At near-grazing incidence, media interfaces appear mirror-like especially due to reflection of the s polarization, despite being poor reflectors at normal incidence. Polarized sunglasses block the s polarization, greatly reducing glare from horizontal surfaces.

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 French engineer and physicist Augustin-Jean Fresnel (/freɪˈnɛl/) who was the first to understand that light is a transverse wave, when no one realized that the waves 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 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.^[1] 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 (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 z 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 xy plane in the derivation below); then the magnetic field is normal to the plane of incidence.

Although the reflection 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).

Configuration

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 θ_i, θ_r and θ_t, respectively. The relationship between these angles is given by the law of reflection:

$${\displaystyle \theta _{\mathrm {i} }=\theta _{\mathrm {r} },}$$

$${\displaystyle n_{1}\sin \theta _{\mathrm {i} }=n_{2}\sin \theta _{\mathrm {t} }.}$$

The behavior of light striking the interface is explained 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.

Power (intensity) reflection and transmission coefficients

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.^[2]

The reflectance for s-polarized light is

$${\displaystyle R_{\mathrm {s} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {i} }-Z_{1}\cos \theta _{\mathrm {t} )){Z_{2}\cos \theta _{\mathrm {i} }+Z_{1}\cos \theta _{\mathrm {t} ))}\right|^{2},}$$

$${\displaystyle R_{\mathrm {p} }=\left|{\frac {Z_{2}\cos \theta _{\mathrm {t} }-Z_{1}\cos \theta _{\mathrm {i} )){Z_{2}\cos \theta _{\mathrm {t} }+Z_{1}\cos \theta _{\mathrm {i} ))}\right|^{2},}$$

We assume that the media are non-magnetic (i.e., μ₁ = μ₂ = μ₀), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies).^[3] Then the wave impedances are determined solely by the refractive indices n₁ and n₂:

$${\displaystyle Z_{i}={\frac {Z_{0)){n_{i))}\,,}$$

$${\displaystyle R_{\mathrm {s} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}\cos \theta _{\mathrm {t} )){n_{1}\cos \theta _{\mathrm {i} }+n_{2}\cos \theta _{\mathrm {t} ))}\right|^{2}=\left|{\frac {n_{1}\cos \theta _{\mathrm {i} }-n_{2}{\sqrt {1-\left({\frac {n_{1)){n_{2))}\sin \theta _{\mathrm {i} }\right)^{2)))){n_{1}\cos \theta _{\mathrm {i} }+n_{2}{\sqrt {1-\left({\frac {n_{1)){n_{2))}\sin \theta _{\mathrm {i} }\right)^{2))))}\right|^{2}\!,}$$

$${\displaystyle R_{\mathrm {p} }=\left|{\frac {n_{1}\cos \theta _{\mathrm {t} }-n_{2}\cos \theta _{\mathrm {i} )){n_{1}\cos \theta _{\mathrm {t} }+n_{2}\cos \theta _{\mathrm {i} ))}\right|^{2}=\left|{\frac {n_{1}{\sqrt {1-\left({\frac {n_{1)){n_{2))}\sin \theta _{\mathrm {i} }\right)^{2))}-n_{2}\cos \theta _{\mathrm {i} )){n_{1}{\sqrt {1-\left({\frac {n_{1)){n_{2))}\sin \theta _{\mathrm {i} }\right)^{2))}+n_{2}\cos \theta _{\mathrm {i} ))}\right|^{2}\!.}$$

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: ^[4]

$${\displaystyle T_{\mathrm {s} }=1-R_{\mathrm {s} ))$$

$${\displaystyle T_{\mathrm {p} }=1-R_{\mathrm {p} ))$$

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:

$${\displaystyle R_{\mathrm {eff} }={\frac {1}{2))\left(R_{\mathrm {s} }+R_{\mathrm {p} }\right).}$$

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, θ_i = θ_t = 0, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to

$${\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2)){n_{1}+n_{2))}\right|^{2}\,.}$$

For common glass (n₂ ≈ 1.5) surrounded by air (n₁ = 1), 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 n₁ to n₂, there is a particular angle of incidence at which R_p 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 n₁ = 1 and n₂ = 1.5 (typical glass).

Total internal reflection

When light travelling in a denser medium strikes the surface of a less dense medium (i.e., n₁ > n₂), beyond a particular incidence angle known as the critical angle, all light is reflected and R_s = R_p = 1. 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 42°.

