174 BELL SYSTEM TECHNICAL JOURNAL 



Since ij and no are reversible, it has been customary to abbreviate the tensor 

 by writing one number in place of the two in the following form: 



11 = 1; 22 = 2; 33 = 3; 12 = 21 = 6; 13 = 31 = 5; 23 = 32 = 4 (52) 



Since the reduced tensor is associated with the engineering strains, it is 

 necessary to investigate the numerical relationships between the four in- 

 dex symbols and the two index symbols. From equation (48), when m 

 7^ n, the change in the impermeability constant i8,„„ is given by 



Since Sr = 2Sij = 2Sji we have the relation that 



tnijmn = mrs(i,j, fti, u = I to 3, f, s, = I to 6) (54) 



In equation (45) we cannot in general interchange the order of ij and no 

 since U does not contain product terms of strains and electric displace- 

 ments and hence in general 



Mrs 7^ nisr. (55) 



Hence in the most general case there are 36 photoelastic constants. Crystal 

 symmetries cut down the number of constants as shown in a later section. 

 The tensor r„,„o defined in equation (45) as 



pfi jj 

 (47r)V^„<, = -■ ,■ ,■ (56) 



OOmOOn OOo 



shows that we can interchange the order of m and n since U contains product 

 terms of bm and 6„ . Hence 



'mno ' nmo I,*-' ' / 



and this is usually replaced by the two index symbols 



rqo = r,nno(ni, n, = I to 3; q = I to 6). 



The so called "true" electro-optic constants are measured at constant 

 strain and for this case the modifications in the impermeability constants 

 are given by the equation 



Em = DnWmn + fmnoDo]. (58) 



Since m and // are interchangeable, the third rank tensor is usually replaced 

 by the two index symbols 



rtino == rqoini, n, o = t to 3; q = \ to 6). (59) 



As discussed in the next sections, these constants can be determined by 

 applying an electric field of a frequency high enough so that the principal 

 resonances and their harmonics cannot be excited by the applied field, and 

 measuring the resulting birefringence along definite directions in the crystal. 

 On the other hand if we apply a static field to the crystal, an additional effect 

 occurs because the crystal is strained by the piezoelectric effect and this 

 causes a photoelastic effect in addition to the "true" electro-optic effect. A 



