Electro-gyration
Encyclopedia
The electrogyration 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 electrogyration effect changes its sign under that operation, contrary to the Faraday effect
.
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.
, 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 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 (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 electrogyration in 1964 (the manuscript received on February 21, 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 (the manuscript received on July 7, 1969).
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.
.
, (1)
or
, (2)
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:
, (3)
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:
, (4)
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:
. (5)
The electrogyration effect may be also induced by spontaneous polarization
appearing in the course of ferroelectric phase transitions
. (6)
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
(with the symmetry of polar vector,
) 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.
and the azimuth 
are defined respectively by the relations
, (7)
, (8)
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 (i.e., the optic axes), the eigenwave become circularly polarized (
), 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:
, (9)
or
, (10)
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.(8). Thus, in order to describe the polarization azimuth at 
and the gyration tensor, simplified relations
, (11)
or
. (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
.
electrogyrations has been studied in the dielectric (
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 electrogyration effect represents the first of the gradient nonlinear optical phenomena ever revealed.
Phenomenon
A phenomenon , plural phenomena, is any observable occurrence. Phenomena are often, but not always, understood as 'appearances' or 'experiences'...
, that consists in the change of optical activity (gyration) of crystals by a constant or time-varying electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
. Being a spatial dispersion
Dispersion (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...
effect, the induced optical activity exhibit different behavior under the operation of wave vector reversal, when compare with the Faraday effect
Faraday effect
In physics, the Faraday effect or Faraday rotation is a Magneto-optical phenomenon, that is, an interaction between light and a magnetic field in a medium...
: the optical activity increment associated with the electrogyration effect changes its sign under that operation, contrary to the Faraday effect
Faraday effect
In physics, the Faraday effect or Faraday rotation is a Magneto-optical phenomenon, that is, an interaction between light and a magnetic field in a medium...
.
The electrogyration effect linear in the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
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 fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
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 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 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 (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 electrogyration in 1964 (the manuscript received on February 21, 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 (the manuscript received on July 7, 1969).
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.
.
Electrodynamics relations
The electric field and the electric displacement vectors of electromagnetic wave propagating in gyrotropic crystals may be written respectively as:, (1)or
, (2)where
is the optical frequency impermeability tensorTensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
,
the dielectric permittivity tensorTensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
,
, the mean refractive index, - induction, , polar third rank tensors, the unit antisymmetric Levi-Civit pseudo-tensor, the wave vectorWave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...
, 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:, (3)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
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
occurred under the action of electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
or/and is written as:, (4)where
and are third- and fourth-rank axial tensors describing the linear and quadratic electrogyration, respectively. In the absence of linear birefringenceBirefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...
, electrogyration increment of the specific rotatory power is given by:
. (5)The electrogyration effect may be also induced by spontaneous polarization
appearing in the course of ferroelectric phase transitions. (6)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 tensorTensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
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 fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
(with the symmetry of polar vector,
) 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 tensorTensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
(
). 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, (7) , (8) where
is the polarization azimuth of the incident light with respect to the principal indicatrix axis, the linear birefringenceBirefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...
,
the phase retardation, , and . In the case of light propagation along optically isotropic directions (i.e., the optic axes), the eigenwave become circularly polarized (), with different phase velocities and different signs of circular polarizationCircular polarization
In electrodynamics, circular polarization of an electromagnetic wave is a polarization in which the electric field of the passing wave does not change strength but only changes direction in a rotary type manner....
(left and right ones). Hence the relation (8) may be simplified so as to describe a pure polarization plane rotation:
, (9)or
, (10)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.(8). Thus, in order to describe the polarization azimuth at and the gyration tensor, simplified relations, (11)or
. (12)are often used.
According to Eq.(11), when the light propagates along anisotropic directions, the gyration
Gyration
In geometry, a gyration is a type of rotation.The center of a rotational symmetry is a rotation point. A rotation point that does not lie on a mirror is called a gyration point. A rotocenter is a rotation point with an integral number of rotational symmetries.-See also:*Terms starting with...
(or the electro-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 [2] as an effect quadratic in the external field. Later on, both the linear and quadraticelectrogyrations has been studied in the dielectric (
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 electrogyration effect represents the first of the gradient nonlinear optical phenomena ever revealed.