178 BELL SYSTEM TECHNICAL JOURNAL 



The birefringence in any direction can be calculated from equation (75) ; 

 since (Sn = ■Vi/V, it equals l//ii where /zi is the index of refraction corre- 

 sponding to a light wave with its electric displacement in the (3'n direction. 

 Similarly, for the second solution at right angle to the first, 



^11 = -o - -^ (76) 



1 " M2 



Hence if we designate the expression under the radical by ivo and half the 

 expression on the right outside the radical by Ki , we have 



(77) 



Since mi and fio are very nearly equal even in the most birefringent crystal, 

 we have nearly 



3 



M2 - Ml = ^ = y VkI . (78) 



For special directions in the crystal, the expression for Ko simplifies very 

 considerably. Along the x, y and z axes, the values are 



3 



X, {<p = 0°,d = 90°) ; 5. = ^ V(/?3. - M' + (2,523)^ 

 F, (^ = 90°, e = 90°); By^^ V(/3n - fe)^ + (2^^)' (79) 



3 



Z, {<p = 0°,d ^ 0°); B, ^^ V(/3ii - fe)^ + (2/3i2)2. 



If any natural birefringence exists along these axes, (2/323)- will be very 

 small compared to this and 



^x = - (^33 - /3o + A;i - Ao) = - ( - - -^ + ^, - A, ) 

 /^, = -;^ (/3i - iS3 + Ai - A.) = - ( - - -1 + Ai - A J 



Z Z \lJia fJ-c / 



i?. = V (^1 - /32 + Ai - Ao) = - ( -2 - - + A, - A, ) . 



Z Z \lla IJ-b / 



Hence, for this case, measurements along the three axes will loll the ditTer- 

 ence between the three effects Ai , A^ and An . To get absolute values requires 

 a direct measurement of the index of refraction along one of the axes and 

 its change with fields or stresses. This is a considerably more difficult meas- 



(80) 



