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where is the position vector (in meters). |
These solutions represent planar waves traveling in the direction of the normal vector . If we define the z direction as the direction of . and the x direction as the direction of , then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation |
Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation. |
This solution is the linearly polarized solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector. |
Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids. This is the basis for the Fourier transform method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form |
The wave vector is related to the angular frequency by |
where is the wavenumber and is the wavelength. |
The electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength. |
Assuming monochromatic fields varying in time as formula_30, if one uses Maxwell's Equations to eliminate , the electromagnetic wave equation reduces to the Helmholtz Equation for : |
with "k = ω/c" as given above. Alternatively, one can eliminate in favor of to obtain: |
A generic electromagnetic field with frequency can be written as a sum of solutions to these two equations. The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions. However, applying this expansion to each vector component of or will give solutions that are not generically divergence-free (), and therefore require additional restrictions on the coefficients. |
The multipole expansion circumvents this difficulty by expanding not or , but or into spherical harmonics. These expansions still solve the original Helmholtz equations for and because for a divergence-free field , . The resulting expressions for a generic electromagnetic field are: |
where formula_35 and formula_36 are the "electric multipole fields of order (l, m)", and formula_37 and formula_38 are the corresponding "magnetic multipole fields", and and are the coefficients of the expansion. The multipole fields are given by |
where "h"l(1,2)("x") are the spherical Hankel functions, "E"l(1,2) and "B"l(1,2) are determined by boundary conditions, and |
are vector spherical harmonics normalized so that |
The multipole expansion of the electromagnetic field finds application in a number of problems involving spherical symmetry, for example antennae radiation patterns, or nuclear gamma decay. In these applications, one is often interested in the power radiated in the far-field. In this regions, the and fields asymptote to |
The angular distribution of the time-averaged radiated power is then given by |
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel () who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the "s" and "p" polarizations incident upon a material interface. |
When light strikes the interface between a medium with refractive index "n"1 and a second medium with refractive index "n"2, both reflection and refraction of the light may occur. The Fresnel equations give the ratio of the "reflected" wave's electric field to the incident wave's electric field, and the ratio of the "transmitted" wave's electric field to the incident wave's electric field, for each of two components of polarization. (The "magnetic" fields can also be related using similar coefficients.) These ratios are generally complex, describing not only the relative amplitudes but also the phase shifts at the interface. |
The equations assume the interface between the media is flat and that the media are homogeneous and isotropic. The incident light is assumed to be a plane wave, which is sufficient to solve any problem since any incident light field can be decomposed into plane waves and polarizations. |
There are two sets of Fresnel coefficients for two different linear polarization components of the incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized (or "randomly polarized") light has an equal amount of power in each of two linear polarizations. |
The s polarization refers to polarization of a wave's electric field "normal" to the plane of incidence (the direction in the derivation below); then the magnetic field is "in" the plane of incidence. The p polarization refers to polarization of the electric field "in" the plane of incidence (the plane in the derivation below); then the magnetic field is "normal" to the plane of incidence. |
Although the reflectivity and transmission are dependent on polarization, at normal incidence ("θ" = 0) there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients (and another special case is mentioned below in which that is true). |
In the diagram on the right, an incident plane wave in the direction of the ray IO strikes the interface between two media of refractive indices "n"1 and "n"2 at point O. Part of the wave is reflected in the direction OR, and part refracted in the direction OT. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as "θ"i, "θ"r and "θ"t, respectively. |
The relationship between these angles is given by the law of reflection: |
