Partial transmission and reflection of a pulse travelling from a low to high refractive index medium.

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 Augustin-Jean Fresnel (/freɪˈnɛl/) 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[edit]

When light strikes the interface of a medium of a given 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 ratio of the reflected and transmitted electric fields to that of the incident beam (the waves' magnetic fields can also be related using similar coefficients). 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.^[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[edit]

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. P polarization refers to polarization of the electric field in the plane of incidence (along the X and Y directions 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 also another special case is mentioned below in which that is true).

Power (intensity) reflection and transmission coefficients[edit]

Variables used in the Fresnel equations

Power coefficients: air to glass

Power coefficients: glass to air

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 of the incident, reflected and refracted waves 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:

θi=θr,displaystyle theta _mathrm i =theta _mathrm r ,

and Snell's law:

n1sin⁡θi=n2sin⁡θt.displaystyle n_1sin theta _mathrm i =n_2sin theta _mathrm t .

The behaviour 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 supplied, 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.^[2]

The reflectance for s-polarized light is

Rs=|Z2cos⁡θi−Z1cos⁡θtZ2cos⁡θi+Z1cos⁡θt|2,frac Z_2cos theta _mathrm i -Z_1cos theta _mathrm t Z_2cos theta _mathrm i +Z_1cos theta _mathrm t right

while the reflectance for p-polarized light is

Rp=|Z2cos⁡θt−Z1cos⁡θiZ2cos⁡θt+Z1cos⁡θi|2,displaystyle R_mathrm p =left

where Z₁ and Z₂ are the wave impedances of media 1 and 2, respectively.

We restrict our scope to non-magnetic media (i.e. materials for which μ₁ = μ₂ = μ₀) which is typically true at optical frequencies (and most other problems)^[3]. Then the wave impedances are determined solely by the refractive indices n₁ and n₂:

Zi=Z0nidisplaystyle textZ_i=frac textZ_0textn_i

where Z₀ is the impedance of free space. Making this substitution, we obtain equations using the refractive indices:

Rs=|n1cos⁡θi−n2cos⁡θtn1cos⁡θi+n2cos⁡θt|2=|n1cos⁡θi−n21−(n1n2sin⁡θi)2n1cos⁡θi+n21−(n1n2sin⁡θi)2|2,^2=left

Rp=|n1cos⁡θt−n2cos⁡θin1cos⁡θt+n2cos⁡θi|2=|n11−(n1n2sin⁡θi)2−n2cos⁡θin11−(n1n2sin⁡θi)2+n2cos⁡θi|2.displaystyle R_mathrm p =left

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]

Ts=1−Rsdisplaystyle T_mathrm s =1-R_mathrm s

and

Tp=1−Rpdisplaystyle 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). For just considering reflection, since cos(θ_i) = cos(θ_r), the ratio of reflected to incident irradiance in the wave's direction is the same as the ratio of the power flow in the normal direction, and this complication can be ignored.

Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be describe 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:

Reff=12(Rs+Rp).displaystyle R_mathrm eff =frac 12left(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[edit]

Normal incidence[edit]

For the case of normal incidence, θi=θt=0displaystyle theta _mathrm i =theta _mathrm t =0, and there is no distinction between s and p polarization. Thus, the reflectance simplifies to

R=|n1−n2n1+n2|2frac n_1-n_2n_1+n_2right.

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[edit]

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

Total internal reflection[edit]

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 is known as total internal reflection. Total internal reflection occurs for incidence angles for which, using Snell's law, the sine of the angle of refraction would exceed unity (whereas sin(θ) ≤ 1 for all real θ). Again assuming glass with n=1.5 surrounded by air, the critical angle is approximately 41°.

Amplitude reflection and transmission coefficients[edit]

The above practical equations relating powers (which could be measured with a photometer for instance) are derived from the actual Fresnel equations which solve the physical problem in terms of electric and magnetic field amplitudes. Those underlying equations supply generally complex-valued ratios of those amplitudes and may take several different forms, depending on formalisms used. The amplitude coefficients are usually represented by lower case r and t (whereas the power coefficients are capitalized).

