nonlinear_optics_and_lasers

## Laser Fundamentals

A laser produces light through stimulated emission, in which a photon passing an excited atom
triggers the emission of a second photon identical in frequency, phase, direction, and
polarization. This copying process is what distinguishes laser light from thermal light: the
photons are coherent, so the output is directional, monochromatic, and capable of being focused to
a spot limited only by diffraction.


nonlinear_optics_and_lasers

Three components are required. A gain medium supplies the excited atoms; a pump supplies the energy
to excite them; and a resonator — typically two mirrors — returns light through the gain medium
repeatedly so that amplification compounds. Oscillation begins when the round-trip gain exceeds the
round-trip loss, which defines the threshold, and above threshold the output rises approximately
linearly with pump power while the internal field clamps the gain at the loss value.


nonlinear_optics_and_lasers

Population inversion — more atoms in the upper state than the lower — is required for net
amplification, and it cannot be achieved in a two-level system by optical pumping, because the same
transition that excites also stimulates emission and the populations equalize at best. Three-level
and four-level schemes solve this by using separate transitions for pumping and lasing, and the
four-level arrangement is markedly better because its lower laser level empties rapidly, so
inversion is achieved at low pump power rather than requiring more than half the population to be
excited.


nonlinear_optics_and_lasers

Laser fundamentals:
  stimulated emission  a photon triggering an identical copy
  coherence            the property distinguishing laser from thermal light
  spatial coherence    directionality and focusability
  temporal coherence   monochromaticity and interference over long paths
  gain medium          the excited atoms supplying amplification
  pump                 optical, electrical, or chemical energy input
  resonator            mirrors returning light through the gain
  threshold            round-trip gain equal to round-trip loss
  gain clamping        above threshold the gain is pinned at the loss value
  population inversion  more atoms above than below
  the two-level impossibility  optical pumping cannot invert a two-level system
  three-level scheme   requires more than half the population excited; inefficient
  four-level scheme    a rapidly emptying lower level; low threshold
  linewidth            set by the gain bandwidth and the cavity; narrower with a
                       longer, more selective cavity
  the efficiency chain  pump to upper state, upper state to photons, photons out
                       of the cavity; each stage has its own loss and the product
                       is what the specification reports


nonlinear_optics_and_lasers

Q: A laser's output power rises steeply above a certain pump power and is essentially zero below
it. This threshold occurs because:
A. The gain medium activates suddenly
B. Below it, round-trip gain is less than round-trip loss, so oscillation cannot build; above it,
   the field grows until gain saturates back down to equal the loss
C. The mirrors change reflectivity
D. The pump is nonlinear
Reasoning: Laser oscillation requires net round-trip gain exceeding unity. Below that condition
spontaneous emission is not amplified into a coherent field. Above it the intracavity intensity
grows until saturation clamps the gain at exactly the loss, and additional pump power appears as
output. Answer: B.


nonlinear_optics_and_lasers

## Cavity Modes and Beam Quality

The resonator supports discrete modes, and they divide into two families. Longitudinal modes are
the standing-wave patterns along the cavity axis, spaced in frequency by the inverse round-trip
time, and a typical cavity supports many of them within the gain bandwidth. Transverse modes
describe the intensity pattern across the beam, and the fundamental one has a Gaussian profile that
is the most useful in nearly every application.


nonlinear_optics_and_lasers

A Gaussian beam has a specific propagation behaviour that must be understood to use lasers
correctly. It has a minimum radius at the waist, expands hyperbolically away from it, and
approaches a constant far-field divergence angle inversely proportional to the waist size. Tight
focusing and low divergence are therefore in direct opposition — a beam focused to a small spot
diverges rapidly, and a collimated beam cannot be focused arbitrarily small.


nonlinear_optics_and_lasers

Beam quality quantifies departure from the ideal Gaussian, expressed as a factor describing how much
worse the beam's focusability is than the diffraction limit. It matters because it determines the
minimum achievable spot size and therefore the achievable intensity, and a high-power laser with
poor beam quality can be less useful for cutting or for coupling into a fibre than a lower-power
one with a clean mode. Power alone is not the specification that matters.


nonlinear_optics_and_lasers

Modes and beams:
  longitudinal modes   standing waves along the axis; spaced by the inverse
                       round-trip time
  single frequency operation  achieved by an intracavity selective element
  transverse modes     the intensity pattern across the beam
  the fundamental mode  Gaussian; the most useful in nearly all applications
  waist                the minimum beam radius
  Rayleigh range       the distance over which the beam stays approximately
                       collimated
  divergence           inversely proportional to waist size
  the fundamental tradeoff  tight focus and low divergence oppose each other
  beam quality factor  how much worse than the diffraction limit
  the practical consequence  focusability, not power, determines achievable
                       intensity
  mode matching        coupling into a fibre or cavity requires matching the mode,
                       not merely aiming
  thermal lensing      pump-induced index gradients acting as a lens; degrades
                       mode quality at high power and shifts the cavity design
                       point with pump level
  stability criterion  which mirror curvature and spacing combinations support a
                       confined mode; a cavity outside the stable region does not
                       oscillate in a useful mode at all


nonlinear_optics_and_lasers

## Pulsed Operation

Continuous operation is limited in peak power by the pump. Pulsed operation concentrates energy in
time, achieving peak powers orders of magnitude above the average, and the two dominant techniques
achieve very different regimes.


nonlinear_optics_and_lasers

Q-switching holds the cavity loss high while the gain medium accumulates inversion, then abruptly
lowers it, releasing the stored energy in a single short pulse. The pulse duration is set by the
cavity round-trip time and the gain, typically nanoseconds, and the peak power is the stored energy
divided by that duration — kilowatts to megawatts from a modest average power. It is the standard
approach for material processing, ranging, and pumping other nonlinear processes.


nonlinear_optics_and_lasers

Mode-locking is different in mechanism and produces much shorter pulses. If the many longitudinal
modes of a cavity are forced into a fixed phase relationship, they interfere constructively at one
point per round trip and destructively elsewhere, producing a single circulating pulse whose
duration is inversely proportional to the number of locked modes. A broad gain bandwidth therefore
permits a short pulse, which is why titanium-doped sapphire with its enormous bandwidth is the
classic ultrafast medium and why femtosecond pulses require materials chosen for bandwidth rather
than for efficiency.


nonlinear_optics_and_lasers

Pulsed operation:
  Q-switching          store inversion, then release abruptly
  pulse duration       nanoseconds; set by cavity time and gain
  peak power           stored energy over pulse duration
  applications         processing, ranging, pumping nonlinear stages
  mode-locking         fixing the phase relationship among longitudinal modes
  the resulting pulse  a single circulating pulse
  duration relation    inversely proportional to the locked bandwidth
  the bandwidth requirement  short pulses need broad gain bandwidth
  active mode-locking  an intracavity modulator at the round-trip frequency
  passive mode-locking  a saturable absorber favouring high intensity; produces
                       the shortest pulses
  Kerr lens mode-locking  self-focusing acting as an effective fast absorber
  chirped pulse amplification  stretch, amplify, recompress; permits amplifying
                       ultrashort pulses without destroying the amplifier
  the reason it was necessary  peak intensity in an amplifier exceeds the damage
                       threshold unless the pulse is stretched first
  repetition rate      set by the cavity length in mode-locked operation
  pulse energy versus rate  a fixed average power divides between them, and the
                       application determines which matters


nonlinear_optics_and_lasers

## Nonlinear Optics and Frequency Conversion

At ordinary intensities a material's polarization response to a field is linear, which is why
optical frequencies do not mix and superposition holds. At the intensities lasers provide the
response acquires higher-order terms, and those terms couple different frequencies, allowing light
to generate light at new frequencies. Nonlinear optics is the study of these processes and it is
what makes laser wavelengths available outside the wavelengths that lasing media naturally provide.


nonlinear_optics_and_lasers

Second-order processes require a material without inversion symmetry, which restricts them to
certain crystal classes. Second harmonic generation converts two photons into one at double the
frequency, and it is the standard way to reach green from infrared. Sum and difference frequency
generation combine two inputs. Parametric processes split one photon into two, and an optical
parametric oscillator built around this is a widely tunable source.


nonlinear_optics_and_lasers

Phase matching is the practical requirement that determines whether any of these work. The generated
wave must stay in phase with the driving polarization over the interaction length, which requires
the refractive indices at the involved frequencies to satisfy a specific relationship. Since
dispersion normally prevents this, the standard techniques exploit birefringence — using different
polarizations whose indices can be matched by choosing the crystal angle — or quasi-phase-matching,
which periodically reverses the crystal's nonlinear coefficient so that the accumulated phase error
resets before it cancels the output.


nonlinear_optics_and_lasers

Nonlinear optics:
  the linear regime    polarization proportional to field; no frequency mixing
  higher-order terms   significant at laser intensities
  second-order processes  require a material without inversion symmetry
  second harmonic generation  two photons into one at double frequency
  sum and difference frequency  combining two inputs
  parametric processes  one photon splitting into two
  optical parametric oscillator  a widely tunable source built on that process
  third-order processes  available in all materials, including glass and gases
  self-phase modulation  intensity-dependent index modulating the pulse's own
                       phase; broadens the spectrum
  four-wave mixing     the third-order frequency mixing process
  Kerr effect          intensity-dependent refractive index
  self-focusing        the beam creating its own lens; leads to filamentation and
                       damage
  phase matching       the generated wave staying in phase with its source
  birefringent phase matching  index matching by polarization and crystal angle
  quasi-phase-matching  periodic reversal of the nonlinear coefficient
  conversion efficiency  scales with intensity and interaction length; why pulsed
                       sources are used for conversion
  the practical design  the crystal choice, its length, the focusing, and the
                       temperature all interact and are optimized together


nonlinear_optics_and_lasers

## Ultrafast Optics and Applications

Ultrashort pulses have properties that follow directly from their bandwidth. A pulse of femtosecond
duration necessarily spans a broad spectrum, and any dispersive element stretches it because
different frequencies travel at different speeds. Managing this dispersion is the central practical
task in ultrafast optics, and it requires compensating elements — prism or grating pairs, chirped
mirrors — that introduce the opposite sign of dispersion.


