/20 & /20 where by using the square root we have

assumed random irregularities. The requirement that a modulator
have an optical distortion of less than /20, not allowing for any im-
perfections in the remainder of the optics, is extremely difficult and will
probably represent a serious problem for some time to come. The high
requirements placed on individual components is a direct result of the
large number of such elements that must be placed in series for the
complete DLD.

The maximum angular aperture of a DLD is limited by a number
of effects: (7) the response of the Wollaston prisms, (2) the walk-off
of the beam as it is deflected to larger and larger angles, (22) the
angular aperture of the modulators, and (iv) the angular aperture of
the output lens. These limitations will now be considered in more
detail.

The deviation angle of a Wollaston prism is not constant but is a
function of the incident angle (see Appendix A) and at some angle
the deviation will vary sufficiently such that the array of angles is no
longer uniformly spaced. Calculations based on the equations in
Appendix A indicate that if the Wollaston prism with the smallest
deviation is placed first in the DLD and the next largest second, etc.,
for a total of 20 stages and a maximum angle of deviation of 8°, the
array 18 uniformly spaced to within 10 percent.

As the beam traverses through the DLD, the deflection angle can
become larger and larger and unless the apertures of the prisms and
modulators are very large, the beam will eventually strike the sides of
the apparatus. It is clear that the Wollaston prism with the largest
deviation angle should be placed last in the DLD in order to minimize
the spreading of the beams. With this arrangement approximately 14
of the beam is intercepted by the apparatus for a 20-stage deflector
with a maximum deviation angle of 8°. This latter figure is not a
result applicable in all cases because it depends on the specific lengths
of the elements in a DLD.

The relative retardation in a modulator is also a function of the
incident angle. How rapidly this function varies depends on the type
of modulator. In KDP and similar crystals the angular aperture is
very small (less than 1 minute of are for 30 dB extinction between
crossed polarizers for a crystal thickness of 0.089 inches!) since these
crystals are uniaxial, with a large birefringence, and will, therefore,
give large relative retardation for even small angles away from the
crystal axis. Techniques are reported that compensate the birefrin-
gence?!2 in KDP but the degree of success is not made clear. Because
of the limited angular aperture, crystals in the class with KDP are

not considered attractive for the DLD. In CuCl and other crystals in
this class, the angular aperture can be very large because they are
cubic crystals in the field-free case and are therefore optically iso-
tropic. Angular apertures of +25° have been reported'* for this
material. KTN is also a cubic crystal, but the electro-optic effect in
this crystal is quadratic in contrast to the linear effect in most other
materials useful as modulators. Therefore, KTN is usually biased by a
de voltage in order to reduce the value of the modulation voltage, and
this bias reduces the angular aperture of the modulator; however, as
shown in Appendix B, the angular aperture can still be +10° for
reasonable bias fields.

The lens at the output of the DLD will focus the beams to points
on the image plane. If this lens is not perfect, the spots will be larger
than that calculated by diffraction theory, and the capacity of the
DLD will be reduced. Since the choice of lens will depend on the
application of the DLD, it is not possible to state very precisely what
the angular aperture could be; however, +10° seems reasonable for
most applications.

The calculations on the capacity of the DLD have been based on a
plane wave of uniform intensity resulting in an Airy pattern in the
far field. By placing a filter in such a beam which attenuates the light
as a function of the radial distance, it is possible to greatly reduce the
energy in the rings at an expense of slightly increasing the size of the
central disk.1* If such a filter could be effectively incorporated in the
DLD, the crosstalk between resolvable positions could be greatly
reduced without significantly reducing the overall capacity.

As an illustrative example, we will calculate the number of re-
solvable positions assuming that the minimum angle is set by diffrac-
tion theory which corresponds to 2.44 A/D for 16.7-dB crosstalk and
to 4.88 4/D for 23.2-dB crosstalk, and that the maximum angle is
+10°. For a wavelength of 6000A and an aperture of 1 cm, this cor-
responds to a two-dimensional array of approximately 4(10)® re-
solvable positions with a crosstalk ratio of 16.7 dB and (10)® positions
with a crosstalk of 23.2 dB. These two cases imply 22 basic units in the
first case and 20 units in the latter. If one can learn to use much
larger angles, then the number of positions will increase; however, on
the other hand, if components are optically imperfect so that diffrac-
tion-limited performance is not possible, then these numbers will be
reduced.

Thus far the DLD has only been considered in conjunction with a
diffraction-limited beam; however, images may also be transmitted

through the deflector. Since it takes a number of diffraction-limited
points to make up an image, the capacity of a DLD in terms of images
will obviously be less.

At the present time it is not possible to construct a large capacity
DLD using electro-optic modulators since these materials are not
available in the quantity and quality required. In order to check the
performance of this system, it is necessary to replace the electro-optic
modulators with half-wave plates and thereby replace electronic acti-
vation with mechanical rotation.

A system as shown in Fig. 3 was constructed consisting of 18 mica
half-wave plates, 7 pair of quartz Wollaston prisms, and 2 pair made
from calcite. The aperture of the system was 18 mm and the wave-
length was 6328 A. The smallest deviation angle in the system was 1
minute and the largest was 4°; the smallest deflection angle cor-
responds to 8.38 A/D, which is a separation somewhat larger than that
considered earlier in this paper. The aperture of the pinhole was (0.001
inches which is larger, by a factor of approximately 4, than that re-
quired to give a diffraction-limited divergence to the wave emerging
from lens 1. This system when used with an aperture of this size
should be considered to be deflecting an image rather than operating
with a diffraction-limited beam. The purpose of using a spot of this
size is that the ratio of light in the central disk to that in the diffrac-
tion rings is much higher than when a diffraction-limited beam is
used, hence the crosstalk between adjacent positions should decrease
shen compared to the diffraction-limited case.

Fig. 4 is a picture of a focal plane taken with this apparatus. It
shows the 218 214(10)® resolvable positions. This picture was taken
by setting each half-wave plate in the halfway position so that light

WOLL ASTON. WOLLASTON
‘ ca PRISM 2
eet aes
Ee On HHH
APERTURE LENS1 LENS 2 FOCAL
HALF WAVE / \ HALE WAVE PLANE
PLATE | “PLATE 2

was divided equally into both polarizations. In this way all 2** po-
sitions are simultaneously illuminated. Fig. 5 is an enlargement of an
arbitrarily-selected subsection of Fig. 4 and shows the resolution
much more clearly.

Fig. 6 is an enlargement of a single position taken under two cases:
(1) the pattern on the top illustrates the focused beam with the DLD

removed from the system, and (72) the pattern on the bottom is the
same focused beam with the DLD in the system. The degradation of
the pattern on the bottom is a result of the optical imperfections in
the many elements that make up the DLD. A comparison of the two
patterns shows that the DLD did not increase the size of the central
disk by an appreciable factor but did make it much more irregular.
Fig. 6 was overexposed in order to show the weaker diffraction rings
much more clearly.

The crosstalk ratio between adjacent positions was measured by
first setting the DLD so that only one position was present at the
focal plane. A 0.001-inch aperture was placed at the focal point, ad-
justed in position for maximum light transmission, and the amount
of light was measured by a detector. The aperture was then moved
to an adjacent position, and the amount of light passing through
the opening was again measured. The ratio of these two numbers
is the crosstalk. The measurements were made for various settings
of the DLD and the values ranged from 20 to 28 dB. The range in the
measurements is presumably due to imperfections in the optical
system which cause the focused spot to be irregular and unsymmetric
in shape. The irregular shape is also evident from Fig. 6.

Fig. 6— The degradation of the focused beam pattern by the DLD. (The
pattern on the top was taken without the DLD in the system and that on the
bottom with the DLD in the system.)

The performance of this system is probably worse than the A/4
tolerance discussed previously in the paper but is probably not much
worse than a wave or so; this latter figure was not measured directly
but was estimated from diagrams which show spot patterns as a
function of various aberrations.*°°

It is anticipated that an electronically activated modulator will be
the weak link in DLD performance for some time to come; it therefore
is important to know how an inefficient modulator will affect the
performance of a DLD. In this study, we assume that the Wollaston
prisms in the DLD are perfect and that the modulator can be charac-
terized by a single term, #, which is defined as

7 = 2 .. light intensity in the desired polarization _
b —— light intensity in the undesired polarization (1)
with

With perfect modulators the image plane would have one bright
spot at the desired position and the remaining 2”-1 positions would
be completely dark. With imperfect modulators, and for simplicity
we assume that they are all imperfect to the same degree, some light
will fall on every position. The resulting intensity distribution on the
focal plane has been studied by otherst®?* and is also given in
Appendix C. It is shown there that the intensities in the focal plane
of an n unit DLD can be generated by the expansion of a”[1
+ (1/E)]|" where the first term a” gives the intensity of the desired
position, the second term n(a"/H) implies n positions with intensity
a"/E, the third term [n(n-1) /2](a"/H?) implies n(n-1)/2 positions
of intensity a"/H?, etc. The sum of all the coefficients in the expansion
of (1 — 1/E)” is equal to 2” so that each position in the focal plane can
be assigned to one of these terms.

It can also be established that the polarization of the even powers of
KE, i.e, a", a"/E?, a"/E*, etc. have the opposite sense of polarization
from that of the odd powers of £, i.e., a"/E, a"/E?, etc. This result
can be determined from the basic definition of the efficiency (1).

The requirement on the modulator efficiency is determined by the
particular application of the DLD. If the deflector will be used to
accomplish localized heating or welding, to supply energy for a switch,
or to be used as a printout or display, then the ratio of the intensity
at the desired location to that at the next highest position is impor-
tant. This ratio must be sufficiently large so that the intensity at the
desired location must be great enough to cause the reaction, and yet
the intensity at the next brightest position must be less than that to
cause the reaction. For example, it is possible that for processes that
depend on heating 10 dB is a sufficient ratio, whereas for a visual
display 30 dB or greater may be necessary so that the eye will not be
confused by multiple images. The intensity in the desired position,
as previously stated, is a” and that in the next highest case is a”/E
for the opposite polarization and a"/EH? for the same polarization.
Therefore, this ratio is 1/H when there is no polarization selection and
1/E? when polarization selection is used. In general, it should always
be possible to eliminate the opposite polarization so that the first
troublesome term will be a”/E?.

The DLD can also be used as a memory device.” In this application
& memory is placed in the focal plane which is constructed such that
at each of the focused points an opaque or transparent spot is present.
This code is suitable for a binary organized memory where, for
example, the opaque position can represent 0 and the transparent

position can represent 1. The nature of the position can be read by
placing a detector behind the memory and then directing a light beam
to the desired location. The presence of an opaque position is de-
termined by no response at the detector, and similarly a transparent
position will result in a positive response.

When the modulators are imperfect, the undesired beams of light
will also strike the memory plane and have some chance of reaching
the detector with the possibility of causing erroneous results. For
error-free operation the detector must receive more light when the
DLD is addressed to a transparent position than when it is addressed
to an opaque position, and for future reference let us call the ratio of
these two intensities the signal-to-background ratio, R. The least light
that can reach the detector when the DLD is addressed to a trans-
parent position is a”, i.e., the main beam alone, while the most light
that can reach the detector when the DLD is addressed to an opaque
position is Its. — a”, ie., all the light except for the main beam. The
minimum signal-to-background ratio is then

The Rmin ratio defined by (2) is not an unreasonable minimum in
that a memory plane can be designed to give ratios very close to the
values calculated by this equation.

Three different ways of interrogating the memory will be discussed,
and for each case calculations will be made for the signal-to-back-
ground ratios. The first case is where all of the light is allowed to
strike the memory plane; the second uses a polarization selection
before the memory so that only light polarized in the same sense as
the main beam will reach the memory plane; and the third is the
reflection mode of operation. To prevent confusion the subscripts 1, 2,
3 will be used on the Ruin ratios for the three cases mentioned.

4.1 Case 1—All Light From The DLD Allowed to Reach The Memory
In this case J... = 1 since a + 6 = 1. Therefore,
a” 1

The signal-to-background ratio for this case is plotted as a function of
modulator efficiency, HZ, for several values of n in Fig. 7. It shows that
for an nm = 20 DLD with an (Ryin)i ratio of 5 dB, a modulator
efficiency of 18.6 dB is required.

If polarization selection is used after the DLD, which would require
an additional modulator and polarizer, the odd powers of H can be
cancelled from (2) and the (Ryin)o ratio becomes

This ratio, (Rmin)2, is plotted as a function of efficiency in Fig. 8. It
shows that for n = 20 and (Rmin)o = 5 dB, a modulator with an
efficiency of 14.0 dB is required. This case represents an improvement
of 4.6 dB over the first case.

