Fresnel equations


The Fresnel equations describe the reflection and transmission of light 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.

Overview

When light strikes the interface between a medium with refractive index n1 and a second medium with refractive index n2, both reflection and refraction of the light may occur. The Fresnel equations describe the ratios of the reflected and transmitted waves' electric fields to the incident wave's electric field. Since these are complex ratios, they describe not only the relative amplitude, but phase shifts between the waves.
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.

S and P 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 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 ; 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 ; then the magnetic field is normal to the plane of incidence.
Although the reflectivity and transmission are dependent on polarization, at normal incidence there is no distinction between them so all polarization states are governed by a single set of Fresnel coefficients.

Power (intensity) reflection and transmission coefficients

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 n1 and n2 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 θi, θr and θt, respectively.
The relationship between these angles is given by the law of reflection:
and Snell's law:
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 amplitudes are obtained, but in practice one is more often interested in formulae which determine power coefficients, since power is what can be directly measured at optical frequencies. The power of a wave is generally proportional to the square of the electric field amplitude.
We call the fraction of the incident power that is reflected from the interface the reflectance R, and the fraction that is refracted into the second medium is called the transmittance 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.
The reflectance for s-polarized light is
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, which is typically a good approximation at optical frequencies. Then the wave impedances are determined solely by the refractive indices n1 and n2:
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 θt using Snell's law and trigonometric identities.
As a consequence of conservation of energy, one can find the transmitted power simply as the portion of the incident power that isn't reflected:
and
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 multiplied by cosθ for a wave at an angle θ to the normal direction. This complication can be ignored in the case of the reflection coefficient, since cosθi = cosθr, 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.

Special cases

Normal incidence

For the case of normal incidence,, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to
For common glass surrounded by air, the power reflectance at normal incidence can be seen to be about 4%, or 8% accounting for both sides of a glass pane.

Brewster's angle

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. This angle is known as Brewster's angle, and is around 56° for n1=1 and n2=1.5.

Total internal reflection

When light travelling in a denser medium strikes the surface of a less dense medium, 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. For glass with n=1.5 surrounded by air, the critical angle is approximately 41°.

Complex amplitude reflection and transmission coefficients

The above equations relating powers 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. 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.
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 on a plane interface, reflected at an angle, and with a transmitted wave at an angle, corresponding to the above figure. Note that in the cases of an interface into an absorbing material 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 is opposite that of a reflected wave ; for s polarization both are the same.
Using these conventions,
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. This results in:
The factor of is the reciprocal of the ratio of the media's wave impedances. 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 just beyond the interface. This is an evanescent field which does not propagate as a wave 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 . 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.

Alternative forms

In the above formula for, if we put 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.

Multiple surfaces

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.

History

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, 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 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.
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.

Theory

Here we systematically derive the above relations from electromagnetic premises.

Material parameters

In order to compute meaningful Fresnel coefficients, we must assume that the medium is 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 permittivity and the permeability of the medium. For a vacuum, these have the values ϵ0 and μ0, respectively. Hence we define the relative permittivity , and the relative permeability.
In optics it is common to assume that the medium is non-magnetic, so that μrel=1. For ferromagnetic materials at radio/microwave frequencies, larger values of μrel must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies, μrel is indeed very close to 1; that is, μ≈μ0.
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.

Electromagnetic plane waves

In a uniform plane sinusoidal electromagnetic wave, the electric field has the form
where is the complex amplitude vector, is the imaginary unit, is the wave vector, 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 , with the result that the 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 known as the phase velocity. This in turn is equal to. 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, 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 the same two equations gives where
This is the intrinsic admittance.
From we obtain the phase velocity. For a vacuum this reduces to. Dividing the second result by the first gives
For a non-magnetic medium, this becomes.
Taking the reciprocal of, we find that the intrinsic impedance is. In a vacuum this takes the value known as the impedance of free space. By division,. For a non-magnetic medium, this becomes

The wave vectors

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. Let and be the unit vectors in the and directions, respectively. Let the plane of incidence be the plane, 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.
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, then the wave vector has magnitude in the first medium 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
Hence:

The ''s'' components

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 the field normal to the plane of incidence. For the s polarization, that means the field. If the incident, reflected, and transmitted fields are in the direction, 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
and
At normal incidence indicated by an additional subscript 0, these results become
and
At grazing incidence, we have hence and.