nonlinear_optics_and_lasers

The peak intensities available from short pulses open regimes inaccessible otherwise. Material
processing with ultrashort pulses removes material before heat diffuses out of the interaction
region, producing clean features with no melted rim, which is why they are used for precision
machining and for surgery on transparent tissue. High harmonic generation drives electrons in a gas
so violently that they radiate at very high harmonic orders, producing coherent extreme ultraviolet
light and attosecond pulses that resolve electron motion.


nonlinear_optics_and_lasers

Frequency combs connect the optical and microwave domains and earned recognition as a
transformative measurement tool. A mode-locked laser's spectrum is a comb of evenly spaced lines
whose spacing is the repetition rate and whose offset can be measured and stabilized. Once both are
locked to a radio-frequency reference, every optical line has a precisely known frequency, which
turns optical frequency measurement from a heroic undertaking into a routine one and is what made
optical atomic clocks practical.


nonlinear_optics_and_lasers

Ultrafast and applications:
  bandwidth-duration relation  short pulses necessarily span broad spectra
  dispersion           different frequencies travelling at different speeds
  chirp                the resulting time-ordering of frequencies within a pulse
  compensation         prism pairs, grating pairs, chirped mirrors
  the practical burden  every optical element in the path contributes dispersion
  ultrashort processing  material removed before heat diffuses; no melted rim
  multiphoton absorption  enables processing inside transparent materials
  high harmonic generation  extreme ultraviolet and attosecond pulses
  pump-probe spectroscopy  one pulse initiates, a delayed pulse observes; the
                       standard method for resolving fast dynamics
  frequency comb       evenly spaced optical lines from a mode-locked laser
  the two parameters   repetition rate and carrier-envelope offset
  self-referencing     measuring the offset by comparing a doubled low-frequency
                       part of the comb with its high-frequency part
  the consequence      optical frequencies measured against a radio reference
  optical clocks       enabled by comb-based frequency counting
  spectroscopy         broadband and high-resolution simultaneously
  the general pattern  laser development has repeatedly turned measurements that
                       were feasibility demonstrations into routine instruments,
                       and the applications that follow are usually not the ones
                       the developers anticipated


nuclear_and_particle

## Nuclear Structure

The nucleus contains protons and neutrons bound by the residual strong force, and its
existence poses an immediate question: the protons repel each other electrostatically with
enormous force at nuclear separations, so something stronger must hold them. The strong
interaction is roughly a hundred times stronger than the electromagnetic at those distances
and is very short-ranged, falling off over about a femtometre, which explains why nuclei exist
and why they cannot grow indefinitely.


nuclear_and_particle

That competition between short-range attraction and long-range repulsion determines nuclear
stability. Each nucleon interacts only with its immediate neighbours because the strong force
saturates, so the binding energy grows roughly linearly with nucleon count. Electrostatic
repulsion, being long-ranged, grows with the square of the proton count. As nuclei get larger
the repulsion grows faster, which is why heavy nuclei need progressively more neutrons to
remain bound and why there is an upper limit to stability.


nuclear_and_particle

Binding energy per nucleon is the quantity that organizes nuclear energetics. It rises steeply
for the lightest nuclei, peaks near iron-56 at about 8.8 million electronvolts per nucleon, and
declines slowly thereafter. The peak is why fusion of light nuclei releases energy and fission
of heavy ones does too, and why iron is the endpoint of stellar nucleosynthesis — beyond it,
neither process releases energy, so a stellar core of iron has no remaining fuel and collapses.


nuclear_and_particle

Nuclear structure:
  nucleons             protons and neutrons; the neutron is slightly heavier
  strong force         binds nucleons; short-ranged, saturating, ~100x
                       electromagnetic at nuclear distances
  binding energy       the mass deficit times c squared
  binding per nucleon  peaks at iron-56, ~8.8 MeV
  fusion below iron    releases energy
  fission above iron   releases energy
  isotopes             same protons, different neutrons
  nuclear radius       ~1.2 fm times the mass number to the one third power
  liquid drop model    volume, surface, Coulomb, asymmetry, and pairing terms;
                       reproduces binding energies well
  shell model          nucleons in orbitals; explains magic numbers
  magic numbers        2, 8, 20, 28, 50, 82, 126; closed shells give extra
                       stability
  valley of stability  the neutron-to-proton ratio for bound nuclei; rises from 1
                       for light nuclei toward about 1.5 for heavy ones
  neutron excess       needed in heavy nuclei to dilute proton repulsion


nuclear_and_particle

The magic numbers are worth noting because they are direct evidence that nucleons occupy
discrete energy levels rather than forming an undifferentiated fluid. Nuclei with a magic
number of protons or neutrons are unusually stable and abundant, doubly magic nuclei such as
helium-4 and lead-208 exceptionally so, in the same way that noble gases are chemically inert
because their electron shells are closed. That the same shell logic applies to two entirely
different force laws is a nontrivial and instructive result.


nuclear_and_particle

Q: Both fusion of hydrogen and fission of uranium release energy. This is because:
A. Both increase the total mass
B. Binding energy per nucleon peaks at iron, so moving toward iron from either direction
   releases energy
C. Fusion releases energy and fission absorbs it
D. Both split heavy nuclei
Reasoning: The binding energy per nucleon curve peaks near iron-56. Fusing light nuclei moves
up the curve toward the peak, and splitting heavy nuclei also moves toward the peak from the
other side. In both cases the products are more tightly bound, so the excess mass is released
as energy. Answer: B.


nuclear_and_particle

## Radioactivity and Decay

An unstable nucleus decays toward a more stable configuration, and the mode depends on which
way it is unstable. Alpha decay emits a helium-4 nucleus and occurs in heavy nuclei where the
Coulomb repulsion is large; it proceeds by quantum tunnelling through the Coulomb barrier,
which is why the half-lives vary over more than twenty orders of magnitude while the emitted
energies vary by only a factor of two — the exponential sensitivity of tunnelling to barrier
height and energy.


nuclear_and_particle

Beta decay converts a neutron to a proton or the reverse, mediated by the weak interaction.
Beta-minus emits an electron and an antineutrino and occurs in neutron-rich nuclei; beta-plus
emits a positron and a neutrino and occurs in proton-rich ones, as does electron capture. The
neutrino was postulated precisely because the emitted electron's energy spectrum is continuous
rather than discrete, which appeared to violate energy conservation until a third undetected
particle carrying the remainder was proposed — a proposal made in 1930 and confirmed
experimentally only in 1956.


nuclear_and_particle

Gamma decay emits a photon from a nucleus in an excited state, changing neither the proton nor
the neutron count. It typically follows alpha or beta decay, which frequently leaves the
daughter nucleus excited, and the discrete gamma energies are characteristic of the emitting
nuclide, which is what makes gamma spectroscopy a precise identification technique.


nuclear_and_particle

Decay and radiation:
  alpha                emits helium-4; heavy nuclei; tunnelling, so half-lives span
                       an enormous range
  beta-minus           neutron to proton plus electron and antineutrino
  beta-plus            proton to neutron plus positron and neutrino
  electron capture     an inner electron absorbed; competes with beta-plus
  gamma                photon from an excited nucleus; characteristic energies
  decay law            N = N_0 exp(-lambda t);  half-life = ln2 / lambda
  activity             decays per second; becquerel
  penetration          alpha stopped by paper, beta by millimetres of aluminium,
                       gamma requires centimetres of lead
  ionizing power       inverse to penetration; alpha is most damaging internally
                       and harmless externally
  absorbed dose        gray; energy per unit mass
  equivalent dose      sievert; weighted by radiation type
  background           ~2-3 mSv per year typical, dominated by radon and by
                       cosmic rays at altitude
  half-life range      from microseconds to billions of years


nuclear_and_particle

The inverse relationship between penetration and ionizing power has a consequence that is
frequently misunderstood. Alpha particles are stopped by a sheet of paper or the dead outer
layer of skin, so an external alpha source is essentially harmless. The same particles deposit
all their energy in a very short path, so an alpha emitter inhaled or ingested delivers an
intense localized dose and is among the most hazardous internal contaminants. The hazard
depends on the exposure pathway, not on the radiation type alone.


nuclear_and_particle

Radiometric dating applies the decay law in reverse, and its reliability rests on decay rates
being independent of temperature, pressure, and chemical environment — which has been verified
across the ranges accessible to experiment. The assumptions that must be checked in any
application are that the system remained closed to gain or loss of parent and daughter, and
that the initial daughter abundance is known or determinable, which isochron methods establish
from the data itself rather than assuming.


nuclear_and_particle

## Fission, Fusion, and Applications

Fission splits a heavy nucleus into fragments, releasing energy and typically two or three
neutrons. Those neutrons can induce further fissions, and whether the reaction sustains itself
depends on the multiplication factor: the average number of subsequent fissions caused by each
one. Below one the reaction dies out, at exactly one it is critical and self-sustaining at
constant rate, and above one it grows.


nuclear_and_particle

Reactor control rests on a detail that makes it possible at all. About 0.65 percent of fission
neutrons are delayed, emitted seconds to minutes after the fission by decaying fragments rather
than promptly. A reactor is operated so that it is subcritical on prompt neutrons alone and
critical only with the delayed contribution, which means the reactor's response time is
governed by the delayed neutron timescale of seconds rather than by the prompt timescale of
microseconds. Without delayed neutrons, mechanical control would be impossible.