Fig. 7— Minimum detector ratio versus modulator efficiency for a DLD where
the detector is placed behind the focal plane.

An alternate way" of reading the memory plane is shown in Fig. 9.
In this case, the light is reflected from a mirror located just behind
the memory and is redirected through the DLD to be detected
after passing through a second aperture. The second aperture elimin-
ates a large part of the background and therefore the ratio, Ruin, for
the same modulator efficiency is considerably improved. The deriva-
tion of the Ryin ratio for this reflecting mode of operation, (Rmin)3, is
given in Appendix D and only the result 1s shown here:

(Rmin)3 18 plotted as a function of # in Fig. 10. With this mode of
operation a modulator with an efficiency of only 9.1 dB is required to
give an (Rmin)g ratio of 5 dB or better. The reflection mode of opera-
tion therefore represents an improvement, when measured by reduced
requirements on the modulator, of 9.5 dB over the first case and 4.9 dB

Fig. 8— Minimum detector ratio vs modulator efficiency for a DLD with
the detector placed behind the focal plane and with polarization selection.

Fig. 9— Arrangement of elements for the DLD using the reflection mode of
operation.

over the second case. It should be emphasized that the improvement
ratios given here are strongly dependent on the choice of the Amin
ratio, which was taken to be 5 dB in this paper. For larger Rin
ratios the improvement in the HE ratio would be even greater and vice
versa.

A disadvantage of the reflection mode is that a low f-number lens
must be used at the output of the DLD. The reason for this is that
the light reflected from the plane mirror must enter the output end of
the DLD, and this requires that the focal plane be approximately 14
the linear dimension of the aperture of the DLD. One can show that
this requires a lens of f:1.5 or so if the angular spread of the DLD 1s
+10°. If a lens is designed to have a spherical focal surface and a
spherical mirror is used as the reflector, then the lens can have any f
number.

Any light reflected from the surfaces in the DLD when used in the
reflection mode can be prevented from entering the aperture near
the detector by giving a slight tilt to the elements that make up the
DLD. It is possible to choose an angle such that no reflection is
centered on the aperture.

V. EXPERIMENTS WITH THE MANUALLY-OPERATED DLD EQUIPPED WITH
POOR MODULATORS

Fig. 10-—- Minimum detector ratio versus modulator efficiency using the re-
flection mode of operation.

Ideal half-wave plates have the property that if the angle between
the polarization direction and the axis of the half-wave plate is 6
then the plane of polarization emerging from the plate will be rotated
by 26 from its original direction. For maximum efficiency, the half-
wave plates are oriented at 6 = 0 if no switching action is desired and
at 6 = 45° if the other polarization is desired. The practical maximum
efficiency for the split mica plates used in this experiment ranged
from 30 to 40 dB, which is high enough to give almost perfect DLD
behavior. In order to simulate poor modulators, the wave plates are
set at 6 = e for the predominantly unswitched case and at 6 = 45° — «
for the predominantly switched case. The angle « can then be adjusted
to achieve any degree of modulator efficiency.

To illustrate the behavior of a DLD under the influence of poor
modulators, the half-wave plates were set for H = 10 dB, and a
picture, which is shown in Fig. 11, was taken at the focal plane. This
picture shows some characteristics which will now be enumerated:

(1) The desired spot is shown as the brightest point in the upper
left-hand quadrant and is vertically polarized. 7

(72) Some of the 18 points whose intensity is 1/H of the main
beam are shown in the lower right-hand quadrant and are horizontally
polarized.

(122) Some of the 153 points whose intensity is 1/H? of the main
beam are shown in the upper left-hand quadrant and are vertically
polarized.

(iv) Some of the 816 points whose intensity is 1/H? of the main
beam are shown in the lower right-hand quadrant and are horizontally
polarized.

(v) The points in the upper left-hand quadrant are vertically
polarized and those in the lower right-hand quadrant are horizontally
polarized. A polarization selector in this case would eliminate the
entire lower side. It will always eliminate the side that is opposite to
the one that contains the main beam.

(v1) There are no points with significant intensity (> 1/EH"/?) in
either the upper-right or lower-left quadrant.

As all of the focal plane is not shown in this figure, some of the
background positions are missing in order that an enlargement could
be presented. In this exposure there is a total of approximately .4.6
times more light energy in the background than in the main beam.

The performance of the reflection mode of operation was compared
to the theoretical calculation by making use of the properly mis-
oriented half wave plates. For two reasons (5) cannot be directly
used: (1) In this experiment a memory was used that contained only
one opaque position and for such a case the signal-to-background ratio
using this equation is not accurate, and (iw) equation (5) assumes
that the opaque positions are also perfectly absorbing which is not
valid ‘for this experiment in that the opaque position reflected 4
percent of the incident power. A signal-to-background ratio for this
particular experiment can be calculated as follows. When the DLD is
addressed to a transparent position the light reaching the mirror is
Tyo and the light striking the mirror when the DLD is addressed to
the opaque position is Ij — (1 — T)a" where T is the power reflec-
tion coefficient. The ratio of these two values is

The calculated and experimental values of (R.x»)3 are summarized
in Table I. The first column lists the modulator efficiency, the second
column contains the calculated (Rexp)3 and the third column lists the
measured values.

The agreement between the calculated and measured quantities
agree very well and indicate that the behavior of the reflection mode
of operation is adequately understood.

In the DLD thus far discussed, only one memory in the output
focal plane is used (see Fig. 3), and therefore the memory is read one
bit at a time. For some applications it may be advantageous to
parallel the output as shown in Fig. 12 in order to increase the bit
capacity of the DLD. With this scheme the number of bits read with
each setting of the DLD is equal to the number of memory planes;
the memory now corresponds to one with word organization. Fig. 12
shows four such memory planes but by going to a three-dimensional
array it is possible to parallel 30 to 40 such planes and still use only
one output lens providing the maximum angular aperture of the DLD
is limited to a total angle of 12° or so. If additional lenses are em-
ployed, then any number of memory planes can be incorporated.

This scheme of paralleling is directly applicable to cases 1 and 2,
which were discussed in Section IV, but will not work for the reflection
mode since there is no way to distinguish between different memory
planes when the light is redirected through the DLD. In order to
parallel the output of the reflection mode, three different schemes have
been devised: (2) to use different wavelengths for each memory plane
and separate the colors before and after the DLD,” (i) to modulate
a monochromatic beam at each memory plane with a different fre-
quency and then to separate the different frequencies after reflection
through the DLD,?° and (12) to arrange the memory planes to have
different distances between the output of the DLD and the memory
plane and to use a short pulse of light; the different planes can now
be read since each plane will return the pulse to the detector at a
different time.**

One difficulty that can arise in the paralleling schemes is that very
low f-number lenses must be used to refocus the output plane into
repeated images (Fig. 13). The first lense placed after the DLD can

have a reasonable f number since the light beams are still confined to
the aperture of the DLD. The second lens must have f:1 or so if the
output of the DLD has a total angle of approximately 12°. Additional
lenses must have even lower f numbers (Fig. 13). The practical solu-
tion to this problem is to perform all of the paralleling within the
first focal length. This scheme will not work with the reflection mode
where the time of flight varies for each memory plane since one lens
implies only one distance between it and the various memory planes.

The problem of reading a memory has been discussed in earlier
sections of this paper; we will now consider the problem, which is
again primarily the result of imperfect modulators, of using the DLD
to write into a memory. Two general types of materials will be
considered for use as a memory medium. One is a medium where the
process is linear in terms of total exposure, i.e., the effect on the
medium of n pulses of light of intensity I/n, each lasting for a time
AT, is the same for any value of n; and the second is one which has a
threshold in terms of the light intensity. An example of the first type
is a photographic film and of the second is a memory based on a
transparent ferrimagnetic garnet at its compensation temperature.”*

A linear medium has the disadvantage, for this application, in that
it integrates the light striking its surface. Therefore, when the photo-
graphic film is being exposed by a DLD with poor modulators, the
problem of the background light must be considered. This problem
is different from that considered in Section IV because in that case
the DLD was set for one address and the question was asked what is
the light intensity distribution over the whole focal plane. In this
case, we ask what is the total amount of light energy striking one
position on the memory plane when the DLD is addressed to all of
the positions. When the DLD is set for one address, the total exposure
over the whole plane is a”[1 + (1/E)]"AT where AT is the duration
of the exposure. This result is evident from the discussion in Section
IV. When one sits at a position and the DLD is addressed to all
positions and dwells at each position for the same time AT’, the total
exposure at that position is also a"[1 + (1/H)]"AT. This latter result
has been previously published?’ and is also proven in Appendix C.

The following calculations, which represent worst cases for writing,
can be performed. The first case that will be considered is the situation
where all of the light is allowed to strike the memory plane and the
second case makes use of polarization selection. In both cases it will be
assumed that all points will be addressed, except for one.

In this case the light striking any of the addressed positions will
have an exposure of nearly a*(1 + 1/H)"AT and the exposure at the
one position that was not addressed will be [a"(1 + 1/4)” — a’JAT,
and the ratio of these two values is

The exposure ratios M, and Mz are plotted as a function of modula-
tor efficiency in Figs. 14 and 15. These plots can be used to determine
the minimum modulator efficiency required for a certain exposure
ratio.

From the exposure ratio and the properties of the medium, e.g.,
photographic film, the density ratios of the positions can be calculated.
These two ratios do not have to be the same, as a material such as
photographic film can be very linear in terms of exposure but the ex-
posure vs film density can be very nonlinear.

For a theshold medium the problem is much simpler. The only re-
quirement is that the most intense beam must be greater than threshold
and that the next highest position be less than threshold. As men-
tioned in Section IV, this ratio is 1/H when no polarization selection
is used and 1/H? when polarization selection is used.

_ Fig. 14— Exposure ratio vs modulator efficiency for a DLD where all of the
light is allowed to reach the memory plane.

2(10)*x along the Y bank for a total of 4(10)°2 total retardation. With
a DLD, a total retardation of 207 can accomplish the same number
of resolvable positions. Therefore, it is evident that the DLD makes
very efficient use of the variable retardation. The reason for this ef-
ficiency is that the DLD makes use of the fixed retardation in the
passive elements whereas the analog deflector must generate all of the
retardation. In addition, the DLD can be designed for any separation
between the beams and still not require any more than the 207 varia-
ble retardation. The analog reflector, on the other hand, cannot
separate the beams any further without supplying additional retarda-
tion.

Fig. 15 — Exposure ratio vs modulator efficiency for a DLD where polarization
selection is used.

resolvable positions that can be attained is reasonably close to that
allowed by diffraction theory. The effect of imperfect modulators on
the performance of the DLD has also been discussed.

The discussion presented here does not mention the problems as-
sociated with the high-speed switching of an electro-optic modulator,
which is a problem that must be solved if the DLD is to have broad
application. This problem has been studied by 8S. K. Kurtz.?*

The author is greatly indebted to H. E. D. Scovil and K. D. Bowers
for valuable discussions involving the applications and overall per-
formance requirements of the digital-light deflector.

The author is also indebted to J. T. Sibilia and J. E. Geusic for their
contributions to many of the solutions associated with the reflection
mode of operation. He also wishes to acknowledge discussions with
R. G. Smith on some of the problems associated with imperfect
modulators and to J. G. Skinner on some of the optical problems.

The formulas necessary to trace the two wave normals through a
Wollaston prism in a direction as shown in Fig. 16 are given below:

A useful approximate formula for calculating the total deviation angle
of a Wollaston prism, A, (A = a + 6b) for perpendicular incidence,
y = Oils

The variation of A with respect to a variation in y at perpendicular
incidence, (dA/dy),-0 , can be calculated from (9) and (10) to be

This calculation is valid for materials that are cubic, and therefore
optically isotropic, in the absence of an electric field and become
uniaxial, with the optic axis parallel to the electric field, in the pres-
ence of an electric field.

Fig. 17 describes the placement of the crystal with respect to the
incident radiation. The vy plane is the first surface of the modulator,
and the second surface is parallel to the first and passes through the
point Z equals —T’. The induced C axis of the crystal is parallel to the
y axis. The light ray makes an angle y with the z axis, and the inter-
section of the plane of incidence with the zy plane makes an angle a
with the z axis. The relative retardation between the extraordinary
and ordinary ray can be calculated to be

reduces the familiar (np — ,) (2x7'/d,) for perpendicular incidence.
Since n, and n, are nearly the same in this case, we will expand

no)? and higher. We will also expand sin y using a power series in y.
The result of these substitutions is

Fig. 17 — Coordinate axes showing the placement of the uniaxial crystal with
respect to the incident radiation.