The ''p'' components

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 . 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 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
Solving for and we find
and
At normal incidence indicated by an additional subscript 0, these results become
and
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.

Power ratios (reflectivity and transmissivity)

The Poynting vector for a wave is a vector whose component in any direction is the irradiance of that wave on a surface perpendicular to that direction. For a plane sinusoidal wave the Poynting vector is where and are due only to the wave in question, and the asterisk denotes complex conjugation. Inside a lossless dielectric, and are in phase, and at right angles to each other and to the wave vector ; so, for s polarization, using the and components of and respectively, the irradiance in the direction of is given simply by which is in a medium of intrinsic impedance. To compute the irradiance in the direction normal to the interface, as we shall require in the definition of the power transmission coefficient, we could use only the component of or or, equivalently, simply multiply by the proper geometric factor, obtaining.
From equations and, taking squared magnitudes, we find that the reflectivity is
for the s polarization, and
for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same cosθ, the impedance and geometric factors mentioned above are identical and cancel out. But in computing the power transmission, these factors must be taken into account.
The simplest way to obtain the power transmission coefficient is to use . In this way we find
for the s polarization, and
for the p polarization.
In the case of an interface between two lossless media, one can obtain these results directly using the squared magnitudes of the amplitude transmission coefficients that we found earlier in equations and. But, for given amplitude, the component of the Poynting vector in the direction is proportional to the geometric factor and inversely proportional to the wave impedance. Applying these corrections to each wave, we obtain two ratios multiplying the square of the amplitude transmission coefficient:
for the s polarization, and
for the p polarization. The last two equations apply only to lossless dielectrics, and only at incidence angles smaller than the critical angle.

Equal refractive indices

From equations and, we see that two dissimilar media will have the same refractive index, but different admittances, if the ratio of their permeabilities is the inverse of the ratio of their permittivities. In that unusual situation we have , so that the cosines in equations,,,, and to cancel out, and all the reflection and transmission ratios become independent of the angle of incidence; in other words, the ratios for normal incidence become applicable to all angles of incidence.

Non-magnetic media

Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing by ) yields
For non-magnetic media we can substitute the vacuum permeability μ0 for μ, so that
that is, the admittances are simply proportional to the corresponding refractive indices. When we make these substitutions in equations to and equations to, the factor cμ0 cancels out. For the amplitude coefficients we obtain:
For the case of normal incidence these reduce to:
The power reflection coefficients become:
The power transmissions can then be found from.

Brewster's angle

For equal permeabilities, if and are complementary, we can substitute for and for so that the numerator in equation becomes which is zero. Hence and only the s-polarized component is reflected. This is what happens at the Brewster angle. Substituting for in Snell's law, we readily obtain
for Brewster's angle.

Equal permittivities?

Although it is not encountered in practice, we can consider the case of two media with a common permittivity, but different refractive indices due to different permeabilities. From equations and, we see that if ϵ is fixed instead of μ, then becomes inversely proportional to, with the result that the subscripts 1 and 2 in equations to are interchanged. Hence, in and, the expressions for and in terms of refractive indices will be interchanged, so that Brewster's angle will give instead of and any beam reflected at that angle will be p-polarized instead of s-polarized. Similarly, Fresnel's sine law will apply to the p polarization instead of the s polarization, and his tangent law to the s polarization instead of the p polarization.
This switch of polarizations has an analog in the old mechanical theory of light waves. One could predict reflection coefficients that agreed with observation by supposing that different refractive indices were due to different densities and that the vibrations were normal to what was then called the plane of polarization, or by supposing that different refractive indices were due to different elasticities and that the vibrations were parallel to that plane. Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.