Light travelling along the optic axis of an optically active uniaxial crystal (such…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Polarisation and anisotropic media · Polarisation and anisotropic media

Optics and Photonics, Ch Birefringence and crystal optics

Problem

Light travelling along the optic axis of an optically active uniaxial crystal (such as quartz) does not split into ordinary and extraordinary waves; instead the two states that travel through the crystal unchanged in form are the left- and right-circular states, with indices

n_L

and

n_R

respectively. A linearly polarised wave at azimuth

\theta

to some fixed reference direction

x

(Jones vector

(\cos\theta, \sin\theta)

) enters a slab of such a crystal of thickness

d

, at vacuum wavelength

\lambda

. The crystal is lossless, the incidence is normal, and

n_L

n_R

are each taken as fixed real numbers at this

\lambda

. ((a)) Using the identity

(\cos\theta,\sin\theta) = \dfrac{1}{\sqrt2}\left[e^{i\theta}\,\dfrac{1}{\sqrt2}(1,-i) + e^{-i\theta}\,\dfrac{1}{\sqrt2}(1,i)\right]

, where

\tfrac1{\sqrt2}(1,-i)

and

\tfrac1{\sqrt2}(1,i)

are the right- and left-circular Jones vectors (CV3), show that on leaving the slab the light is again exactly linearly polarised, whatever the value of

\theta

. ((b)) Show that the azimuth of the emerging state differs from

\theta

by an amount independent of

\theta

, of magnitude

\theta_{\mathrm{rot}} = \pi(n_L-n_R)d/\lambda

. ((c)) Quartz along its axis has

n_L - n_R = 7\times10^{-5}

at 589 nm. Use your formula to find

\theta_{\mathrm{rot}}

for

d = 1.00

mm and compare it with the quoted rotation of quartz, 21.7 degrees per millimetre. ((d)) In one sentence, say why the result of part (b), that every input azimuth is turned by the same fixed angle, is exactly what makes the two-measurement test of Section 9.7 work for optical activity, in a way it does not for a birefringent plate.

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