Electro gyration

Electro - gyration - the effect of spatial dispersion, consisting in the appearance or change of optical activity ( gyration ) in crystals under the influence of a constant or alternating electric field .

As a phenomenon of spatial dispersion, electro-gyration differs from the Faraday effect in the behavior of the increase in optical activity when the sign of the wave vector is changed, that is, when the electro-gyration effect, the increase in optical activity changes sign when the sign of the wave vector changes, but not in the Faraday effect.

The electro- gyration effect proportional to the electric field strength ( linear electro-gyration ) is allowed in crystals that belong to all point symmetry groups , with the exception of three cubic groups - m3m, 432 і ${\overline {4}}3m$ ${\ displaystyle {\ overline {4}} 3m}$ $\ overline {4} 3m$ and the effect, proportional to the square of the electric field strength ( quadratic electrogyration ), is allowed by symmetry only in acentric crystals.

Historical background

A change in the sign of optical activity induced by an electric field was first observed in LiH3 (SeO4) 2 ferroelectric crystals by G. Futamoy and R. Pepinsky in 1961 ^[1] during the repolarization of ferroelectric domains (a change in the point symmetry group during the 2 / m - m phase transition ) The observed phenomenon was explained by the peculiarity of the domain structure (interchange of optical axes during the repolarization of the domain structure), and not by the electrogyration induced by spontaneous polarization. The first description of the electro-gyration effect induced by an electric field and spontaneous polarization during ferroelectric phase transitions was apparently proposed by K. Aizu in 1963 ^[2] (the article was received on September 9, 1963). Probably K. Aizu was the first to define the electro-gyration effect as: "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 " . " The term "electrogyration" was also first proposed by K. Aizu. Simultaneously with K. Aizu, I. S. Zheludev proposed a description of electrogirading in 1964 based on the symmetry approach and tensor relations ^[3] (the article was received on February 21, 1964). In this article, electro-gyration was called electro-optical activity. In 1969, O. G. Vloch first experimentally discovered the electro-gyration effect induced by an electric field in quartz crystals and determined the coefficients by quadratic electro-gyration. ^[4] (the article was received on July 7, 1969).
Thus, the electro-gyration effect was foreseen and described at the same time by the Japanese scientist K. Aizu and the Russian scientist I. S. Zheludev in 1963-1964. and was first experimentally discovered by Ukrainian scientist O. G. Vloch in 1969 ^[4] ^[5] ^[6] ^[7] .

Description of the phenomenon

Electrodynamic description

The vector of electric field strength (or induction ) of an electromagnetic wave propagating in a gyrotropic crystal can be represented as:

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

or

$D_{i}=\epsilon _{ij}^{0}E_{j}+\delta _{ijk}{\frac {\partial E_{j}}{\partial x_{k}}}=\epsilon _{ij}^{0}E_{j}+(ie_{ijl}{g}_{lk}k_{k})E_{j}$ ${\ displaystyle D_ {i} = \ epsilon _ {ij} ^ {0} E_ {j} + \ delta _ {ijk} {\ frac {\ partial E_ {j}} {\ partial x_ {k}}} = \ epsilon _ {ij} ^ {0} E_ {j} + (ie_ {ijl} {g} _ {lk} k_ {k}) E_ {j}}$ , (2)