nuclear_and_particle

Fusion combines light nuclei and releases more energy per unit mass than fission, with fuel
that is abundant and products that are not long-lived waste. The difficulty is the Coulomb
barrier: nuclei must approach within the range of the strong force against their mutual
repulsion, which requires temperatures of order a hundred million kelvin. At those temperatures
matter is a plasma that must be confined, and the two approaches are magnetic confinement,
holding the plasma in a magnetic field for long periods at low density, and inertial
confinement, compressing a small target so rapidly that it fuses before it disassembles.


nuclear_and_particle

Fission and fusion:
  fission              heavy nucleus splits; ~200 MeV per event
  chain reaction       neutrons induce further fissions
  multiplication factor  below 1 subcritical, 1 critical, above 1 supercritical
  delayed neutrons     ~0.65%; what makes reactor control possible
  moderator            slows neutrons to thermal energies where fission cross
                       sections are far larger; water, heavy water, graphite
  control rods         neutron absorbers; adjust reactivity
  enrichment           raising the uranium-235 fraction; ~3-5% for power reactors
  breeder reactor      produces more fissile material than it consumes
  fission products     highly radioactive; the waste problem
  fusion, D-T          deuterium plus tritium; the easiest reaction, and it
                       produces a fast neutron that activates the structure
  Lawson criterion     the density, temperature, and confinement time product
                       required for net energy
  magnetic confinement tokamaks and stellarators; long confinement, low density
  inertial confinement laser or beam compression; short confinement, high density
  aneutronic fuels     avoid neutron activation; require much higher temperatures


nuclear_and_particle

The waste question is worth stating accurately because the discussion is frequently
imprecise. Fission products are intensely radioactive and mostly short-lived, decaying to
background levels over a few hundred years. Transuranic actinides produced by neutron capture
are far less intensely radioactive and persist for tens of thousands of years, and they are what
drives the long-term disposal requirement. Reprocessing separates them and they can in principle
be transmuted in fast reactors, which converts a hundred-thousand-year problem into a
several-hundred-year one — a technically demonstrated capability whose deployment is limited by
economics and by proliferation concerns rather than by physics.


nuclear_and_particle

## The Standard Model

The standard model describes all known fundamental particles and three of the four forces, and
it is the most precisely tested theory in science. Matter consists of fermions in three
generations: quarks that feel the strong force and leptons that do not. Forces are mediated by
gauge bosons. The Higgs field gives mass to the fermions and to the weak bosons through
spontaneous symmetry breaking.


nuclear_and_particle

The three generations are a genuine puzzle. The second and third generations are heavier
copies of the first with otherwise identical properties, they decay rapidly into first
generation particles, and ordinary matter consists entirely of the first. Why there are three,
and why the masses span such an enormous range — the top quark is roughly three hundred
thousand times heavier than the electron — is unexplained by the theory, which takes the masses
as input parameters rather than predicting them.


nuclear_and_particle

Quarks are never observed in isolation because of colour confinement. The strong force between
them does not weaken with distance as other forces do; the field energy grows, so separating
two quarks eventually supplies enough energy to create a new quark-antiquark pair, and the
result is two hadrons rather than two free quarks. The complementary property, asymptotic
freedom, is that the interaction weakens at very short distances, so quarks behave as nearly
free particles inside a hadron — which is why high-energy scattering experiments can probe them
at all.


nuclear_and_particle

The standard model:
  quarks               up, down, charm, strange, top, bottom; fractional charge,
                       carry colour
  leptons              electron, muon, tau and their neutrinos
  three generations    heavier copies; unexplained
  photon               electromagnetic; massless, infinite range
  W and Z bosons       weak; massive, hence very short range
  gluons               strong; carry colour themselves, which is why the force
                       behaves so differently
  Higgs boson          the excitation of the field giving mass; confirmed 2012
  confinement          quarks are never free; the field energy grows with
                       separation
  asymptotic freedom   the strong interaction weakens at short distance
  hadrons              baryons of three quarks, mesons of quark-antiquark
  mass origin          most of a proton's mass is binding energy, not quark mass
  antimatter           same mass, opposite charge; annihilates with matter
  neutrino oscillation  neutrinos change flavour, which requires nonzero mass;
                       the first confirmed departure from the original standard
                       model


nuclear_and_particle

Neutrino oscillation deserves emphasis because it is established physics beyond the standard
model as originally formulated. Neutrinos produced in one flavour are detected as another,
which requires that the flavour states be superpositions of states with different masses — and
therefore that at least two neutrino masses are nonzero. The original theory assumed them
massless. The absolute mass scale, the ordering of the masses, and whether neutrinos are their
own antiparticles all remain open, and each is the target of substantial current experimental
effort.


nuclear_and_particle

## Accelerators, Detectors, and Open Questions

Particle accelerators probe short distances by achieving high energy, and the connection is
direct: the de Broglie wavelength is inversely proportional to momentum, so resolving smaller
structures requires higher energy. They also create heavy particles, since energy converts to
mass, which is why discovering a particle of a given mass requires a collider capable of
supplying it.


nuclear_and_particle

Colliding beams rather than striking a fixed target is what makes the highest energies
accessible. In a fixed-target collision much of the energy goes into the momentum of the
products rather than into the interaction, and the useful centre-of-mass energy grows only as
the square root of the beam energy. In a symmetric collider the centre-of-mass frame is the
laboratory frame and the full energy is available, which is why every high-energy frontier
machine is a collider.


nuclear_and_particle

Detectors reconstruct what happened from what emerges, and modern designs are layered
cylinders each identifying different particles. Tracking detectors in a magnetic field measure
charged particle trajectories, with the curvature giving momentum and the sign of the charge.
Electromagnetic calorimeters absorb electrons and photons and measure their energy. Hadronic
calorimeters do the same for hadrons. Muon chambers on the outside detect the only charged
particles that penetrate everything else. Neutrinos escape entirely and are inferred from
missing transverse momentum.


nuclear_and_particle

Accelerators, detectors, and open problems:
  wavelength and energy  higher energy resolves smaller structures
  fixed target vs collider  collider energy is fully available; fixed target
                         wastes most of it to momentum
  synchrotron radiation  charged particles radiate when accelerated; the reason
                         electron rings are limited and proton rings are not
  luminosity             collision rate; determines how fast rare processes
                         accumulate
  tracking detector      curvature in a magnetic field gives momentum and charge
  calorimeter            absorbs and measures energy
  muon system            the outermost layer; muons penetrate everything else
  missing momentum       infers neutrinos and any invisible particle
  trigger                selects which of billions of collisions per second to
                         record; most data is discarded in microseconds and is
                         unrecoverable, so trigger design determines what can be
                         discovered
  five sigma             the discovery convention in particle physics; a
                         one-in-3.5-million false positive rate, chosen because
                         the number of searches is enormous
  open: dark matter      no standard model candidate
  open: matter asymmetry  standard model CP violation is far too small
  open: neutrino mass    origin, scale, and ordering
  open: hierarchy        why the Higgs mass is so far below the Planck scale
  open: gravity          not included in the standard model at all
  open: parameters       around nineteen free parameters taken from experiment


nuclear_and_particle

The trigger deserves the last technical word because it determines what is discoverable.
A modern collider produces collisions faster than any system could record, so hardware and
software triggers select a tiny fraction within microseconds based on coarse criteria, and
everything else is discarded permanently. This means the experiment can only find what the
trigger was designed to keep, and a signature nobody anticipated is thrown away before anyone
sees it. Considerable effort goes into designing triggers that preserve unanticipated
signatures, and it is a genuine limit on discovery rather than a technicality.


nuclear_and_particle

The applications of nuclear and particle physics extend far beyond the field. Medical imaging
uses positron emission from radiotracers and nuclear magnetic resonance. Radiotherapy uses
beams shaped to deposit dose in a tumour, with proton and carbon ion therapy exploiting the
Bragg peak to concentrate the dose at a controllable depth rather than along the whole path.
Carbon dating and other radiometric methods date archaeological and geological material.
Neutron scattering probes material structure. Accelerator-produced isotopes are used in
industry and medicine. And the World Wide Web was created at a particle physics laboratory to
share data among collaborators — the most economically consequential outcome of the field, and
one that no funding case could have predicted.


nuclear_engineering

## Nuclear Reactions and Cross Sections

Nuclear energy comes from the difference in binding energy per nucleon across the chart of nuclides.
That curve peaks near iron, which means that combining light nuclei releases energy and splitting
heavy nuclei releases energy, while the reverse of each consumes it. This single curve explains why
fusion powers stars, why fission powers reactors, and why iron is the endpoint of stellar
nucleosynthesis.


nuclear_engineering

The cross section quantifies the probability of a reaction, expressed as an effective area presented
by a target nucleus. It depends strongly on the incident particle's energy, and for neutrons the
dependence has structure that dominates reactor design: cross sections generally rise as neutron
energy falls, with sharp resonances at specific energies where the compound nucleus has an excited
state. Fission cross sections for the common fissile isotopes are much larger for slow neutrons than
for fast ones, by roughly two orders of magnitude.


nuclear_engineering

That energy dependence is why most power reactors are thermal reactors. Fission produces fast
neutrons; slowing them to thermal energies through collisions with a moderator dramatically raises
the probability that they cause further fission, which permits a chain reaction with fuel that is
only slightly enriched. A fast reactor, operating without moderation, needs much higher enrichment
and in exchange can breed fissile material from the abundant fertile isotope, which is the tradeoff
underlying the entire design space.


