2 BELL SYSTEM TECHNICAL JOURNAL 



parently didn't consider the general case of six simultaneous stress com- 

 ponents, six strain components, three electric field components and three 

 induction components. The fact that the mentioned relation is true only 

 to the first order of small quantities seems to have escaped the attention of 

 some experimenters who have sought to show non-linearity of the piezo- 

 electric effect by demonstrating non-linearity in the converse effect. 



As a basis for light valve problems, we handle the propagation of light 

 through crystals, then the electro optic effect and the piezo optic effect. 



SECTION 2 

 A Linear Vector Function 



For almost every physical constant of an isotropic medium a crystalline 

 medium has several constants. For instance, a piece of glass has a co- 

 efficient of 'thermal expansion but a crystal has many coefficients of thermal 

 expansion, the coefficient depending on direction. It might be thought 

 that there were no necessary relations between the coefficients in different 

 directions but there are necessary relations. 



As an example of the simplifying relations between the values of physical 

 constants in different directions let us consider a crystal in an electric field. 

 Measurements show that the dielectric constant varies with direction in a 

 crystal. If the field is not in the direction of greatest dielectric constant, 

 the displacement current might veer over a little, much as a nail tries to 

 follow the grain of the wood. We shall assume that for any electric field 

 vector E there corresponds an electric induction vector D which may not 

 coincide with E. Also we assume that the magnitude of D is proportional 

 to £, that is, if E results in D, then nE results in nD. Lastly, we assume 

 that if El results in A, -E2 in Do . . . and £,„ results in Dm, then £1 + £2 • • • 

 Em results m. Di -\- Do -\- . . . Dm. If these assumptions hold, then as any 

 arbitrary field E can be expressed as the vector sum of its three components 

 El, E2, Ez along three arbitrary unit vectors ?', /', k, the induction vector 

 resulting from E can be computed from the induction vectors resulting from 

 £1, £2* and £3. For, let £1 result in Dni + A17 + Ai^, £2 result in Dni + 

 D22J + D^ojz and £3 result in D^i -f As./ + Dzzk, then £1 4- £2 + £3 = £ 

 results in the induction vector: 



D - {EiDii + E2D10 + E,Dn)i 



-f (£iAi + £2A2 + £3A3)y (2.1) 



+ (£iAi + £2A2 + £3A3)^ 



It is seen then, that not more than 9 constants are needed to describe 

 the dielectric properties of a crystal. The energy required to establish the 



