Fresnel equations

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

At near-grazing incidence, media interfaces can be mirror-like, despite being poor reflectors at normal incidence, θ_i = 0

The Fresnel equations (or Fresnel conditions), deduced by Augustin-Jean Fresnel (/freɪˈnɛl/), describe the behaviour of light when moving between media of differing refractive indices. The reflection of light that the equations predict is known as Fresnel reflection.

Overview

When light moves from a medium of a given refractive index, n₁, into a second medium with refractive index, n₂, both reflection and refraction of the light may occur. The Fresnel equations describe what fraction of the light is reflected and what fraction is refracted (i.e., transmitted). They also describe the phase shift of the reflected light.

The equations assume the interface between the media is flat and that the media are homogeneous. The incident light is assumed to be a plane wave, and effects of edges are neglected.

S and p polarizations

The behavior depends on the polarization of the incident ray, which can be separated into 2 cases:

s-polarized (perpendicular) or TE: The incident light is polarized with its electric field perpendicular to the plane containing the incident, reflected, and refracted rays. This plane is called the plane of incidence; it is the plane of the diagram below. The light is said to be s-polarized, from the German, senkrecht, meaning perpendicular.
p-polarized (parallel) or TM: The incident light is polarized with its electric field parallel to the plane of incidence. Such light is described as p-polarized, from parallel.

Power or intensity equations

Variables used in the Fresnel equations

In the diagram on the right, an incident light ray, IO, strikes the interface between two media of refractive indices n₁ and n₂ at point, O. Part of the ray is reflected as ray, OR, and part refracted as ray, 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:

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

and Snell's law:

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

The fraction of the incident power that is reflected from the interface is given by the reflectance or reflectivity, R, and the fraction that is refracted is given by the transmittance or transmissivity, T, (unrelated to the transmission through a medium).^[1]

The reflectance for s-polarized light is

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}=\left|{\frac {{\sqrt {\frac {\mu _{2}}{\epsilon _{2}}}}\cos \theta _{\mathrm {i} }-{\sqrt {\frac {\mu _{1}}{\epsilon _{1}}}}\cos \theta _{\mathrm {t} }}{{\sqrt {\frac {\mu _{2}}{\epsilon _{2}}}}\cos \theta _{\mathrm {i} }+{\sqrt {\frac {\mu _{1}}{\epsilon _{1}}}}\cos \theta _{\mathrm {t} }}}\right|^{2},

while the reflectance for p-polarized light is

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}=\left|{\frac {{\sqrt {\frac {\mu _{2}}{\epsilon _{2}}}}\cos \theta _{\mathrm {t} }-{\sqrt {\frac {\mu _{1}}{\epsilon _{1}}}}\cos \theta _{\mathrm {i} }}{{\sqrt {\frac {\mu _{2}}{\epsilon _{2}}}}\cos \theta _{\mathrm {t} }+{\sqrt {\frac {\mu _{1}}{\epsilon _{1}}}}\cos \theta _{\mathrm {i} }}}\right|^{2},

where Z₁ and Z₂ are the wave impedances of media 1 and 2, respectively, μ₁ and μ₂ are the magnetic permeability of the two materials, and ε₁ and ε₂ are the electric permittivity of the two materials (at the frequency of the light wave).

For non-magnetic media (i.e. materials for which μ₁ ≈ μ₂ ≈ μ₀, where μ₀ is the permeability of free space), we have

Z_{1}={\frac {Z_{0}}{n_{1}}},\ Z_{2}={\frac {Z_{0}}{n_{2}}}.

Then, the reflectance for s-polarized light becomes

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}\!,

while the reflectance for p-polarized light becomes

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 the conservation of energy, the transmittances are given by^[2]

T_{\mathrm {s} }=1-R_{\mathrm {s} }

and

T_{\mathrm {p} }=1-R_{\mathrm {p} }

These relationships hold only for power or intensity, not for complex amplitude transmission and reflection coefficients as defined below.

If the incident light is unpolarised (containing an equal mix of s and p polarisations), the reflectance is

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

For common glass, with n₂ around 1.5, the reflectance at θ_i = 0 is about 4%. Note that reflection by a window is from the front side as well as the back side, and that some of the light bounces back and forth a number of times between the two sides. The combined reflectance for this case is 2R/(1 + R), when interference can be neglected (see below).

The discussion given here assumes that the permeability, μ, is equal to the vacuum permeability, μ₀, in both media, embodying the assumption that the material is non-magnetic. This is approximately true for most dielectric materials, but not for some other types of material. The completely general Fresnel equations are more complicated.

For low-precision applications where polarization may be ignored, such as computer graphics, Schlick's approximation may be used.

