Electro-gyration


The electrogyration effect is the spatial dispersion phenomenon, that consists in the change of optical activity 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 compared with the Faraday effect: the optical activity increment associated with the electrogyration effect changes its sign under that operation, contrary to the Faraday effect. Formally, it is a special case of gyroelectromagnetism obtained when the magnetic permeability tensor is diagonal.
The electrogyration effect linear in the electric field occurs in crystals of all point groups of symmetry except for the three cubic – m3m, 432 and. 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 electrogyration

The changes in the optical activity sign induced by the external electric field have been observed for the first time in ferroelectric crystals LiH32 by H. Futama and R. Pepinsky in 1961, while switching enantiomorphous ferroelectric domains. The observed phenomenon has been explained as a consequence of specific domain structure, rather than the electrogyration induced by spontaneous polarization.
The first description of electrogyration 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 . Probably, K. Aizu has been the first who defined the electro-gyration effect and introduced the term “electrogyration” itself.
Almost simultaneously with K. Aizu, I.S. Zheludev has suggested tensor description of the electrogyration in 1964 . In this paper the electrogyration has been referred to as “electro-optic activity”.
In 1969, O.G. Vlokh has measured for the first time the electrogyration effect induced by external biasing field in the quartz crystal and determined the coefficient of quadratic electro-gyration effect .

Thus, the electrogyration 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.
Later in 2003, the gyroelectricity has been extended to gyroelectromagnetic media, which account for ferromagnetic semiconductors and engineered metamaterials, for which gyroelectricity and gyromagnetism may occur at the same time.

Description

Electrodynamics relations

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

where is the optical frequency impermeability tensor, the dielectric permittivity tensor, , the mean refractive index, - induction,, polar third rank tensors, the unit antisymmetric Levi-Civit pseudo-tensor, the wave vector, and, the second rank gyration pseudo-tensors. The specific rotation angle of the polarization plane caused by the natural optical activity is defined by the relation:
,

where is the refractive index, the wavelength, , the transformation coefficients between the Cartesian and spherical coordinate systems , and the pseudo-scalar gyration parameter.
The electro-gyration increment of gyration tensor occurred under the action of electric field or/and is written as:
,
where and are third- and fourth-rank axial tensors describing the linear and quadratic electrogyration, respectively. In the absence of linear birefringence, electrogyration increment of the specific rotatory power is given by:
.
The electrogyration effect may be also induced by spontaneous polarization appearing in the course of ferroelectric phase transitions
.

Explanation on the basis of symmetry approach

The electrogyration 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 a result, the gyration tensor possessing a symmetry of second-rank axial tensor - 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 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 electrogyration effect, the symmetry of the action should be considered as that of the dyad product or, what is the same, the symmetry of a polar second-rank tensor . Such a centrosymmetric action cannot lead to lowering of centrosymmetric symmetry of crystal to acentric states. This is the reason why the quadratic electrogyration exists only in the acentric crystals.

Eigenwaves in the presence of electrogyration

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

,
where is the polarization azimuth of the incident light with respect to the principal indicatrix axis, the linear birefringence, the phase retardation,, and. In the case of light propagation along optically isotropic directions, the eigenwave become circularly polarized, with different phase velocities and different signs of circular polarization. Hence the relation may be simplified so as to describe a pure polarization plane rotation:
,
or
,
where - is the sample thickness along the direction of light propagation.
For the directions of light propagation far from the optic axis, the ellipticity is small and so one can neglect the terms proportional to in Eq.. Thus, in order to describe the polarization azimuth at and the gyration tensor, simplified relations
,
or
.
are often used.
According to Eq., when the light propagates along anisotropic directions, the gyration effects manifest themselves as oscillations of the azimuth of polarization ellipse occurring with changing phase retardation .

Experimental results

The electrogyration effect has been revealed for the first time in quartz crystals as an effect quadratic in the external field. Later on, both the linear and quadratic
electrogyrations has been studied in the dielectric 2, Pb5SiO42, Pb5SeO42, Pb5GeO42, alums
etc.) semiconductor, ferroelectric
, as well as the photorefractive materials
. The electro-gyration effect induced by a powerful laser radiation 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 electrogyration effect represents the first of the gradient nonlinear optical phenomena ever revealed.