Electro-gyration

The electro-gyration effect is the spatial dispersion phenomenon, that consists in the change of optical activity (gyration) of crystals by a constant or time-varying electric field. Being a spatial dispersion effect, the induced optical activity exhibit different behavior under the operation of wave vector reversal, when compare with the Faraday effect: the optical activity increment associated with the electro-gyration effect changes its sign under that operation, contrary to the Faraday effect.

The electro-gyration effect linear in the electric field occurs in crystals of all point groups of symmetry except for the three cubic – m3m, 432 and $\overline{4}3m\,$. The effect proportional to the square of the electric field can exist only in crystals belonging to acentric point groups of symmetry.

The historical background of discovery of electro-gyration

The changes in the optical activity sign induced by the external electric field have been observed for the first time in ferroelectric crystals LiH₃(SeO₄)₂ by H. Futama and R. Pepinsky in 1961 , while switching enantiomorphous ferroelectric domains (the change in the point symmetry group of the crystal being 2/m«m). The observed phenomenon has been explained as a consequence of specific domain structure (a replacement of optic axes occurred under the switching), rather than the electrogyration induced by spontaneous polarization. The first description of electro-gyration effect induced by the biasing field and spontaneous polarization at ferroelectric phase transitions has been proposed by K. Aizu in 1963 on the basis of third-rank axial tensors (the manuscript received on September 9, 1963). Probably, K. Aizu has been the first who defined the electro-gyration effect (”the rate of change of the gyration with the biasing electric field at zero value of the biasing electric field is provisionally referred to as “electrogyration””) and introduced the term “electrogyration” itself. Almost simultaneously with K. Aizu, I.S. Zheludev has suggested tensor description of the electro-gyration in 1964 (the manuscript received on February 21, 1964). In this paper the electro-gyration has been referred to as “electro-optic activity”. In 1969, O.G. Vlokh has measured for the first time the electro-gyration effect induced by external biasing field in the quartz crystal and determined the coefficient of quadratic electro-gyration effect (the manuscript received on July 7, 1969).
Thus, the electro-gyration effect has been predicted simultaneously by Aizu K. and Zheludev I.S. in 1963–1964 and revealed experimentally in quartz crystals by Vlokh O.G. in 1969. .

Description

Electrodynamics relations

The electric field and the electric displacement vectors of electromagnetic wave propagating in gyrotropic crystals may be written respectively as:

$E_{i}=B_{ij}^{0}D_{j}+\tilde\delta_{ijk}\frac{\partial D_{j}}{\partial x_{k}}=B_{ij}^{0}D_{j}+(ie_{ijl}\tilde{g}_{lk}k_k)D_{j}\,$,                (1)

or

$D_{i}=\epsilon_{ij}^{0}E_{j}+\delta_{ijk}\frac{\partial E_{j}}{\partial x_{k}}=\epsilon_{ij}^{0}E_{j}+(ie_{ijl}{g}_{lk}k_k)E_{j}\,$,              (2)
where   B_ij⁰  is the optical frequency impermeability tensor,   ϵ_ij⁰ the dielectric permittivity tensor,  $\tilde{g}_{lk}\overline{n}=g_{kl}$,  $\overline{n}$ the mean refractive index,  D_j  - induction, δ_ijk , δ̃_ijk  polar third rank tensors,  e_ijl  the unit antisymmetric Levi-Civit pseudo-tensor,   k_k  the wave vector, and g_lk ,  g̃_lk the second rank gyration pseudo-tensors. The specific rotation angle of the polarization plane   ρ  caused by the natural optical activity is defined by the relation:
$\rho=\frac{\pi}{\lambda n}g_{lk}l_{l}l_{k}=\frac{\pi}{\lambda n}G\,$,                         (3)
where  n   is  the refractive index,  λ  the wavelength,   l_l , l_k  the transformation coefficients between the Cartesian and spherical coordinate systems  (l₁ = sin Θcos φ , l₂ = sin Θsin φ, l₃ = cos Θ), and  G  the pseudo-scalar gyration parameter.

The electro-gyration increment of gyration tensor occurred under the action of electric field E_m  or/and E_n  is written as:

Δg_lk = γ_lkmE_m + β_lkmnE_mE_n ,                      (4)

where γ_lkm  and β_lkmn  are third- and fourth-rank axial tensors describing the linear and quadratic electro-gyration, respectively. In the absence of linear birefringence, electro-gyration increment of the specific rotatory power is given by:

$\Delta \rho=\frac{\pi}{\lambda n}g_{lk}l_{l}l_{k}=\frac{\pi}{\lambda n}\Delta G=\frac{\pi}{\lambda n}(\gamma _{lkm}E_{m}+\beta _{lkmn}E_{m}E_{n})l_{l}l_{k}$.          (5)

The electro-gyration effect may be also induced by spontaneous polarization P_m^sP_n^s  appearing in the course of ferroelectric phase transitions :