In order that a modulator switch the sense of polarization, it is nec-
essary that the retardation be changed by x. In general, the retarda-
tion will be changed from Nx to (N + 1) with the application of an
electric field. The retardation as a function of incident angle for the
largest retardation, (NV + 1)z, using terms only up to y?, becomes

The angular aperture of a modulator is determined by the value of
y where the change in retardation from + becomes serious enough to
cause unwanted behavior. If we call this change in retardation AR,
then the value of y that corresponds to this AR can be calculated from
(16), 1.e.,

n.AR |
_ lax + 1)r(cos’ a — 4) ]° uo)
V’rom (17) one can see that the angular aperture for a given material
is inversely proportional to ~/N + 1 so that we can write

Ybiased = —tonblased ’ (18)
VN+1
i.e., to the same degree of performance the angular aperture of a mod-
ulator biased to Nz is decreased by the factor 1/WN +1 of the
unbiased case.

For a material with a linear electro-optic effect, it is most sensible
to use N = 0 in order to use the lowest voltages; this will result in the
maximum angular aperture. For a quadratic material like KTN, it is
sometimes more efficient to use a biasing de voltage in order to reduce
the modulation voltage. In that case, an N of 10 or 20 might be used.
If we decide that the maximum retardation error is AR = 0.02067,
corresponding to a minimum extinction of 30 dB between polarizers,
then the angular aperture of KTN is +26° in the unbiased case, +-7.8°
for N = 10, and +5.7° for N = 20.

The angular aperture of biased KT'N can be increased by placing
a properly oriented positive uniaxial crystal such as quartz in series
with the biased KTN modulator. This technique can be used to elim-
inate the terms in y* from the total retardation and thereby increase
the angular aperture to approximately that of the unbiased case.

Assume that we have a DLD consisting of n stages made up of
modulators with an efficiency, EH = a/b, a + b = 1. We again assume
that only the modulators are imperfect, and that every modulator can
be characterized by the same efficiency.

Any light beam incident to a modulator is broken up into two
beams, an “a” beam for the desired polarization and a “b” beam for
the oppositely polarized position. A table can be made up which lists
the total number of paths through the DLD (Table II). Since we have
a choice at each modulator as to whether an a or b path is taken, the
different paths are characterized by all possible combinations of the

a and 6 terms. Let us consider the paths containing an r number of 6
terms and an (n — r) number of a terms. The total number of pos-
sible ways of grouping these terms is given by

n!
seal aa a Tae 19
(n — r)!r! 1)
and the associated intensity of these beams is a~"b". The total
number of all paths is then given by summing r through its range

where the intensity terms have been included. These terms can also be
generated by

To derive (22), the DLD was set for one address and the intensity
at each point was determined. We now ask what are the different in-
tensities that arrive at a particular position when the DLD is ad-
dressed to all possible positions.

In order to address the DLD to every position, the state of the
modulators can be arranged according to Table III. In this table, 0
means no change in the state of polarization and 1 refers to a change

to the opposite sense of polarization. Table III is a partial listing. The
complete table is made by first writing the nth column which con-
sists of alternating 0’s and 1’s for a total number of 2” entries; the
(n — 1)th column is written by entering pairs (2') of the 0’s and 1’s
for a total of 2”; the (n — 2)th column by entering 2? of 0’s and 1’s,
etc. The sequence of addresses in Table III would place the main beam
once at each location on the focal plane. The 0’s and 1’s that appear
in any horizontal row is the address of that beam.

We must now be able to compare intensities between that of the
main beam, which we shall call A,, and some arbitrary position, which
we shall call A,. Table IV illustrates the technique. Table IV was
constructed by using a rule that sets the intensity ratio at 1 for
modulators that have the same setting and b/a at modulators that
have different settings.

We now wish to determine all of the intensities x some arbitrary
position, say A, = ...010 while the DLD is addressed to all positions.
Using Table III which lists all of the addresses and Table IV which
illustrates the comparison rule, we can construct a table (Table V)
which lists the intensity ratios at A, = ...010.

Table V is similar to Table III in appearance in that for any
vertical column a 0, 1 in Table III is changed into 1, b/a or b/a, 1 to
make Table V.

Table V is one that lists all possible combinations of the entries 1,
b/a and therefore is calculable from the same general formula as that
deduced for Table IT, 1.e.,

It is evident that Table V would not change, except for a different
ordering of the horizontal rows, no matter what the particular address
of A,. Therefore, (23) is the same for all points A,.

Equation (28) will list the intensity ratios between A, and some
point A,. If we multiply though by the intensity of A, = a”, then (22)
becomes

TABLE V— List or INTENSITY Ratios at Point ---010 WHEN
DLD 1s ADDRESSED TO ALL POSITIONS

A beam of light traveling through the DLD breaks up into 2” exit
beams due to the imperfect modulators. These intensities are given by
the terms in the expansion of (a + b)” as shown in Appendix C. We
now need to ask how much of the light comes back through the
second aperture after being reflected from the focal plane (see Fig. 9).

Let us consider one term of the expansion of (a + b)", say a”-"b’.
We state that this system is reciprocal and that if a”-"b" of the
incident beam exits the DLD, then if unity power were directed
through the DLD in exactly the opposite direction the same fraction
of power, i.e., a®~"b", will pass through the aperture.

Thus, for an n unit DLD there will be 2” exit terms and each of
these terms, for example a”b", will generate one term that contributes

* This appendix represents the results of calculations performed jointly by
J.T. Sibilia and the author.

to the intensity at the second aperture; and for the example above that
corresponds to (a*-"b")*. The total number of terms that exit through
the second aperture is then the sum of the squares of the exit terms
and can be generated by (a? + b?)”.

Before we can add up the 2” terms in the second aperture, we must
know something about the relative phases of the terms. Each of the
2” exit terms in the expansion of (a + 6b)” traverses through a dif-
ferent optical path length in the DLD. The reason for this is because
light traverses through some of the prisms as an ordinary ray and
others as an extraordinary ray, and the combinations of such paths
are different for each of the 2” exit beams. Thus, unless the DLD has
been specifically designed to the contrary, each path has a different
phase delay in passing through the DLD.

A term in the expansion (a? + b?)” such as a?(-")b* represents an
E field of [a?-"b*"]? and a phase factor yg. Consider the sum of all
the n!/(n — r) tr! terms of the type a?-” 6?"

[arb F@, Rs ee aie): (25)
The intensity of the sum of all a?-"b?" terms is given by the
square of (25)

f 2
= ae ; for the same phase.
(near
As explained earlier, all of the phases are, in general, different, and so
we will use the random phase addition. The intensity at the second
aperture is then the sum of all of the terms in (a? + 67)”.
The £# ratio, the ratio of the light from the main beam, a?”, to the

. Schmidt, U. J., Optical Processing of Information, Spartan Books, Ince., Balti-

. Nelson, T. J., Digital Light Deflection, B.S.T.J., 43, May, 1964, p. 821.
. Kulcke, W., Harris, T. J., Kosanke, K., and Max, E., IBM J. Res. Develop., &,

. Born, M. and Wolf, E., Principles of Optics, Pergamon Press, 1959, p. 394.
. Cagnet, M., Francon, M., and Thrierr, J. C., Atlas of Optical Phenomena,

Kulcke, W., Kosanke, K., Max, E., and Fleisher, H., Use of Optical Masers
in Displays and Printers, Third Quarterly Report, February 24 through May
23, 1964, Contract No. DA36-039-AMC-00118(E).

Skinner, J. G., Increasing the Memory Capacity of the Digital Light Deflector
by “Color Coding,” B.S.T.J., 48, April, 1966, pp. 597-608.
Seidel, H., unpublished work.

Kurtz, 8. K., Design of an Electro-Optic Polarization Switch for a High-
Capacity High-Speed Digital Light Deflection System, B.S.TWJ., 45, Octo-
ber, 1966, pp. 1209-1246.

Transistor Distortion Analysis Using
Volterra Series Representation

Intermodulation distortion due to nonlinear elements in transistors is
analyzed using Volterra series representation. It is shown that this technique
1s well suited for the analysis of transistor distortzon where the nonlinearities
are small but frequency dependent. An ac transistor model incorporating
four nonlinearities 1s briefly described. The nonlinear nodal equations of
the model are successively solved by expressing nodal voltages in terms of
the Volterra series expansion of the input voltage. Based on this analysis,
a digital computer program has been developed which computes the second
and the third harmonic distortion for a given set of input frequencies and
transistor parameters. The results compare favorably with measured values.
This method also enables the derwation of closed form ac expressions for a
simplified model; these expressions show the dependence of distortion on
frequency, load and source wmpedances, bias currents and voltages, and
the parameters of the transistor. The technique 1s also extended to cascaded
transistors, and simplified expressions for the overall distortion in terms
of the distortion and gain of individual transistors are derived. Finally, a
few pertinent practical applications are discussed.

Solid-state long-haul analog communication systems are being de-
signed for higher frequencies to meet the growth in demand. One of
the more critical and significant problems facing the system designer
is intermodulation noise arising from transistor nonlinearities. Thus,
an analysis of transistor distortion at higher frequencies is a practical
problem; this paper investigates the transistor distortion using the
Volterra series as an analysis tool.

Transistor distortion has been investigated in some detail previously.
Many authors have considered the exponential nonlinear relation be-
tween emitter current and emitter-to-base voltage which is important

at low currents.''’’***"? The effect of frequency on this nonlinear source
alone has been reported.” Three nonlinearities (exponential, avalanche,
and hr, at de) have been examined by Riva, Beneteau and Dallavolta.°
For currents up to 20 mA and frequencies up to 100 kHz, Meyer’’*’®
has developed a more accurate and complex model obtaining the non-
linearities from h-parameters. However, he takes into account the fre-
quency dependence by assuming that the h-parameters can be written
as h' +- jwh’’. Moreover, he does not take into account avalanche dis-
tortion, nor has he extended the model to higher currents (100 mA) and
frequencies (20 MHz). The model described here considers four non-
linearities; they are, exponential, avalanche, hy, , and collector capaci-
tance nonlinearities. These nonlinearities are superimposed on a linear
ac equivalent circuit.’°’’* Much of the initial development of the model
with three nonlinearities was done by Thomas.’°

The transistor model is analyzed using a Volterra series representa-
tion; this series is a generalization of the power series. In a now classic
report, Wiener applied this analysis technique” to find the response
of a nonlinear device to noise."* Bose has carried the theory further.’
Following a series of lectures by Wiener,’’ the theoretical framework,
higher-dimensional transforms, and optimization with Gaussian inputs
were considered by Brilliant,’° George,’’ and Chesler,’® respectively.
Barrett’” has treated statistical inputs. The synthesis problem has been
examined by Van Trees,’ who also applied the method to phase-locked
loops.” The technique has been extended to discrete systems,”’”’”* and
a class of time-variant systems.”*’’’ More recently the theory of the
convergence of the series has been treated.”® This work relies more on
George’s work on the higher-dimensional transform theory.”

Even though much work has been done in this area, the Volterra
series has not found a wide application in solving nonlinear system
problems due to several reasons; if the rate of convergence is not rapid,
the higher-degree terms, which are cumbersome to handle, cannot be
neglected; hence, it cannot conveniently represent gross nonlinearities.
It is not simple to invert the multidimensional transforms to the time
domain, and it is not a useful technique to determine the stability of a
nonlinear differential equation.

The Volterra series method does, however, offer certain distinct
advantages in analyzing transistor distortion. Since transistor distortion
is frequency dependent, the power series is inadequate to characterize
it; the Volterra series does indeed represent frequency dependent sys-
tems. The nonlinearities in the transistors under consideration are
extremely small so that the second- and third-degree terms suffice to

characterize them. Since the output corresponding to sinusoidal input
signals is of interest, there is no need to find the inverse of the higher-
dimensional transforms; the output can be expressed in terms of the
transform of the kernel. The higher-dimensional transforms of the
kernel are complex numbers when s; = jw; , where s; is the complex
variable in the transform domain; hence, these kernels can be numeri-
cally evaluated using the computer (see Section IV). Moreover, for a
slightly simpler model closed form ac expressions can be derived. Since
the kernels retain phase information, this approach will be useful for
the AM-to-PM conversion problem at IF frequencies. Finally, in an
amplifier two or more transistors are cascaded; the nonlinear behavior
of such cascaded transistors is a significant problem. The Volterra series
approach can be easily extended to study such cascaded transistors.

A brief exposition of Volterra series with pertinent reference to the
problem under consideration is presented below. For further details
the reader is referred to the references cited.

where ¢, , C2 , C3 are constants. For a time-invariant system with memory
(capacitors and inductors in an electrical network), the linear term
(¢,2(t)} is replaced by the convolution integral (a(t) = 0; 4 < 0)

This transform domain representation of the system [C,(s)] has been
an invaluable aid to the communication engineers since it brings into
focus the frequency behavior of the system.