Where $B_{ij}^{0}$ ${\ displaystyle B_ {ij} ^ {0}}$ - tensor of optical polarization constants, $\epsilon _{ij}^{0}$ ${\ displaystyle \ epsilon _ {ij} ^ {0}}$ - dielectric constant tensor , ${\tilde {g}}_{lk}{\overline {n}}=g_{kl}$ ${\ displaystyle {\ tilde {g}} _ {lk} {\ overline {n}} = g_ {kl}}$ , ${\overline {n}}$ ${\ displaystyle {\ overline {n}}}$ - the average value of the refractive indices , $D_{j}$ ${\ displaystyle D_ {j}}$ - induction $\delta _{ijk}$ ${\ displaystyle \ delta _ {ijk}}$ , ${\tilde {\delta }}_{ijk}$ ${\ displaystyle {\ tilde {\ delta}} _ {ijk}}$ - polar tensor of the third rank, $e_{ijl}$ ${\ displaystyle e_ {ijl}}$ - completely antisymmetric, single Levi-Civita pseudo-tensor, $k_{k}$ ${\ displaystyle k_ {k}}$ - wave vector $g_{lk}$ ${\ displaystyle g_ {lk}}$ and ${\tilde {g}}_{lk}$ ${\ displaystyle {\ tilde {g}} _ {lk}}$ - axial tensors of the second rank (gyration tensors). Specific angle of rotation of the plane of polarization $\rho$ ${\ displaystyle \ rho}$ associated with natural optical activity is determined by the ratio:

$\rho ={\frac {\pi }{\lambda n}}g_{lk}l_{l}l_{k}={\frac {\pi }{\lambda n}}G$ ${\ displaystyle \ rho = {\ frac {\ pi} {\ lambda n}} g_ {lk} l_ {l} l_ {k} = {\ frac {\ pi} {\ lambda n}} G}$ , (3)

de $n$ ${\ displaystyle n}$ - refractive index , $\lambda$ ${\ displaystyle \ lambda}$ - wavelength of optical radiation, $l_{l}$ ${\ displaystyle l_ {l}}$ and $l_{k}$ ${\ displaystyle l_ {k}}$ - transformation relations between Cartesian and spherical coordinate systems ( $l_{1}=\sin \Theta \cos \varphi$ ${\ displaystyle l_ {1} = \ sin \ Theta \ cos \ varphi}$ , $l_{2}=\sin \Theta \sin \varphi ,l_{3}=\cos \Theta$ ${\ displaystyle l_ {2} = \ sin \ Theta \ sin \ varphi, l_ {3} = \ cos \ Theta}$ ), $G$ ${\ displaystyle G}$ - pseudoscalar gyration parameter. Electro-gyration increment of the gyration tensor under the influence of an electric field $E_{m}$ ${\ displaystyle E_ {m}}$ and / or $E_{n}$ ${\ displaystyle E_ {n}}$ can be represented as:

$\Delta g_{lk}=\gamma _{lkm}E_{m}+\beta _{lkmn}E_{m}E_{n}$ ${\ displaystyle \ Delta g_ {lk} = \ gamma _ {lkm} E_ {m} + \ beta _ {lkmn} E_ {m} E_ {n}}$ , (four)

Where $\gamma _{lkm}$ ${\ displaystyle \ gamma _ {lkm}}$ and $\beta _{lkmn}$ ${\ displaystyle \ beta _ {lkmn}}$ axial tensors of the third and fourth ranks, describing linear and quadratic electrogyration, respectively. In the absence of linear birefringence, the electro-gyration increment of the specific rotation of the plane of polarization of light is written as:

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

The electro-gyration effect can be induced by spontaneous polarization during ferroelectric phase transitions ^[8] :

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

The enantiomorphism of ferroelectric domains is manifested precisely due to the electro-gyration effect induced by spontaneous polarization.

Symmetry Description

The electro-gyration effect can be quite simply explained on the basis of the symmetric approach, that is, on the basis of the symmetry principles of Curie and Neumann. In crystals with a center of symmetry, optical activity ( gyration ) is forbidden, since, according to the Neumann principle, the point group of medium symmetry must be a subgroup of the point group of the effect, which is a property of this medium. Since the gyration tensor , which possesses the symmetry of the axial tensor of the second rank, $\infty 2$ ${\ displaystyle \ infty 2}$ , does not represent a subgroup of the symmetry group of a centrosymmetric medium - natural optical activity cannot exist in such a medium. According to the Curie symmetry principle, under the influence of external influence on the medium, the symmetry of the medium decreases to the symmetry group, which is the intersection of the sets of symmetry groups of the action and the medium. Thus, the influence of an electric field with the symmetry of the polar vector (the symmetry group is $\infty mm$ ${\ displaystyle \ infty mm}$ ) on a crystal with a center of symmetry leads to a decrease in the symmetry of the crystal to an acentric symmetry group, which allows the appearance of optical activity. However, with a quadratic electro-gyration effect, the symmetry of the action should be considered as the symmetry of the dyadic product of two polar vectors of electric field strength $E_{m}E_{n}$ ${\ displaystyle E_ {m} E_ {n}}$ , i.e., as the symmetry of the polar tensor of the second rank (symmetry group - $\infty /mmm$ ${\ displaystyle \ infty / mmm}$ ) Such a centrosymmetric effect is not able to lower the symmetry of the medium to the acentric group. It is this fact that is the reason that quadratic electrogyration can exist only in acentric media.