45° incidence

Reflection at 45° incidence is very commonly used for making 90° turns. For the case of light traversing from a less dense medium into a denser one at 45° incidence (θ = 45°), it follows algebraically from the above equations that R_p equals the square of R_s:

$${\displaystyle R_{\text{p))=R_{\text{s))^{2))$$

This can be used to either verify the consistency of the measurements of R_s and R_p, or to derive one of them when the other is known. This relationship is only valid for the simple case of a single plane interface between two homogeneous materials, not for films on substrates, where a more complex analysis is required.

Measurements of R_s and R_p at 45° can be used to estimate the reflectivity at normal incidence.^{[citation needed]} The "average of averages" obtained by calculating first the arithmetic as well as the geometric average of R_s and R_p, and then averaging these two averages again arithmetically, gives a value for R₀ with an error of less than about 3% for most common optical materials.^{[citation needed]} This is useful because measurements at normal incidence can be difficult to achieve in an experimental setup since the incoming beam and the detector will obstruct each other. However, since the dependence of R_s and R_p on the angle of incidence for angles below 10° is very small, a measurement at about 5° will usually be a good approximation for normal incidence, while allowing for a separation of the incoming and reflected beam.

Complex amplitude reflection and transmission coefficients

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 shifts in addition to their amplitudes. Those underlying equations supply generally complex-valued ratios of those EM fields and may take several different forms, depending on the formalism used. The complex amplitude coefficients for reflection and transmission are usually represented by lower case r and t (whereas the power coefficients are capitalized). As before, we are assuming the magnetic permeability, µ of both media to be equal to the permeability of free space µ₀ as is essentially true of all dielectrics at optical frequencies.

In the following equations and graphs, we adopt the following conventions. For s polarization, the reflection coefficient r is defined as the ratio of the reflected wave's complex electric field amplitude to that of the incident wave, whereas for p polarization r is the ratio of the waves complex magnetic field amplitudes (or equivalently, the negative of the ratio of their electric field amplitudes). The transmission coefficient t is the ratio of the transmitted wave's complex electric field amplitude to that of the incident wave, for either polarization. The coefficients r and t are generally different between the s and p polarizations, and even at normal incidence (where the designations s and p do not even apply!) the sign of r is reversed depending on whether the wave is considered to be s or p polarized, an artifact of the adopted sign convention (see graph for an air-glass interface at 0° incidence).

The equations consider a plane wave incident on a plane interface at angle of incidence ${\displaystyle \theta _{\mathrm {i} ))$, a wave reflected at angle ${\displaystyle \theta _{\mathrm {r} }=\theta _{\mathrm {i} ))$, and a wave transmitted at angle ${\displaystyle \theta _{\mathrm {t} ))$. In the case of an interface into an absorbing material (where n is complex) or total internal reflection, the angle of transmission does not generally evaluate to a real number. In that case, however, meaningful results can be obtained using formulations of these relationships in which trigonometric functions and geometric angles are avoided; the inhomogeneous waves launched into the second medium cannot be described using a single propagation angle.

Using this convention,^[5][6]

$${\displaystyle {\begin{aligned}r_{\text{s))&={\frac {n_{1}\cos \theta _{\text{i))-n_{2}\cos \theta _{\text{t))}{n_{1}\cos \theta _{\text{i))+n_{2}\cos \theta _{\text{t)))),\\[3pt]t_{\text{s))&={\frac {2n_{1}\cos \theta _{\text{i))}{n_{1}\cos \theta _{\text{i))+n_{2}\cos \theta _{\text{t)))),\\[3pt]r_{\text{p))&={\frac {n_{2}\cos \theta _{\text{i))-n_{1}\cos \theta _{\text{t))}{n_{2}\cos \theta _{\text{i))+n_{1}\cos \theta _{\text{t)))),\\[3pt]t_{\text{p))&={\frac {2n_{1}\cos \theta _{\text{i))}{n_{2}\cos \theta _{\text{i))+n_{1}\cos \theta _{\text{t)))).\end{aligned))}$$

One can see that t_s = r_s + 1^[7] and n₂/n₁t_p = r_p + 1. One can write very similar equations applying to the ratio of the waves' magnetic fields, but comparison of the electric fields is more conventional.