nuclear_engineering

Reactions and cross sections:
  binding energy per nucleon  peaks near iron; explains fusion and fission both
                       releasing energy
  cross section        an effective area representing reaction probability
  energy dependence    generally rising as neutron energy falls
  resonances           sharp peaks at compound nucleus excited states
  fissile              fissions with slow neutrons
  fertile              becomes fissile after capture and decay
  thermal neutrons     slowed to equilibrium with the medium
  the two-order factor  thermal fission cross sections vastly exceed fast ones
  moderation           slowing neutrons through collisions
  good moderators      light nuclei that scatter without absorbing; hydrogen,
                       deuterium, carbon
  the moderator tradeoff  ordinary water moderates well and absorbs some
                       neutrons, so it requires enriched fuel; heavy water absorbs
                       far less and permits natural uranium
  fast reactors        no moderation; higher enrichment, breeding capability
  neutron economy      the accounting of where every neutron goes; the central
                       design consideration
  delayed neutrons     a small fraction emitted seconds after fission by decay
                       products, and they are what makes a reactor controllable


nuclear_engineering

Q: Delayed neutrons constitute under one percent of fission neutrons. Why do they dominate reactor
control?
A. They have higher energy
B. Reactors are operated so that the prompt neutrons alone are slightly subcritical, making the
   chain reaction depend on the delayed fraction; this stretches the characteristic response time
   from microseconds to seconds, which is what makes mechanical control possible
C. They cause more fissions
D. They are easier to detect
Reasoning: With prompt neutrons alone, the multiplication timescale is the prompt neutron lifetime,
far too short for any control system. Operating in the range where criticality requires the delayed
contribution ties the period to the precursor decay times. Exceeding prompt criticality removes this
margin entirely and is the condition every design and procedure is arranged to prevent. Answer: B.


nuclear_engineering

## Reactor Physics and Control

Criticality is the condition where each generation of neutrons exactly reproduces itself, quantified
by the multiplication factor. Below one the chain reaction dies out; above one it grows; at exactly
one it is steady. A reactor is not held at criticality by a fixed configuration but by continuous
adjustment, because the reactivity changes as fuel depletes, as fission products accumulate, and as
temperature varies.


nuclear_engineering

Reactivity feedback is the mechanism that makes a reactor stable, and it must be negative for a safe
design. When temperature rises, several effects change the reactivity: thermal expansion reduces
density, the moderator becomes less effective, and resonance absorption broadens with temperature so
that more neutrons are captured without fission. A design where the net effect is negative responds
to a power increase by reducing reactivity, which limits the excursion without any control action.
This is inherent safety and it is a design requirement rather than a bonus.


nuclear_engineering

Fission product poisoning is the operational effect that most surprises people new to the field.
Xenon-135 has an enormous absorption cross section and is produced partly by the decay of a
precursor, so after a power reduction its concentration keeps rising for hours before falling. A
reactor shut down or reduced in power may become impossible to restart for many hours until the
xenon decays, and the transient after a return to power runs the other way. Managing this is a
routine part of operation and it constrains how quickly a plant can change output.


nuclear_engineering

Reactor physics:
  multiplication factor  neutrons per generation relative to the previous
  criticality          exactly one; steady power
  subcritical and supercritical  decaying and growing
  reactivity           the fractional departure from criticality
  control rods         strong absorbers moved to adjust reactivity
  chemical shim        a dissolved absorber for slow adjustment
  burnable poisons     absorbers that deplete as fuel does, flattening the
                       reactivity over the cycle
  temperature feedback  the inherent stability mechanism
  Doppler broadening   resonance absorption widening with temperature; a prompt
                       negative effect
  void coefficient     the reactivity change when coolant boils or is lost; must
                       be negative in a safe design
  the design requirement  net negative feedback, not merely stable operation
  xenon poisoning      a strong absorber accumulating after power reduction
  the restart constraint  hours of inability to restart after a shutdown
  power maneuvering    limited by poisoning dynamics as well as by thermal stress
  decay heat           several percent of full power immediately after shutdown,
                       falling slowly; it must be removed indefinitely, and every
                       major reactor accident has involved a failure to do so


nuclear_engineering

## Thermal Hydraulics and Safety

A power reactor is a heat source and the plant around it is a heat engine, so the limits are as much
thermal as nuclear. Heat is generated in the fuel, conducted through the fuel and cladding,
transferred to the coolant, and carried to the steam system. Each step has a limit, and the binding
one is usually the heat flux at which boiling changes character from efficient nucleate boiling to
film boiling, where a vapour layer insulates the surface and the temperature jumps abruptly.


nuclear_engineering

That transition — the critical heat flux — is the governing safety limit in water-cooled reactors,
because exceeding it causes cladding temperature to rise sharply within seconds and can lead to
failure. Operating margins are defined relative to it, and the instrumentation and protection system
exists to keep the plant away from the conditions where it could be approached.


nuclear_engineering

Safety design follows defence in depth: multiple independent barriers between the fuel and the
environment, multiple independent means of achieving each safety function, and a design basis that
assumes failures occur. The fundamental safety functions are controlling reactivity, removing heat,
and containing radioactive material, and every safety system serves one of them. Passive systems
that work by gravity, natural circulation, or stored energy are preferred over active ones because
they do not depend on power or on operator action, and the trend in newer designs is toward
achieving the safety functions passively for an extended period.


nuclear_engineering

Thermal hydraulics and safety:
  the heat path        fuel, cladding, coolant, steam system
  nucleate boiling     efficient heat transfer with bubbles at the surface
  film boiling         an insulating vapour layer; temperature jumps
  critical heat flux   the transition; the governing thermal limit
  operating margin     defined relative to it
  the three safety functions  control reactivity, remove heat, contain material
  defence in depth     independent barriers and independent means
  the barriers         fuel matrix, cladding, pressure boundary, containment
  single failure criterion  safety functions achievable with any one component
                       failed
  redundancy and diversity  multiple copies and multiple different mechanisms
  passive systems      gravity, natural circulation, stored energy
  the preference       no dependence on power or operator action
  emergency core cooling  injecting coolant on loss of the primary inventory
  containment          the final barrier; designed for the pressure and
                       temperature of the worst analysed release
  the accident lesson  the events that mattered involved decay heat removal
                       failing, often for reasons outside the design basis, which
                       is why extended station blackout coping capability became
                       a design requirement afterward


nuclear_engineering

## Radiation Interaction, Shielding, and Dosimetry

Radiation interacts with matter differently by type, and shielding design follows directly. Alpha
particles are stopped by paper or the outer skin layer and are hazardous only if the source is
inhaled or ingested. Beta particles penetrate millimetres of tissue and are shielded by low-atomic-
number materials — using lead produces secondary photons through braking radiation, so plastic
first and lead second is the correct order. Gamma rays are attenuated exponentially with no definite
range, requiring dense high-atomic-number material and thickness proportional to the logarithm of
the required reduction.


nuclear_engineering

Neutrons require a different approach entirely because they interact with nuclei rather than
electrons. Shielding them means first slowing them with hydrogen-rich material, then absorbing the
slowed neutrons with a material having a high thermal capture cross section, and finally attenuating
the gamma rays that capture produces. A layered shield of water or polyethylene, boron, and lead
addresses all three, and omitting any layer leaves a hazard the other layers do not cover.


nuclear_engineering

Dose quantities are distinguished carefully because they answer different questions. Absorbed dose
is energy deposited per unit mass. Equivalent dose weights it by radiation type, since alpha
particles cause more damage per unit energy than gammas. Effective dose further weights by the
sensitivity of the tissues irradiated, giving a single number for whole-body risk comparison. Using
the wrong one — comparing an absorbed dose to an effective dose limit, for instance — produces errors
of an order of magnitude.


nuclear_engineering

Radiation and dosimetry:
  alpha                stopped by paper; hazardous internally
  beta                 millimetres of tissue; shield with low atomic number first
  braking radiation    secondary photons from stopping betas in dense material
  gamma                exponential attenuation, no definite range
  half-value layer     the thickness halving the intensity; the practical shield
                       unit
  neutrons             interact with nuclei; require a layered shield
  the neutron layers   moderate, capture, then attenuate capture gammas
  time, distance, shielding  the three protective factors; distance is free and
                       most effective
  the inverse square   halving intensity requires only a modest distance increase
                       close to a source
  absorbed dose        energy per unit mass
  equivalent dose      weighted by radiation type
  effective dose       further weighted by tissue sensitivity
  the comparison error  mixing the quantities changes answers by an order of
                       magnitude
  contamination versus exposure  material present versus radiation received; the
                       responses are entirely different
  detection            ionization chambers, scintillators, semiconductors, and
                       thermoluminescent dosimeters, each suited to a different
                       combination of radiation type, energy range, and whether
                       real-time or integrated measurement is needed


nuclear_engineering

## Fuel Cycle and Fusion

The nuclear fuel cycle runs from mining through conversion, enrichment, fabrication, irradiation,
storage, and eventual disposal or recycling. Enrichment increases the fraction of the fissile
isotope, and the same technology at higher output produces weapons material, which is why enrichment
capability is the central concern of nonproliferation policy and why the fuel cycle is a political
subject as much as a technical one.


nuclear_engineering

Spent fuel contains unfissioned fuel, fissile material produced during irradiation, and fission
products of widely varying half-life. The short-lived products dominate the heat and radiation for
decades; the long-lived ones dominate the disposal timescale. Reprocessing separates the reusable
material from the waste, reducing the volume requiring long-term disposal and producing separated
fissile material, which is the proliferation concern that makes the choice contested.


nuclear_engineering

Fusion offers a fuel supply that is effectively unlimited and no long-lived fission product waste,
and it has proven far harder than fission. Confining a plasma hot enough for the nuclei to overcome
their electrostatic repulsion, dense enough for collisions to be frequent, and for long enough for
the released energy to exceed the input is the requirement, and the product of those three
quantities must exceed a threshold. Magnetic confinement in a toroidal geometry and inertial
confinement by compressing a fuel pellet are the two main approaches, and both have now demonstrated
energy release exceeding the energy delivered to the fuel while remaining well short of a net-
positive power plant.