Natural waves during electro-

In the general case, when light propagates in optically anisotropic directions, in the presence of electro-gyration, the natural waves of the medium become elliptically polarized with the rotation of the azimuth of the axis of polarization ellipse. Ellipticity and azimuth are determined by the relations:
$\kappa ={\frac {\Delta G}{2\Delta n{\overline {n}}}}$ ${\ displaystyle \ kappa = {\ frac {\ Delta G} {2 \ Delta n {\ overline {n}}}}}$ , (7)
$\tan 2(\alpha -\chi )={\frac {2\kappa }{1+\kappa ^{2}}}\tan {\boldsymbol {\Gamma }}\left(1+{\frac {P\tan 2\alpha +(1-R)}{R+\tan ^{2}2\alpha }}\right)$ ${\ displaystyle \ tan 2 (\ alpha - \ chi) = {\ frac {2 \ kappa} {1+ \ kappa ^ {2}}} \ tan {\ boldsymbol {\ Gamma}} \ left (1 + {\ frac {P \ tan 2 \ alpha + (1-R)} {R + \ tan ^ {2} 2 \ alpha}} \ right)}$ , (eight)
respectively, where $\alpha$ ${\ displaystyle \ alpha}$ - orientation of the azimuth of linearly polarized light entering the medium relative to the axes of the optical indicatrix, $\Delta n$ ${\ displaystyle \ Delta n}$ - linear birefringence, ${\boldsymbol {\Gamma }}$ ${\ displaystyle {\ boldsymbol {\ Gamma}}}$ - phase difference $P={\frac {(1-\kappa ^{2})^{2}}{2\kappa (1+\kappa ^{2})}}$ ${\ displaystyle P = {\ frac {(1- \ kappa ^ {2}) ^ {2}} {2 \ kappa (1+ \ kappa ^ {2})}}}$ , $R=\left({\frac {2\kappa }{1+\kappa ^{2}}}\right)^{2}+\left({\frac {1-\kappa ^{2}}{1+\kappa ^{2}}}\right)^{2}$ ${\ displaystyle R = \ left ({\ frac {2 \ kappa} {1+ \ kappa ^ {2}}} \ right) ^ {2} + \ left ({\ frac {1- \ kappa ^ {2} } {1+ \ kappa ^ {2}}} \ right) ^ {2}}$ . In the case of light propagation in an optically isotropic direction, the eigenwaves become circularly polarized with different phase velocities and different signs of circular polarization (right and left). Then relation (8) can be simplified to describe the rotation of the plane of polarization of light: $2(\alpha -\chi )={\boldsymbol {\Gamma }}$ ${\ displaystyle 2 (\ alpha - \ chi) = {\ boldsymbol {\ Gamma}}}$ , (9)
or $\rho d=\alpha -{\frac {\boldsymbol {\Gamma }}{2}}$ ${\ displaystyle \ rho d = \ alpha - {\ frac {\ boldsymbol {\ Gamma}} {2}}}$ , (ten)
Where $d$ ${\ displaystyle d}$ - the length of the sample in the direction of light propagation. For light propagation directions far from the optical axis, ellipticity $\kappa$ ${\ displaystyle \ kappa}$ is a small quantity and in (8) we can neglect the terms with $\kappa ^{2}$ ${\ displaystyle \ kappa ^ {2}}$ . Then, to describe the orientation of the azimuth of the polarization ellipse and the gyration tensor, we can use simplified relations:

$\tan 2\chi =-2\kappa \sin {\boldsymbol {\Gamma }}$ ${\ displaystyle \ tan 2 \ chi = -2 \ kappa \ sin {\ boldsymbol {\ Gamma}}}$ , (eleven)
or $g_{kl}=2\chi \Delta n{\overline {n}}$ ${\ displaystyle g_ {kl} = 2 \ chi \ Delta n {\ overline {n}}}$ . (12)

According to relation (11), when the light propagates in anisotropic directions, the gyration (or electro-gyration) effect is manifested in oscillations of the azimuth of the polarization ellipse with a change in the phase difference.

Experimental Results

The electro-gyration effect was first observed in a quadratic form in quartz crystals. Later, both linear and quadratic electro gyration ^{[9] were} studied in dielectric (HIO ₃ ^[10] , LiIO ₃ ^[11] , PbMoO ₄ ^[12] , NaBi (MoO ₄ ) ₂ , Pb ₅ SiO ₄ (VO ₄ ) ₂ , Pb ₅ SeO ₄ (VO ₄ ) ₂ , Pb ₅ GeO ₄ (VO ₄ ) ₂ ^[13] , alum ^[14] ^[15] ^[16] , etc.) semiconductor (AgGaS ₂ , CdGa ₂ S ₄ ) ^[17] , ferroelectric (crystals of the TGS family, Ferric salt, Pb ₅ Ge ₃ O ₁₁ , KDP, etc.) ^[18] ^[19] ^[20] ^[21] and photorefractive (BiSiO ₂₀ , BiGeO ₂₀ , Bi ₁₂ TiO ₂₀ ) materials ^{[ 22]} ^[23] ^[24] . The electro-gyration effect induced by high-power laser radiation (self-induced electro-gyration) was studied in ^[25] ^[26] . The effect of electrogyration on photorefractive recording was studied in ^[27] ^[28] . Electrogyration, in fact, is the first detected effect of gradient nonlinear optics, since from the point of view of nonlinear electrodynamics, taking into account frequency permutations, the existence of a gradient of the electric field of a light wave within small lengths (for example, a constant lattice) corresponds to a macroscopic gradient of an external electric field ^[29] .