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: ^[8]

$${\displaystyle R=|r|^{2}.}$$

On the other hand, calculation of the power transmission coefficient T 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 (irradiance) is given by the square of the electric field amplitude divided by the characteristic impedance of the medium (or by the square of the magnetic field multiplied by the characteristic impedance). This results in:^[9]

$${\displaystyle T={\frac {n_{2}\cos \theta _{\text{t))}{n_{1}\cos \theta _{\text{i))))|t|^{2))$$

In the case of total internal reflection where the power transmission T is zero, t 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 T = 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 r_p and r_s (whose magnitudes are unity in this case). 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 r_s‍, if we put ${\displaystyle n_{2}=n_{1}\sin \theta _{\text{i))/\sin \theta _{\text{t))}$ (Snell's law) and multiply the numerator and denominator by 1/n₁ sin θ_t‍, we obtain ^[10][11]

$${\displaystyle r_{\text{s))=-{\frac {\sin(\theta _{\text{i))-\theta _{\text{t)))}{\sin(\theta _{\text{i))+\theta _{\text{t))))).}$$

If we do likewise with the formula for r_p‍, the result is easily shown to be equivalent to ^[12][13]

$${\displaystyle r_{\text{p))={\frac {\tan(\theta _{\text{i))-\theta _{\text{t)))}{\tan(\theta _{\text{i))+\theta _{\text{t))))).}$$

These formulas ^[14][15][16] are known respectively as Fresnel's sine law and Fresnel's tangent law.^[17] Although at normal incidence these expressions reduce to 0/0, one can see that they yield the correct results in the limit as‍ θ_i → 0.

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 ^[18] 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.^[19] 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.^[20] But the reason for that dependence was such a deep mystery that in late 1817, Thomas Young was moved to write:

[T]he great difficulty of all, which is to assign a sufficient reason for the reflection or nonreflection of a polarised ray, will probably long remain, to mortify the vanity of an ambitious philosophy, completely unresolved by any theory.^[21]

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.^[22] 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.^[23]

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.^[24] 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.^[25] The first derivation from electromagnetic principles was given by Hendrik Lorentz in 1875.^[26]

In the same memoir of January 1823,^[24] Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients (r_s and r_p) 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.^[27] The verification involved

• calculating the angle of incidence that would introduce a total phase difference of 90° between the s and p components, for various numbers of total internal reflections at that angle (generally there were two solutions),
• subjecting light to that number of total internal reflections at that angle of incidence, with an initial linear polarization at 45° to the plane of incidence, and
• checking that the final polarization was circular.^[28]

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.^[29]

Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoir ^[30] in which he introduced the needed terms linear polarization, circular polarization, and elliptical polarization,^[31] 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.^[32]

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).

Derivation

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 (approximately) linear and homogeneous. If the medium is also isotropic, the four field vectors‍ E, B, D, H  are related by

$${\displaystyle {\begin{aligned}\mathbf {D} &=\epsilon \mathbf {E} \\\mathbf {B} &=\mu \mathbf {H} \,,\end{aligned))}$$

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 ≈ 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 (c) 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 Z, which is the ratio of the amplitude of E to the amplitude of H. It is therefore desirable to express n and Z in terms of ϵ and μ, and thence to relate Z to n. The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave admittance Y, which is the reciprocal of the wave impedance Z.

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 E has the form

${\displaystyle \mathbf {E_{k)) e^{i(\mathbf {k\cdot r} -\omega t)},}$

(1)

where E_k is the (constant) complex amplitude vector, i is the imaginary unit, k is the wave vector (whose magnitude k is the angular wavenumber), r is the position vector, ω is the angular frequency, t is time, and it is understood that the real part of the expression is the physical field.^{[Note 1]}  The value of the expression is unchanged if the position r varies in a direction normal to k; hence k is normal to the wavefronts.