Amplitude coefficients: air to glass

Amplitude coefficients: glass to air

In the following, the reflection coefficient r is the ratio of the reflected wave's complex electric field amplitude to that of the incident wave. The transmission coefficient t is the ratio of the transmitted wave's electric field 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 θidisplaystyle theta _mathrm i on a plane interface, reflected at an angle θrdisplaystyle theta _mathrm r , and with a transmitted wave at an angle θtdisplaystyle theta _mathrm t , 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). ^{[Note 1]}

Using these conventions,.^[5][6]

rs=n1cos⁡θi−n2cos⁡θtn1cos⁡θi+n2cos⁡θt,ts=2n1cos⁡θin1cos⁡θi+n2cos⁡θt,rp=n2cos⁡θi−n1cos⁡θtn2cos⁡θi+n1cos⁡θt,tp=2n1cos⁡θin2cos⁡θi+n1cos⁡θt.displaystyle beginalignedr_texts&=frac n_1cos theta _texti-n_2cos theta _texttn_1cos theta _texti+n_2cos theta _textt,\[3pt]t_texts&=frac 2n_1cos theta _textin_1cos theta _texti+n_2cos theta _textt,\[3pt]r_textp&=frac n_2cos theta _texti-n_1cos theta _texttn_2cos theta _texti+n_1cos theta _textt,\[3pt]t_textp&=frac 2n_1cos theta _textin_2cos theta _texti+n_1cos theta _textt.endaligned

One can see that t_s = r_s + 1^[7] and n₂/n₁t_p=r_p+1. 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: ^[8]

R=|r|2.^2.

On the other hand, calculation of the power transmission coefficient T is less straight-forward, 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^[9]:

T=n2cos⁡θtn1cos⁡θi|t|2.^2.

The factor of n₂/n₁ is the reciprocal of the ratio of the media's wave impedances (since we assume μ=1). The factor of cos(θ_t)/cos(θ_i) 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 T=0, 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). 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[edit]

In the above formula for r_s‍, if we put n2=n1sin⁡θi/sin⁡θtdisplaystyle n_2=n_1sin theta _texti/sin theta _textt (Snell's law) and multiply the numerator and denominator by 1/n₁ sin θ_t‍, we obtain ^[10][11]

rs=−sin⁡(θi−θt)sin⁡(θi+θt).displaystyle r_texts=-frac sin(theta _texti-theta _textt)sin(theta _texti+theta _textt).

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

rp=tan⁡(θi−θt)tan⁡(θi+θt).displaystyle r_textp=frac tan(theta _texti-theta _textt)tan(theta _texti+theta _textt).

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[edit]

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[edit]

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 of the Fresnel rhomb — which 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).

Theory[edit]

Here we systematically derive the above relations from electromagnetic premises.

Material parameters[edit]

In order to compute meaningful Fresnel coefficients, we must assume the medium is (approximately) linear and homogeneous. If the medium is also isotropic then the four field quantities are related through multiplication by the scalar quantities ϵ and μ:   B = μH  and  D = ϵE‍,‍ where μ is the permeability of the medium and ϵ is the permittivity of the medium.

The permittivity ϵ sets the ratio of D to E. More often one refers to the relative permittivity also known as the dielectric constant ϵ_rel which, when multiplied by the permittivity of free space ϵ₀ yields the permittivity ϵ. In optics, one usually is supplied with the index of refraction n which is just the square root of ϵ_rel as long as μ_rel =1 (the usual case):

ϵ=ϵ0ϵrel=ϵ0nμreldisplaystyle epsilon =epsilon _0epsilon _rel=epsilon _0frac sqrt nmu _rel

It is possible for a material to have a magnetic polarization, though in most cases this is not true and one just assumes the relative permeability μ_rel=1. In general though
μ=μ0μreldisplaystyle mu =mu _0mu _textrel
establishes the ratio of B to H in an isotropic medium. At optical frequencies all materials have μ_rel=1 (except for possible metamaterials). This is usually true at lower frequencies but for the cases of ferromagnetic materials at radio/microwave frequencies such a material's μ_rel must be taken into account.