The behavior of light striking the interface is solved by considering the electric and magnetic fields that constitute an electromagnetic wave, and the laws of electromagnetism, as shown below. The ratio of waves' electric field (or magnetic field) amplitudes are obtained, but in practice one is more often interested in formulae which determine "power" coefficients, since power (or irradiance) is what can be directly measured at optical frequencies. The power of a wave is generally proportional to the square of the electric (or magnetic) field amplitude. |
We call the fraction of the incident power that is reflected from the interface the reflectance (or "reflectivity", or "power reflection coefficient") "R", and the fraction that is refracted into the second medium is called the transmittance (or "transmissivity", or "power transmission coefficient") "T". Note that these are what would be measured right "at" each side of an interface and do not account for attenuation of a wave in an absorbing medium "following" transmission or reflection. |
while the reflectance for p-polarized light is |
where and are the wave impedances of media 1 and 2, respectively. |
We assume that the media are non-magnetic (i.e., "μ"1 = "μ"2 = "μ"0), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies). Then the wave impedances are determined solely by the refractive indices "n"1 and "n"2: |
where is the impedance of free space and =1,2. Making this substitution, we obtain equations using the refractive indices: |
The second form of each equation is derived from the first by eliminating "θ"t using Snell's law and trigonometric identities. |
As a consequence of conservation of energy, one can find the transmitted power (or more correctly, irradiance: power per unit area) simply as the portion of the incident power that isn't reflected: |
Note that all such intensities are measured in terms of a wave's irradiance in the direction normal to the interface; this is also what is measured in typical experiments. That number could be obtained from irradiances "in the direction of an incident or reflected wave" (given by the magnitude of a wave's Poynting vector) multiplied by cos"θ" for a wave at an angle "θ" to the normal direction (or equivalently, taking the dot product of the Poynting vector with the unit vector normal to the interface). This complication can be ignored in the case of the reflection coefficient, since cos"θ"i = cos"θ"r, so that the ratio of reflected to incident irradiance in the wave's direction is the same as in the direction normal to the interface. |
Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be described as unpolarized. That means that there is an equal amount of power in the "s" and "p" polarizations, so that the "effective" reflectivity of the material is just the average of the two reflectivities: |
For low-precision applications involving unpolarized light, such as computer graphics, rather than rigorously computing the effective reflection coefficient for each angle, Schlick's approximation is often used. |
For the case of normal incidence, formula_11, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to |
For common glass ("n"2 ≈ 1.5) surrounded by air ("n"1=1), the power reflectance at normal incidence can be seen to be about 4%, or 8% accounting for both sides of a glass pane. |
At a dielectric interface from to , there is a particular angle of incidence at which goes to zero and a p-polarised incident wave is purely refracted, thus all reflected light is s-polarised. This angle is known as Brewster's angle, and is around 56° for "n"1=1 and "n"2=1.5 (typical glass). |
When light travelling in a denser medium strikes the surface of a less dense medium (i.e., ), beyond a particular incidence angle known as the "critical angle", all light is reflected and . This phenomenon, known as total internal reflection, occurs at incidence angles for which Snell's law predicts that the sine of the angle of refraction would exceed unity (whereas in fact sin"θ" ≤ 1 for all real "θ"). For glass with "n"=1.5 surrounded by air, the critical angle is approximately 41°. |
The above equations relating powers (which could be measured with a photometer for instance) are derived from the Fresnel equations which solve the physical problem in terms of electromagnetic field complex amplitudes, i.e., considering phase in addition to power (which is important in multipath propagation for instance). Those underlying equations supply generally complex-valued ratios of those EM fields and may take several different forms, depending on formalisms used. The complex amplitude coefficients are usually represented by lower case "r" and "t" (whereas the power coefficients are capitalized). |
In the following, the reflection coefficient is the ratio of the reflected wave's electric field complex amplitude to that of the incident wave. The transmission coefficient is the ratio of the transmitted wave's electric field complex amplitude to that of the incident wave. We require separate formulae for the "s" and "p" polarizations. In each case we assume an incident plane wave at an angle of incidence formula_13 on a plane interface, reflected at an angle formula_14, and with a transmitted wave at an angle formula_15, corresponding to the above figure. Note that in the cases of an interface into an absorbing material (where "n" is complex) or total internal reflection, the angle of transmission might not evaluate to a real number. |