To advance the phase by the angle ϕ, we replace ωt by ωt + ϕ  (that is, we replace −ωt by −ωt − ϕ), with the result that the (complex) field is multiplied by e^−iϕ. So a phase advance is equivalent to multiplication by a complex constant with a negative argument. This becomes more obvious when the field (1) is factored as‍ E_k e^ik⋅re^−iωt, where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t. time corresponds to multiplication by −iω. ^{[Note 2]}

If ℓ is the component of r in the direction of k , the field (1) can be written‍ E_k e^{i(kℓ−ωt)}.  If the argument of e^i(⋯) is to be constant,  ℓ must increase at the velocity‍ ${\displaystyle \omega /k\,,\,}$ known as the phase velocity  (v_p). This in turn is equal to‍ ${\displaystyle c/n}$. Solving for k gives

${\displaystyle k=n\omega /c\,.}$

(2)

As usual, we drop the time-dependent factor e^−iωt, 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

${\displaystyle \mathbf {E_{k)) e^{i\mathbf {k\cdot r} }.}$

(3)

For fields of that form, Faraday's law and the Maxwell-Ampère law respectively reduce to ^[33]

$${\displaystyle {\begin{aligned}\omega \mathbf {B} &=\mathbf {k} \times \mathbf {E} \\\omega \mathbf {D} &=-\mathbf {k} \times \mathbf {H} \,.\end{aligned))}$$

Putting  B = μH  and  D = ϵE‍,‍ as above, we can eliminate B and D to obtain equations in only E and H:

$${\displaystyle {\begin{aligned}\omega \mu \mathbf {H} &=\mathbf {k} \times \mathbf {E} \\\omega \epsilon \mathbf {E} &=-\mathbf {k} \times \mathbf {H} \,.\end{aligned))}$$

$${\displaystyle {\begin{aligned}\mu cH&=nE\\\epsilon cE&=nH\,,\end{aligned))}$$

${\displaystyle n=c\,{\sqrt {\mu \epsilon ))\,.}$

(4)

Dividing (or cross-multiplying) the same two equations gives  H = YE‍, where

${\displaystyle Y={\sqrt {\epsilon /\mu ))\,.}$

(5)

This is the intrinsic admittance.

From (4) we obtain the phase velocity‍ ${\displaystyle c/n=1{\big /}\!{\sqrt {\mu \epsilon \,))}$.  For a vacuum this reduces to‍ ${\displaystyle c=1{\big /}\!{\sqrt {\mu _{0}\epsilon _{0))))$.  Dividing the second result by the first gives

$${\displaystyle n={\sqrt {\mu _{\text{rel))\epsilon _{\text{rel))))\,.}$$

${\displaystyle n={\sqrt {\epsilon _{\text{rel))))}$

${\textstyle Z={\sqrt {\mu /\epsilon ))}$.

${\textstyle Z_{0}={\sqrt {\mu _{0}/\epsilon _{0))}\,\approx 377\,\Omega \,,}$

${\textstyle Z/Z_{0}={\sqrt {\mu _{\text{rel))/\epsilon _{\text{rel))))}$.

${\displaystyle Z=Z_{0}{\big /}\!{\sqrt {\epsilon _{\text{rel))))=Z_{0}/n.}$

Wave vectors

In Cartesian coordinates (x, y,‍z), let the region‍ y < 0‍ have refractive index n₁ , intrinsic admittance Y₁ , etc., and let the region‍ y > 0‍ have refractive index n₂ , intrinsic admittance Y₂ , etc. Then the xz plane is the interface, and the y axis is normal to the interface (see diagram). Let i and j (in bold roman type) be the unit vectors in the x and y directions, respectively. Let the plane of incidence be the xy plane (the plane of the page), with the angle of incidence θ_i measured from j towards i. Let the angle of refraction, measured in the same sense, be θ_t , where the subscript t stands for transmitted (reserving r for reflected).

In the absence of Doppler shifts, ω does not change on reflection or refraction. Hence, by (2), the magnitude of the wave vector is proportional to the refractive index.

So, for a given ω, if we redefine k as the magnitude of the wave vector in the reference medium (for which n =‍1), then the wave vector has magnitude n₁k in the first medium (region‍ y < 0‍ in the diagram) and magnitude n₂k in the second medium. From the magnitudes and the geometry, we find that the wave vectors are

$${\displaystyle {\begin{aligned}\mathbf {k} _{\text{i))&=n_{1}k(\mathbf {i} \sin \theta _{\text{i))+\mathbf {j} \cos \theta _{\text{i)))\\[.5ex]\mathbf {k} _{\text{r))&=n_{1}k(\mathbf {i} \sin \theta _{\text{i))-\mathbf {j} \cos \theta _{\text{i)))\\[.5ex]\mathbf {k} _{\text{t))&=n_{2}k(\mathbf {i} \sin \theta _{\text{t))+\mathbf {j} \cos \theta _{\text{t)))\\&=k(\mathbf {i} \,n_{1}\sin \theta _{\text{i))+\mathbf {j} \,n_{2}\cos \theta _{\text{t)))\,,\end{aligned))}$$