From the material's ϵ and μ we form two constants directly determining the form of a plane wave in that material. The index of refraction n inversely affects the phase velocity of a plane wave and inversely affects the magnitude of the wave vector k (below):

n=ϵμcdisplaystyle n=frac sqrt epsilon mu c
=ϵrelμreldisplaystyle =sqrt epsilon _textrelmu _textrel

Secondly, the wave impedance of a material defines the ratio of the magnitudes of the electric and magnetic field in a plane wave:

Z=μϵdisplaystyle Z=sqrt frac mu epsilon

In free space Z takes the value of the impedance of free space:

Z0=μ0ϵ0≈377Ωdisplaystyle Z_0=sqrt frac mu _0epsilon _0approx 377Omega

Consequently the wave impedance can be found simply in terms of the refractive index for non-magnetic media as Z₀/n or in general:

Z=μrelϵrelZ0displaystyle Z=sqrt frac mu _textrelepsilon _textrelZ_0
=Z0nμrel.displaystyle =frac Z_0textnmu _textrel.

In the derivation below, we use rather the reciprocal of Z, the wave admittance Y=1/Z.

Electromagnetic plane waves[edit]

In a uniform plane sinusoidal electromagnetic wave, the electric field E has the form

Ekei(k⋅r−ωt),displaystyle mathbf E_k e^i(mathbf kcdot 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 2]}  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 3]}

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 ω/k,displaystyle omega /k,,, known as the phase velocity. This in turn is equal to‍ c/n,displaystyle c/n,,, where c is the speed of light in vacuum and n is the refractive index. Solving for k gives

k=nω/cdisplaystyle k=nomega /c,.

(2)

As usual, we drop the time-dependent factor e^−iωt which is understood to multiply every complex field quantity. Then Faraday's law and the Maxwell-Ampère law respectively reduce to ^[33]

ωB=k×EωD=−k×H.displaystyle beginalignedomega mathbf B &=mathbf k times mathbf E \omega mathbf D &=-mathbf k times mathbf H ,.endaligned

Using the material properties discussed above, we can eliminate B and D to obtain equations in only E and H:

ωμH=k×EωϵE=−k×H.displaystyle beginalignedomega mu mathbf H &=mathbf k times mathbf E \omega epsilon mathbf E &=-mathbf k times mathbf H ,.endaligned

In the case of real material parameters ε and μ (true in a lossless dielectric), k , E , H  form a right-handed orthogonal triad. Thus the above vector multiplications can be reduced to scalar multiplication. Solving the two equations for consistency, we find again the original constraint on the magnitude of k and the refractive index n:

|k|=ωμϵdisplaystyle

n=cμϵdisplaystyle n=c,sqrt mu epsilon ,

(4)

which determines the phase velocity  vp=c/n=1/μϵdisplaystyle textv_textp=textc/n=1big /!sqrt mu epsilon .

And using the two equations, dividing the magnitude of H by E again yields the wave admittance (reciprocal of the wave impedance):

Y=ϵ/μdisplaystyle Y=sqrt epsilon /mu ,.

(5)

The wave vectors[edit]

Incident, reflected, and transmitted wave vectors (k_i‍, k_r‍, and k_t ), for incidence from a medium with refractive index n₁ to a medium with refractive index n₂. The red arrows are perpendicular to the wave vectors.

In Cartesian coordinates (x, y,‍z), let the region‍ y < 0‍ have refractive index n₁ , intrinsic impedance Y₁ , etc., and let the region‍ y > 0‍ have refractive index n₂ , intrinsic impedance 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

ki=n1k(isin⁡θi+jcos⁡θi)kr=n1k(isin⁡θi−jcos⁡θi)kt=n2k(isin⁡θt+jcos⁡θt)=k(in1sin⁡θi+jn2cos⁡θt),displaystyle beginalignedmathbf k _texti&=n_1k(mathbf i sin theta _texti+mathbf j cos theta _texti)\[.5ex]mathbf k _textr&=n_1k(mathbf i sin theta _texti-mathbf j cos theta _texti)\[.5ex]mathbf k _textt&=n_2k(mathbf i sin theta _textt+mathbf j cos theta _textt)\&=k(mathbf i ,n_1sin theta _texti+mathbf j ,n_2cos theta _textt),,endaligned