Notes

↑ [1] Futama H. and Pepinsky R. (1962), “Optical activity in ferroelectric LiH ₃ (SeO ₃ ) ₂ ”, J.Phys.Soc.Jap., 17, 725.
↑ [2] Aizu K. (1964) "Reversal in optical rotatory power -" gyroelectric "crystals and" hypergyroelectric "crystals", Phys. Rev. 133 (6A), A1584-A1588
↑ [3] Zheludev I.S. (1964). Crystallography . 9 , 501-505.
↑ ¹ ² [4] Vlokh OG (1970). "Electrooptical activity of quartz crystals", Ukr.Fiz.Zhurn. 15 (5), 758-762. [Blokh OG (1970). "Electrooptical activity of quartz crystals", Sov.Phys. Ukr.Fiz.Zhurn. 15 , 771.]
↑ [5] Vlokh OG (1971) "Electrogyration effects in quartz crystals", Pis.ZhETF. 13 , 118-121 [Blokh OG (1971), “Electrogyration effects in quartz crystals”, Sov.Phys. Pis.ZhETF. 13 , 81-83.]
↑ [6] Vlokh OG (1987), “Electrogyration properties of crystals” Ferroelectrics 75 , 119-137.
↑ [7] Vlokh OG (2001) “The historical background of the finding of electrogyration”, Ukr.J.Phys.Opt. , 2 (2), 53-57
↑ [8] Vlokh OG, Kutniy IV, Lazko LA, and Nesterenko V.Ya. (1971) "Electrogyration of crystals and phase transitions", Izv.AN SSSR, ser.fiz. XXXV (9), 1852-1855.
↑ [9] Vlokh OG, Krushel'nitskaya TD (1970). Axial four-rank tensors and quadratic electro-gyration, Kristallografiya 15 (3), 587-589 [Vlokh OG, Krushel'nitskaya TD (1970). “Axial four-rank tensors and quadratic electro-gyration,” Sov.Phys. Crystalallogr. , 15 (3)]
↑ [10] Vlokh OG, Lazko LAand Nesterenko V.Ya. (1972). "Revealing of the linear electro-gyration effect in $\alpha$ ${\ displaystyle \ alpha}$ HIO ₃ crystals ”, Kristallografiya , 17 (6), 1248-1250. [ Sov.Phys. Crystalallogr. , 17 (6)]
↑ [11] Vlokh OG, Laz'ko LA, Zheludev IS (1975). “Effect of external factors on gyrotropic properties of LiIO ₃ crystals”, Kristallografiya 20 (3), 654–656 [ Sov.Phys. Crystalallogr. , 20 (3), 401]
↑ [12] Vlokh OG, Zheludev ISand Klimov IM (1975), “Optical activity of the centrosymmetric crystals of lead molibdate - PbMoO ₄ , induced by electric field (electro-gyration)”, Dokl. AN SSSR. 223 (6), 1391-1393.
↑ [13] Vloch O. G. (1984) Effects of spatial dispersion in parametric crystal optics. Lviv: Higher school.
↑ [14] Archived on August 13, 2011. Weber HJ and Haussuhl S. (1974), "Electric-Field-Induced Optical Activity and Circular Dichroism of Cr-Doped KAl (SO ₄ ) ₂ · 12H ₂ O" Phys. Stat. Sol. (B) 65 , 633-639.
↑ [15] Weber HJ and Haussuhl S. (1979), “Electrogyration and piezogyration in NaClO ₃ ” Acta Cryst. A35 225-232.
↑ [16] Weber HJ, Haussuhl S. (1976) “Electrogyration effect in alums”, Acta Cryst. A32 892-895
↑ [17] Vlokh OG, Zarik AV, Nekrasova IM (1983), “On the electro-gyration in AgGaS ₂ and CdGa ₂ S ₄ crystals”, Ukr.Fiz.Zhurn. 28 (9), 1334–1338.
↑ [18] Kobayashi J., Takahashi T., Hosakawa T. and Uesu Y. (1978). "A new method for measuring the optical activity of crystals and the optical activity of KH ₂ PO ₄ ", J.Appl. Phys. 49 , 809-815.
↑ [19] Kobayashi J., Uesu Y. and Sorimachi H. (1978), “Optical activity of some non-enantiomorphous ferroelectrics”, Ferroelectrics . 21 , 345-346.
↑ [20] Uesu Y., Sorimachi H. and Kobayashi J. (1979), "Electrogyration of a Nonenantiomorphic Crystal, Ferroelectric KH ₂ PO ₄ " Phys. Rev. Lett. 42 , 1427-1430.
↑ [21] Archived on December 11, 2012. Vlokh OG, Lazgko LA, Shopa YI (1981), “Electrooptic and Electrogyration Properties of the Solid Solutions on the Basis of Lead Germanate,” Phys. Stat Sol. (a) 65 : 371-378.
↑ [22] Vlokh OG, Zarik AV (1977), “The effect of electric field on the polarization of light in the Bi ₁₂ SiO ₂₀ , Bi ₁₂ GeO ₂₀ , NaBrO ₃ crystals”, Ukr.Fiz.Zhurn. 22 (6), 1027-1031.
↑ [23] Deliolanis NC, Kourmoulis IM, Asimellis G., Apostolidis AG, Vanidhis ED, and Vainos NA (2005), "Direct measurement of the dispersion of electrogyration coefficient of photorefractive Bi ₁₂ GeO ₂₀ ", J. Appl. Phys. 97 , 023531.
↑ [24] Deliolanis NC, Vanidhis ED, and Vainos NA (2006), "Dispersion of electogyration in sillenite crystals", Appl. Phys. B 85 (4), 591-596.
↑ [25] Akhmanov SA, Zhdanov BV, Zheludev NI, Kovrigin NI, Kuznetsov VI (1979). "Nonlinear optical activity in crystals", Pis.ZhETF . 29 , 294-298.
↑ [26] Zheludev NI, Karasev V.Yu., Kostov ZM Nunuparov MS (1986) “Giant exciton resonance in nonlinear optical activity”, Pis.ZhETF , 43 (12), 578-581.
↑ [27] Brodin MS, Volkov VI, Kukhtarev NV and Privalko AV (1990), "Nanosecond electrogyration selfdiffraction in Bi12TiO20 (BTO) crystal", Optics Communications , 76 (1), 21-24.
↑ [28] Kukhtarev NV, Dovgalenko GE (1986) "Self-diffraction electrogyration and electroellipticity in centrosymmetric crystals", Sov.J. Quantum Electron. ., 16 (1), 113-114.
↑ [29] Archived on December 16, 2012. Vlokh RO (1991). “Nonlinear medium polarization with account of gradient invariants.”, Phys. Stat.Sol (b) , 168 , k47-K50.