We consider the sign of a wave's electric field in relation to a wave's direction. Consequently, for "p" polarization at normal incidence, the positive direction of electric field for an incident wave (to the left) is "opposite" that of a reflected wave (also to its left); for "s" polarization both are the same (upward). |
One can see that and . One can write similar equations applying to the ratio of magnetic fields of the waves, but these are usually not required. |
Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the power reflection coefficient R is just the squared magnitude of "r": |
On the other hand, calculation of the power transmission coefficient is less straightforward, since the light travels in different directions in the two media. What's more, the wave impedances in the two media differ; power is only proportional to the square of the amplitude when the media's impedances are the same (as they are for the reflected wave). This results in: |
The factor of is the reciprocal of the ratio of the media's wave impedances (since we assume "μ"="μ"0). The factor of is from expressing power "in the direction" normal to the interface, for both the incident and transmitted waves. |
In the case of total internal reflection where the power transmission is zero, nevertheless describes the electric field (including its phase) just beyond the interface. This is an evanescent field which does not propagate as a wave (thus =0) but has nonzero values very close to the interface. The phase shift of the reflected wave on total internal reflection can similarly be obtained from the phase angles of and (whose magnitudes are unity). These phase shifts are different for "s" and "p" waves, which is the well-known principle by which total internal reflection is used to effect polarization transformations. |
In the above formula for , if we put formula_19 (Snell's law) and multiply the numerator and denominator by , we obtain |
If we do likewise with the formula for , the result is easily shown to be equivalent to |
These formulas are known respectively as "Fresnel's sine law" and "Fresnel's tangent law". Although at normal incidence these expressions reduce to 0/0, one can see that they yield the correct results in the limit as . |
When light makes multiple reflections between two or more parallel surfaces, the multiple beams of light generally interfere with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than the light's coherence length, which for ordinary white light is few micrometers; it can be much larger for light from a laser. |
An example of interference between reflections is the iridescent colours seen in a soap bubble or in thin oil films on water. Applications include Fabry–Pérot interferometers, antireflection coatings, and optical filters. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference. |
The transfer-matrix method, or the recursive Rouard method can be used to solve multiple-surface problems. |
In 1808, Étienne-Louis Malus discovered that when a ray of light was reflected off a non-metallic surface at the appropriate angle, it behaved like "one" of the two rays emerging from a doubly-refractive calcite crystal. He later coined the term "polarization" to describe this behavior. In 1815, the dependence of the polarizing angle on the refractive index was determined experimentally by David Brewster. But the "reason" for that dependence was such a deep mystery that in late 1817, Thomas Young was moved to write: |
In 1821, however, Augustin-Jean Fresnel derived results equivalent to his sine and tangent laws (above), by modeling light waves as transverse elastic waves with vibrations perpendicular to what had previously been called the plane of polarization. Fresnel promptly confirmed by experiment that the equations correctly predicted the direction of polarization of the reflected beam when the incident beam was polarized at 45° to the plane of incidence, for light incident from air onto glass or water; in particular, the equations gave the correct polarization at Brewster's angle. The experimental confirmation was reported in a "postscript" to the work in which Fresnel first revealed his theory that light waves, including "unpolarized" waves, were "purely" transverse. |
Details of Fresnel's derivation, including the modern forms of the sine law and tangent law, were given later, in a memoir read to the French Academy of Sciences in January 1823. That derivation combined conservation of energy with continuity of the "tangential" vibration at the interface, but failed to allow for any condition on the "normal" component of vibration. The first derivation from "electromagnetic" principles was given by Hendrik Lorentz in 1875. |