where the last step uses Snell's law. The corresponding dot products in the phasor form (3) are

ki⋅r=n1k(xsin⁡θi+ycos⁡θi)kr⋅r=n1k(xsin⁡θi−ycos⁡θi)kt⋅r=k(n1xsin⁡θi+n2ycos⁡θt).displaystyle beginalignedmathbf k _textimathbf cdot r &=n_1k(xsin theta _texti+ycos theta _texti)\mathbf k _textrmathbf cdot r &=n_1k(xsin theta _texti-ycos theta _texti)\mathbf k _texttmathbf cdot r &=k(n_1xsin theta _texti+n_2ycos theta _textt),.endaligned

(6)

Hence:

At  y=0,   ki⋅r=kr⋅r=kt⋅r=n1kxsin⁡θidisplaystyle y=0,,~~~mathbf k _textimathbf cdot r =mathbf k _textrmathbf cdot r =mathbf k _texttmathbf cdot r =n_1kxsin theta _texti,.

(7)

The s components[edit]

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

Ei=eiki⋅r,displaystyle E_texti=e^imathbf k _textimathbf cdot r ,

(8)

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

Er=rseikr⋅rEt=tseikt⋅r.displaystyle beginalignedE_textr&=r_s,e^imathbf k _textrmathbf cdot r \E_textt&=t_s,e^imathbf k _texttmathbf cdot r .endaligned

(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 the field normal to the plane of incidence. For the s (TE) 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,

Hi=Y1eiki⋅rHr=Y1rseikr⋅rHt=Y2tseikt⋅r.displaystyle beginalignedH_texti&=,Y_1e^imathbf k _textimathbf cdot r \H_textr&=,Y_1r_s,e^imathbf k _textrmathbf cdot r \H_textt&=,Y_2t_s,e^imathbf k _texttmathbf cdot r .endaligned

(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,

Ei+Er=EtHicos⁡θi−Hrcos⁡θi=Htcos⁡θt  }   at  y=0displaystyle left.beginalignedE_texti+E_textr&=E_textt\H_texticos theta _texti-H_textrcos theta _texti&=H_texttcos theta _texttendaligned~~right~~~textat~~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

1+rs=tsY1cos⁡θi−Y1rscos⁡θi=Y2tscos⁡θt,displaystyle beginaligned1+r_texts&=,t_texts\Y_1cos theta _texti-Y_1r_textscos theta _texti&=,Y_2t_textscos theta _textt,,endaligned

(12)

which are easily solved for r_s and t_s‍, yielding

rs=Y1cos⁡θi−Y2cos⁡θtY1cos⁡θi+Y2cos⁡θtdisplaystyle r_texts=frac Y_1cos theta _texti-Y_2cos theta _texttY_1cos theta _texti+Y_2cos theta _textt

(13)

and

ts=2Y1cos⁡θiY1cos⁡θi+Y2cos⁡θtdisplaystyle t_texts=frac 2Y_1cos theta _textiY_1cos theta _texti+Y_2cos theta _textt,.

(14)

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

rs0=Y1−Y2Y1+Y2displaystyle r_texts0=frac Y_1-Y_2Y_1+Y_2

(15)

and

ts0=2Y1Y1+Y2displaystyle t_texts0=frac 2Y_1Y_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[edit]

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

Ei=eiki⋅rEr=rpeikr⋅rEt=tpeikt⋅r.displaystyle beginalignedE_texti&=e^imathbf k _textimathbf cdot r \E_textr&=r_p,e^imathbf k _textrmathbf cdot r \E_textt&=t_p,e^imathbf k _texttmathbf cdot r .endaligned

(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 (TM) 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,

