Parametric crystal optics of nonmagnetic ferroics

a0ij = FijklHkith
2
P ! ! = ! !E!E! corresponds to the linear electrooptic e ect.
1
1.2. Some tensor relations, symmetric aspects
It is known 3, 4] that optical properties of crystals are described by the material equations taking into account a spatial dispersion 2]: Ei = a0ij Dj = (aij + i ijlkl )Dj ; where ijl = eijk gkl is an antisymmetric polar tensor, gkl is an axial gyration tensor dual to the above one, eijk is a Levi-Civita tensor, kl is a wave vector, aij is a tensor of the optical polarization constants without consideration of spatial dispersion (aij is a reciprocal tensor with respect to "ij tensor). The parametric optical e ects are determined by expanding aij or "ij tensors and ijl or gkl tensors into power series with respect to di erent elds, namely: 1. aij = aij + rijk Ek + Rijkl Ek El +... - electrooptics 5, 6]; (1) 2. gij = gij + ijkEk + ijklEk El +... - electrogyration 2, 7-10]; (2) 3. aij = aij + ijkl kl +... - elastooptics 11]; (3) 4. gij = gij + ijkl kl +... - elastogyration 2]; (4) 5. aij = aij + qijklp kl Ep - elasto-electrooptics; 6. gij = gij + Qijklp kl Ep - elasto-electrogyration. In inhomogeneous elds, gradient parametric optical e ects are possible 12], for example, a torsional-gyration e ect 13]: @ gij = ijklm @xkl : m The restrictions as to the form of polar (aij , rijk , Rijkl, ijkl, qijklp) and axial (gij , ijk, ijkl, ijkl, Qijklp) tensors are due to the point symmetry of crystals: rijk = gij = ijkl = ijkl = qijklp = 0 - in the crystals having inversion symmetry; aij , Rijkl, ijk, ijkl, Qijklp 6= 0 - in acentrically and centrally symmetric crystals. The possibility for appearing the mentioned parametric optical e ects agrees with the following principles of the point symmetry: 1. The Curie principle { a superposition of the elements of the symmetry of crystals and suitable elds.
Received May 6, 1998
Proceeding from the symmetry principles of crystal physics and thermodynamics we analyze parametrical optical phenomena induced by external fields of different kind and structural phase transitions in ferroelectrics and ferroelastics. Special attention is paid to the phenomena of spatial dispersion (electro- and piezogyration). A phenomenological approach to the description of these phenomena is illustrated by the most expressive experimental results obtained for the following crystals: KH 2(1?x) D 2x PO 4 (KDP, DKDP), K 2 H 2 AsO 4 (CDA), RbH 2 PO 4 (RDP), Pb 5 (Ge (1?x) Si x ) 3 O 11, Pb 3 (PO 4 ) 2 , K 2 Cd 2 (SO 4 ) 3 , (NH 3 CH 2 COOH) 3 H 2 SO 4 (TGS), and NaKC 4 H 4 O 6 4H 2 O (RS). Apart from new effects in the crystals of the A 2 BX 4 group with incommensurately modulated structure, in particular, crystals [N(CH 3 ) 4 ] 2 ZnCl 4 , [N(CH 3 ) 4 ] 2 FeCl 4 , K 2 ZnCl 4 , K 2 ZnCl 4 , Co 2+ are considered. Key words: crystal optics, ferroelectrics, optical activity, ferroics, phase transitions PACS: 42.30.Lr, 42.70.a
F = k EE + k E E k Aaf Aaf + : : : + EEE + EE E + EH E + : : :
1 2 0 0 3 1 2 0 3 0
c O.G.Vlokh
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O.G.Vlokh
+ EEEE + EE E E + : : :
1 2 0 0
The formulas determining polarization P, magnetization M and mechanical stress X can be deduced di erentiating the above expression with respect to corresponding variables (E , H , A ). The terms with the third and the forth rank tensors correspond to nonlinear optics. For example, the expression with describes the phenomena of wave generation with di erential (subtractive) and summable frequencies. In particular, the phenomena of optical detection and second harmonic generation are described by the formula:
0 0 0 0 0 0 0 0
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Parametric crystal optics of nonmagnetic ferroics
2. The Neumann principle { obedience of the symmetry of the crystal to the symmetry of the property. The e ects taking place in the magnetic eld are additionally restricted not only by the symmetry of the crystal but also by that of the kinetic coe cients in non-dissipative media 14]: 1. The Onsager principle: aij (H ) = aji(?H ). 2. The principle of Hermitian tensor: a0ij (H ) = a0ji(H ) - for the real part of the tensor, a00ij (H ) = ?a00ji (H ) - for the imaginary part of the tensor. Because of these features the following magneto-optic e ects are possible in the non-dissipative nonmagnetic crystals: 1. Magneto-optic change of optical birefringence (Kotton-Moutton):