In the same memoir of January 1823, Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients ( and ) gave complex values with unit magnitudes. Noting that the magnitude, as usual, represented the ratio of peak amplitudes, he guessed that the argument represented the phase shift, and verified the hypothesis experimentally. The verification involved |
Thus he finally had a quantitative theory for what we now call the "Fresnel rhomb" — a device that he had been using in experiments, in one form or another, since 1817 (see "Fresnel rhomb §History"). |
The success of the complex reflection coefficient inspired James MacCullagh and Augustin-Louis Cauchy, beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a complex refractive index. |
Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoir in which he introduced the needed terms "linear polarization", "circular polarization", and "elliptical polarization", and in which he explained optical rotation as a species of birefringence: linearly-polarized light can be resolved into two circularly-polarized components rotating in opposite directions, and if these propagate at different speeds, the phase difference between them — hence the orientation of their linearly-polarized resultant — will vary continuously with distance. |
Thus Fresnel's interpretation of the complex values of his reflection coefficients marked the confluence of several streams of his research and, arguably, the essential completion of his reconstruction of physical optics on the transverse-wave hypothesis (see "Augustin-Jean Fresnel"). |
Here we systematically derive the above relations from electromagnetic premises. |
In order to compute meaningful Fresnel coefficients, we must assume that the medium is (approximately) linear and homogeneous. If the medium is also isotropic, the four field vectors are related by |
where "ϵ" and "μ" are scalars, known respectively as the (electric) "permittivity" and the (magnetic) "permeability" of the medium. For a vacuum, these have the values "ϵ"0 and "μ"0, respectively. Hence we define the "relative" permittivity (or dielectric constant) , and the "relative" permeability . |
In optics it is common to assume that the medium is non-magnetic, so that "μ"rel=1. For ferromagnetic materials at radio/microwave frequencies, larger values of "μ"rel must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies (except possible metamaterials), "μ"rel is indeed very close to 1; that is, "μ"≈"μ"0. |
In optics, one usually knows the refractive index "n" of the medium, which is the ratio of the speed of light in a vacuum () to the speed of light in the medium. In the analysis of partial reflection and transmission, one is also interested in the electromagnetic wave impedance , which is the ratio of the amplitude of to the amplitude of . It is therefore desirable to express "n" and in terms of "ϵ" and "μ", and thence to relate to "n". The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave "admittance" , which is the reciprocal of the wave impedance . |
In the case of "uniform plane sinusoidal" waves, the wave impedance or admittance is known as the "intrinsic" impedance or admittance of the medium. This case is the one for which the Fresnel coefficients are to be derived. |
In a uniform plane sinusoidal electromagnetic wave, the electric field has the form |
where is the (constant) complex amplitude vector, is the imaginary unit, is the wave vector (whose magnitude is the angular wavenumber), is the position vector, "ω" is the angular frequency, is time, and it is understood that the "real part" of the expression is the physical field. The value of the expression is unchanged if the position varies in a direction normal to ; hence "is normal to the wavefronts". |
To advance the phase by the angle "ϕ", we replace by (that is, we replace by ), with the result that the (complex) field is multiplied by . So a phase "advance" is equivalent to multiplication by a complex constant with a "negative" argument. This becomes more obvious when the field () is factored as where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t. time corresponds to multiplication by . |
If "ℓ" is the component of in the direction of the field () can be written . If the argument of is to be constant, "ℓ" must increase at the velocity formula_22 known as the "phase velocity" . This in turn is equal to formula_23. Solving for gives |
As usual, we drop the time-dependent factor which is understood to multiply every complex field quantity. The electric field for a uniform plane sine wave will then be represented by the location-dependent "phasor" |
For fields of that form, Faraday's law and the Maxwell-Ampère law respectively reduce to |
Putting and as above, we can eliminate and to obtain equations in only and : |
If the material parameters "ϵ" and "μ" are real (as in a lossless dielectric), these equations show that form a "right-handed orthogonal triad", so that the same equations apply to the magnitudes of the respective vectors. Taking the magnitude equations and substituting from (), we obtain |
where and are the magnitudes of and . Multiplying the last two equations gives |
