Brewster Polarizer Design
Design pile-of-plates polarizers at Brewster's angle: transmittance and degree of polarization of m glass plates, with the multiple reflections between surfaces.
θB = arctan n, rp(θB) = 0, rs(θB) = ((n² − 1)/(n² + 1))²
m plates (2m surfaces, incoherent): T = (1 − r)/(1 + (2m − 1) r) for each polarization
P = (Tp − Ts)/(Tp + Ts) = m/(m + (2n/(n² − 1))²) at θB
Lossless, plane-parallel plates in air, spaced so the reflections add in power (no interference). r is the reflectance of one surface; glass to air reflects the same at the refracted angle.
Brewster angle
56.66°
one surface reflects rs = 15.67 %
Pile transmittance at θB
Tp 100.0 %
Ts 40.22 %
Degree of polarization (unpolarized in)
0.4264
ignoring multiple reflections: 0.5926
Extinction Tp/Ts
3.96 dB
ratio 2.49 : 1
Transmittance and polarization against the angle of incidence (4 plates)
Degree of polarization at θB against the number of plates
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Lossless plane-parallel plates in air, Fresnel reflectance per surface, multiple reflections added in power: T = (1 − r)/(1 + (2m − 1)r). Not modelled: absorption, interference, beam offset.
- Goldstein D. H. (2011). Polarized Light, 3rd ed., ch. Mueller Matrices for Dielectric Plates, pp. 177–199. CRC Press. doi:10.1201/b10436
- Shurcliff W. A. (1962). Polarized Light: Production and Use, ch. 6 Reflection polarizers, pp. 78–86. Harvard University Press. doi:10.4159/harvard.9780674424135.c7
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., §1.5 Reflection and refraction of a plane wave (Fresnel formulae, total reflection). Cambridge University Press. doi:10.1017/CBO9781139644181.010