Past exam of the optics and electrodynamics course of the University of Cambridge / 2022 / Section B / 4 / a

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Solution

Birefringence means that two independent polarization eigenmodes have different refractive indices at the same frequency and propagation direction. The rotatory-power formula relevant here requires circular birefringence: the eigenmodes are opposite circular polarizations. General linear birefringence instead introduces phase retardation between linearly polarized components and need not rotate every incident linear polarization through one common angle.

Use the complex amplitude convention

Re [E e^{- iω t}]

and transverse unit vectors

e_{1}, e_{2}

u_{\pm} = \frac{e _{1} \mp i e _{2}}{2}, e_{1} = \frac{u _{+} + u _{-}}{2} .

Propagation through thickness

d

adds phases

ϕ_{\pm} = (2 π / λ_{0}) n_{\pm} d

, where

λ_{0}

is the vacuum wavelength. Recombining the two components gives

\frac{e ^{i ϕ_{+}} u _{+} + e ^{i ϕ_{-}} u _{-}}{2} = e^{i \overset{ˉ}{ϕ}} (e_{1} cos θ + e_{2} sin θ), θ = \frac{ϕ _{+} - ϕ _{-}}{2} .

Thus the rotation and the specific rotatory power per unit length are

θ = \frac{π d}{λ _{0}} (n_{+} - n_{-}), ρ \equiv \frac{θ}{d} = \frac{π}{λ _{0}} (n_{+} - n_{-}) = \frac{ω}{2 c} (n_{+} - n_{-}) .

The units of

ρ

are

rad m^{- 1}

. The sign follows the specified circular basis and viewing convention. Equal attenuation of the two eigenmodes has been assumed; unequal attenuation also produces elliptical polarization.

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