${\displaystyle {\begin{aligned}\mathbf {k} _{\text{i))\mathbf {\cdot r} &=n_{1}k(x\sin \theta _{\text{i))+y\cos \theta _{\text{i)))\\\mathbf {k} _{\text{r))\mathbf {\cdot r} &=n_{1}k(x\sin \theta _{\text{i))-y\cos \theta _{\text{i)))\\\mathbf {k} _{\text{t))\mathbf {\cdot r} &=k(n_{1}x\sin \theta _{\text{i))+n_{2}y\cos \theta _{\text{t)))\,.\end{aligned))}$

(6)

Hence:

At  ${\displaystyle y=0\,,~~~\mathbf {k} _{\text{i))\mathbf {\cdot r} =\mathbf {k} _{\text{r))\mathbf {\cdot r} =\mathbf {k} _{\text{t))\mathbf {\cdot r} =n_{1}kx\sin \theta _{\text{i))\,.}$

(7)

The s components

For the s polarization, the E field is parallel to the z axis and may therefore be described by its component in the z direction. Let the reflection and transmission coefficients be r_s and t_s , respectively. Then, if the incident E field is taken to have unit amplitude, the phasor form (3) of its z-component is

${\displaystyle E_{\text{i))=e^{i\mathbf {k} _{\text{i))\mathbf {\cdot r} },}$

(8)

and the reflected and transmitted fields, in the same form, are

${\displaystyle {\begin{aligned}E_{\text{r))&=r_{s\,}e^{i\mathbf {k} _{\text{r))\mathbf {\cdot r} }\\E_{\text{t))&=t_{s\,}e^{i\mathbf {k} _{\text{t))\mathbf {\cdot r} }.\end{aligned))}$

(9)

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 E field. If the incident, reflected, and transmitted E fields (in the above equations) are in the z-direction ("out of the page"), then the respective H fields are in the directions of the red arrows, since  k , E , H  form a right-handed orthogonal triad. The H fields may therefore be described by their components in the directions of those arrows, denoted by  H_i , H_r‍, H_t .  Then, since  H = YE‍,

${\displaystyle {\begin{aligned}H_{\text{i))&=\,Y_{1}e^{i\mathbf {k} _{\text{i))\mathbf {\cdot r} }\\H_{\text{r))&=\,Y_{1}r_{s\,}e^{i\mathbf {k} _{\text{r))\mathbf {\cdot r} }\\H_{\text{t))&=\,Y_{2}t_{s\,}e^{i\mathbf {k} _{\text{t))\mathbf {\cdot r} }.\end{aligned))}$

(10)

At the interface, by the usual interface conditions for electromagnetic fields, the tangential components of the E and H fields must be continuous; that is,

${\displaystyle \left.{\begin{aligned}E_{\text{i))+E_{\text{r))&=E_{\text{t))\\H_{\text{i))\cos \theta _{\text{i))-H_{\text{r))\cos \theta _{\text{i))&=H_{\text{t))\cos \theta _{\text{t))\end{aligned))~~\right\}~~~{\text{at))~~y=0\,.}$

(11)

When we substitute from equations (8) to (10) and then from (7), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations

${\displaystyle {\begin{aligned}1+r_{\text{s))&=\,t_{\text{s))\\Y_{1}\cos \theta _{\text{i))-Y_{1}r_{\text{s))\cos \theta _{\text{i))&=\,Y_{2}t_{\text{s))\cos \theta _{\text{t))\,,\end{aligned))}$

(12)

which are easily solved for r_s and t_s‍, yielding

${\displaystyle r_{\text{s))={\frac {Y_{1}\cos \theta _{\text{i))-Y_{2}\cos \theta _{\text{t))}{Y_{1}\cos \theta _{\text{i))+Y_{2}\cos \theta _{\text{t))))}$

(13)

and

${\displaystyle t_{\text{s))={\frac {2Y_{1}\cos \theta _{\text{i))}{Y_{1}\cos \theta _{\text{i))+Y_{2}\cos \theta _{\text{t))))\,.}$

(14)

At normal incidence‍ (θ_i = θ_t = 0), indicated by an additional subscript 0, these results become

${\displaystyle r_{\text{s0))={\frac {Y_{1}-Y_{2)){Y_{1}+Y_{2))))$

(15)

and

${\displaystyle t_{\text{s0))={\frac {2Y_{1)){Y_{1}+Y_{2))}\,.}$

(16)

At grazing incidence‍ (θ_i → 90°), we have  cos θ_i → 0‍, hence  r_s → −1  and  t_s → 0.

