Fresnel equations

The Fresnel equations give the fractions of light reflected and transmitted where a wave meets the boundary between two media of refractive indices $n_1$ and $n_2$, as functions of the angle of incidence and the polarization. At normal incidence both polarizations reflect the same fraction,

$$R={\left(\frac{n_1-n_2}{n_1+n_2}\right)}^2,$$

which is 4.0% for glass of index 1.5 in air, 3.4% for the cleaved end of a silica fiber, and 30.6% for silicon at 1550 nm. Away from normal incidence, the reflectance of s-polarized light rises steadily to 100% at grazing incidence, while that of p-polarized light first falls to zero at the Brewster angle. Going from a denser medium into a less dense one, both polarizations are fully reflected beyond the critical angle, where total internal reflection begins.

The equations

Light arrives at angle ${\theta }_1$ from the normal and is refracted to ${\theta }_2$, given by Snell's law, $n_1\sin {\theta }_1=n_2\sin {\theta }_2$. The two polarizations are treated separately: s-polarized light has its electric field perpendicular to the plane of incidence (the plane containing the ray and the surface normal), and p-polarized light has it in that plane. For the amplitudes of the reflected and transmitted fields,

$$\begin{array}{rl}{\displaystyle r_s}& {\displaystyle =\frac{n_1\cos {\theta }_1-n_2\cos {\theta }_2}{n_1\cos {\theta }_1+n_2\cos {\theta }_2},}\\ {\displaystyle t_s}& {\displaystyle =\frac{2n_1\cos {\theta }_1}{n_1\cos {\theta }_1+n_2\cos {\theta }_2},}\end{array}$$ $$\begin{array}{rl}{\displaystyle r_p}& {\displaystyle =\frac{n_2\cos {\theta }_1-n_1\cos {\theta }_2}{n_2\cos {\theta }_1+n_1\cos {\theta }_2},}\\ {\displaystyle t_p}& {\displaystyle =\frac{2n_1\cos {\theta }_1}{n_2\cos {\theta }_1+n_1\cos {\theta }_2}\mathrm{.}}\end{array}$$

The reflectances are $R_s=\mathrm{\mid }{r}_s{\mathrm{\mid }}^2$ and $R_p=\mathrm{\mid }{r}_p{\mathrm{\mid }}^2$. The transmittances carry a factor for the change in beam width and speed, $T=(n_2\cos {\theta }_2\mathrm{/}{n}_1\cos {\theta }_1)\mathrm{\mid }t{\mathrm{\mid }}^2$, so that $R+T=1$ for each polarization when neither medium absorbs. Unpolarized light reflects the average, $(R_s+R_p)\mathrm{/}2$. Using Snell's law, the reflection coefficients can also be written with angles alone:

$$r_s=-\frac{\sin ({\theta }_1-{\theta }_2)}{\sin ({\theta }_1+{\theta }_2)},r_p=\frac{\tan ({\theta }_1-{\theta }_2)}{\tan ({\theta }_1+{\theta }_2)}\mathrm{.}$$

Textbooks differ in the sign convention for $r_p$ (the form here follows Hecht); the reflectances do not depend on it.

Reflectance against angle

For light going from air into glass of index 1.5:

The reflectance changes little up to about 30°, so the normal-incidence value serves for most lens and window surfaces; near grazing incidence, even an uncoated glass surface reflects most of the light. The s reflectance is the larger at every angle. The Fresnel Equations Calculator gives the reflection coefficients, reflectances and phases of both polarizations at any angle, with the Brewster and critical angles, for transparent and absorbing media. The Thin-Film Reflectance Calculator plots $R_s$ and $R_p$ against angle for a bare interface, or for a coated one through the characteristic matrix method.

Reflectance at normal incidence

Air to each material at 1550 nm, with the index values from the refractive index entry. Return loss is $-10{\log }_{10}R$:

Between two materials the reflection depends only on their index ratio, so it drops quickly as the indices approach each other: a fiber core (1.45) against an optical epoxy of 1.56 reflects 0.13% (28.7 dB), and silicon against silica 17.1%. The reflectance changes with wavelength only through dispersion of the index; fused silica reflects 3.5% at 589 nm, where its index is 1.458.

Phase on reflection

At normal incidence, the field reflected from a medium of higher index is inverted, a phase shift of π, and the field reflected from a medium of lower index is not. In a quarter-wave anti-reflection coating of magnesium fluoride on glass, both reflections are inverted, and the extra half wave of round trip through the layer puts them out of phase, so they cancel. A single such layer lowers the reflectance of BK7 at 550 nm from 4.3% to 1.3%, and multilayer stacks hold it below about 0.5% across the visible.