A generalization of the second-degree term, c.[x(t)]’, is the double
convolution integral

The output depends on the past values of the input; the above expression
involves a product of the input with itself, thus representing a quadratic
system. ¢2(¢ — 7; , £ — 72) is known as the second-degree Volterra kernel.

A two-dimensional Laplace transform can be defined for (4) after
introducing dummy variables ¢, and ¢,. As shown in Appendix A, (4)
becomes

When two sinusoidal signals at frequencies f, and f, are applied
(Appendix A), the output at the harmonic frequency f, + f, is given
by {| Co(fa + fo) | cos (27(fa & fo)é + dazs)]. Since in general C2(f. , fz)
will not be equal to C.(f, , —f,), different values of distortion at different
harmonic frequencies are directly reflected in the kernel. Moreover, as
in the power series case, the 2f product is less by a factor of two.

Likewise, the third-degree term [c3(x(r))*] can be generalized to a
triple convolution integral;

The magnitude of the signal at the harmonic frequency f, + f, — f.
due to the three fundamental signals at f, , f, and f, is given by | Cs(f, ,
fs, —f-) |. The constants like 1/4 for a ‘8f,’ product are the same as
obtained from the power series approach. |

Later in the paper (in Section IV) the cascade relations in the trans-
form domain are frequently used; their physical significance is discussed
in detail in Section VI. (See also Fig. 1.) The cascade formulae and the
procedure for deriving them are given in Appendix A.

The second and third harmonic distortion are defined as the second
and third harmonic power in dBm, respectively, when the fundamental
power at the output of the transistor is at zero dBm (one milliwatt).
In the analysis of the model in Section IV, the output voltage is ex-
pressed in terms of a Volterra series of the input voltage. Thus, the
kernels C,(s:), Co(s:, S2), and C3(s,, S, Ss) are the voltage transfer
ratios; for a given load R&, , the second and the third harmonic distortion
in dBm are given by the following expressions:

A model is a simple but realistic representation of a physical phe-
nomenon in terms of measurable parameters such that the phenomenon
can be analyzed, and controlled if possible. The linear equivalent circuit
of a transistor is one such example. In reality, several elements of the
transistor equivalent circuit are not linear but are linearized versions
of nonlinear functions; they are the first-degree terms of the Taylor’s
series expansion of the nonlinear functions. Hence, a logical way to
develop the nonlinear model is to consider the second- and third-degree
terms of the Taylor’s series expansion; thus, the emitter resistance
(exponential nonlinearity), current gain (hp, and avalanche nonlin-
earity), and the collector capacitance (collector capacitance nonlin-
earity) have been represented by nonlinear voltage dependent current
generators whose parameters are higher-degree Taylor’s series terms.
This approach has another advantage in that it is difficult to measure
the nonlinearities since they are small; but, it is not too difficult to
measure the overall functions and to curve fit with the known theoretical

The emitter current, J; , is related to the emitter voltage, V., by
the exponential relation

and A and B are constants which depend on the transistor parameters
(Ref. 27; p. 181, p. 249). An experimental curve of the emitter current
I, and the emitter-to-base voltage V,. 1s shown in Fig. 3. This non-
linearity is expressed as a voltage-dependent current generator by a
Taylor’s series expansion of (10) as follows:

where the Taylor’s series coefficients are derived in terms of known
parameters, the emitter resistance r,, and the emitter bias current J; ;
1.€.,

The collector current is a nonlinear function of the emitter current
at higher values of current (hr, nonlinearity) and of the collector-to-base
voltage at higher values of voltage (avalanche nonlinearity).”’ hy, ,
the ratio of Ig to Ig, is plotted as a function of collector current J,
in Fig. 4. It is seen that the following empirical relation” matches the
experimental result (Fig. 4):

where herp max 18 the maximum value of hrz , Ic max 18 the value of Ig
at which hyg max occurs, and a is a constant.

The avalanche nonlinearity is due to avalanche multiplication which
occurs at higher collector-to-base voltage. It is determined from the
collector characteristic which is a plot of collector current (J,) and
collector-to-emitter voltage (Vex), (Fig. 5). The empirical Miller’s
avalanche multiplication factor is given by

where V¢zo is the sustained voltage, and the exponent n is determined
by experiment. From expressions (13) and (14), the ratio IgusIz is
given by

The ac 7, can be expressed in terms of 2, and v,,[v3 — v,] by a Taylor’s
series expansion of (15). Since 7, is a function of emitter voltage v, , 7,
is represented by a current generator g(v2 , v3 — v1); for convenience in
notation it is separated into a linear term g,(v2., v3 — v,), a second-
degree term g2(v2, v3 — 0) and a third-degree term g3(v2, v3 — 2).
The linear term equals M o(arte )2 + M, (v3; — v,). The second-degree
term is given by a.M/,K? (v2)" + mo(v3 — 1)? + (a,M,)Kyvo(v3 — 04).
The coefficients a, , a2, MZ,, m2, etc., and the third-degree term are
given in Appendix B.

The collector capacitance is a nonlinear function of collector-to-base
voltage (Veg) since the depletion layer width is a function of Vo, .
The exact functional relationship is determined by plotting the common-
base imaginary part of h.. as a function of collector-to-base voltage
(Vez) as shown in Fig. 6.° It is evident from the figure that C, follows
the 1/3 voltage law (Ref. 19; Equation 5-96);

Fig. 6— Collector capacitance nonlinearity — calculated and measured curves.

This nonlinearity is represented as a frequency (differentiation) and
voltage-dependent current generator as follows:

The above nonlinear current generators are incorporated in the linear
equivalent circuit as shown in Fig. 2. The linear equivalent circuit
parameters are obtained from the equivalent circuit characterization.
They can, for example, be computed from the h-parameters at different
frequencies. In general, the distortion is not a critical function of the
linear parameters. (Figs. 14 to 17).

All the nonlinear coefficients (K, , a: , m2 , etc.) are easily obtained
from a simple computer program. The parameters to be specified along
with typical values for transistor type A-2486 are listed in Appendix C.

The Volterra series method is applied to the model to compute the
second and the third harmonic distortion. The voltage at each node is
a nonlinear frequency-dependent function of the input voltage. Each
nodal voltage is expressed by a Volterra series expansion of the generator
voltage; since the nonlinearities are small only three terms are con-
sidered. The kernels at each node are determined from Kirchoff’s current
equations.

The Kirchoff’s current law is applied at each node; the currents are
next expressed in terms of the generator voltage v, , the three nodal
voltages v, , ve , and v3 , and the known linear and nonlinear parameters.
The impedances are represented by their transforms and o denotes
that it operates on the voltage across it. The nodal equations are given
below.

Since each nodal voltage is to be expressed in terms of three Volterra
kernels, there are nine unknown Volterra kernels to be determined from
the three equations. The problem of solving for nine unknowns from
three equations is resolved by noting that the polynomials x, x” and
z are linearly independent; hence, each degree term is separately
and successively solved. The linear kernels are first determined; then
the second-degree kernels are determined in terms of the linear kernels;
lastly, the third-degree kernels are evaluated in terms of the first- and
second-degree kernels.

Let A,(s), Bi(s), Ci(s) denote the transforms of the linear kernels
at nodes one, two and three, respectively. From the nodal equations
(18) to (20), the following vector matrix equation is derived.

Equation (21) is solved by inverting matrix Pz(s) and post-multiply-
ing by the vector
1
Z,(s)

The second-degree terms are equated next in (18) to (20). There are
two types of second-degree terms; those arising from the unknown
second-degree kernels [for example, (s; + s,)C,A2(s;, S2)] and those
arising from the known nonlinear coefficients and the known linear
kernels [for example, K. T{_, B,(s;)]. The terms associated with the un-
known second-degree kernels are the same as were associated with the
unknown linear kernels in (21), but at the harmonic frequency (s; + $2).
The following vector matrix equation is obtained for the second-degree

where g. and fz represent the second harmonic contribution due to
Jo(V2 , V3 — ¥,) and y.(v3 — v2); hence,

The vector on the left side of (23) is known. Thus, the unknown kernels
are determined by inverting the matrix P;(s, + s,) and post-multiplying
by the vector on the left-hand side of (28). When s, = jw, , the inversion
of the matrix and the post multiplication by the vector can be done
numerically.

The procedure for obtaining the third-degree kernels is almost the
same; the significant difference is that the vector on the left side not
only contains terms arising from the third-degree nonlinear parameters
but also includes second-degree coefficients which give rise to third-
degree terms by the interaction of the first- and the second-degree
kernels. These interaction terms are denoted by Koz , Jos » Yo3 , respec-
tively. For example, K3(B,) = K; II}_, Bi(s;), whereas K.3 = 2k.B,(s,)
B.(s2, 83) which shows the interaction of the first- and the second-
degree kernels. The third-degree kernels are derived from the following
equations:

A computer program has been developed which calculates the kernels
and the second and the third harmonic distortion. It uses existing pro-
grams to invert the matrix P,(s). The nonlinear coefficients are com-
puted from the known and measured parameters. Computed and meas-
ured results at different currents are given in Fig. 7. The program has
been extended to common-base and common-collector configurations.

V. SIMPLIFIED DISTORTION EXPRESSIONS, THEIR PHYSICAL SIGNIFICANCE
AND COMPARISON WITH EXPERIMENTAL RESULTS

Another advantage of the Volterra series method is that it permits
derivation of closed-form expressions for second and third harmonic
distortion. These equations show the interaction between the various
nonlinear parameters and the effect of frequency.

The model includes the base resistance (r,), the emitter resistance
(r.,), the diffusion capacitance (C,), the load (R,) and the source im-
pedances Zg(s), and three nonlinearities, namely, exponential, ava-
Janche, and hy, nonlinearities. In the computer program Cy, Ci. ,

r., C,, m, and collector capacitance nonlinearity have been taken into
account. The expressions given below are for the common-emitter
configuration.

In the third harmonic distortion term given below, the interaction
terms due to the first- and the second-degree kernels have not been in-
cluded mainly to reduce the complexity; in certain cases, they may be
significant.

The interaction of different nonlinearities and their dependence on
load impedance, source impedance, bias currents, bias voltage and
frequency is indeed somewhat complex. However, the closed form ex-
pressions derived above give a general qualitative picture which will
be discussed now.

It is important to know the effect of frequency on distortion. The
distortion depends not only on the frequencies of the fundamental tones
but also on the harmonic frequency of interest. As shown in Fig. 8,

M, due to a + b product is better than M, of a — b product by 10 dB
with the two tones at 8.382 and 7.266 MHz. These measurements were
made with the transistor biased at 100 mA, 10V and with R, = 50Q
and R, = 50. In curve (a) of Fig. 8, the fundamental tone was in-
creased from 2 MHz to 10.5 MHz and signals at 2f and 3f were measured.
It is seen that both M7, and M; improved with increase in frequency.
A theoretical explanation on the basis of dominant terms (in this range
of parameter values) in (28) and (29) is given below. In (28) as well as
in (29) the terms in brackets are multiplied by a frequency-dependent
term

In this range of frequency (s = harmonic frequency), if K,(1 — a,) S
| sC, | and if | (r, + R,)sC, | > 1 but K, > | (sC2) |, then the above
term reduces to K,/sC, which decreases with increase in frequency.
However, the avalanche terms (M 2, M3, etc.) involve the terms sC,
[in (28) and (29)] and S;C, in the numerator. Thus, if the avalanche
terms are dominant, as at higher voltages, there should be no net con-
tribution due to avalanche terms alone. The exponential terms (K.,/(K,)°
and K;/(K;)*] are multiplied by the factor

This term is independent of frequency if (sC.(r, + #,) + 1) > 1.
Thus the above discussion shows that distortion will improve with
increase in frequency at lower voltages and if | sC.(r, + R,) +1 |< 1.
To verify this statement, the voltage was increased to 20 volts and the
input resistance changed to 225. The plots of M, and M; with fre-
quency, as measured, are given in curves labeled (b) in Fig. 8. It is
seen that 17, and M; do not improve with increase in frequency. The
small improvement can be attributed to the hz terms.

In general, increase in frequency increases distortion; this is especially
true for the common base configuration. But as shown above, for certain
ranges of frequency and certain values of source impedance, distortion
can improve with frequency.