Hi=Y1eiki⋅rHr=Y1rpeikr⋅rHt=Y2tpeikt⋅r.displaystyle beginalignedH_texti&=,Y_1e^imathbf k _textimathbf cdot r \H_textr&=,Y_1r_p,e^imathbf k _textrmathbf cdot r \H_textt&=,Y_2t_p,e^imathbf k _texttmathbf cdot r .endaligned

(18)

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

Eicos⁡θi−Ercos⁡θi=Etcos⁡θtHi+Hr=Ht  }   at  y=0displaystyle left.beginalignedE_texticos theta _texti-E_textrcos theta _texti&=E_texttcos theta _textt\H_texti+H_textr&=H_texttendaligned~~right~~~textat~~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

cos⁡θi−rpcos⁡θi=tpcos⁡θtY1+Y1rp=Y2tp.displaystyle beginalignedcos theta _texti-r_textpcos theta _texti&=,t_textpcos theta _textt\Y_1+Y_1r_textp&=,Y_2t_textp,.endaligned

(20)

Solving for r_p and t_p‍, we find

rp=Y2cos⁡θi−Y1cos⁡θtY2cos⁡θi+Y1cos⁡θtdisplaystyle r_textp=frac Y_2cos theta _texti-Y_1cos theta _texttY_2cos theta _texti+Y_1cos theta _textt

(21)

and

tp=2Y1cos⁡θiY2cos⁡θi+Y1cos⁡θtdisplaystyle t_textp=frac 2Y_1cos theta _textiY_2cos theta _texti+Y_1cos theta _textt,.

(22)

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

rp0=Y2−Y1Y2+Y1displaystyle r_textp0=frac Y_2-Y_1Y_2+Y_1

(23)

and

tp0=2Y1Y2+Y1displaystyle t_textp0=frac 2Y_1Y_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)[edit]

The Poynting vector for a plane 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. The Poynting vector is‍ 1/2‍ReE × H^∗ where E and H are due only to the wave in question. Inside a lossless dielectric (the usual case) E and H are in phase and at right angles, and at right angles to the wave vector k, so for s polarization, using the Z and X-Y components of E and H respectively (or the X-Y and -Z components of E and H for p polarization) the irradiance in the direction of k is just given by EH/2 or E²/2Z in a medium of characteristic 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, one could use only the X component (rather than the full X-Y component) of H or E; or equivalently just 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

Rs=|Y1cos⁡θi−Y2cos⁡θtY1cos⁡θi+Y2cos⁡θt|2^2

(25)

for the s polarization, and

Rp=|Y2cos⁡θi−Y1cos⁡θtY2cos⁡θi+Y1cos⁡θt|2displaystyle R_textp=left

(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. They are thus not taken into account, as they must be in computing the power transmission, below.

The simplest way to obtain the power transmission coefficient (or transmissivity , ratio of transmitted power to incident power in the direction normal to the interface, Y) is to simply use R+T=1 due to conservation of energy. We find for the two polarization cases:

Ts=1−Rs==4Re(Y1Y2cos⁡θicos⁡θt)|Y1cos⁡θi+Y2cos⁡θt|2displaystyle T_texts=1-R_texts==frac 4Re(Y_1Y_2cos theta _texticos theta _textt)^2

(27)

for the s polarization, and

Tp=1−Rp==4Re(Y1Y2cos⁡θicos⁡θt)|Y2cos⁡θi+Y1cos⁡θt|2displaystyle T_textp=1-R_textp==frac 4Re(Y_1Y_2cos theta _texticos theta _textt)Y_2cos theta _texti+Y_1cos theta _texttright

(28)

for the p polarization.

