Microscope Polarizer Calculator
Extinction of a polarizing microscope: polarizer extinction ratio, the loss of extinction with NA at uncoated surfaces, and the smallest retardance you can see.
two sheets, k2/k1 = 1/ER: H(θ) = H0 cos²θ + H90 sin²θ, H0 = (k1² + k2²)/2, H90 = k1k2
an oblique ray through m surfaces keeps G and leaks L of its power into the crossed analyzer (Fresnel Tp, Ts); L ∝ NA⁴ at small NA, steeper above
smallest visible retardance: sin²(Γ/2) = 1/ERsys
m counts the uncoated surfaces the cone crosses; a mounted slide in air has 2 (slide underside, cover-glass top). The illumination and the collected cone are limited by the smaller NA, so NAeff = 0.50. Polarizer extinction ER = 10000.
System extinction ratio
4.03e+3
36.1 dB
Polarizers alone
5.00e+3
(ER + 1/ER)/2
Surfaces alone
2.09e+4
ideal polarizers, G/L
Smallest visible retardance
2.74 nm
at 546 nm, specimen at 45°; signal = background
Crossed transmission
1.14e-4
of the unpolarized source; lossless sheets (k₁ = 1)
Parallel transmission
0.458
of the unpolarized source; lossless sheets (k₁ = 1)
Transmission at 0° from crossed
1.14e-4
Transmission against analyzer angle
Relative to the unpolarized source; 0° is crossed and ±90° is parallel. The floor at 0° is the system leak.
System extinction ratio against NA
The leak from oblique rays grows as NA⁴ at small NA and faster above: once it exceeds the polarizers' own leak, the surfaces set the extinction ratio.
Uncoated flat surfaces turn the polarization of oblique rays (Inoué & Hyde 1957); lens surfaces are not modelled.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Sheet pair H(θ) = H₀cos²θ + H₉₀sin²θ plus the polarization rotation at uncoated flat air–glass surfaces, averaged over a filled pupil. Absolute values are illustrative: real lens surfaces and coatings dominate.
- Shurcliff W. A. (1962). Polarized Light: Production and Use, ch. 3 Polarizers: classes and performance parameters, pp. 32–42 (principal transmittances k₁, k₂). Harvard University Press. doi:10.4159/harvard.9780674424135.c4
- Inoué S., Hyde W. L. (1957). Studies on depolarization of light at microscope lens surfaces. J. Biophys. Biochem. Cytol. 3(6), 831–838: rotation of polarization by oblique rays at uncoated surfaces. doi:10.1083/jcb.3.6.831
- 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