The load resistance is an external parameter which the circuit designer
can vary; hence, it is useful to know its effect on distortion. The second
and the third harmonic terms are multiplied by 1/V R, and 1/R, ~

terms, respectively; it shows that the distortion can be reduced by in-
creasing R,. However, the avalanche terms M@,, M., and I, are
multiplied by the R; term, so that increasing R, will increase the con-
tribution from the avalanche terms. Thus, an increase in R, may in-
crease distortion or reduce it due to cancellation. (The contribution from
the collector capacitance terms also increases with increase in load (fz).)
Because of the above interaction, for a given set of parameters and
input frequencies and the harmonic frequency of interest there exists
an optimum load RP, ; this, of course, can be determined using the com-
puter program. In Fig. 9, the measured values of MM, and M; at different
values of R; are plotted; in both cases increasing R, reduces distortion
until the optimum value is reached and then distortion increases with
increase in Rf, .

Source impedance is another important external parameter. The
source impedance affects the exponential nonlinearities K,/K{ in (28)
and K;/K? in (29) by the factor

At low frequencies, this nonlinearity is reduced by the factor 1/[(1 — a)
(R, + r,)K, + 1]. Thus, an increase in R, will reduce distortion from
this source. However, the contribution from other nonlinearities are
increased by |
1+kK iE, a 1's) .
i he, ad ep

It is seen from this expression that if K,(R, + 7,)(1 — a,) is greater
than 1, the other nonlinearities are not affected by the increase in R, .
Thus, if the exponential term is dominant, increasing #, reduces dis-
tortion at low frequencies. At higher harmonic frequencies if
| sC.(Z,(s) + 7r,) | > 1, the distortion terms are independent of the
source impedance since the [r, -+ Z,(s)] term in the numerator and
denominator cancel. This is well illustrated in the measured results of
Fig. 10. The second harmonic frequency being 0.7 MHz, | sC.(R, + 7) |
is not much greater than one up to R, = 1000; hence, the second har-
monic distortion improves with increase in source resistance up to
140Q. Further increase in ?, does not cause much change in distortion.
The third harmonic frequency 1s 17.3 MHz; hence, a change in R, does
not affect M7; appreciably. (| (sC.)-(R, + 7.) | > 1)

Increase in bias current usually reduces distortion due to the following
reasons. The increase in emitter bias current reduces the exponential

it becomes zero at Ig = Ig max/e, and then becomes negative, and
increases with further increase in J, . The coefficient a; decreases with
bias current J,,. Thus, in general, an increase in bias current has the
effect of reducing both the second and third harmonic distortion (Fig. 7)
(at least until a. = 0).

Whereas exponential and hp, terms are functions of bias current, the
avalanche and collector capacitance nonlinearities are affected by the
bias voltage. The coefficient 7, increases with the voltage; but 17, and
M, increase much more rapidly (Fig. 12). (Both the collector capacitance
nonlinear coefficients y. , y3 decrease with the increase in bias voltage.)
The effects of change in bias voltage are especially pronounced at higher
load resistance since avalanche (and collector capacitance) terms be-
come dominant. The third harmonic distortion decreases more with
the increase in voltage (Fig. 18) than the second harmonic distortion
does.

The physical significance of the closed form expressions has been
qualitatively discussed. Precise quantitative estimates can and have
been obtained using the computer program. For example, the effect
of varying linear parameters by fifty percent of their original values was
studied. The results show that the distortion does not critically depend
on the linear parameters (Figs. 14 to 17). The other transistor parameters
such as Ig wax, Vezo, 1, etc., can also be varied.

It is often stated that in a multi-stage amplifier, the output stage
alone determines the over-all distortion. Even though this statement
is true to some extent, it is frequently found in practice that the effects

of the previous stages cannot be ignored and sometimes the previous
stage is dominant. This is especially true if both minimum noise figure
(which requires lower bias current) and modulation requirements are
to be met by a two stage amplifier. Two analysis tools based on Volterra
series are presented here which enable the study of such cascaded stages.

The first approach makes use of the cascaded formulae mentioned
earlier; this method illustrates the cascade phenomenon and permits
derivation of simple cascade rules.

Consider two cascaded transistors (Fig. 1); let the output voltage
(v2) of the first transistor be denoted by D(v,); the output voltage (v3)
of the second stage by H(v.) and F(v,). The aim is to compute the kernels
Fi(s1), Fo(s, , 82), F'3(s; , 82, 83) knowing D and E. To calculate D(s;),
etc., it is necessary to know the load impedance of the first stage which

is the input impedance of the second transistor. This can be computed;
thus, for a given generator impedance and bias conditions, D(s,),
D2(s1 , 82), D3(s; , 82 , 83) can be determined. H(s,), H2(s; , 82.) and £3(s; ,
So , 83) can be computed for a given load and bias conditions with R, = 0
(voltage v. is directly impressed across the second). Now expression
v3 in terms of v, is given by

It is seen that F is related to H and D by the cascade formulae whose
physical significance is discussed below.

which states that the overall gain in dB is the gain of the first stage
in dB plus the gain of the last stage in dB.

The first term of the formula states that a given harmonic product
from the first transistor D,(jw, + jw,) is amplified by the second transis-
tor at the harmonic frequency E,(jw, - jw,). The second term shows
that the two fundamental tones are amplified by the first transistor
[D1 (jw,)D,(+jw,)] and then the second transistor acts on these tones
to produce distortion F,(jw, , jo).

The second term is the second harmonic distortion of the last stage.
The first term expresses the contribution from the first stage; it approxi-

This shows that if the gain of the last stage is high, the contribution
from the first stage is small. Equation (85) is approximate in two re-
spects; it neglects the frequency effects and the phase addition of the
contributions from the first and the second stage. In (85), the second
stage gain in question is actually the ratio

which involves the two fundamental and the harmonic frequencies.
As an example, a shaping network which was introduced increased the
gain (18 dB) at the harmonic frequency (0.7) MHz) and decreased the
gain at fundamental tones 15.2 MHz (8 dB) and 14.5 MHz (8 dB)
with the result the overall distortion was poorer by 34 dB.

The first term shows that the third harmonic product of the first stage
[D3(si , S2 , S3)] is amplified by the last stage at the harmonic frequency
[E,(s, + se + 83)]. The second term is the interaction term; it arises
when the second-degree kernel of the last stage [E.(s, , s. + s3)] acts
on the sum of the fundamental [D,(s,)] and the second harmonic output
of the first stage [D.(s2 , s3)]. The last term shows that the second stage
third-degree kernel [E3(s, , s2 , 83)] acts on the fundamental tones am-
plified at the respective frequencies by the first stage [D,(s,)D,(s2)D, (ss) |.

From (86), the overall third harmonic distortion is related to that
of the individual transistors by

The first term is the contribution from the third harmonic term of
the first stage; it is given approximately by

beers harmonic | -- pew harmonic erage
of the first stage in dBm of the second stage in dBm

The third term in (87) is the third harmonic distortion of the last stage
in dBm.

It is seen that the effect of the first stage and the interaction term
can be reduced by increasing the gain of the last stage. Equation (389)
illustrates that the second harmonic distortion of each stage should be
good. This may become a limitation if the first stage is biased at lower
currents.

In the above simplified expressions [(88) and (39)] phase addition and
frequency effects have not been considered. In (88), 2 (gain in dB)
actually represents

In (39) the second harmonic distortion is to be measured with two
tones, one at the fundamental and the other at the harmonic frequency

Moreover, the kernel must be made symmetrical by taking the average of
three possible combinations.

Thus, the simplified expressions (35), (88), and (39) are exact if the
transistor performance is not frequency dependent; in general, they

can be used to get a qualitative picture. Equations (34) and (87) are
indeed exact and take into account the frequency dependence. The
computer program is being extended to calculate (34) and (387).

An alternate approach to calculate the distortion of cascaded stages
is to analyze the nonlinear equivalent circuit of cascaded transistors
using the nodal technique illustrated in Section IV. The nodal equations
are derived first; next each nodal voltage is expressed in terms of the
Volterra series of the input voltage; the resulting vector matrix equations
are successively solved. Since the procedure is similar, the details are
omitted.

Two common-collector stages were cascaded using this approach.
(Fig. 18). The measured values at 120 mA, 10 V with 75 ohm source and
load impedances were —87 dBm and —112 dBm for the second and
the third harmonic distortion, respectively. The computed distortion
values are —88.7 dBm for second and —116.6 dBm for third harmonic
distortion. Thus, good agreement with experimental result 1s obtained.

The cascade formulas are simple, physically meaningful and yield
rules of thumb to judge the effect of the first stages. The nodal approach
is more complicated. However, the advantage of the nodal approach
is that it is general and can be used for an amplifier. For example, a
cascade of common-emitter and common-collector stages involves five
nodes; if shunt feedback is used at the input and at the output, the
same program can be used to analyze this amplifier. (Cascade formulas
do not take feedback into account.) In general, the nodal approach can
be extended to study frequency-dependent nonlinear network with
n nodes, if the nonlinearities are small.

In the initial design of L4 repeater a common-emitter—common-
emitter—common-collector configuration”® was used in the power am-
plifier. The third harmonic modulation performance was not as good as
desired. This led first to the study of the output common-collector stage.
As shown in Fig. 19, the increase in source impedance increases the
distortion of the common-collector stage. Since the preceding common-
emitter stage output impedance is high, the common-collector per-
formance was not optimum. Secondly, the preceding common-emitter
stage was studied because the gain of the common-collector stage is low.
(see Section VI) As shown in Fig. 20, increase in load impedance beyond
optimum R, degrades its performance radically. Since the common-
collector input impedance is high, the common-emitter stage perform-
ance was not optimum either. Thus, in the redesign work by Ken
Tantarelli, the common-collector output stage is not being used.

Another interesting application feature was the Improvement in
modulation performance of the common-emitter stage with increase
in voltage. As shown in Fig. 8, it is a function of load impedance, and
at about 1500, maximum improvement was obtained.

New coaxial systems are currently being studied to operate at higher
frequencies. Different configurations have been examined for the output
stage. The model showed that common-collector and common-base
performance is poorer with an increase in frequency and thus the use
of these stages as output stages was questioned (unless transistors with

higher f,’s are available). Recently when a new, high-frequency modula-
tion test set was built, experiments confirmed the prediction. The third
harmonic coefficient (173) for a + b — ec product was 8 dB poorer at
36.5 MHz (due to signals at 36.5 MHz, 40.1 MHz, and 43.1 MHz)
compared to the value at 17.3 MHz (due to signals at 14.5, 15.2, and
16.6 MHz). The common-emitter configuration modulation performance
suffered only about one dB degradation.

This paper has presented a useful analysis tool for investigating the
frequency-dependent nonlinear behavior of transistors. A digital pro-
gram for all the three configurations has been prepared. The results
obtained compare favorably with experimental results. The closed-form
expressions yield a qualitative picture of distortion. The Volterra series
proves useful in examining cascaded transistors; a few rules of thumb
are derived and a general nodal analysis which can be extended to
cascaded stages with feedback is developed. The practical applications
cited show that the technique can be useful in the computer-aided
optimal design of linear transistor feedback amplifiers.

The author acknowledges gratefully the cooperation he received from
Lee C. ‘Thomas who is responsible for much of the initial development
of the model; he has been particularly concerned about the cancellation
mechanism which he observed in the analog computer simulation. The
author wishes to thank Mr. Jack Huang, who first suggested the model,
Miss J. A. Nicosia, who wrote the program, Mr. F. Kelcourse for many
useful discussions, and Mr. R. E. Maurer for reading the draft. The
author would also like to thank Dr. F. H. Blecher for his encouragement
and continued interest and guidance.

If the system is physically realizable, c.(¢ — 7,,¢— 72) = 0, for 7; > 1.
Hence, the limits can be extended to o.

Introducing dummy variables ¢, and t,, the two-dimensional trans-
form is taken

For the second-degree case, consider two sinusoidal signals at fre-
quencies f, and f,. The input x(7) equals,
_ | exp (wat) + exp (— jw.) exp (Jw,7) + exp (—jos7)
AG Pe | sowecrerane= geek a= ae te eG
(45)
From (41)

This term occurs twice as does its complex conjugate.
Hence, the output due to the a + b term alone is

The 2m, term and its conjugate occur only once in (46); hence, it is
6 db better. The response of the third harmonic kernel to three sinusoidal
inputs is similarly treated.