Brewster angle

The p reflection vanishes when ${\theta }_1+{\theta }_2=90\mathrm{°}$, that is, when the reflected and refracted rays would be perpendicular. This happens at the Brewster angle,

$$\tan {\theta }_B=\frac{n_2}{n_1},$$

which is 56.3° for glass of index 1.5, 53.1° for water, 55.5° for fused silica at 633 nm and 74.0° for silicon at 1550 nm. At the Brewster angle the reflected light is purely s-polarized; for glass of index 1.5 it carries 14.8% of the incident s power. Brewster windows on gas laser tubes use this to pass p-polarized light without reflection loss, which makes the laser oscillate in p polarization, and light reflected from water or a road near this angle is mostly horizontally polarized, which is the glare that polarizing sunglasses block. A stack of plates at the Brewster angle is a simple polarizer of modest extinction: it transmits all the p-polarized light and removes part of the s-polarized light at every surface.

Total internal reflection

When light goes from a higher to a lower index, $\sin {\theta }_2=(n_1\mathrm{/}{n}_2)\sin {\theta }_1$ reaches 1 at the critical angle,

$$\sin {\theta }_c=\frac{n_2}{n_1},$$

41.8° for glass of index 1.5 against air, 43.8° for fused silica against air, and 24.5° for silicon against silica at 1550 nm. Beyond it, $\cos {\theta }_2$ becomes imaginary, both $\mathrm{\mid }{r}_s\mathrm{\mid }$ and $\mathrm{\mid }{r}_p\mathrm{\mid }$ equal 1, and all the light is reflected, while an evanescent field reaches into the second medium, typically by a fraction of a wavelength and further close to the critical angle. This is the confinement mechanism of optical fibers and waveguides. The internal reflectance climbs steeply just before the critical angle: inside glass of index 1.5 it is 4.0% at normal incidence, 10.6% (s) at 30°, 39% (s) at 40°, and 100% from 41.8°.

Under total internal reflection the two polarizations acquire different phase shifts. In glass of index 1.51 at an angle of 54.6° they differ by 45°, so two reflections give a quarter-wave retardance, the principle of the Fresnel rhomb waveplate.

Absorbing materials and metals

For an absorbing material the refractive index is complex, $\tilde{n}=n+i\kappa$, and the same equations apply with complex quantities. At normal incidence from air,

$$R=\frac{(n-1{)}^2+{\kappa }^2}{(n+1{)}^2+{\kappa }^2}\mathrm{.}$$

Metals, whose extinction coefficient $\kappa$ is large, reflect most of the incident light; a weakly absorbing material, with $\kappa$ much smaller than $n$, reflects almost as if it were transparent.

Surfaces, plates and facets

A plate has two surfaces. When their reflections add incoherently, a lossless plate transmits $(1-R)\mathrm{/}(1+R)$, 92.3% for glass of index 1.5. When they interfere, as in a thin, flat window under a laser, the plate is an etalon whose transmission swings between 85.2% and 100% with wavelength and thickness, averaging the same 92.3%.

Reflections at fiber ends and chip facets send light back into lasers and form unwanted cavities. The usual remedies are anti-reflection coatings, angled facets and angle-polished connectors (8° in fiber), and index-matching gels or adhesives. In an OTDR trace, every open end and poorly mated connector appears as a Fresnel reflection spike, and in a link budget these reflections are counted as optical return loss. The same reflection is useful in a semiconductor laser: the cleaved facets of the chip reflect roughly 30% and serve as its mirrors without any coating.

Common questions

Why does glass reflect about 4% of light?

Because $((1.5-1)\mathrm{/}(1.5+1){)}^2=0.04$: the Fresnel reflectance at normal incidence for an index of 1.5 in air. A window has two surfaces and reflects about 8% in total, unless it is anti-reflection coated.

What is the difference between s and p polarization?

s-polarized light has its electric field perpendicular to the plane of incidence and p-polarized light parallel to it; the letters come from the German senkrecht (perpendicular) and parallel. The s polarization reflects more strongly at every angle between normal and grazing incidence, and only the p polarization has a Brewster angle, where its reflectance is zero.

How much light is reflected at the Brewster angle?

None of the p-polarized light. For glass of index 1.5, the s-polarized light reflects 14.8%, so unpolarized light reflects 7.4%, all of it s-polarized.

Do the Fresnel equations apply to coated surfaces?

They apply at each interface of the coating. A multilayer is handled by combining the Fresnel coefficients of all its interfaces with the phase delays of the layers between them, which is what the characteristic matrix method does.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017). M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).