[1] [1] Futama H. and Pepinsky R. (1962), “Optical activity in ferroelectric LiH ₃ (SeO ₃ ) ₂ ”, J.Phys.Soc.Jap., 17, 725.

[2] [2] Aizu K. (1964) "Reversal in optical rotatory power -" gyroelectric "crystals and" hypergyroelectric "crystals", Phys. Rev. 133 (6A), A1584-A1588

[3] [3] Zheludev I.S. (1964). Crystallography . 9 , 501-505.

[ifo.lviv.ua-4] ¹ ² [4] Vlokh OG (1970). "Electrooptical activity of quartz crystals", Ukr.Fiz.Zhurn. 15 (5), 758-762. [Blokh OG (1970). "Electrooptical activity of quartz crystals", Sov.Phys. Ukr.Fiz.Zhurn. 15 , 771.]

[5] [5] Vlokh OG (1971) "Electrogyration effects in quartz crystals", Pis.ZhETF. 13 , 118-121 [Blokh OG (1971), “Electrogyration effects in quartz crystals”, Sov.Phys. Pis.ZhETF. 13 , 81-83.]

[6] [6] Vlokh OG (1987), “Electrogyration properties of crystals” Ferroelectrics 75 , 119-137.

[7] [7] Vlokh OG (2001) “The historical background of the finding of electrogyration”, Ukr.J.Phys.Opt. , 2 (2), 53-57

[8] [8] Vlokh OG, Kutniy IV, Lazko LA, and Nesterenko V.Ya. (1971) "Electrogyration of crystals and phase transitions", Izv.AN SSSR, ser.fiz. XXXV (9), 1852-1855.

[9] [9] Vlokh OG, Krushel'nitskaya TD (1970). Axial four-rank tensors and quadratic electro-gyration, Kristallografiya 15 (3), 587-589 [Vlokh OG, Krushel'nitskaya TD (1970). “Axial four-rank tensors and quadratic electro-gyration,” Sov.Phys. Crystalallogr. , 15 (3)]

[10] [10] Vlokh OG, Lazko LAand Nesterenko V.Ya. (1972). "Revealing of the linear electro-gyration effect in $\alpha$ ${\ displaystyle \ alpha}$ HIO ₃ crystals ”, Kristallografiya , 17 (6), 1248-1250. [ Sov.Phys. Crystalallogr. , 17 (6)]

[11] [11] Vlokh OG, Laz'ko LA, Zheludev IS (1975). “Effect of external factors on gyrotropic properties of LiIO ₃ crystals”, Kristallografiya 20 (3), 654–656 [ Sov.Phys. Crystalallogr. , 20 (3), 401]

[12] [12] Vlokh OG, Zheludev ISand Klimov IM (1975), “Optical activity of the centrosymmetric crystals of lead molibdate - PbMoO ₄ , induced by electric field (electro-gyration)”, Dokl. AN SSSR. 223 (6), 1391-1393.