For the system shown in Fig. 1, the cascade formula are given below.
The cascade relations can be symbolically written as

F'4(s, , S2 , 83) = Ey(s, + 82. + 83)D3(s, , Se , 83)
+ 2E.(s, , 82 + 83)Di(81)De(Se , 83) + E3(s: , 82 , 83) UJ Dy,(s;). (58)
A physical interpretation of the formula for cascaded transistors is
given in Section VI. The procedure for deriving the cascade relation is

as follows: the output Z(t) of the last stage is expressed in terms of the
Volterra series of its input. (Only two terms are considered)

+f [Pet—n,t— 2) ued dr. (54)
The output of the first stage y(t) is related to its input by
u(r) = [alr = a)2(e) do

Substituting (55) in (54), terms of the same degree are collected; as an
example, the first second-degree term equals

2
fa ex(t = T) ll d,(r So Oy t — To) I] x(a;) do, . (56)
t=1
Taking the two-dimensional transforms yields
2
Ey (8, + 82)Da(s, , 82) II X(s;). (57)

A two-dimensional Taylor’s series expansion of (58) is taken; 7, is
expressed by K(v.) and vez = vz — 0, . Hence,

The coefficients m, , mz , and m, equal m; = Ig(M;/M,); 7 = 1, 2, 8,
where I is the collector de bias current.
The hrz coefficients are given below:

From (62) for g3(v2; v3 — v1), g3 18 obtained by replacing B,(s;) for
v, and C,(s;) — A,(s;) for (v3 — v,); moreover, the kernel must also be
symmetrical. Since the procedure is the same as for g, it is omitted.
The interaction terms are given below:

A2486 is an n-p-n silicon transistor with overlay type of construction.
Its fr ranges from 800 to 1000 MHz. It is a power transistor with current
capability of 1 amp and can handle 2.2 watts of power.

. Lotsch, H., Survey. of Nonlinear Distortions in Transistor Stages Including

. Reynolds, J., Nonlinear Distortions and Their Cancellation in Transistors,

Meyer, N. I., Nonlinear Distortion and Small Signal Parameters of Alloyed
Junction T'ransistors, Danish Science Press Ltd., Copenhagen, 1960.

Thomas, L. C., Broadband linearization of Transistor Amplifiers, Presented at
International Solid State Circuits Conference, February, 1967.

Narayanan, S., Transistor Distortion Analysis Using Volterra Series Repre-
sentation, (Oral presentation at International Conference on Communica-
tions, June, 1967).

Volterra, V., Theory of Functionals and of Integral and Integrodifferential
Equations, Dover Publications, New York, 1959.

Wiener, N., Response of a Nonlinear Device to Noise, MIT, Radiation
Laboratory, Cambridge, Mass., Report No. 129, (V-16), April, 1942.

Bose, A. G., A Theory of Nonlinear Systems, Technical Report 309, Research
Laboratory of Electronics, MIT, May 15, 1956.

Wiener, N., Nonlinear Problems in Random Theory, MIT Press, Cambridge,
Mass., 1958.
Brilliant, M. B., Theory of the Analysis of Nonlinear Systems, Technical
Report 345, Research Laboratory of Electronics, MIT, March 3, 1958.
George, D. A., Continuous Nonlinear Systems, Technical Report 355, Re-
search Lab of Electronics, MIT, July 22, 1959.

Barrett, J. F., The Use of Functionals in the Analysis of Nonlinear Physical
Systems, Statistical Advisory Unit, Report No. 1/57, Ministry of Supply,
Great Britain, 1957.

Bush, A. M., Some Techniques for the Synthesis of Nonlinear Systems, Sc-D
Thesis, Dept. of Electrical Engineering, MIT, Cambridge, Mass., May, 1965.

Narayanan, §., Transform Methods for Special Nonlinear Systems, Ph.D.
Thesis, Carnegie Tech., Pittsburgh, Pa., May, 1965. .

Flake, R. H., Volterra Series Representation of Time-Varying Nonlinear
Systems, Proc. Second International Congress of IFAC on Automatic
Control Based, Switzerland, Paper No. 408/1, 1963.

Ku, Y. H. and Wolf, A. A., Volterra-Wiener Functionals for the Analysis for
Nonlinear Systems, J. Franklin Inst., 281, No. 1, January, 1966, pp. 9-26.

Kelcourse, F. C. and Labbe, L. P., Transistor Feedback Amplifiers for 0.5
m/c-20 m/e Long Haul Coaxial Cable Transmission System, IEEE NEREM
Conference, 1964.

This paper considers the effectiveness of error-correcting codes for the
transmission of numerical data. In such a situation, errors in the nu-
merically most significant positions of a message are of greater con-
sequence than are errors in the less significant positions. A measure
of transmission fidelity based upon the average magnitude by which the
numbers delivered to the destination differ from the transmitted numbers
1s developed and 1s referred to as the average numerical error (ANE). Codes
are compared by comparing the ANE that results from their use.

Significant-bit codes are defined and the ANE resulting from their use
is determined. For constant-symbol-rate transmission, the relative effect-
weness of various coding schemes is analyzed when the error probability
in the channel ts small, The ANE resulting from the use of certain specific
codes is numerically evaluated and compared.

The usual approach to coding is to ignore the actual meaning of the
transmitted symbols and to represent them in a purely statistical
manner. As a result, all message errors are assumed to be equally
costly and codes have been sought that simply reduce the probability
that a message is received in error.

While this may.be appropriate for the transmission of some types
of data, there are situations in which other criteria of goodness are
of greater merit. If, for example, one is interested in the transmission
of the temperature of a satellite, the probability that a particular
observation is transmitted incorrectly may have little direct relation
to system performance whereas a measure of the average magnitude
by which the received data differ from the data actually transmitted
could prove useful.

* The material presented i in this paper is based upon the dissertation, Coding
for Numerical Data Transmission, submitted by the author to The Johns Hopkins

University in conformity with the requirements for the degree Doctor of
Philosophy.

This paper develops a criterion of transmission fidelity for numeri-
cal data transmitted over a binary symmetric channel based upon the
average numerical error which occurs. Significant-bit codes are de-
fined and the average numerical error resulting from their use is de-
termined for a binary symmetric channel with independent errors. For
constant-symbol-rate transmission, the relative effectiveness of various
coding schemes is analyzed when the probability that a symbol is
received in error is small. In order to obtain a feeling for the utility
of coding, the average numerical error resulting from certain specific
codes is numerically evaluated.

Throughout this paper, the channel is taken to include all operations
performed upon the symbols during transmission. A binary symmetric
channel is defined to be a binary channel such that

(z) the channel always gives one of the binary symbols at its output,
(2) the probability that any particular sequence of errors occurs is
independent of the symbols transmitted.

In some sections, we shall consider a binary symmetric channel with
independent errors. This is a binary symmetric channel for which
the errors occur independently with probability p where 0 S p S 4
and p = 1 — 4g.

The elements of the Galois field of two elements are denoted by 0
and 1. Let the symbol G denote component by component modulo 2
addition of vectors (or n-tuples) whose components are field elements.
The set of all such vectors forms a vector space I of dimension n over
the field of two elements. Because a field element can be viewed as a
vector with one component, @ will also be used to denote the addition
of field elements.

A binary group code V is a subset of I which forms a group. Over
the field of two elements, any set of n-tuples that forms a group is
indeed a vector space. Therefore, a binary group code V forms a sub-
space of I. The dimension of V is k.

The implementation of a binary group code can be viewed in the
following manner. The encoder receives & binary information symbols
(called a message) from the source and determines from the message
(n — k) binary parity check symbols (called an ending). The message
and ending may be interleaved or transmitted sequentially to form a
block of length n (called a code vector). The decoder operates upon

the blocks of n binary symbols coming from the channel in an attempt
to correct transmission errors and provides k binary symbols at its
output. The notation (7,k) is used to denote such a code.

Consider the message (m,, ™Mz-1, *** , ™,). The code vector used
to transmit this message will have m, , m,-1, -°* , m, 1n the k informa-
tion positions. The (n — k) parity check positions that form the ending
are denoted by é; , €2, ++ , €n,-,- Lhe order in which the information
positions and the parity check positions are arranged for transmission
is arbitrary.

Let H denote the parity check matrix for a binary group code. H
is an (n — k) X n matrix whose entries are field elements. An n-tuple
v is a code vector if and only if

where H denotes the transpose of H. H can be written in a form such
that each column of H that corresponds to a parity check position in a
code vector is a distinct weight* one (n — k)-tuple. When this is done,
let C,(1 S | S k) denote the column in H that is in the position that
corresponds to position m, in a code vector.

For a binary symmetric channel, the order in which symbols are
transmitted can affect code performance. For the binary symmetric
channel with independent errors, the order in which symbols are trans-
mitted does not affect performance. In the latter case, we can write
H as

A system for transmitting observations performed upon some physical
process over a binary channel is shown in Fig. 1. So that the relation-
ship between the observed numbers and the code will be clear, a general
formulation will be presented.

If each quantization step is of uniform size, the quantizer output
can be represented as A -++ Bz where A and B are constants and the
integer 7 indicates the quantization level. The ‘‘source scale-to-binary
converter” receives A + Biz from the quantizer and transmits 7 to
the encoder. The ‘binary-to-source scale converter’? receives some
integer 7 from the decoder and delivers A + Bj to the destination.

* The weight of a vector v is the number of nonzero components in v and is denoted
by wiv]. The distance between two vectors wu and v is wlu @ v]I.

Let Pr {7 | 7} be the probability of receiving 7 at the decoder output
when 7 served as the encoder input and let Pr {z} be the probability
that 7 is sent. The average numerical error (ANE) that occurs is

If all values of 7 are equally likely to be observed and if the range
forzisO Si < 2° — 1, Pr {z} = 2. The range for 7 is thus0 Sj S
2" — 1 and (3) becomes

Br ok
ANE = & > yi é| Prt tj | a.
7=( j=
Because B is a constant not dependent upon the particular coding
scheme implemented, B may be set equal to 1 when comparing the
effectiveness of different codes. Accordingly, we shall consider the
expression

ANE = laa eee. (4)
For a specified value of k, a given coding scheme is considered perferable
to some other coding scheme if the ANE resulting from the implementa-
tion of the given code is less than the ANE resulting from the alternative
code.

The code enters (4) through the terms Pr {j | 7}. Thus, for a binary
symmetric channel, the ANE will, in general, be dependent not only
upon the error statistics of the channel but also upon the order in which
the symbols are transmitted.

It is possible to simplify (4) to an expression that involves terms of
the form Pr {7 | 0} exclusively. This reduces the number of terms by a
factor of 2* and demonstrates that knowledge of the error probabilities
conditional upon zero being sent is sufficient to evaluate the ANE.
However, it is necessary to develop some notation and to present two
lemmas before proceeding to simplify (4). The proofs of the lemmas are
omitted because the lemmas follow from the group property of the
code.

When the integer 7 is to be sent, let us assume that the message
ultilized is the k-bit binary representation of 2 (which is denoted by
B(z)) such that

The ending H; = (€,, €2, °** » €x-x) required to encode B(z) 1s chosen
so that the resulting code vector C'(z) satisfies (1).

Lemma 2: Let B(l) = Bit) B BY) as in Lemma 1. For fixed 1(0 S
t < 2° — 1), as j successively takes on the values 0, 1, 2, --- , 2° — 1,1
takes on each of the values in the range 0 S 1 S 2" — 1 once and only once.

Theorem 1: Let all messages be equally likely to be transmitted and let
the channel be binary symmetric (but not necessarily with independent
errors). For these conditions, the average numerical error ts

Proof: By Lemmas 1 and 2, for each value of 7 and for a specified
value of 1, there will be a unique integer 7, such that Pr {j, | 7} = Pr {/| 0}
where B(l) = Bit) @ B(j,). From (4),

jx i) +H —e | = 2-2"
Because of the symmetries involved,
Qk—1 2k—1
2Dli-étl= Dll + lH 7 | = 22"
Thus, (6) becomes

In (5), notice that Pr {0 | 0} does not appear and that the terms Pr {7 | 0}
are not weighted linearly in 7 but that the weighting coefficients go
in steps as powers of 2 with several conditional probabilities having
the same weighting coefficient. Notice that the weighting coefficient
for Pr {z | 0} is 2’-* where (j — 1) is the largest power of 2 in z. All errors
with the same coefficient are of the same seriousness and a good code
must reduce these sets of probabilities rather than simply minimize the
probability that a few very large errors occur.

Because the set of messages B(z) (2° S$ i S 2’ — 1) gives rise to
the set of conditional probabilities whose weighting coefficient in the
ANE expression is 2*~*, we shall call these messages the j-level messages

and the corresponding conditional probabilities, Pr {2** | 0} through
Pr {27 — 1 | 0}, the j-level conditional probabilities. The 0-level message
is defined to be B(O) and the 0-level conditional probability to be
Pr {0 | 0}.

(2) Component m; in each message is 1.
(22) Components m,;(j + 1 S 7 S k) in each message are 0.
(2722) Kivery possible (7 — 1)-tuple occurs once and only once as com-
ponents m, through m,;_, of some j-level message.

