170 BELL SYSTEM TECHNICAL JOURNAL 



III. Location of Optic Axes in a Crystal 



When the expression in the radical of equation (28) vanishes the two 

 velocities are equal and an optic axis exists. Since the expression inside the 

 radical can be written 



Ua' - h')(co^' dcos^ ^ + sin^so) - (b'- - c'-)sm^ 6]^ 



(37) 

 — 4(a- — b-) (c- — b-) sin- ^sin- tp = () 



then, since the square is always positive and since (a- — b-) > and 

 (b^ — c^) > 0, the equation can vanish only if ^ = 0. But ^ = indicates 

 that the two optic axes always lie in a plane perpendicular to the inter- 

 mediate velocity b. With (p = Q then the square vanishes when 



If (a- — b'-} < ib- — c~) the value of the tan 6 is less than unity and the 

 crystal is called a positive crystal. For this case the two axes approach more 

 closely the Z axis having the velocity c than they do the X axis. If 

 (a^ — b^) > (b- — c~) the crystal is negative. 



li a — b or b = c the crystal has a single optic axis and is respectively a 

 positive or negative uniaxial crystal. For the first case the two velocities 

 are given by 



vi = a = b, V2 = Va- cos^ 6 -}- c- sin^ 6. (39) 



The first velocity is that of the ordinary ray while that of the second is that 

 of the extraordinary ray. Since a > c, the ordinary ray will have a velocity 

 greater than the extraordinary ray except along the optic axis where they 

 are equal. Since c < a, the maximum axis for any ellipse, formed by inter- 

 secting the Fresnel ellipsoid at an angle to the optic axis, will lie in the plane 

 formed by the normal and the c axis and hence the direction of polarization 

 of the extraordinary ray will lie in the c, n plane. The polarization of the 

 ordinary ray will be perpendicular to this plane. 



\i h = c the a axis is the optic axis and the velocities of the two rays are 

 again 



vi = c and V2 = a (1-sin" 0cos" <p) + c"'(sin" ^cos' <p) (40) 



Hence, when d— 90°, (p = 0°, the two velocities are equal and a is the optic 

 axis. In this case the velocity of the extraordinary ray is greater than that 

 of the ordinary ray except along the a axis, and the crystal is a negative 

 uniaxial crystal. The polarization of the extraordinary ray lies again in the 