[13] [13] Vloch O. G. (1984) Effects of spatial dispersion in parametric crystal optics. Lviv: Higher school.

[14] [14] Archived on August 13, 2011. Weber HJ and Haussuhl S. (1974), "Electric-Field-Induced Optical Activity and Circular Dichroism of Cr-Doped KAl (SO ₄ ) ₂ · 12H ₂ O" Phys. Stat. Sol. (B) 65 , 633-639.

[15] [15] Weber HJ and Haussuhl S. (1979), “Electrogyration and piezogyration in NaClO ₃ ” Acta Cryst. A35 225-232.

[16] [16] Weber HJ, Haussuhl S. (1976) “Electrogyration effect in alums”, Acta Cryst. A32 892-895

[17] [17] Vlokh OG, Zarik AV, Nekrasova IM (1983), “On the electro-gyration in AgGaS ₂ and CdGa ₂ S ₄ crystals”, Ukr.Fiz.Zhurn. 28 (9), 1334–1338.

[18] [18] Kobayashi J., Takahashi T., Hosakawa T. and Uesu Y. (1978). "A new method for measuring the optical activity of crystals and the optical activity of KH ₂ PO ₄ ", J.Appl. Phys. 49 , 809-815.

[19] [19] Kobayashi J., Uesu Y. and Sorimachi H. (1978), “Optical activity of some non-enantiomorphous ferroelectrics”, Ferroelectrics . 21 , 345-346.

[20] [20] Uesu Y., Sorimachi H. and Kobayashi J. (1979), "Electrogyration of a Nonenantiomorphic Crystal, Ferroelectric KH ₂ PO ₄ " Phys. Rev. Lett. 42 , 1427-1430.

[21] [21] Archived on December 11, 2012. Vlokh OG, Lazgko LA, Shopa YI (1981), “Electrooptic and Electrogyration Properties of the Solid Solutions on the Basis of Lead Germanate,” Phys. Stat Sol. (a) 65 : 371-378.

[22] [22] Vlokh OG, Zarik AV (1977), “The effect of electric field on the polarization of light in the Bi ₁₂ SiO ₂₀ , Bi ₁₂ GeO ₂₀ , NaBrO ₃ crystals”, Ukr.Fiz.Zhurn. 22 (6), 1027-1031.

[23] [23] Deliolanis NC, Kourmoulis IM, Asimellis G., Apostolidis AG, Vanidhis ED, and Vainos NA (2005), "Direct measurement of the dispersion of electrogyration coefficient of photorefractive Bi ₁₂ GeO ₂₀ ", J. Appl. Phys. 97 , 023531.

[24] [24] Deliolanis NC, Vanidhis ED, and Vainos NA (2006), "Dispersion of electogyration in sillenite crystals", Appl. Phys. B 85 (4), 591-596.

[25] [25] Akhmanov SA, Zhdanov BV, Zheludev NI, Kovrigin NI, Kuznetsov VI (1979). "Nonlinear optical activity in crystals", Pis.ZhETF . 29 , 294-298.

[26] [26] Zheludev NI, Karasev V.Yu., Kostov ZM Nunuparov MS (1986) “Giant exciton resonance in nonlinear optical activity”, Pis.ZhETF , 43 (12), 578-581.

[27] [27] Brodin MS, Volkov VI, Kukhtarev NV and Privalko AV (1990), "Nanosecond electrogyration selfdiffraction in Bi12TiO20 (BTO) crystal", Optics Communications , 76 (1), 21-24.

[28] [28] Kukhtarev NV, Dovgalenko GE (1986) "Self-diffraction electrogyration and electroellipticity in centrosymmetric crystals", Sov.J. Quantum Electron. ., 16 (1), 113-114.

[29] [29] Archived on December 16, 2012. Vlokh RO (1991). “Nonlinear medium polarization with account of gradient invariants.”, Phys. Stat.Sol (b) , 168 , k47-K50.