For a perfect error-correcting code used with a binary symmetric
channel with independent errors, it is possible to compute the j-level
conditional probabilities and thus the ANE from a knowledge of the
weight distribution of the code vectors on each level (these weight
distributions have been referred to as level weight structures.)' The
problem of efficiently computing the level weight structures from knowl-
edge of the parity check matrix has been discussed previously.’

In order to permit the error-correcting capabilities of a code to
correspond somewhat to the significance of the information positions,
it is possible to formulate a type of code which uses a subcode to protect
the (k — k,) most significant positions of a message and simply transmits
the remaining symbols unprotected. The name significant-bit code
(SB code) is used for this type of code. An SB code is specified by the
parity check matrix H., and the ANE resulting from the use of an
SB code is ANE sg, .

The code utilized to protect the (k — k)) most significant informa-
tion positions will be named the base code. Because it is confined to
the (k — ko) most significant positions, we can abstract the base code
and study it as a separate entity. Accordingly, the base code vectors
are (n — ky)-tuples of which the first (k — k,) positions are the base
messages.

Although the concept of SB codes is applicable to any binary sym-
metric channel, we shall assume independent errors in the following
analysis. Thus, from (2), the base code is specified by the base parity
check matrix Hg, where

In this case, the code vector C(i) = B(z) | EH, where the symbol |
indicates that C(z) can be partitioned into the k-tuple B(z) and the

(n — k)-tuple LE; . Let B(z) be partitioned so that B(z) = B’(2’) | B’(7’’)
where B’(2’) denotes the (k — ky) most significant positions of B(:)
and B’'(2’’) denotes the k, least significant positions of B(z). Then

The range for 7’ isO S 7’ S 2°°** — 1 and for?” isO S 7” S 2" — 1.
Let Pr, {2’ | j’} denote the probability of receiving 7’ when j’ is sent
using the base code. By Theorem 1, the ANE for the base code (ANE;,) is

k—ko oi-]
ANU =: D2" D3 Pig ao}. (9)
j=1 i’ =2Qin1
Because the base code is used exclusively to protect the (k — kp)
most significant information positions, Hs, must have the form

where 0 is used to represent an all-zero column of Hs, and where
the C/(1 <= 1 S k — k,) are the columns of H,. The coset leaders”
in the standard array” for the SB code must be obtained from the
coset leaders in the standard array for the base code by expanding the
base coset leaders in length to n-tuples by inserting k, zeros in informa-
tion positions 1 through k, of the expanded vectors. Because all vectors
in column 7 of the standard array for the SB code will have B’’(7’’)
in information positions 1 through fy ,

We shall now show that ANI.s, can be expressed in terms of the
properties of the base code.

Theorem 2: Let the base code be defined as above. For a binary symmetric
channel with independent errors and when all messages are equally likely
to be transmitted,

I'rom Theorem 1, ANEs, = ANH’ + ANE”.
Let us first analyze ANE’. For 1 S 7 S k), the sum of the 7-level
conditional probabilities is

yi]
» Pr {4 | 0} _ » poe Oe Lr; {0 | 0}
i=Qin2 itt egint
where we have used (10) and realized that 7’ = 0 for all messages on
this level. Because every (7 — 1)-tuple occurs as components m, through
m;-, of some j-level message and m; = 1 in every j-level message,

2ko—j
reggae vane Bleqgee -
= a pr» (i ge w | (t’")} Pr, {oF 1 +. 5 | O}.
i’’=0
As 7” runs through the range 0 S 7” S 2°° — 1, each possible kp-
tuple occurs once and only once. Therefore,

Because of the manner in which the sets were chosen, ANE” can be
expanded as

Notice that the situation k = k, can be included in this formulation
if we define ANE, = 0 and Pr, {0|0} = 1 whenk = k, . Thus, uncoded
transmission can be regarded as an SB code in which k = ky.

The interpretation of (11) is interesting. The quantity >>", 277-"pq"
is the ANE that results from the uncoded transmission of ko-tuples.
Thus, ANI. gz is the ANE for uncoded transmission of ko-tuples weighted
by Prz {0 | 0} plus 2”° times ANE, .

(11) enables the computation of ANEgs,; from the properties of the
base code. Because the base code involves messages of length (k — ky),
it is easier to analyze than the entire SB code.

Consider two error-correcting codes which are denoted as V, and
V.. Let V, be an (m,, &) code and V. be an (n,, k) code where n,
may or may not be equal to n,. Let ¢, denote the minimum weight
of the n,-tuples that are not coset leaders in the standard array for
V,. Similarly, let «, denote the minimum weight of the n.-tuples that
are not coset leaders in the standard array for V,.

where 7,;; is the number of n,-tuples of weight j in the column headed
by C(z) in the standard array for V,. Thus, for V,, the average nu-

Thus, for p sufficiently small, if e, > e,, ANE, < ANE, and V,
results in less ANE than V,.

The minimum weight of the vectors that are not coset leaders in an
SB code is 1. Thus, consider two SB codes denoted by V sp; and V gp
where V sp; 1s an (n, , k) code and V spo 18 an (N2, k) code. V gp protects
the (k — ko,) most significant positions and V gg2 protects the (kK — ko.)
most significant positions of a message. By reasoning analogous to
that above, for p small, if kp, < ko2 and if the base codes used in V gp,
and V spe correct all weight one errors, then V s,, results in less ANE
than V spe.

We thus have the following ranking of codes for p small. The ranking
(in order of increasing effectiveness) assumes that the schemes are
compared for the same value of k.

(it) An SB code protecting (k — ko) positions where k # ky.
(277) An SB code protecting (k — ky + k’) positions where k’ > 0.
(iv) An e-error-correcting code where e 2 1.

To obtain a feeling for the utility of coding for numerical data trans-
mission over a binary symmetric channel with independent errors, the
ANE resulting from certain codes for k = 26 will be evaluated for

constant-symbol-rate transmission. Ref. 3 contains similar information
fork = 1,4 and 11.

Let ANEyc denote the ANID when no coding is used. Contrary to
the concept of code equivalence that is obtained under the assumption
that all errors are equally costly (i.e., when probability of message
error is used as the measure of code performance), the ordering of the
columns of the parity check matrix can affect code performance. Thus,
for the (31, 26) perfect single error-correcting code (PSEC code), every
ordering of the columns of the parity check matrix could yield a distinct
ANE. Upper and lower bounds on the ANE for this code are obtained
in Ref. 3 and are denoted herein as ANEys; and ANE;z , respectively.

By numerical computation, the ordering in (14) was found to result
in as small an ANE as any other ordering tried. The number actually
tried was by necessity a small fraction of all possible orderings of the
26 columns. However, notice that C,. through C2, each have a one in
the same position thus assuring us that the number of weight three
code vectors on levels 12 through 26 will be the theoretical minimum for
this code (by Theorem 9 in Ref. 3). For values of p that are of primary
interest (less than 107°), this assures us that it is not possible to find
a different ordering that will result in a significantly better performance
(although there are other orderings that in fact give equal performance).
Let ANEp denote the ANE that results from the code specified in (14).

Ap = )111100001111000111100011107Z;)- (14)
L1IDODLLIOOLLOOILOLIOOIIOILIIO!LI
LOLOLODOLOLOLIOLIOLIIOLIOLOLIOII

If the columns of (14) are regarded as the 5-bit binary representations
of integers, then the ordering from left to right corresponds to decreasing
integer value (with powers of two omitted because they appear in J;).
Similar ordering was observed to be preferable for the (15, 11) PSEC
code’ and, by exhaustive search, actually found to be as good as any
other ordering for the (7, 4) PSEC code’.

Table I compares ANE;; , ANEyg, and ANE,. For convenience
(and so that the values given will agree with the data plotted in Figs.
2, 8, and 4), the ANE has been normalized by dividing by 2° — 1
(i.e., the full-scale value).

The following SB codes are considered. For each, Hy and the nota-
tion used for the resulting ANE in Figs. 2, 3, and 4 is given. Theorem
2 permits the computation of the ANE for these codes from a knowl-
edge of the base code.

The ANE is denoted as ANE,3,1) .
Base Code 2: (5, 1) perfect double error-correcting code.

Base Code 3: This base code uses independent (38, 1) PSEC codes to
protect the two most significant information positions.
1 0
1 0
las 01 i;

0 |
Because the codes are used independently, the required conditional
probabilities for the base code can be readily calculated. The ANE
is denoted as ANE¢s3 31) ,¢3,1) -

Base Code 5: This base code uses a (8, 1) PSEC code to protect the
most significant information position and a (7, 4) PSEC code to protect

Figs. 2, 38, and 4 present ANEyo , ANEys; , ANE,: , ANE,p, and
the ANE of the SB codes considered. In each case, the ANE has been
normalized by dividing by 2°° — 1. For clarity, logarithmic scales are
used as p decreases from 107° until p becomes sufficiently small so
that the results for small p apply.

The following observations can be made for constant-symbol-rate
transmission.

(2) Improvements in transmission fidelity are obtainable by the
utilization of codes. It should be noted that no one code is the most
desirable for all p (0 < p < 4) and in some cases the codes that are
best for small p turn out to be less effective than uncoded transmission
for the larger values of p.

(21) For k = 26, it can be shown that the probability that a message
is received in error when the PSEC code is used is less (for 0 < p < 3)
than the probability that a message is received in error using any of
the SB codes considered. Thus, under the criterion of minimizing the
probability that a message is received in error, the PSEC code is pref-
erable to any of the SB codes considered.

However, when the ANE is used as a measure of code effectiveness
for numerical data transmission, we observe that the SB codes are
preferable to the PSEC code for certain values of p. Thus, when com-
paring codes, the ranking obtained using probability of message error
as the performance index may not correspond to the ranking obtained
using ANE as an index. We can conclude that probability of message
error and ANE are not equivalent measures of code performance and

that, in some cases, the ANE can be reduced by using a code whose
probability of message error is not minimal.

(71) For k = 26, consider the relative performance of the PSEC
code and the SB codes. When p is small, the PSEC code will be effective
because it can correct all single errors (the only type that have much
probability of occurring) whereas a single error in certain positions
of an SB code will result in a message error. For larger values of p,
there is an increasing chance that an error pattern will occur which the
PSEC code cannot correct. The SB codes become effective in this situa-

tion. If multiple errors occur during transmission such that the errors
occurring in the (k — ko) most significant information positions and the
check positions form an error pattern correctable by the base code,
this will be corrected leaving any errors in the ky least significant
information positions uncorrected. Therefore, the most costly portion
of a large number of error patterns can be corrected. As p increases,
the number of positions in the base code must decrease so that un-
correctable error patterns in the positions covered by the base code
have a sufficiently small probability of occurrence so that the base code
can operate effectively. In other words, as p increases, more and more
protection must be provided for the significant bits so that the most
costly errors are prevented.

(wv) For p small, the ANE from uncoded transmission is approxi-
mately (2° — 1)p. For small p, the ANE as a fraction of full scale for
uncoded transmission is thus very nearly independent of k.

1. Buchner. M. M., Jr.. Computing the Spectrum of a Binary Group Code.
BS.T.J.. 45, March, 1966. pp. 441-449.

3. Buchner, M. M.. Jr.. Coding for Numerical Data Transmission, Ph.D.
Dissertation, The Johns Hopkins University, Baltimore, Maryland, 1965.

VAcuav E. Brens8, A.B., 1950, Harvard College; M.A. and Ph.D., 1953,
Princeton University; Bell Telephone Laboratories, 1953—. Mr. Benes
has been engaged in mathematical research on stochastic processes,
traffic theory, and servomechanisms. In 1959-60 he was visiting lec-
turer in mathematics at Dartmouth College. He is the author of
General Stochastic Process in the Theory of Queues (Addison-Wesley,
1963), and of Mathematical Theory of Connecting Networks and
Telephone Traffic (Academic Press, 1965). Member, American Mathe-
matical Society, Association for Symbolic Logic, Institute of Mathe-
matical Statistics, SIAM, Mind Association, Phi Beta Kappa.

Morcan M. Bucuner, Jr., B.E.S., 1961, Ph.D., 1965, The Johns
Hopkins University; Bell Telephone Laboratories, 1965—. At Bell
Telephone Laboratories, Mr. Buchner was engaged in a study of im-
pulse noise in an effort to understand its characteristics and its effects
upon data communications. At present, he is on a military leave of
absence and is serving as a Lieutenant in the U. 8. Army Electronics
Command, Fort Monmouth, N. J. Member, IEEE, Tau Beta Pi, Sigma
Xi, Eta Kappa Nu.

