How it is calculated
Snell's law gives the refraction angle, \(n_1\sin\theta_i = n_2\sin\theta_t\). The Fresnel amplitude reflection coefficients for s (TE, perpendicular to the plane of incidence) and p (TM, parallel) polarization are
with power reflectances \(R_s = |r_s|^2\), \(R_p = |r_p|^2\), and \(R = (R_s+R_p)/2\) for unpolarized light. For lossless media \(T = 1 - R\) in each polarization. (Sign conventions for \(r_p\) differ between textbooks, but the reflectances do not.)
At the Brewster angle \(R_p = 0\): p-polarized light passes with no reflection loss, and the reflected beam is purely s-polarized. Laser tubes, Brewster windows, and polarizing plates use this. Above the critical angle, going from high to low index, there is no transmitted wave and \(R = 1\) (total internal reflection).
An uncoated window has two surfaces. Summing the incoherent multiple reflections between them gives a transmission of \((1-R)/(1+R)\) per polarization, slightly better than the single-pass estimate \((1-R)^2\).
Reference values
| Material (in air) | n | R₀ per surface | Window T | θB |
|---|---|---|---|---|
| Water, 589 nm | 1.333 | 2.04 % | 96.0 % | 53.1° |
| CaF₂, 587.6 nm | 1.4338 | 3.18 % | 93.8 % | 55.1° |
| Fused silica, 587.6 nm | 1.4585 | 3.48 % | 93.3 % | 55.6° |
| N-BK7, 587.6 nm | 1.5168 | 4.22 % | 91.9 % | 56.6° |
| Sapphire (o-ray), 587.6 nm | 1.768 | 7.70 % | 85.7 % | 60.5° |
| ZnSe, 10.6 µm | 2.403 | 17.0 % | 70.9 % | 67.4° |
| Silicon, 1550 nm | 3.476 | 30.6 % | 53.1 % | 74.0° |
| Germanium, 10.6 µm | 4.003 | 36.0 % | 47.0 % | 76.0° |
This is why infrared optics made of Si, Ge, or ZnSe are almost always anti-reflection coated. An uncoated germanium window loses more than half the beam.
Worked example
Light in air (\(n_1 = 1\)) hits N-BK7 (\(n_2 = 1.5168\)) at 45°. Snell's law gives \(\theta_t = 27.79°\). Then \(R_s = 9.60\) %, \(R_p = 0.92\) %, and unpolarized \(R = 5.26\) %, compared with 4.22 % at normal incidence. The Brewster angle is 56.60°. Going the other way, from BK7 into air, light is totally internally reflected beyond \(\theta_c = 41.25°\). That is why a 45° BK7 right-angle prism works as a mirror.
Assumptions and limits
- Non-absorbing, isotropic media. Metals and strongly absorbing materials need a complex index \(\tilde n = n + ik\), which this tool does not support. Birefringent crystals (sapphire, quartz, calcite) have different indices for different polarizations.
- Single, uncoated, smooth interface. Coatings, thin films, and surface roughness change the reflectance completely. Use the coating vendor's curves for coated optics.
- Window transmission assumes incoherent addition: a thick window, or a beam with short coherence length. Thin plates with coherent light show etalon fringes between 1 − 4R and 1.
References
- E. Hecht, Optics, 5th ed., Pearson (2017), §4.6.
- M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press (1999), §1.5.
- I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965).