OPTICAL PROPERTIES IN CRYSTALS 173 



effects, however, have to be measured at the same frequency before equahty 

 exists. 



To obtain the changes in the optical properties caused by the strain and 

 the electric displacement we have to determine the fields and displacements 

 occurring at the high frequencies of optics. Even for piezoelectric vibrations 

 occurring at as high frequencies as they can be driven by the piezoelectric 

 effect, these frequencies are small compared to the optic frequencies / and 

 can be considered to be static displacements or strains. Hence, writing 



Em = Eli- Eme^-", Dn = dI + D^e^"', 



w 



here co = lirf, the first of equation (47) can be written in the form 



Jlim — Umij >^ ij 1" J-^ n 



Ptmi r Wijmn •J ij I n 



jat 





no x^O I 



(48) 



Tf we develop one of the fields, say £i , this can be written in the form 



E.e''^' = [^11 + mar^Sii + rnM + rn2Dl + r.uDWD.e'-" 



+ [^12 + MimSij + n.M + ruoDl + ri2zDl]D2e'''' (49) 



+ [/3i3 + MimSij + r^^lDl + rn2Dl + ruzDl]D^e'''' 



where the first number of r refers to the field, the second to the optical value 

 of D and the third to the static value of D. Hence, for the general case, 



-Em^^"' = Dne^'^^[(3mn + mijmnSij + rmnoE>^. (50) 



From the definition of the two tensors niijno and r,„„o given by equation 

 (45), we can show that there are relations between the various components 

 of the tensors. For the first tensor niijno , since Sij = Sji is a symmetrical 

 tensor, then 



niijno ^ nijino \p^) 



r) / ri^ TI \ 



dSij \d8n d8o/ 



it is obvious that we can interchange the order of 5„ and 80 so that 



niiino ^^ niijon 



