Polarizing Beamsplitter (PBS) Design
Design a MacNeille polarizing beamsplitter cube: a quarter-wave high/low-index stack at Brewster's angle between glass prisms, with its extinction against wavelength and angle.
MacNeille cube: (HL)NH quarter-wave stack on the hypotenuse, glass on both sides
H/L interfaces at Brewster: tan θH = nL/nH ⇔ ng² sin²θg = nH²nL²/(nH² + nL²)
layer thickness d = λ₀/(4√(n² − ng²sin²θg)), response by the characteristic-matrix method
Indices are held constant with wavelength (no dispersion) and the layers are lossless. The cube's outer faces are not included. A 45° cube is used at normal incidence on its faces.
Transmitted (p) at λ₀
Tp 98.19 %
s leaking: Ts = 3.28e-5
Reflected (s) at λ₀
Rs 99.997 %
p leaking: Rp = 1.81 %
Extinction ratios
Tp/Ts 44.8 dB
Rs/Rp 17.4 dB
Brewster match at the H/L interfaces
Rp per interface 2.0e-13
needs ng = 1.6829 at this angle
Wanted outputs against wavelength
Leakage against wavelength (log)
Extinction ratios at λ₀ against the angle in air from the cube-face normal (log)
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
(HL)ᴺH quarter-wave stack between glass by the characteristic-matrix method: coherent, lossless, constant indices. Not modelled: the cube's outer faces, dispersion, absorption.
- MacNeille S. M. (1946). Beam splitter. US patent 2,403,731 (Eastman Kodak): quarter-wave layers at Brewster's angle between glass prisms. patents.google.com
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 9 Tilted Coatings (MacNeille polarizer), pp. 403–469. CRC Press. doi:10.1201/9781420073034-c9
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 2 Basic Theory, pp. 13–71. CRC Press. doi:10.1201/9781420073034-c2