$\Delta \rho=\frac{\pi}{\lambda n}g_{lk}l_{l}l_{k}=\frac{\pi}{\lambda n}\Delta G=\frac{\pi}{\lambda n}(\tilde\gamma _{lkm}P_{m}^{s}+\tilde\beta _{lkmn}P_{m}^{s}P_{n}^{s})l_{l}l_{k}$.          (6)

Explanation on the basis of symmetry approuch

The electro-gyration effect can be easy explained on the basis of Curie and Neumann symmetry principles. In the crystals that exhibit centre of symmetry, natural gyration can not exist, since, due to the Neumann principle, the point symmetry group of the medium should be a subgroup of the symmetry group that describes the phenomena, which are properties of this medium. As aresult, the gyration tensor possessing a symmetry of second-rank axial tensor - ∞2  is not a subgroup of centrosymmetric media and so the natural optical activity cannot exist in such media. According to the Curie symmetry principle, external actions reduce the symmetry group of the medium down to the group defined by intersection of the symmetry groups of the action and the medium. When the electric field (with the symmetry of polar vector, ∞mm ) influences the crystal which possess the inversion centre, the symmetry group of the crystal should be lowered to the acentric one, thus permitting the appearance of gyration. However, in case of the quadratic electro-gyration effect, the symmetry of the action should be considered as that of the dyad product E_mE_n  or, what is the same, the symmetry of a polar second-rank tensor (∞/mmm ). Such a centrosymmetric action cannot lead to lowering of centrosymmetric symmetry of crystal to acentric states. This is the reason why the quadratic electro-gyration exists only in the acentric crystals.

Eigenwaves in the presence of electro-gyration

In a general case of light propagation along optically anisotropic directions, the eigenwaves become elliptically polarized in the presence of electro-gyration effect, including rotation of the azimuth of polarization ellipse. Then the corresponding ellipticity κ  and the azimuth χ  are defined respectivelly by the relations

$\kappa =\frac{\Delta G}{2\Delta n\overline{n}}\,$,                                (7)
 $\tan 2(\alpha -\chi )=\frac{2\kappa }{1+\kappa ^2}\tan \boldsymbol{\Gamma } \left ( 1+\frac{P\tan 2\alpha +(1-R)}{R+\tan ^22\alpha } \right )\,$ ,            (8)     

where α  is the polarization azimuth of the incident light with respect to the principal indicatrix axis, Δn  the linear birefringence, Γ  the phase retardation, $P=\frac{(1-\kappa ^2)^2}{2\kappa (1+\kappa ^2)}\,$ , and $R=\left (\frac{2\kappa }{1+\kappa ^2}\right )^2+\left (\frac{1-\kappa ^2}{1+\kappa ^2}\right )^2\,$. In the case of light propagation along optically isotropic directions (i.e., the optic axes), the eigenwave become circularly polarized (κ = 1 ), with different phase velocities and different signs of circular polarization (left and right ones). Hence the relation (8) may be simplified so as to describe a pure polarization plane rotation:

2(α−χ) = Γ ,                               (9)

or

$\rho d=\alpha -\frac{\boldsymbol\Gamma}{2}\,$,                            (10)

where d  - is the sample thickness along the direction of light propagation. For the directions of light propagation far from the optic axis, the ellipticity κ  is small and so one can neglect the terms proportional to κ²  in Eq.(8). Thus, in order to describe the polarization azimuth at α = 0  and the gyration tensor, simplified relations

tan 2χ =  − 2κsin Γ ,                           (11)

or

$g_{kl}=2\chi \Delta n\overline{n}\,$.                             (12)

are often used. According to Eq.(11), when the light propagates along anisotropic directions, the gyration (or the electro-gyration) effects manifest themselves as oscillations of the azimuth of polarization ellipse occurring with changing phase retardation Γ  .

Experimental results

The electro-gyration effect has been revealed for the first time in quartz crystals [2] as an effect quadratic in the external field. Later on, both the linear and quadratic electro-gyrations has been studied in the dielectric (α− HIO₃ , LiIO₃ , PbMoO₄ , NaBi(MoO₄)₂, Pb₅SiO₄(VO₄)₂, Pb₅SeO₄(VO₄)₂, Pb₅GeO₄(VO₄)₂ , alums etc.) semiconductor (AgGaS₂, CdGa₂S₄) , ferroelectric (TGS, Rochelle Salt, Pb₅Ge₃O₁₁ and KDP families etc.) , as well as the photorefractive (Bi₁₂SiO₂₀, Bi₁₂GeO₂₀, Bi₁₂TiO₂₀) materials . The electro-gyration effect induced by a powerful laser radiation (a so called self-induced or dynamic electro-gyration) has been studied in the works . The influence of electro-gyration on the photorefraction storage has been investigated in , too. From the viewpoint of nonlinear electrodynamics, the existence of gradient of the electric field of optical wave in the range of the unit cell corresponds to macroscopic gradient of the external electrical field, if only the frequency transposition is taken into account. In that sense, the electro-gyration effect represents the first of the gradient nonlinear optical phenomena ever revealed.

See also

Piezo-gyration
Faraday effect

ru:Электрогирация