The p components

For the p polarization, the incident, reflected, and transmitted E 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  E_i , E_r‍, E_t  (redefining the symbols for the new context). Let the reflection and transmission coefficients be r_p and t_p. Then, if the incident E field is taken to have unit amplitude, we have

${\displaystyle {\begin{aligned}E_{\text{i))&=e^{i\mathbf {k} _{\text{i))\mathbf {\cdot r} }\\E_{\text{r))&=r_{p\,}e^{i\mathbf {k} _{\text{r))\mathbf {\cdot r} }\\E_{\text{t))&=t_{p\,}e^{i\mathbf {k} _{\text{t))\mathbf {\cdot r} }.\end{aligned))}$

(17)

If the E fields are in the directions of the red arrows, then, in order for  k , E , H  to form a right-handed orthogonal triad, the respective H fields must be in the −z 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 H 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.^[34]

So, for the incident, reflected, and transmitted H fields, let the respective components in the −z direction be  H_i , H_r‍, H_t .  Then, since  H = YE‍,

${\displaystyle {\begin{aligned}H_{\text{i))&=\,Y_{1}e^{i\mathbf {k} _{\text{i))\mathbf {\cdot r} }\\H_{\text{r))&=\,Y_{1}r_{p\,}e^{i\mathbf {k} _{\text{r))\mathbf {\cdot r} }\\H_{\text{t))&=\,Y_{2}t_{p\,}e^{i\mathbf {k} _{\text{t))\mathbf {\cdot r} }.\end{aligned))}$

(18)

At the interface, the tangential components of the E and H fields must be continuous; that is,

${\displaystyle \left.{\begin{aligned}E_{\text{i))\cos \theta _{\text{i))-E_{\text{r))\cos \theta _{\text{i))&=E_{\text{t))\cos \theta _{\text{t))\\H_{\text{i))+H_{\text{r))&=H_{\text{t))\end{aligned))~~\right\}~~~{\text{at))~~y=0\,.}$

(19)

When we substitute from equations (17) and (18) and then from (7), the exponential factors again cancel out, so that the interface conditions reduce to

${\displaystyle {\begin{aligned}\cos \theta _{\text{i))-r_{\text{p))\cos \theta _{\text{i))&=\,t_{\text{p))\cos \theta _{\text{t))\\Y_{1}+Y_{1}r_{\text{p))&=\,Y_{2}t_{\text{p))\,.\end{aligned))}$

(20)

Solving for r_p and t_p‍, we find

${\displaystyle r_{\text{p))={\frac {Y_{2}\cos \theta _{\text{i))-Y_{1}\cos \theta _{\text{t))}{Y_{2}\cos \theta _{\text{i))+Y_{1}\cos \theta _{\text{t))))}$

(21)

and

${\displaystyle t_{\text{p))={\frac {2Y_{1}\cos \theta _{\text{i))}{Y_{2}\cos \theta _{\text{i))+Y_{1}\cos \theta _{\text{t))))\,.}$

(22)

At normal incidence‍ (θ_i = θ_t = 0) indicated by an additional subscript 0, these results become

${\displaystyle r_{\text{p0))={\frac {Y_{2}-Y_{1)){Y_{2}+Y_{1))))$

(23)

and

${\displaystyle t_{\text{p0))={\frac {2Y_{1)){Y_{2}+Y_{1))}\,.}$

(24)

At grazing incidence (θ_i → 90°), we again have  cos θ_i → 0‍, hence  r_p → −1  and  t_p → 0.

Comparing (23) and (24) with (15) and (16), 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 (power per unit area) of that wave on a surface perpendicular to that direction. For a plane sinusoidal wave the Poynting vector is‍ 1/2‍Re{E × H^∗}, where E and H are due only to the wave in question, and the asterisk denotes complex conjugation. Inside a lossless dielectric (the usual case), E and H are in phase, and at right angles to each other and to the wave vector k ; so, for s polarization, using the z and xy components of E and H respectively (or for p polarization, using the xy and -z components of E and H), the irradiance in the direction of k is given simply by EH/2 ,‍ which is‍ E²/2Z  in a medium of intrinsic impedance Z = 1/Y. 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 x component (rather than the full xy component) of H or E or, equivalently, simply multiply EH/2 by the proper geometric factor, obtaining (E²/2Z ) cos θ.