nuclear_engineering

Fuel cycle and fusion:
  the cycle stages     mining, conversion, enrichment, fabrication, irradiation,
                       storage, disposal or recycling
  enrichment           raising the fissile fraction
  the dual use problem  the same capability at higher output produces weapons
                       material
  spent fuel composition  unfissioned fuel, bred fissile material, fission
                       products
  the two timescales   decades of heat and radiation, millennia of disposal
                       concern
  interim storage      pools then dry casks; a mature and low-risk technology
  geological disposal  the technical consensus solution and a persistent
                       political difficulty
  reprocessing         separating reusable material from waste
  the proliferation tension  separated fissile material is the concern
  the fusion conditions  temperature, density, and confinement time above a
                       product threshold
  magnetic confinement  a toroidal field configuration holding the plasma
  inertial confinement  compressing a pellet faster than it disassembles
  tritium breeding     required for a deuterium-tritium plant; the fuel is
                       generated from lithium in the surrounding blanket
  materials            the hardest unsolved engineering problem, since the
                       structure surrounding the plasma is subjected to a neutron
                       flux that damages and activates it over the plant's life


optics_and_photonics

## Geometric Optics

Geometric optics treats light as rays travelling in straight lines through homogeneous
media, bending at interfaces. It is the short-wavelength limit of wave optics, valid when
the structures involved are much larger than the wavelength, and it is entirely adequate for
designing lenses, mirrors, and imaging systems until diffraction becomes the limiting factor.


optics_and_photonics

Refraction at an interface follows Snell's law, with the product of refractive index and the
sine of the angle from the normal conserved across the boundary. Light entering a denser
medium bends toward the normal and slows, since the refractive index is the ratio of the
speed of light in vacuum to that in the medium. Going from dense to less dense, the
refracted ray bends away from the normal, and beyond a critical angle it cannot escape at
all — total internal reflection, which is the mechanism of optical fiber, of prisms used as
mirrors in binoculars, and of the sparkle of a cut diamond, whose high index gives it a
critical angle of only 24 degrees.


optics_and_photonics

The thin lens equation relates object distance, image distance, and focal length, with sign
conventions that must be stated because they differ between texts. Magnification is the
negative ratio of image to object distance, so a real image formed beyond the focal point is
inverted. The distinction between real and virtual images is worth precision: a real image
can be projected onto a screen because light actually converges there, while a virtual image
only appears to exist because diverging rays are extrapolated backward, which is what a
magnifying glass and a plane mirror produce.


optics_and_photonics

Geometric optics relations:
  refractive index      n = c / v;  air ~1.0003, water 1.33, glass 1.5-1.9
  Snell's law           n1 sin(theta1) = n2 sin(theta2)
  critical angle        sin(theta_c) = n2/n1,  n1 > n2
  thin lens             1/f = 1/d_o + 1/d_i
  magnification         m = -d_i / d_o
  lensmaker's equation  1/f = (n-1)(1/R1 - 1/R2)
  f-number              N = f / D;  light gathering scales as 1/N^2
  depth of field        increases with f-number, decreases with magnification
  numerical aperture    NA = n sin(theta_max);  resolution and light collection
  mirror equation       same form as thin lens; f = R/2


optics_and_photonics

Aberrations are the departures from ideal imaging and each has a distinct signature.
Spherical aberration arises because a spherical surface does not focus all annuli to the
same point, producing a soft image with no sharp focus; it is corrected by aspheric surfaces
or by combining elements. Coma makes off-axis points image as comet shapes. Astigmatism
gives different focal positions for rays in perpendicular planes. Field curvature means the
sharp image surface is not flat. Distortion changes magnification with field position,
producing barrel or pincushion shapes without blurring. Chromatic aberration comes from the
refractive index varying with wavelength and is corrected by combining glasses with
different dispersions in an achromatic doublet — a defect mirrors do not have at all, which
is one reason large telescopes use them.


optics_and_photonics

Q: A ray passes from glass (n = 1.5) into air. Total internal reflection occurs above an
incidence angle of approximately:
A. 30 degrees   B. 42 degrees   C. 48 degrees   D. It cannot occur
Reasoning: The critical angle satisfies sin(theta_c) = n_air/n_glass = 1/1.5 = 0.667, so
theta_c = 41.8 degrees. Total internal reflection requires going from higher to lower index,
which is satisfied here. Answer: B.


optics_and_photonics

## Wave Optics

Wave optics accounts for interference and diffraction, phenomena geometric optics cannot
describe. Interference occurs when coherent waves superpose, adding constructively where
they are in phase and destructively where they are opposed. The requirement is coherence —
a fixed phase relationship — which is why interference from independent light sources is not
observed and why demonstrations split a single beam.


optics_and_photonics

Young's double slit produces fringes whose spacing depends on wavelength, slit separation,
and screen distance, and it is the canonical demonstration that light is a wave. The same
experiment performed with single photons, or with electrons, produces the same fringe
pattern built up one detection at a time, which is the cleanest statement of wave-particle
duality and remains the most direct demonstration of quantum superposition.


optics_and_photonics

Thin film interference explains the colours of oil slicks, soap bubbles, and anti-reflection
coatings. Light reflecting from the top and bottom surfaces of a thin film interferes, with
the path difference determined by film thickness and index, and a half-wavelength phase shift
occurring on reflection from a higher-index medium. Anti-reflection coatings exploit this
deliberately: a quarter-wave layer with index equal to the geometric mean of the two
surrounding media makes the two reflections cancel, and multilayer stacks extend the
cancellation across a broad band, which is why quality optics appear coloured in reflection
rather than white.


optics_and_photonics

Diffraction is the spreading of waves at apertures and edges, and it sets the fundamental
resolution limit of every optical instrument. A circular aperture produces an Airy pattern
with a central disc surrounded by rings, and two point sources are conventionally resolvable
when the centre of one falls on the first minimum of the other — the Rayleigh criterion,
giving an angular resolution of 1.22 times wavelength divided by aperture diameter.


optics_and_photonics

Wave optics relations:
  double slit maxima     d sin(theta) = m lambda
  single slit minima     a sin(theta) = m lambda
  Rayleigh criterion     theta = 1.22 lambda / D
  diffraction grating    d sin(theta) = m lambda;  resolving power R = mN
  thin film constructive 2 n t = (m + 1/2) lambda, with one phase-inverting
                         reflection
  anti-reflection        n_coat = sqrt(n1 n2),  thickness = lambda/(4 n_coat)
  coherence length       L = c / delta_nu ≈ lambda^2 / delta_lambda
  Abbe diffraction limit d = lambda / (2 NA);  microscope resolution
  Fresnel number         determines near-field versus far-field regime


optics_and_photonics

Coherence length is the practical parameter governing whether interference is observable in
a given configuration. It is the path difference over which a source maintains a fixed phase
relationship, and it is inversely proportional to spectral bandwidth: a white light source
has a coherence length of a few micrometres, an LED perhaps tens, and a stabilized laser can
exceed kilometres. Interferometric measurement requires the path difference to stay within
the coherence length, which is exactly why laser interferometry can measure over long
baselines and white-light interferometry only works within a very narrow range — a property
turned into an advantage in optical coherence tomography, where the short coherence length
provides depth sectioning.


optics_and_photonics

## Polarization

Polarization describes the orientation of the electric field oscillation in an
electromagnetic wave. Unpolarized light, such as that from thermal sources, has rapidly
randomizing orientation. Linear polarization confines the field to one plane; circular
polarization results from two perpendicular components of equal amplitude with a quarter-wave
phase difference, so the field vector rotates; elliptical is the general case.


optics_and_photonics

A linear polarizer transmits the component along its axis. Malus's law gives the transmitted
intensity as the incident intensity times the squared cosine of the angle between the
polarization and the axis. For unpolarized input, a polarizer transmits half regardless of
orientation. The three-polarizer demonstration is the standard surprise: crossed polarizers
transmit nothing, and inserting a third at 45 degrees between them restores transmission,
because each polarizer projects the field onto a new axis rather than merely filtering a
pre-existing component.


optics_and_photonics

Reflection polarizes light, which has direct practical consequence. Light reflecting from a
dielectric surface is preferentially polarized parallel to the surface, and at Brewster's
angle the reflected light is completely polarized. This is why polarized sunglasses reduce
glare from water and roads by blocking the horizontal component, and why photographers use
rotating polarizing filters to control reflections and to darken skies, since scattered
skylight is also partially polarized.


optics_and_photonics

Polarization components and effects:
  polarizer            transmits one linear component; Malus's law
  Brewster's angle     tan(theta_B) = n2/n1;  reflected light fully polarized
  birefringence        different index for different polarizations; calcite,
                       quartz, stressed plastic
  wave plate           introduces a phase difference; quarter-wave converts linear
                       to circular, half-wave rotates linear polarization
  optical activity     rotation of polarization by chiral molecules; used to
                       measure sugar concentration
  Faraday rotation     magnetic-field-induced rotation; non-reciprocal, which is
                       what makes optical isolators possible
  photoelastic         stress-induced birefringence; visualizes stress fields
  LCD                  voltage-controlled birefringence between crossed polarizers
  Kerr and Pockels     electric-field-induced birefringence; fast optical modulators


optics_and_photonics

The non-reciprocity of Faraday rotation deserves emphasis because it enables a component
with no passive equivalent. Most optical effects are reciprocal — light traversing a path
backward experiences the same effect — so a wave plate rotating polarization one way rotates
it back on return. Faraday rotation depends on the magnetic field direction rather than the
propagation direction, so the rotation adds rather than cancels on a return pass. Combining
it with polarizers produces an optical isolator that passes light one way and blocks the
reverse, which is essential in laser systems where back-reflection into the cavity
destabilizes or damages the source.