Special cases

Normal incidence

For the case of normal incidence, $\theta _{\mathrm {i} }=\theta _{\mathrm {t} }=0$ , and there is no distinction between s and p polarization. Thus, the reflectance simplifies to

R=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}

Brewster's angle and total internal reflection

At one particular angle for a given n₁ and n₂, the value of R_p goes to zero and a p-polarised incident ray is purely refracted. This angle is known as Brewster's angle, and is around 56° for a glass medium in air or vacuum. Note that this statement is only true when the refractive indices of both materials are real numbers, as is the case for materials like air and glass. For materials that absorb light, like metals and semiconductors, n is complex, and R_p does not generally go to zero.

When moving from a denser medium into a less dense one (i.e., n₁ > n₂), above an 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. The critical angle occurs when $\theta _{\mathrm {t} }=\pi /2$ , and is approximately 41° for glass in air.

Magnetic materials

For magnetic materials there exists the special case where the real refractive indices of the two media are equal, n₁ = n₂, but the magnetic permeabilities are unequal. In this case the reflected ray is independent of the incident polarization and of the angle of incidence, and the magnitude of the reflection is constant except at grazing incidence.^[3]

Amplitude or field equations

Equations for coefficients corresponding to ratios of the electric field complex-valued amplitudes of the waves (not necessarily real-valued magnitudes) are also called Fresnel equations. These take several different forms, depending on the choice of formalism and sign convention used. The amplitude coefficients are usually represented by lower case r and t.

Amplitude ratios: air to glass

Amplitude ratios: glass to air

Conventions used here

In this treatment, the coefficient $r$ is the ratio of the reflected wave's complex electric field amplitude to that of the incident wave. The coefficient $t$ is the ratio of the transmitted wave's electric field amplitude to that of the incident wave. The light is split into s and p polarizations as defined above. (In the figures to the right, s polarization is denoted " $\scriptstyle \bot$ " and p is denoted " $\scriptstyle \parallel$ ".)

For s polarization, a positive $r$ or $t$ means that the electric fields of the incoming and reflected or transmitted wave are parallel, while negative means anti-parallel. For p polarization, a positive $r$ or $t$ means that the magnetic fields of the waves are parallel, while negative means anti-parallel.^[4] It is also assumed that the magnetic permeability, $µ$ , of both media is equal to the permeability of free space $µ 0$ .

(Some authors use the opposite sign convention for $r p$ , so that $r p$ is positive when the incoming and reflected magnetic fields are anti-parallel, and negative when they are parallel. This latter convention has the convenient advantage that the s and p sign conventions are the same at normal incidence. However, either convention, when used consistently, gives the right answers.)

Formulas

Using the arbitrary sign conventions above,^[4]

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

Notice that $t p \neq r p + 1$ .^[5] Instead $n 2 / n 1 t p = r p +1$ .

Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the amplitude reflection coefficient is related to the reflectance $R$ by^[6]

R=|r|^{2}.

The transmittance $T$ is generally not equal to $| t | 2$ , since the light travels with different direction and speed in the two media. The transmittance is related to $t$ by:^[7]

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

The factor of $n 2 / n 1$ occurs from the ratio of intensities (closely related to irradiance). The factor of $cos(θ t)/cos(θ i)$ represents the change in area $m$ of the pencil of rays, needed since $T$ , the ratio of powers, is equal to the ratio of (intensity × area). In terms of the ratio of refractive indices,

\rho ={\frac {n_{2}}{n_{1}}},

and of the magnification $m$ of the beam cross section occurring at the interface,

T=\rho m|t|^{2}.

In the case of total internal reflection, $t p$ and $t s$ are complex numbers representing the evanescent wave. This accounts for non-zero values of $t p$ and $t s$ seen in the graph 'Amplitude ratios: glass to air'.

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^[8] can be used to solve multiple-surface problems.

Notes

↑ Hecht 1987, p. 100.
↑ Hecht 1987, p. 102.
↑ Giles, C. Lee; Wild, Walter J. (1982). "Fresnel reflection and transmission at a planar boundary from media of equal refractive indices" (PDF). Applied Physics Letters. 40 (3): 210–212.
1 2 Lecture notes by Bo Sernelius, main site, see especially Lecture 12.
↑ Hecht 2002, p. 116, eq.(4.49)-(4.50).
↑ Hecht 2002, p. 120, eq.(4.56).
↑ Hecht 2002, p. 120, eq.(4.57).
↑ Heavens, O. S. (1955). Optical Properties of Thin Films. Academic Press. chapt. 4.

References

Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. ISBN 0-201-11609-X.
Hecht, Eugene (2002). Optics (4th ed.). Addison Wesley. ISBN 0-321-18878-0.

External links

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 – 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.

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