In the case of an interface between two lossless media (real ε and μ >0) one can obtain these results directly using the squared magnitude of the amplitude transmission coefficients we found earlier in equations (14) and (22). Then, as noted above, to find the Poynting vector in the Y direction, each power must be multiplied by the geometric factor cos θ‍ and divided by the wave impedance Z. Applying these corrections to each wave we obtain two ratios multiplying the square of the amplitude transmission coefficient:

Ts=(2Y1cos⁡θiY1cos⁡θi+Y2cos⁡θt)2Y2Y1cos⁡θtcos⁡θi=4Y1Y2cos⁡θicos⁡θt(Y1cos⁡θi+Y2cos⁡θt)2displaystyle T_texts=left(frac 2Y_1cos theta _textiY_1cos theta _texti+Y_2cos theta _texttright)^2frac ,Y_2,Y_1,frac cos theta _texttcos theta _texti=frac 4Y_1Y_2cos theta _texticos theta _texttleft(Y_1cos theta _texti+Y_2cos theta _texttright)^2

(27)

for the s polarization, and

Tp=(2Y1cos⁡θiY2cos⁡θi+Y1cos⁡θt)2Y2Y1cos⁡θtcos⁡θi=4Y1Y2cos⁡θicos⁡θt(Y2cos⁡θi+Y1cos⁡θt)2displaystyle T_textp=left(frac 2Y_1cos theta _textiY_2cos theta _texti+Y_1cos theta _texttright)^2frac ,Y_2,Y_1,frac cos theta _texttcos theta _texti=frac 4Y_1Y_2cos theta _texticos theta _texttleft(Y_2cos theta _texti+Y_1cos theta _texttright)^2

(28)

for the p polarization. In addition to only applying to lossless dielectrics, the latter equations are valid only at incidence angles smaller than the critical angle (beyond which T=0 of course!).

Equal refractive indices[edit]

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. It 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.

Non-magnetic media[edit]

Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Since

Y=ncμdisplaystyle Y=frac n,cmu ,.

(from dividing (4) by (5))
we can substitute the vacuum permeability μ₀ for μ, so that

Y1=n1cμ0  ;   Y2=n2cμ0displaystyle Y_1=frac n_1,cmu _0~~;~~~Y_2=frac n_2,cmu _0,.

When we make these substitutions in equations (13) to (16) and equations (21) to (26), the factor cμ₀ cancels out, with the result that the admittances are simply proportional to the corresponding refractive indices. Thus we obtain

rs=n1cos⁡θi−n2cos⁡θtn1cos⁡θi+n2cos⁡θtdisplaystyle r_texts=frac n_1cos theta _texti-n_2cos theta _texttn_1cos theta _texti+n_2cos theta _textt

(29)

ts=2n1cos⁡θin1cos⁡θi+n2cos⁡θtdisplaystyle t_texts=frac 2n_1cos theta _textin_1cos theta _texti+n_2cos theta _textt,

(30)

rp=n2cos⁡θi−n1cos⁡θtn2cos⁡θi+n1cos⁡θtdisplaystyle r_textp=frac n_2cos theta _texti-n_1cos theta _texttn_2cos theta _texti+n_1cos theta _textt

(33)

tp=2n1cos⁡θin2cos⁡θi+n1cos⁡θtdisplaystyle t_textp=frac 2n_1cos theta _textin_2cos theta _texti+n_1cos theta _textt,

(34)

For the case of normal incidence these reduce to:

rs0=n1−n2n1+n2displaystyle r_texts0=frac n_1-n_2n_1+n_2

(31)

ts0=2n1n1+n2displaystyle t_texts0=frac 2n_1n_1+n_2

(32)

rp0=n2−n1n2+n1displaystyle r_textp0=frac n_2-n_1n_2+n_1

(35)

tp0=2n1n2+n1displaystyle t_textp0=frac 2n_1n_2+n_1

(36)

The power reflection coefficients become:

Rs=|n1cos⁡θi−n2cos⁡θtn1cos⁡θi+n2cos⁡θt|2^2

(37)

Rp=|n2cos⁡θi−n1cos⁡θtn2cos⁡θi+n1cos⁡θt|2^2.

(38)

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

Brewster's angle[edit]

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 (33) 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

θi=arctan⁡(n2/n1)displaystyle theta _texti=arctan(n_2/n_1)

(39)

for Brewster's angle.