optics_and_photonics

## Lasers and Light Sources

A laser produces coherent, monochromatic, directional light through stimulated emission. An
excited atom stimulated by a photon of matching energy emits a second photon identical in
frequency, phase, direction, and polarization — this cloning is the origin of coherence and
is what distinguishes laser light from thermal emission.


optics_and_photonics

Three ingredients are required. A gain medium provides the atoms or molecules with a usable
transition. A pump supplies energy to create population inversion, meaning more atoms in the
upper state than the lower, which is necessary because otherwise absorption exceeds
stimulated emission. And an optical cavity, typically two mirrors, feeds light back through
the gain medium so it is amplified repeatedly, with one mirror partially transmitting to let
the beam out.


optics_and_photonics

Population inversion cannot be achieved in a two-level system, because pumping the
transition also stimulates the downward transition and the populations equalize at best.
Three-level and four-level schemes solve this by pumping to a higher state that decays
rapidly into the upper laser level, and in the four-level case having the lower laser level
also empty rapidly so it stays nearly unpopulated. Four-level systems have much lower
threshold pump power for this reason and dominate practical designs.


optics_and_photonics

Laser properties and types:
  monochromatic        narrow linewidth; a stabilized laser can reach sub-hertz
  coherent             long coherence length; enables interferometry and holography
  directional          low divergence set by diffraction from the aperture
  brightness           very high power per unit area per unit solid angle; this,
                       not total power, is what distinguishes a laser
  gas (HeNe, CO2)      good beam quality; CO2 at 10.6 um for material processing
  solid state (Nd:YAG) high power, 1064 nm, frequency-doubled to 532 nm green
  fiber                gain in a doped fiber; efficient, excellent beam quality,
                       now dominant for industrial cutting
  semiconductor        compact, efficient, directly modulated; communications and
                       consumer devices; poorer beam quality without correction
  Ti:sapphire          broadly tunable, ultrashort pulses
  mode locking         fixed phase across cavity modes produces femtosecond pulses
  Q-switching          stores energy then releases it in a nanosecond pulse of
                       enormous peak power


optics_and_photonics

The distinction between average and peak power is essential when working with pulsed lasers
and is a frequent source of both underestimated capability and underestimated hazard. A
mode-locked laser emitting one nanojoule pulses of 100 femtoseconds at 80 megahertz has an
average power of 80 milliwatts and a peak power of 10 kilowatts. The peak is what drives
nonlinear processes and what damages optics; the average is what determines thermal load.
Specifying only one of them describes the source incompletely.


optics_and_photonics

[user] I'm coupling a 976 nm diode into a single-mode fiber and only getting 30% through.
[assistant] tool_call optics.fiber_coupling({"source": "laser_diode",
"wavelength_nm": 976, "fiber_mfd_um": 6.2, "fiber_na": 0.14,
"source_na_fast": 0.55, "source_na_slow": 0.12, "lens_na": 0.25,
"measured_efficiency": 0.30})
[tool] {"na_mismatch_fast_axis": true, "captured_fraction_fast": 0.45,
"mode_overlap": 0.71, "theoretical_max": 0.32, "verdict": "beam_asymmetry_dominant",
"recommended": ["fast_axis_collimator", "anamorphic_pair", "or_multimode_fiber"]}
[assistant] You are actually at the theoretical limit for this setup — 30% measured against
a 32% maximum. The problem is not alignment; it is that your diode's beam is highly
asymmetric and single-mode fiber accepts only a symmetric mode.


optics_and_photonics

A laser diode emits from a rectangular aperture, so it diverges very differently in the two
axes: your fast axis has an NA of 0.55 and your slow axis 0.12. Your fiber's NA is 0.14, so
most of the fast-axis light is emitted outside the acceptance cone and simply cannot be
captured no matter how well you align. Even the captured light has poor mode overlap because
its spatial profile is elliptical and the fiber mode is circular.


optics_and_photonics

Three ways forward, depending on what you need.


optics_and_photonics

If you need single-mode output, circularize the beam. A fast-axis collimator — a short-focal
cylindrical microlens placed close to the facet — collimates the divergent axis first, after
which an anamorphic prism pair or a pair of crossed cylindrical lenses equalizes the two
axes. Done well this reaches 60-75% coupling. This is standard practice for diode-to-fiber
coupling and the components are commercially available.


optics_and_photonics

If you only need the power delivered and not a single spatial mode, use multimode fiber. A
105 micron core with 0.22 NA will take essentially everything with simple aspheric coupling,
often above 85%. For pumping a fiber amplifier this is the usual answer, since pump light
does not need to be single mode.


optics_and_photonics

If you need both high power and single mode, buy a fiber-pigtailed diode. The manufacturer
does the beam shaping at the wafer level with far better tolerances than bench assembly
achieves, and the specified coupled power is what you get.


optics_and_photonics

One diagnostic worth running first: measure the free-space beam profile at two distances to
confirm the divergence in each axis matches the datasheet. If the fast axis is worse than
specified the diode may be damaged, and no coupling optics will fix that.


optics_and_photonics

## Fiber, Detectors, and Nonlinear Optics

Optical fiber guides light by total internal reflection between a higher-index core and a
lower-index cladding. Single-mode fiber has a core small enough — around nine micrometres —
that only one spatial mode propagates, eliminating modal dispersion and enabling the
bandwidth-distance products that make long-haul communication possible. Multimode fiber has
a larger core, is far easier to couple into and to splice, and suffers modal dispersion that
limits it to short reaches.


optics_and_photonics

Loss in silica fiber has a minimum near 1550 nanometres, around 0.2 decibels per kilometre,
which is why that wavelength band carries essentially all long-haul traffic. At that loss a
signal travels roughly 100 kilometres before needing amplification, and erbium-doped fiber
amplifiers happen to have gain in exactly that band, which is an accident of materials
science that shaped the entire architecture of global communications.


optics_and_photonics

Dispersion limits how fast data can be sent. Chromatic dispersion arises because different
wavelengths travel at different speeds, so a pulse containing a range of wavelengths spreads
as it propagates, eventually overlapping its neighbours. It is compensated by fiber with
opposite dispersion sign or by digital signal processing at the receiver, which in coherent
systems can undo it almost entirely. Polarization mode dispersion, from residual
birefringence, is stochastic and harder to compensate.


optics_and_photonics

Detectors and fiber quantities:
  photodiode           reverse-biased junction; fast, linear, the workhorse
  PIN vs avalanche     APDs provide internal gain at the cost of excess noise and
                       high bias voltage
  photomultiplier      extremely sensitive, single-photon capable, bulky, fragile
  SPAD / SiPM          solid-state single-photon detection; timing resolution in
                       tens of picoseconds
  CCD / CMOS           imaging arrays; CMOS now dominant on speed and integration
  bolometer            thermal detector; broadband, slow, used in the far infrared
  responsivity         amps per watt; wavelength dependent
  dark current         signal with no light; doubles roughly every 7 degrees C
  NEP                  noise equivalent power; the sensitivity figure of merit
  fiber loss minimum   ~0.2 dB/km at 1550 nm
  dispersion           ~17 ps/(nm km) in standard single-mode fiber at 1550 nm


optics_and_photonics

Nonlinear optics becomes relevant at high intensity, where the material's response is no
longer proportional to the field. Second harmonic generation converts two photons into one
of twice the frequency, which is how green laser pointers produce 532 nanometres from a 1064
nanometre source. It requires a material without inversion symmetry and phase matching, so
that the generated waves from different depths add constructively rather than cancelling —
achieved by exploiting birefringence to equalize the phase velocities, or by periodically
poling the material to reset the phase.


optics_and_photonics

Third-order effects occur in any material, including glass, and dominate fiber behaviour at
high power. The Kerr effect makes the refractive index intensity-dependent, causing
self-phase modulation that broadens a pulse's spectrum, self-focusing that can damage optics,
and cross-phase modulation between wavelength channels. Four-wave mixing generates new
frequencies from combinations of existing ones, which is a nuisance in
wavelength-multiplexed systems and a tool for wavelength conversion and parametric
amplification. Supercontinuum generation, where an intense short pulse in a suitable fiber
broadens into a spectrum spanning an octave or more, combines several of these effects and
produces a source that is simultaneously as bright as a laser and as broadband as a lamp —
which is what made frequency comb metrology possible and earned a Nobel Prize for connecting
optical frequencies to microwave standards.


plasma_physics

## What Makes a Plasma

A plasma is an ionized gas in which the free charges produce collective behaviour, and the
collective aspect is the defining feature rather than the ionization. A gas with a few stray ions
is a gas; a plasma is a medium in which the long-range electromagnetic interaction between
charges dominates over the short-range collisions that govern neutral gases, so the constituents
move in response to fields they collectively generate.


plasma_physics

Debye shielding is the mechanism producing quasineutrality. A charge introduced into a plasma
attracts opposite charges and repels like ones, so its field is screened over a characteristic
distance — the Debye length — beyond which the plasma appears neutral. This is why a plasma is
quasineutral on scales larger than the Debye length while permitting substantial charge
separation on smaller scales, and it is the first of three criteria a medium must satisfy to
behave as a plasma.


plasma_physics

The three criteria are that the Debye length be much smaller than the system, so shielding
operates; that many particles occupy a Debye sphere, so the shielding is a statistical rather
than a two-body effect; and that the plasma frequency exceed the collision frequency with
neutrals, so collective behaviour dominates over collisional damping. A medium satisfying all
three exhibits plasma behaviour, and the parameter space they define spans from interstellar space
at one particle per cubic centimetre to the interior of a star.


