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Intensity ratio of e and o rays

Short answer

Intensity ratio I_e/I_o = cot²θ. With θ = 30°, cot²30° = 3, so I_e : I_o = 3 : 1.

If the plane of vibration of the incident beam makes an angle of 30° with the optic axis, compare the intensities of extraordinary and ordinary light.

When plane polarized light (amplitude A) falls on a doubly refracting crystal with its plane of vibration at angle θ to the optic axis, it splits into:

  • e-ray with vibration along the optic axis: amplitude A cos θ
  • o-ray with vibration perpendicular to the axis: amplitude A sin θ
        optic axis (e)
         |   A
         |  /
         | /θ
         |/_______ o

Intensity is proportional to the square of the amplitude:

I_e / I_o = (A cos θ)² / (A sin θ)² = cot² θ

Given: θ = 30°.

I_e / I_o = cot² 30° = (√3)² = 3

Answer: I_e : I_o = 3 : 1. The extraordinary ray is three times as intense as the ordinary ray. (Defining θ as the angle to the axis, the e-component is the larger one.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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