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Quarter wave plate thickness numerical

Short answer

For a quarter-wave plate, t = λ/[4(μ_e−μ_o)]. With λ=5893 Å, μ_e−μ_o=0.009, t = 1.64×10⁻⁵ m (16.4 µm).

Calculate the thickness of quarter wave plate for light of wavelength 5893 Å. Given refractive indices of ordinary and extraordinary ray 1.544 and 1.553 respectively.

A quarter-wave plate is a doubly refracting crystal plate (such as quartz or calcite) cut with its optic axis parallel to its faces. It introduces a path difference of λ/4 (a phase difference of π/2) between the ordinary and extraordinary rays.

Condition: (μ_e − μ_o) t = λ/4 (for a positive crystal such as quartz, μ_e > μ_o), so

t = λ / [4 (μ_e − μ_o)]

Given: λ = 5893 Å = 5.893 × 10⁻⁷ m, μ_o = 1.544, μ_e = 1.553.

μ_e − μ_o = 1.553 − 1.544 = 0.009

t = 5.893×10⁻⁷ / (4 × 0.009)
  = 5.893×10⁻⁷ / 0.036
  = 1.637×10⁻⁵ m

Answer: t ≈ 1.64 × 10⁻⁵ m = 16.4 µm (1.64 × 10⁻³ cm).

This is the minimum thickness; any odd multiple, (2m + 1)λ / 4(μ_e − μ_o), also works.

✓ Answer checked by Claude Opus 5.5 (high effort)

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