Dividing (or cross-multiplying) the same two equations gives where |
From () we obtain the phase velocity formula_27. For a vacuum this reduces to formula_28. Dividing the second result by the first gives |
For a "non-magnetic" medium (the usual case), this becomes formula_30. |
Taking the reciprocal of (), we find that the intrinsic "impedance" is formula_31. In a vacuum this takes the value formula_32 known as the impedance of free space. By division, formula_33. For a "non-magnetic" medium, this becomes formula_34 |
In Cartesian coordinates , let the region have refractive index intrinsic admittance etc., and let the region have refractive index intrinsic admittance etc. Then the plane is the interface, and the axis is normal to the interface (see diagram). Let and (in bold roman type) be the unit vectors in the and directions, respectively. Let the plane of incidence be the plane (the plane of the page), with the angle of incidence measured from towards . Let the angle of refraction, measured in the same sense, be where the subscript stands for "transmitted" (reserving for "reflected"). |
In the absence of Doppler shifts, "ω" does not change on reflection or refraction. Hence, by (), the magnitude of the wave vector is proportional to the refractive index. |
So, for a given "ω", if we "redefine" as the magnitude of the wave vector in the "reference" medium (for which ), then the wave vector has magnitude in the first medium (region in the diagram) and magnitude in the second medium. From the magnitudes and the geometry, we find that the wave vectors are |
where the last step uses Snell's law. The corresponding dot products in the phasor form () are |
For the "s" polarization, the field is parallel to the axis and may therefore be described by its component in the direction. Let the reflection and transmission coefficients be and respectively. Then, if the incident field is taken to have unit amplitude, the phasor form () of its component is |
and the reflected and transmitted fields, in the same form, are |
Under the sign convention used in this article, a positive reflection or transmission coefficient is one that preserves the direction of the "transverse" field, meaning (in this context) the field normal to the plane of incidence. For the "s" polarization, that means the field. If the incident, reflected, and transmitted fields (in the above equations) are in the direction ("out of the page"), then the respective fields are in the directions of the red arrows, since form a right-handed orthogonal triad. The fields may therefore be described by their components in the directions of those arrows, denoted by . Then, since |
At the interface, by the usual interface conditions for electromagnetic fields, the tangential components of the and fields must be continuous; that is, |
When we substitute from equations () to () and then from (), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations |
which are easily solved for and yielding |
At "normal incidence" indicated by an additional subscript 0, these results become |
At "grazing incidence" , we have hence and . |
For the "p" polarization, the incident, reflected, and transmitted fields are parallel to the red arrows and may therefore be described by their components in the directions of those arrows. Let those components be (redefining the symbols for the new context). Let the reflection and transmission coefficients be and . Then, if the incident field is taken to have unit amplitude, we have |
If the fields are in the directions of the red arrows, then, in order for to form a right-handed orthogonal triad, the respective fields must be in the direction ("into the page") and may therefore be described by their components in that direction. This is consistent with the adopted sign convention, namely that a positive reflection or transmission coefficient is one that preserves the direction of the transverse field the field in the case of the "p" polarization. The agreement of the "other" field with the red arrows reveals an alternative definition of the sign convention: that a positive reflection or transmission coefficient is one for which the field vector in the plane of incidence points towards the same medium before and after reflection or transmission. |
So, for the incident, reflected, and transmitted fields, let the respective components in the direction be . Then, since |
At the interface, the tangential components of the and fields must be continuous; that is, |
When we substitute from equations () and () and then from (), the exponential factors again cancel out, so that the interface conditions reduce to |
At "grazing incidence" , we again have hence and . |
Comparing () and () with () and (), we see that at "normal" incidence, under the adopted sign convention, the transmission coefficients for the two polarizations are equal, whereas the reflection coefficients have equal magnitudes but opposite signs. While this clash of signs is a disadvantage of the convention, the attendant advantage is that the signs agree at "grazing" incidence. |
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