Polarization, Phase Shift & Transmission

A birefringent medium has two axes with different refractive indices: the ordinary axis (nₒ) and extraordinary axis (nₑ). An input p-polarized beam at angle ψ to these axes has projections onto both. The two components travel at different speeds, accumulating phase difference Δφ = (2π/λ)(nₑ − nₒ)d, which changes the polarization state. A subsequent Brewster surface transmits only the p-component — the transmission is: T = 1 − sin²(2ψ) · sin²(Δφ/2)

Phase shift Δφ 0°

Input polarization angle ψ 45°

ψ = 0° → aligned with birefringence axis, no loss | ψ = 45° → maximum depolarization loss

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Polarization ellipse (nₒ vs nₑ axes)

E along nₒ (blue) & nₑ (orange) vs time

Transmission through polarizer vs Δφ

E along nₒ (ordinary axis)

E along nₑ (extraordinary axis)

Transmission curve

Current operating point

p-axis direction

Δφ	Polarization state	Transmission (ψ=45°)	Waveplate
0°	Linear (45°)	100%	—
90°	Circular (right-hand)	50%	Quarter-wave (λ/4)
180°	Linear (−45°) rotated 90°	0%	Half-wave (λ/2)
270°	Circular (left-hand)	50%	Three-quarter-wave
360°	Linear (45°) — restored	100%	Full-wave