Myron 8. Guass, M.S. in Physics, University of Chicago, 1926; Bell
Telephone Laboratories, 1926—. Mr. Glass has been engaged in the
development of electron devices and in applied magnetics and optics.
He is currently supervisor of a group in the Optical Device Depart-
ment engaged in the development of gas lasers and laser mirrors.
Member, AAAS; senior member, IEEE.

Ira Jacoss, B.S. Physics, 1950, City College of New York; MS.,
1952, Ph.D., 1955, Purdue University; Bell Telephone Laboratories,
1955—. Mr. Jacobs has been engaged in studies of electromagnetic
wave propagation in nonuniform and anisotropic media, radar cross-
section and antenna analyses, and in missile guidance and detection
systems. He is currently Head of the Military Communications Re-

search Department in the Detection Systems Laboratory. His current
activities are largely in the field of communication theory. During
the summers of 1964 and 1966 he was a member of Institute for
Defense Analyses study groups considering satellite multiple access
and signal processing. He has served as Project Engineer on a study
of weak-signal communication techniques and is presently Project
Kingineer on a deep-space communication study. Senior member, IEEE;
member, American Physical Society, American Association for the
Advancement of Science, Phi Beta Kappa, Sigma Xi, Sigma Pi Sigma.

T. T. Kapota, B.S., 1953, Yokahama National University (Japan) ;
M.S., 1956, Ph.D., 1960, University of California (Berkeley); Bell
Telephone Laboratories, 1960—. Mr. Kadota has been engaged in the
study of noise theory with application to optimum detection theory.
Member, Sigma Xi.

SUNDARAM NARAYANAN, B. Tech., 1960, Indian Institute of Tech-
nology, Kharagpur (India); M.S., 1963, Ph.D., 1965, Carnegie Insti-
tute of Technology; Bell Telephone Laboratories, 1965—. Mr.
Narayanan is with the Coaxial Systems Studies group and is primarily
concerned with nonlinear distortion mechanism in transistors and in
transistor feedback amplifiers. Member, Sigma Xi.

IF’. N. H. Ropinson, B.A., Cambridge, 1946; M.A., Oxford, 1950;
D. Phil., Oxford, 1955. At Oxford since 1950, where he teaches physics,
Mr. Robinson’s research has mostly been in the fields of very low
temperature and nuclear orientation. As a visitor to Bell Telephone
Laboratories in 1954-55, he worked on noise in electron beams. Since
then he has been a frequent visitor to the Laboratories and spent the
year 1965-66 there when he worked on non-linear optics.

W. SHOCKLEY, B.Sc., 1932, California Institute of Technology;
Ph.D., 1936, Massachusetts Institute of Technology; Bell Telephone
Laboratories, 1936-1955, 1965—. Dr. Shockley is best known as the
inventor of the junction transistor. For this and other contributions,
to transistor physics, he received the 1956 Nobel Prize in Physics
jointly with his two former colleagues at Bell Telephone Laboratories,
John Bardeen and Walter H. Brattain. During World War II, on
leave of absence from Bell, he served as Director of Research for
the Navy’s Anti-Submarine Warfare Operations Research Group and
as expert consultant for the Office of the Secretary of War. He re-
turned to Bell Laboratories after the war and became director of the

solid state physics research program. In 1953, he was named Director
of Transistor Physics Research. During this period he made many
contributions to solid-state physics particularly in connection with
the transistor. In 1955 he left Bell Telephone Laboratories to join
Beckman Instruments Ine. where he established the Shockley Semi-
conductor Laboratory in Palo Alto, California, for research, develop-
ment and production of new transistor and other semiconductor
devices. In 1965 Dr. Shockley returned to Bell Telephone Labora-
tories in the capacity of Executive Consultant. He presently holds
the position of Alexander M. Poniatoff Professor of Engineering
Science at Stanford University. More than 70 United States Patents
have been granted for his inventions. Medal for Merit, Office of
the Secretary of War, 1946; Air Force Association Citation of
Honor, 1951; Morris Liebmann Memorial Prize, IRE, 1952; Oliver
E. Buckley Solid State Physics Prize, American Physical Society,
1953; U.S. Army Certificate of Appreciation, 1953; Comstock Award,
National Academy of Sciences, 1954; Holley Medal, American So-
ciety of Mechanical Engineers, 1963; Wilhelm-Exner Medal, Oester-
reichischer Gewerbeverein of Austria, 1963. Honorary Doctorates
from the University of Pennsylvania, 1955; Rutgers University, 1956;
and Gustavus Adolphus College, 1963. Consultant, Scientific Advisory
Panel of U. 8. Army, Air Force Scientific Advisory Board; Fellow,
IEEE, American Physical Society, American Academy of Arts and
Sciences; Member, President’s Science Advisory Committee Panel on
Scientific and Technical Manpower, American Institute of Physics,
Sigma Xi, Tau Beta Pi.

Smion M. 8zx, B.S., 1957, National Taiwan University, Taiwan,
China; M.S., 1960, University of Washington; Ph.D., 1968, Stanford
University; Bell Telephone Laboratories, 1968—. Mr. Sze has been
concerned with the study of semiconductor device physics. At present
he is engaged in studies of metal-insulator-semiconductor devices and
interface states. Member, Sigma Xi, IEEE.

WiuuiAM J. Tasor, B.S. (Chemistry), 1953, Rensselaer Polytechnic
Institute; A.M. (Physics), 1954, Ph.D. (Chemical Physics), 1957,
Harvard University; U. 8. Army 1957-1959; Bell Telephone Labora-
tories, 1959—. His work at Bell Laboratories included research and
development of microwave masers, the design of the maser for the
Telstar® ground station, and investigation of light deflection tech-
niques. He is currently involved in a study of domain wall motion
in magnetic media.

In a recent brief’ in the B.S.T.J., Zador presents, without proof,
realizability conditions for the input impedance of the lossless tapered
transmission line terminated in unit resistance. Upon a careful examina-
tion of the brief, it appears that the conditions are not accurate. The
following analysis clarifies this point and, incidentally, provides alterna-
tives to Zador’s necessary conditions.

Consider a nonuniform line (Fig. 1) with inductance per unit length
£(x) and capacitance per unit length C(x) such that (to follow Zador)

Let V(a,s) and I(z,s) be the voltage and current along the line with
polarities as indicated in Fig. 1. The equations of the line are

dV(z,s)
ae s£(x)I(x,s)
dI(x,s)
we s(x) V(ax,s).
Eliminating J(z,s) and taking into account that £7) = 1/C(x) we get

respectively. From the reference polarities of the voltages and currents
in Tig. 1, we see that for a unit resistance termination at x = 0 we must

have
V(0,s) = —J(0,s).
Hence, if we impose the condition (following Zador)
y(0,s) = V(O,s) = -2
then for unit resistance termination we should have
dV(0,s) _sV@O,s) as

_ _s_ yllys)
49) = oH y'(is)
Thus, the signs are wrong in Ref. 1. This is not the crucial error however.
In this paper, we will show that the difficulties in Zador’s paper
arise from the following facts:

(1) He does not consider the matched line. Unmatched lines tend
to have almost periodic behavior for large real frequencies and hence

the network functions do not have limits at infinity. This point will
be made more precise in the sequel.

(it) Multiplication of Z(jw) by exp (~—2jlw) in property (277) of the
necessity statement introduces periodic behavior at infinity even in the
matched case.

(4722) Physical meaning has not been attached to the N; and D,.
These should obviously be identified with the well-known ABCD
parameters to correct (27) of the necessity conditions.

Property (222) in the necessity statement does not appear to be true
as stated. One can easily construct many counter examples.

Example 1: The uniform line with (following Zador’s notation) c(z) = 1
and length 7 = 1, terminated in a 1-ohm resistor. Obviously c(x) satisfies
the conditions stipulated by Zador, i.e., c(x) is positive and continuously
differentiable in the interval 0 S x S 1. Clearly the driving point
impedance is

Clearly cos 2w does not have a limit fora — +o.
Consider now a less trivial counter example.

Example 2: The exponential line terminated in a unit resistance. With
Zador’s notation c(z) = exp 27, and! = 1. In this case by solving Zador’s
(1) with the subsequent boundary conditions (appropriately corrected)
we find

Example 3: Consider now the class of transmission lines which have
a, positive bounded and twice differentiable c(z) in the interval 0 S x S l.
It can be shown (see e.g., Ref. 2) that the ABCD parameters satisfy
the following asymptotic relations, for w large:*

These results follow from the classical theory of the asymptotic
behavior of the eigenfunctions of Sturm-Liouville problems.’ The
WKBJ method is a related subject. Schelkunoff has discussed these

* The line is driven at the point z = |. The product of the inductance per unit
length and the capacitance per unit length is assumed to be unity.

matters in an elementary way in at least one of his textbooks (he does
not include the O(1/w) term).

If the line is terminated at « = O with a resistance Ry, we have for
the driving point impedance
RoA(jw) + BUjw) |
RoC (jw) nae o, (joa)

Similarly, fw) = Re exp (—2jlw)Z(jw) does not have a limit for
w — +0, When R&,c(0) = 1, 1.e., when the line is “locally matched”’
at « = 0, we have

Clearly, f(w) does not have the asymptotic behavior stipulated by
Zador; it does not even have a limit (because of the cos 2/w term).
Note that the asymptotic formulas (10), (11), (12), and (13) are
also valid for a continuous positive c(x) which is piecewise twice dif-
ferentiable. This can be proven by partitioning the line at the dis-
continuity points and finding the overall ABCD matrix by multiplying

the ABCD matrices of the sections of the line which now have a twice
differentiable c(z).

Hence, property (277) of Zador’s necessity statement could be replaced
by the following: If (z) c(z) is a positive continuous and piecewise
twice differentiable function of the real variable x, (27) the line is term-
inated in a unit resistance and c(0) = 1, then the following relation
is valid for large w:

voltage reflection coefficient at x = / for the unit resistance terminated
line, then

we can see, using Schelkunoff’s results on wave propagation in stratified
media,’ that for w large

To generalize (following Schelkunoff’) if c(0) = 1 and the first n
derivatives of c(x) are continuous functions of z and vanish at the
boundaries then for large w

B.S.T.J. BRIEFS 1053
and therefore,
ais) = 4. + ota): (29)
Property (zz) in the necessity statement of Zador is also wrong.

Proof: The input impedance of the unit-resistance terminated line may
be written, in terms of the ABCD parameters, as follows:

Consider a line with a twice differentiable c(z). In this case A(s),
B(s), C(s), and D(s) are entire functions of order 1 and type I (see
Ref. 2), 1.e.,

In order to find Zador’s representation with the N;, D; (¢ = 1,2)
functions we should be able to find an entire function ¢(s) # 0 such
that when we multiply both the numerator and denominator of Z(s)
in (30) by this entire function, we get functions V;, D; (@ = 1,2) with
the properties stipulated by Zador.

From (34), (85), (36), and (37) it follows that the functions (g(s) +
y(—s))/2 and (g(s) — e(—s))/2 should be of type O in order that
Zador’s N,; and D; be of type J. Consequently, the functions ¢(s) and
¢(—s) themselves are of type 0. Therefore, it is impossible to find
an ¢(s) such that o(s)v(—s) = exp 2ls as Zador stipulates. So property
(12) in Zador’s necessity statement could be replaced by

where k is a constant. Then NV, , D; (¢ = 1,2) are proportional to the
ABCD parameters with proportionality factor k.

From the above it follows that the sufficiency part as stated is in-
accurate. It might be possible to alter the sufficiency conditions to
make them valid. In this case a proof must be given. The author has
done related work® on realizability conditions for nonuniform RC lines
and is familiar with the difficulties involved in proving sufficiency
conditions of this form.

Finally, Zador’s conjectures do not have an obvious physical in-
terpretation and hence they should be justified.

1. Zador, P. L., Realizability Conditions for the Impedance Function of the Lossless
Tapered Transmission Line, B.8S.T.J., 45, November, 1966, pp. 1667-1669.

2. Protonotarios, E. N. and Wing, O., ‘Analysis and Intrinsic Properties of the
General Nonuniform Transmission Line, To appear [HEE Trans. Microwave
Theor. Tech. March, 1967.

: mee E. L., Ordinary Differential Equations, Dover Publications, Inc., New York,
1956.

; Schelkunoff, S. A., Remarks Concerning Wave Propagation i in Stratified Media,
Commun. Pure ‘Appl. Math. June, 1951, pp. 117-128.

. Protonotarios, E. N., On the analysis and Synthesis of the Nonuniform RC Dis-
tributed Network, Doctoral Dissertation, Columbia University, May, 1966.

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