From equations (13) and (21), taking squared magnitudes, we find that the reflectivity (ratio of reflected power to incident power) is

${\displaystyle R_{\text{s))=\left|{\frac {Y_{1}\cos \theta _{\text{i))-Y_{2}\cos \theta _{\text{t))}{Y_{1}\cos \theta _{\text{i))+Y_{2}\cos \theta _{\text{t))))\right|^{2))$

(25)

for the s polarization, and

${\displaystyle R_{\text{p))=\left|{\frac {Y_{2}\cos \theta _{\text{i))-Y_{1}\cos \theta _{\text{t))}{Y_{2}\cos \theta _{\text{i))+Y_{1}\cos \theta _{\text{t))))\right|^{2))$

(26)

for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same  cos θ, the impedance and geometric factors mentioned above are identical and cancel out. But in computing the power transmission (below), these factors must be taken into account.

The simplest way to obtain the power transmission coefficient (transmissivity, the ratio of transmitted power to incident power in the direction normal to the interface, i.e. the y direction) is to use R + T = 1‍ (conservation of energy). In this way we find

${\displaystyle T_{\text{s))=1-R_{\text{s))=\,{\frac {4\,{\text{Re))\{Y_{1}Y_{2}\cos \theta _{\text{i))\cos \theta _{\text{t))\)){\left|Y_{1}\cos \theta _{\text{i))+Y_{2}\cos \theta _{\text{t))\right|^{2))))$

(25T)

for the s polarization, and

${\displaystyle T_{\text{p))=1-R_{\text{p))=\,{\frac {4\,{\text{Re))\{Y_{1}Y_{2}\cos \theta _{\text{i))\cos \theta _{\text{t))\)){\left|Y_{2}\cos \theta _{\text{i))+Y_{1}\cos \theta _{\text{t))\right|^{2))))$

(26T)

for the p polarization.

In the case of an interface between two lossless media (for which ϵ and μ are real and positive), one can obtain these results directly using the squared magnitudes of the amplitude transmission coefficients that we found earlier in equations (14) and (22). But, for given amplitude (as noted above), the component of the Poynting vector in the y direction is proportional to the geometric factor cos θ‍ and inversely proportional to the wave impedance Z. Applying these corrections to each wave, we obtain two ratios multiplying the square of the amplitude transmission coefficient:

${\displaystyle T_{\text{s))=\left({\frac {2Y_{1}\cos \theta _{\text{i))}{Y_{1}\cos \theta _{\text{i))+Y_{2}\cos \theta _{\text{t))))\right)^{2}{\frac {\,Y_{2}\,}{Y_{1))}\,{\frac {\cos \theta _{\text{t))}{\cos \theta _{\text{i))))={\frac {4Y_{1}Y_{2}\cos \theta _{\text{i))\cos \theta _{\text{t))}{\left(Y_{1}\cos \theta _{\text{i))+Y_{2}\cos \theta _{\text{t))\right)^{2))))$

(27)

for the s polarization, and

${\displaystyle T_{\text{p))=\left({\frac {2Y_{1}\cos \theta _{\text{i))}{Y_{2}\cos \theta _{\text{i))+Y_{1}\cos \theta _{\text{t))))\right)^{2}{\frac {\,Y_{2}\,}{Y_{1))}\,{\frac {\cos \theta _{\text{t))}{\cos \theta _{\text{i))))={\frac {4Y_{1}Y_{2}\cos \theta _{\text{i))\cos \theta _{\text{t))}{\left(Y_{2}\cos \theta _{\text{i))+Y_{1}\cos \theta _{\text{t))\right)^{2))))$

(28)

for the p polarization. The last two equations apply only to lossless dielectrics, and only at incidence angles smaller than the critical angle (beyond which, of course, T = 0 ).

For unpolarized light:

$${\displaystyle T={1 \over 2}(T_{s}+T_{p})}$$

$${\displaystyle R={1 \over 2}(R_{s}+R_{p})}$$

${\displaystyle R+T=1}$

Equal refractive indices

From equations (4) and (5), 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  θ_t = θ_i‍ (that is, the transmitted ray is undeviated), so that the cosines in equations (13), (14), (21), (22), and (25) to (28) 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.^[35] When extended to spherical reflection or scattering, this results in the Kerker effect for Mie scattering.