Equal permittivities?[edit]

Although not encountered in practice, we can consider the case of refractive indices due only to magnetic permeability of the media, with ε_rel=1. From equations (4) and (5), we see that if ϵ is fixed instead of μ, then n becomes proportional to Z, not to Y. Hence, if all the reflection and transmission ratios are expressed in terms of refractive indices, the subscripts will match the impedance forms rather than the admittance forms; that is, the subscripts 1 and 2 in equations (29) to (38) will be interchanged. Hence, in (29) and (33), 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.^[35] 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 (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.^[36] Thus the condition of equal permittivities and unequal permeabilities, although not seen in practice, is of some historical interest.

See also[edit]

• Polarization mixing


• Index-matching material


• Field and power quantities


• 
  Fresnel rhomb, Fresnel's apparatus to produce circularly polarised light [1]
  


• Specular reflection


• Schlick's approximation


• Snell's window


• X-ray reflectivity


• Plane of incidence


• Reflections of signals on conducting lines

Notes[edit]

References[edit]

• M. Born and E. Wolf, 1970, Principles of Optics, 4th Ed., Oxford: Pergamon Press.


• J.Z. Buchwald, 1989, The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century, University of Chicago Press,
  ISBN 0-226-07886-8.


• R.E. Collin, 1966, Foundations for Microwave Engineering, Tokyo: McGraw-Hill.


• O. Darrigol, 2012, A History of Optics: From Greek Antiquity to the Nineteenth Century, Oxford,
  ISBN 978-0-19-964437-7.


• A. Fresnel, 1866  (ed.  H. de Senarmont, E. Verdet, and L. Fresnel), Oeuvres complètes d'Augustin Fresnel, Paris: Imprimerie Impériale (3 vols., 1866–70), vol. 1 (1866).


• E. Hecht, 1987, Optics, 2nd Ed., Addison Wesley,
  ISBN 0-201-11609-X.


• E. Hecht, 2002, Optics, 4th Ed., Addison Wesley,
  ISBN 0-321-18878-0.


• F.A. Jenkins and H.E. White, 1976, Fundamentals of Optics, 4th Ed., New York: McGraw-Hill,
  ISBN 0-07-032330-5.


• H. Lloyd, 1834, "Report on the progress and present state of physical optics", Report of the Fourth Meeting of the British Association for the Advancement of Science (held at Edinburgh in 1834), London: J. Murray, 1835, pp. 295–413.


• W. Whewell, 1857, History of the Inductive Sciences: From the Earliest to the Present Time, 3rd Ed., London: J.W. Parker & Son, vol. 2.


• E.T. Whittaker, 1910, A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century, London: Longmans, Green, & Co.

Further reading[edit]

This article's further reading may not follow Wikipedia's content policies or guidelines. Please improve this article by removing less relevant or redundant publications with the same point of view; or by incorporating the relevant publications into the body of the article through appropriate citations. (October 2014) (Learn how and when to remove this template message)

• 
  Woan, G. (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
  


• 
  Griffiths, David J. (2007). Introduction to Electrodynamics (3rd ed.). Pearson Education. ISBN 81-7758-293-3.
  


• 
  Band, Y. B. (2010). Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers. John Wiley & Sons. ISBN 978-0-471-89931-0.
  


• 
  Kenyon, I. R. (2008). The Light Fantastic – Introduction to Classic and Quantum Optics. Oxford University Press. ISBN 978-0-19-856646-5.
  


• 
  Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3


• 
  McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994,
  ISBN 0-07-051400-3
  

External links[edit]

• 
  Fresnel Equations – Wolfram.


• Fresnel equations calculator


• 
  FreeSnell – Free software computes the optical properties of multilayer materials.


• 
  Thinfilm – Web interface for calculating optical properties of thin films and multilayer materials (reflection & transmission coefficients, ellipsometric parameters Psi & Delta).


• 
  Simple web interface for calculating single-interface reflection and refraction angles and strengths.


• 
  Reflection and transmittance for two dielectrics^{[permanent dead link]} – Mathematica interactive webpage that shows the relations between index of refraction and reflection.


• 
  A self-contained first-principles derivation of the transmission and reflection probabilities from a multilayer with complex indices of refraction.