OPTICAL PROPERTIES IN CRYSTALS 171 



plane of the normal and the optic axis while the ordinary ray is perpendicu- 

 lar to it. 



IV. Derivation or the Electro-optic and Photoelastic Effects 



In a previous paper and in the book "Piezoelectric Crystals and Their 

 Application to Ultrasonics", D. Van Nostrand, 1950, it was shown that the 

 electro-optic and photoelastic effects can be expressed as third derivatives 

 of one of the thermodynamic potentials. Probably the most fundamental 

 way of developing these properties is to express them in terms of the strains, 

 electric displacements and the entropy. For viscoelastic substances it has 

 been shown that the photoelastic effects are directly related to the strains. 

 In terms of the electric displacements, the electro-optic constants do not 

 vary much with temperature whereas, if they are expressed in terms of the 

 fields, the constants of a ferroelectric type of crystal such as KDP increase 

 many fold near the Curie temperature. The entropy is chosen as the funda- 

 mental heat variable, since most measurements are carried out so rapidly 

 that the entropy does not vary. 



The thermodynamic potential which has the strains, electric displace- 

 ments and entropy as the independent variables is the internal energy U, 

 given by 



dU = Tij dSij i- Em^ + & da (41) 



where Sij are the strains, Tij the stresses, £„, the fields. Dm the electric dis- 

 placements, the temperature and a the entropy. In this equation the 

 strains Sa are defined in the tensor form 



1 /dUi dtij\ 



2 \dXj dXiJ 



2 \dXj dXiJ 



where the m's are the displacements along the three axis. In the case of a 

 shearing strain occurring when i ^ j, the strain is only half that usually 

 used in engineering practice. In order to avoid writing the factor l/47r, we 

 use the variable 5„,= Dm/^T. Then, from (41), 



Since, for most conditions of interest, adiabatic conditions prevail, we can 

 set dcr equal to zero and can develop the dependent variables, the fields and 



■* "First and Second Order Equations for Piezoelectric Crystals Expressed in Tensor 

 Form," W. P. Mason, B.S.T.J., Vol. 26, pp. 80-138, Jan., 1947. 



