A biaxial crystal has index ellipsoid in its principal axes

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

What an Instrument Can See, Ch Anisotropic media: the dielectric tensor and the index ellipsoid

Problem

A biaxial crystal has index ellipsoid

\dfrac{x^2}{n_x^2}+\dfrac{y^2}{n_y^2}+\dfrac{z^2}{n_z^2}=1

in its principal axes, with

n_x < n_y < n_z

. A direction

\hat{\mathbf k}

is an optic axis when the plane through the origin perpendicular to

\hat{\mathbf k}

cuts the ellipsoid in a circle, because then the two eigenmode indices for that direction coincide and the crystal is momentarily isotropic for waves travelling that way. You may use without proof one property of a triaxial ellipsoid: every circular central section contains its intermediate axis, here the

y

-axis. ((a)) Consider only planes that contain the

y

-axis, with unit normal

\hat{\mathbf k}=(\sin\theta,0,\cos\theta)

lying in the

xz

-plane. Show that the curve cut by such a plane through the ellipsoid is an ellipse with one semi-axis fixed at

n_y

, and find the length of the other semi-axis as a function of

\theta

. ((b)) Hence derive the value(s) of

\theta

for which that plane curve becomes a circle, and, using the property given above, show that a biaxial crystal therefore has exactly two optic axes, lying in the

xz

-plane and symmetric about the

z

-axis. ((c)) Show that your result gives a single optic axis along

z

in the limit

n_x\to n_y

, and a single optic axis along

x

in the limit

n_y\to n_z

, recovering the uniaxial crystal in each case. ((d)) State in one sentence what happens to a narrow beam launched exactly along one of the two optic axes of a biaxial crystal.

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