plasma_physics

Plasma fundamentals:
  ionization           free charges; a necessary and insufficient condition
  collective behaviour  long-range interaction dominates; the defining property
  Debye length         the shielding distance; sqrt(eps0 k T / n e^2)
  quasineutrality      neutral on scales above the Debye length
  Debye sphere         many particles required for statistical shielding
  plasma frequency     the natural oscillation of electrons about the ions;
                       depends only on density, and light below it cannot
                       propagate
  plasma criteria      Debye length small, many particles per sphere, plasma
                       frequency above the neutral collision frequency
  degree of ionization  from a fraction of a percent to fully ionized; even
                       weakly ionized gases behave as plasmas
  temperature units    conventionally electronvolts; 1 eV is about 11,600 K
  non-thermal plasma   electrons far hotter than ions; the basis of most
                       industrial processing
  the fourth state     the most common state of visible matter in the universe


plasma_physics

The plasma frequency deserves specific attention because it determines how a plasma interacts
with electromagnetic waves. Electrons displaced from the ions oscillate at a frequency depending
only on the electron density, and a wave below that frequency cannot propagate — it is reflected.
This is why the ionosphere reflects radio below a critical frequency, enabling long-distance
shortwave propagation, and why that frequency varies with solar activity and with day and night.
It is also the basis of a common density diagnostic, since measuring the cutoff frequency measures
the density directly.


plasma_physics

Q: A radio wave at 5 MHz reflects off the ionosphere while a 50 MHz wave passes through. The
reason is:
A. Atmospheric absorption
B. The plasma frequency corresponding to the ionospheric electron density lies between them, and
   waves below it cannot propagate
C. The higher frequency has more energy
D. Refraction by the neutral atmosphere
Reasoning: An electromagnetic wave below the local plasma frequency cannot propagate in the
plasma and is reflected. The ionospheric plasma frequency depends on electron density and
typically falls in the range of several to tens of megahertz, varying with solar activity and time
of day, which is exactly what determines the maximum usable frequency for shortwave
communication. Answer: B.


plasma_physics

## Single Particle Motion and Descriptions

Understanding plasma behaviour begins with a single charged particle in electric and magnetic
fields, because the collective behaviour is built from it. A charge in a uniform magnetic field
gyrates in a circle perpendicular to the field at the cyclotron frequency, with a radius
proportional to the perpendicular velocity and inversely proportional to the field. Motion along
the field is unaffected, so the trajectory is a helix.


plasma_physics

Adding a perpendicular electric field produces a drift. The particle accelerates and decelerates
around its gyration, so the orbit's guiding centre moves perpendicular to both fields at a
velocity independent of charge and mass — which means electrons and ions drift together and no
current results. Other drifts, arising from gradients in the magnetic field, from curvature of
the field lines, and from gravity, do depend on charge, so they separate the species and drive
currents. Predicting whether a drift produces a current is one of the more useful pieces of
plasma intuition.


plasma_physics

Magnetic mirroring follows from an adiabatic invariant. As a particle moves into a region of
stronger field, its perpendicular energy increases and its parallel energy decreases to conserve
total energy, so a particle with enough perpendicular velocity is reflected before reaching the
strongest field. This traps particles between two regions of strong field, which is what confines
the radiation belts and what motivated an entire class of fusion device.


plasma_physics

Descriptions and single particle motion:
  gyration             circular motion perpendicular to B at the cyclotron
                       frequency
  Larmor radius        the gyration radius; proportional to perpendicular velocity
                       over field strength
  guiding centre       the gyration's centre; the useful coordinate when the
                       gyration is fast compared to the motion of interest
  E cross B drift      perpendicular to both fields; independent of charge and
                       mass, so it drives no current
  gradient and curvature drifts  charge dependent; separate species and drive
                       currents
  magnetic moment      an adiabatic invariant; conserved when the field varies
                       slowly compared to the gyration
  magnetic mirror      particles reflected from strong field regions; confines the
                       radiation belts
  loss cone            the velocity range that escapes a mirror; the fundamental
                       limitation of mirror confinement
  single particle picture  valid when collective effects are weak
  fluid description    treat the plasma as one or more charged fluids; simpler,
                       and loses velocity-space information
  MHD                  magnetohydrodynamics; one conducting fluid with a magnetic
                       field; the workhorse for large-scale behaviour
  kinetic description  the full velocity distribution; the Vlasov equation;
                       necessary where wave-particle resonance matters


plasma_physics

The choice of description is a modelling decision determining what phenomena can appear.
Magnetohydrodynamics treats the plasma as a single conducting fluid and captures equilibrium,
large-scale instabilities, and wave propagation, and it cannot describe anything depending on the
velocity distribution's shape. Kinetic theory retains the full distribution and captures
wave-particle resonance, Landau damping, and instabilities driven by distribution features, at
enormously greater computational cost. Using a fluid model where kinetic effects matter produces
results that are internally consistent and wrong.


plasma_physics

## Waves, Instabilities, and Magnetohydrodynamics

Plasmas support a rich variety of waves because they have several restoring forces — pressure,
electrostatic attraction, and magnetic tension — and several characteristic frequencies. The
resulting wave zoo is one of the subject's characteristic difficulties, and the organizing
principle is which restoring force dominates and whether the motion is parallel or perpendicular
to the magnetic field.


plasma_physics

Alfvén waves are the most important for large-scale dynamics. Magnetic field lines behave like
elastic strings under tension, and a transverse displacement propagates along them at the Alfvén
speed, which depends on the field strength and the mass density. This is the mechanism carrying
energy and information along magnetic field lines, and it appears throughout solar and
magnetospheric physics.


plasma_physics

Landau damping is the phenomenon most distinctive to plasmas and has no fluid analogue. A wave
can be damped in a collisionless plasma by transferring energy to particles moving at nearly the
wave's phase velocity, and whether it damps or grows depends on whether there are more particles
slightly slower than the wave — which gain energy — or slightly faster, which lose it. For a
Maxwellian distribution the former dominates and the wave damps; for a distribution with a bump
at high velocity the latter dominates and the wave grows, which is an instability.


plasma_physics

Instabilities are the practical obstacle in nearly every plasma application. Any free energy
source — a pressure gradient, a current, a velocity shear, a non-thermal distribution — can drive
a mode unstable, and the resulting turbulence transports particles and energy far faster than
collisions would. This anomalous transport is why magnetic confinement is difficult and why the
performance of a fusion device depends far more on turbulence than on classical collisional
transport.


plasma_physics

Waves and instabilities:
  Langmuir waves       electron oscillations near the plasma frequency
  ion acoustic waves   the plasma analogue of sound; ions provide inertia and
                       electrons provide pressure
  Alfvén waves         transverse propagation along field lines; magnetic tension
                       as the restoring force
  magnetosonic         compressional; combines magnetic and gas pressure
  whistlers            dispersive electron waves; descending audio tones from
                       lightning, which is how they were discovered
  Landau damping       collisionless damping by resonant particles; no fluid
                       analogue
  inverse Landau       a distribution with a positive slope drives growth; the
                       bump-on-tail instability
  two-stream           counterstreaming populations; a fundamental instability
  Rayleigh-Taylor      a heavy fluid supported by a light one; the interchange
                       mode in magnetized plasmas
  kink and sausage     current-driven distortions of a plasma column
  tearing mode         reconnection of field lines; changes the topology and
                       releases magnetic energy
  anomalous transport  turbulence transporting far faster than collisions; the
                       central obstacle in confinement
  frozen-in flux       in a perfectly conducting plasma the field is tied to the
                       fluid; the fundamental MHD result


plasma_physics

Magnetic reconnection deserves emphasis because it violates the frozen-in condition and is where
much of plasma physics' most consequential energy release occurs. Ideal magnetohydrodynamics has
field lines frozen into the fluid so their topology cannot change, and reconnection is the process
by which finite resistivity in a thin current sheet permits them to break and reconnect,
converting stored magnetic energy into kinetic and thermal energy explosively. It powers solar
flares, drives magnetospheric substorms, and limits confinement in fusion devices — and the rate
at which it occurs is far faster than simple resistive estimates predict, a discrepancy that took
decades to resolve and involves the current sheet becoming unstable and fragmenting.


plasma_physics

## Fusion and Magnetic Confinement

Controlled fusion requires confining a plasma hot enough and dense enough for long enough that
the fusion power exceeds the losses. The Lawson criterion expresses this as a requirement on the
product of density, temperature, and confinement time, and the two mainstream approaches occupy
opposite corners of that product: magnetic confinement holds a low-density plasma for seconds,
and inertial confinement compresses a high-density target for nanoseconds.


plasma_physics

The tokamak is the most developed magnetic configuration. A toroidal field confines particles
along field lines, and because a purely toroidal field is stronger on the inside of the torus than
the outside, gradient and curvature drifts separate the charges and produce an electric field
that expels the plasma. A poloidal field, generated by driving a current through the plasma
itself, twists the field lines into helices so that a particle spends time on both the inner and
outer sides and the drifts cancel. The plasma current is therefore essential and it is also a free
energy source driving instabilities, which is the configuration's central tension.


plasma_physics

The stellarator achieves the twist with external coils instead, eliminating the plasma current
and the instabilities it drives, at the cost of coils with complicated three-dimensional geometry
that were impractical to design and build until computational optimization and precision
manufacturing matured. The renewed interest in stellarators follows directly from those
capabilities becoming available.


