Glossary entry

Fresnel equations

Expressions for the reflection and transmission of light at an interface between two media. Derived from Maxwell's equations with boundary conditions.

The Fresnel equations describe how light splits between reflected and transmitted components at an interface between two media of refractive indices $n_1$ (incident side) and $n_2$ (transmitted side). The results depend on incidence angle $\theta_1$ , refraction angle $\theta_2$ (from Snell's law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$ ), and polarization.

Amplitude reflection coefficients (s-polarization perpendicular to the plane of incidence; p-polarization parallel to it):

r_s \;=\; \frac{n_1 \cos\theta_1 - n_2 \cos\theta_2}{n_1 \cos\theta_1 + n_2 \cos\theta_2}, \qquad r_p \;=\; \frac{n_2 \cos\theta_1 - n_1 \cos\theta_2}{n_2 \cos\theta_1 + n_1 \cos\theta_2}.

Power reflectivities:

R_s \;=\; |r_s|^2, \qquad R_p \;=\; |r_p|^2.

Transmission by conservation: $T = 1 - R$ for non-absorbing media.

Normal incidence ( $\theta_1 = 0$ ) simplification:

R \;=\; \left| \frac{n_2 - n_1}{n_2 + n_1} \right|^2 \;=\; \left( \frac{n_2 - n_1}{n_2 + n_1} \right)^2.

For light going from air ( $n_1 = 1$ ) into a material at normal incidence:

Material	$n$ at 1550 nm	$R$ at normal incidence
Fused silica	1.444	3.3%
Silica fiber core (Δn small)	$\sim$ 1.46	3.5%
Sapphire	1.747	7.5%
Silicon	3.480	30.6%
InP	3.165	27.2%
Diamond	2.397	16.8%

Brewster's angle. At $\tan\theta_B = n_2/n_1$ , $R_p = 0$ — p-polarized light is fully transmitted. Used to create high-extinction polarizers (Brewster's-angle plate stacks).

Total internal reflection. When light goes from high to low index ( $n_1 > n_2$ ) at $\theta_1 > \theta_c = \arcsin(n_2/n_1)$ , the wave is fully reflected. This is the mechanism of waveguide confinement (in optical fibers and slab waveguides).

Practical implications:

Fiber facets and PIC facets at normal incidence reflect 3.3% (silica) to 30% (silicon) of input power, creating Fabry–Pérot etalon effects and back-reflection problems. Mitigation: AR coatings, angled facets (typically 7° or 8°), index-matching fluids.
Lens AR coatings typically reduce surface reflection from 4% per surface to $<$ 0.5% per surface over a specified wavelength range.
Optical isolators rely on Brewster-angle elements or Faraday rotators to suppress back-reflections that would otherwise destabilize lasers.