Non-magnetic media

Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing (4) by (5)) yields

$${\displaystyle Y={\frac {n}{\,c\mu \,))\,.}$$

$${\displaystyle Y_{1}={\frac {n_{1)){\,c\mu _{0))}~~;~~~Y_{2}={\frac {n_{2)){\,c\mu _{0))}\,;}$$

${\displaystyle r_{\text{s))={\frac {n_{1}\cos \theta _{\text{i))-n_{2}\cos \theta _{\text{t))}{n_{1}\cos \theta _{\text{i))+n_{2}\cos \theta _{\text{t))))}$

(29)

${\displaystyle t_{\text{s))={\frac {2n_{1}\cos \theta _{\text{i))}{n_{1}\cos \theta _{\text{i))+n_{2}\cos \theta _{\text{t))))\,}$

(30)

${\displaystyle r_{\text{p))={\frac {n_{2}\cos \theta _{\text{i))-n_{1}\cos \theta _{\text{t))}{n_{2}\cos \theta _{\text{i))+n_{1}\cos \theta _{\text{t))))}$

(31)

${\displaystyle t_{\text{p))={\frac {2n_{1}\cos \theta _{\text{i))}{n_{2}\cos \theta _{\text{i))+n_{1}\cos \theta _{\text{t))))\,.}$

(32)

For the case of normal incidence these reduce to:

${\displaystyle r_{\text{s0))={\frac {n_{1}-n_{2)){n_{1}+n_{2))))$

(33)

${\displaystyle t_{\text{s0))={\frac {2n_{1)){n_{1}+n_{2))))$

(34)

${\displaystyle r_{\text{p0))={\frac {n_{2}-n_{1)){n_{2}+n_{1))))$

(35)

${\displaystyle t_{\text{p0))={\frac {2n_{1)){n_{2}+n_{1))}\,.}$

(36)

The power reflection coefficients become:

${\displaystyle R_{\text{s))=\left|{\frac {n_{1}\cos \theta _{\text{i))-n_{2}\cos \theta _{\text{t))}{n_{1}\cos \theta _{\text{i))+n_{2}\cos \theta _{\text{t))))\right|^{2))$

(37)

${\displaystyle R_{\text{p))=\left|{\frac {n_{2}\cos \theta _{\text{i))-n_{1}\cos \theta _{\text{t))}{n_{2}\cos \theta _{\text{i))+n_{1}\cos \theta _{\text{t))))\right|^{2}\,.}$

(38)

The power transmissions can then be found from T = 1 − R.

Brewster's angle

For equal permeabilities (e.g., non-magnetic media), if θ_i and θ_t are complementary, we can substitute‍ sin θ_t‍ for‍ cos θ_i‍, and‍ sin θ_i‍ for‍ cos θ_t‍, so that the numerator in equation (31) becomes‍ n₂‍sin θ_t − n₁‍sin θ_i‍,  which is zero (by Snell's law). Hence  r_p = 0  and only the s-polarized component is reflected. This is what happens at the Brewster angle. Substituting‍ cos θ_i‍ for‍ sin θ_t‍ in Snell's law, we readily obtain

${\displaystyle \theta _{\text{i))=\arctan(n_{2}/n_{1})}$

(39)

for Brewster's angle.

Equal permittivities

Although it is not encountered in practice, the equations can also apply to the case of two media with a common permittivity but different refractive indices due to different permeabilities. From equations (4) and (5), if ϵ is fixed instead of μ, then Y becomes inversely proportional to n, with the result that the subscripts 1 and 2 in equations (29) to (38) are interchanged (due to the additional step of multiplying the numerator and denominator by n₁n₂). Hence, in (29) and (31), the expressions for r_s and r_p in terms of refractive indices will be interchanged, so that Brewster's angle (39) will give  r_s = 0  instead of  r_p = 0‍,‍ and any beam reflected at that angle will be p-polarized instead of s-polarized.^[36] 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 (see § History, above). One could predict reflection coefficients that agreed with observation by supposing (like Fresnel) 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 (like MacCullagh and Neumann) that different refractive indices were due to different elasticities and that the vibrations were parallel to that plane.^[37] Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.