plasma_physics

Fusion and confinement:
  Lawson criterion     the density, temperature, and confinement time product
                       required for net energy
  D-T reaction         the easiest; the largest cross section at accessible
                       temperature
  neutron production   D-T produces a 14 MeV neutron; activates the structure and
                       is also how the energy is extracted and tritium is bred
  tritium breeding     tritium does not occur naturally; the neutron must breed it
                       in a lithium blanket, and the breeding ratio must exceed one
  magnetic confinement  low density, long confinement
  inertial confinement  high density, nanosecond confinement
  tokamak              toroidal plus plasma-driven poloidal field
  the drift problem    a purely toroidal field separates charges; the twist
                       cancels the drift
  plasma current       essential in a tokamak and a free energy source for
                       instabilities
  disruption           a sudden loss of confinement; deposits the stored energy on
                       the wall and is a primary engineering concern at scale
  stellarator          external coils provide the twist; no plasma current, harder
                       geometry
  divertor             directs the edge plasma to a target; handles the exhaust
                       heat flux, which is among the hardest engineering problems
  H-mode               a spontaneous transition to improved confinement with an
                       edge transport barrier; discovered empirically and still
                       not fully explained
  ELMs                 edge instabilities releasing bursts of energy; damaging at
                       reactor scale and requiring active mitigation
  ignition             the fusion power sustains the temperature without external
                       heating; achieved transiently by inertial confinement in
                       2022


plasma_physics

The H-mode transition deserves note because it is a genuine empirical discovery that theory did
not predict and still does not fully explain. Above a threshold heating power, a tokamak plasma
spontaneously forms a transport barrier near its edge, and confinement roughly doubles. The
discovery was essential to the viability of magnetic fusion, since the required device size falls
substantially with better confinement. Its mechanism involves sheared flow suppressing turbulence,
and the details of the threshold and the barrier structure remain active research — which is a
reminder that this is an experimental science in which theory has not been able to predict the
most consequential behaviours in advance.


plasma_physics

## Laboratory, Industrial, and Astrophysical Plasmas

Low-temperature plasmas are the field's largest application by economic value and receive the
least attention in accounts focused on fusion. Plasma processing is essential to semiconductor
manufacturing: plasma etching removes material with a directionality that wet chemistry cannot
achieve, permitting the vertical sidewalls that dense circuits require, and plasma-enhanced
deposition grows films at temperatures the substrate can tolerate.


plasma_physics

The property making these applications work is that the plasma is far from thermal equilibrium.
Electrons, being light, are accelerated efficiently by the applied field and reach temperatures of
several electronvolts, while the heavy ions and neutrals remain near room temperature. The
energetic electrons drive the chemistry — dissociating molecules and creating reactive radicals —
while the gas stays cool enough not to damage the substrate. This separation of electron and gas
temperature is what makes plasma processing possible and has no equilibrium analogue.


plasma_physics

Astrophysical plasmas span an enormous parameter range and share the characteristic that
magnetic fields organize the dynamics. The solar corona, the solar wind, planetary magnetospheres,
the interstellar medium, accretion discs, and jets from compact objects are all plasmas, and in
each the field is dynamically important despite being weak by laboratory standards, because the
scales are so large that the magnetic energy is comparable to or exceeds the kinetic and thermal
energies.


plasma_physics

Applications:
  plasma etching       directional material removal; enables vertical sidewalls
                       and therefore dense circuits
  plasma deposition    film growth at low substrate temperature
  surface treatment    activation, cleaning, adhesion promotion
  non-equilibrium      hot electrons with cool gas; the enabling property
  lighting             fluorescent and high-intensity discharge
  plasma displays      historical; superseded
  arc welding and cutting  thermal plasma; high power density
  plasma medicine      sterilization and wound treatment; reactive species at low
                       temperature
  thrusters            Hall and ion; high specific impulse for spacecraft
  solar corona         a million kelvin above a six thousand kelvin surface; the
                       heating mechanism is unsettled
  solar wind           a continuously expanding corona
  magnetosphere        the field deflecting the wind; reconnection couples them
  interstellar medium  weakly ionized, magnetized, turbulent
  accretion discs      the magnetorotational instability provides the angular
                       momentum transport that makes accretion possible
  astrophysical jets   collimated relativistic outflows; magnetically launched and
                       collimated


plasma_physics

Plasma diagnostics deserve a closing note because measuring a plasma without perturbing it is a
distinctive difficulty. Inserting a probe into a hot plasma both perturbs it and destroys the
probe, so most diagnostics are non-invasive and infer properties from emitted or transmitted
radiation. Spectroscopy gives temperature, density, and composition from line intensities and
widths. Interferometry gives density from the phase shift of a transmitted beam, since the
refractive index depends on electron density. Thomson scattering gives the electron temperature
and density from the spectrum of scattered laser light, and it is the most direct measurement
available. Magnetic probes outside the plasma measure the field and infer the current
distribution. Each measures a different quantity with different assumptions, and cross-checking
several is standard practice precisely because each inference chain has assumptions that the
others do not share.


quantum_information

## Qubits, States, and Measurement

A classical bit is zero or one; a qubit is a superposition of both, described by two complex
amplitudes whose squared magnitudes give the probabilities of each outcome on measurement. The
amplitudes carry a relative phase as well as magnitudes, and that phase has no classical analogue —
it is what makes interference possible and it is where quantum computing's advantage originates.


quantum_information

The state of n qubits requires two to the power n amplitudes to describe, which is the source of both
the promise and the persistent misunderstanding. The exponential state space does not mean a quantum
computer evaluates all possibilities in parallel and reads out the best one. Measurement collapses
the state and returns a single outcome sampled from the probability distribution, so the algorithmic
work is arranging interference so that amplitudes for wrong answers cancel and those for right
answers reinforce. Algorithms that fail to do this gain nothing from superposition.


quantum_information

Measurement is destructive and basis-dependent. Measuring in the computational basis returns zero or
one and destroys the superposition; measuring in a different basis asks a different question and
returns a different distribution. This is not ignorance being revealed — the pre-measurement state
genuinely lacks a definite value, as the violation of classical correlation bounds establishes
experimentally.


quantum_information

Qubits and measurement:
  superposition        both states with complex amplitudes
  probability          the squared magnitude of an amplitude
  relative phase       no classical analogue; enables interference
  the state space      exponential in the qubit count
  the misunderstanding  parallel evaluation with the best answer read out
  measurement collapse  a single sampled outcome
  the algorithmic requirement  interference cancelling wrong answers
  the consequence      superposition alone provides no advantage
  basis dependence     measurement in a different basis asks a different question
  the no-cloning theorem  an unknown quantum state cannot be copied; this
                       forbids the obvious error correction strategy and enables
                       quantum key distribution
  global phase         physically irrelevant; relative phase is not
  the Bloch sphere     a geometric picture of a single qubit's state space
  mixed states         classical uncertainty over quantum states; described by a
                       density matrix rather than a state vector


quantum_information

Q: A quantum algorithm places a register in an equal superposition of all inputs and evaluates a
function. Does this find the desired input efficiently?
A. Yes; all inputs are evaluated simultaneously
B. No — measurement returns one uniformly random outcome, so this gains nothing over classical
   random guessing; an advantage requires subsequent operations that concentrate amplitude on the
   desired outcome through interference
C. Only for small inputs
D. Yes, with enough measurements
Reasoning: Superposition and function evaluation are the easy parts. The hard and essential part is
arranging the amplitudes so the measurement is likely to return what you want. Every quantum
algorithm with a proven advantage does something specific here, and the absence of that step is what
distinguishes a genuine algorithm from the naive picture. Answer: B.


quantum_information

## Gates, Circuits, and Entanglement

Quantum gates are unitary operations, which means they are reversible and preserve total probability.
Single-qubit gates rotate the state on the Bloch sphere; two-qubit gates create correlations between
qubits. A small set of gates suffices to approximate any unitary operation to arbitrary accuracy,
which is the quantum analogue of classical universality, and the standard sets combine a few
single-qubit rotations with one entangling two-qubit gate.


quantum_information

Entanglement is the correlation between qubits that has no classical description: a joint state that
cannot be written as a product of individual states. Measuring one qubit of an entangled pair
immediately determines the correlated outcome for the other regardless of separation, and this does
not permit signalling because the individual outcome is random and only the correlation — visible
after classical communication — carries the structure.


quantum_information

Circuits are read as time flowing left to right, with wires as qubits and boxes as gates. Circuit
depth is the number of sequential gate layers and is the quantity that matters for a noisy machine,
since decoherence accumulates with time rather than with gate count. Compilation maps an abstract
circuit onto a physical device's available gates and connectivity, and on a device where only
neighbouring qubits interact, this inserts swap operations that can multiply the depth substantially
— which is why device connectivity is as important a specification as qubit count.


quantum_information

Gates and entanglement:
  unitary operations   reversible and probability preserving
  single-qubit gates   rotations on the Bloch sphere
  two-qubit gates      create correlations
  universality         a small gate set approximates any unitary
  entanglement         a joint state not decomposable into individual states
  measurement correlation  determined regardless of separation
  no signalling        individual outcomes are random; correlation needs classical
                       communication to observe
  Bell inequality violation  experimental proof that no local classical
                       description reproduces the correlations
  circuit depth        sequential layers; what matters under decoherence
  gate count           less relevant than depth on a noisy device
  compilation          abstract circuit to physical gates and connectivity
  connectivity         which qubits can interact directly
  swap overhead        routing on limited connectivity multiplying depth
  the specification implication  connectivity matters as much as qubit count
  gate fidelity        the error per operation; two-qubit gates are typically an
                       order of magnitude worse than single-qubit ones and are
                       therefore the limiting resource


quantum_information

## Algorithms and Where the Advantage Lies

The known quantum algorithms with proven advantages fall into a few families and the list is shorter
than public discussion suggests. Period-finding underlies the factoring algorithm, which would break
widely deployed public key cryptography and provides an exponential speedup over the best known
classical methods. Amplitude amplification gives a quadratic speedup for unstructured search, which
is provably optimal for that problem and is a much more modest advantage than exponential.


quantum_information

Simulation of quantum systems is the application with the clearest physical motivation and possibly
the earliest practical value. Simulating the dynamics of a many-body quantum system on a classical
computer costs exponentially in the system size for general cases, and a quantum computer does it
naturally, which is the original motivation for the field. Chemistry, materials, and nuclear physics
are the target domains.