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| {{Expert-subject|Physics|date=February 2009}}
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| {{Refimprove|date=July 2007}}
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| 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 (optics)|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. Formally, it is a special case of gyroelectromagnetism obtained when the [[magnetic permeability]] tensor is diagonal.<ref name="ingentaconnect.com">[http://www.ingentaconnect.com/content/vsp/jew/2003/00000017/00000008/art00011] Prati E. (2003) "Propagation in gyroelectromagnetic guiding systems", ''J. of Electr. Wav. and Appl.'' '''17, 8''', 1177</ref> | |
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| 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 <math>\overline{4}3m\, </math>. The effect proportional to the square of the [[electric field]] can exist only in crystals belonging to acentric [[point groups]] of symmetry.
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| == The historical background of discovery of electrogyration ==
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| The changes in the optical activity sign induced by the external electric field have been observed for the first time in ferroelectric crystals LiH<sub>3</sub>(SeO<sub>4</sub>)<sub>2</sub> by H. Futama and R. Pepinsky in 1961
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| ,<ref>[http://www.ipap.jp/jpsj/current/index.htm] Futama H. and Pepinsky R. (1962) "Optical activity in ferroelectric LiH3(SeO3)2", ''J. Phys. Soc. Jap.'' '''17''', 725</ref> 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.
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| 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 <ref>[http://prola.aps.org/abstract/PR/v133/i6A/pA1584_1] Aizu K. (1964) “Reversal in optical rotatory power – “gyroelectric” crystals and “hypergyroelectric” crystals”, Phys. Rev. 133 (6A) A1584–A1588</ref> (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.
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| Almost simultaneously with K. Aizu, I.S. Zheludev has suggested tensor description of the electrogyration in 1964 <ref>[http://www.ifo.lviv.ua/institute/personal/publ/001.pdf] Zheludev I.S. (1964), "Axial tensors of the third rank and the physical effects they describe", ''Kristallografiya'' '''9''', 501-505.[(1965). ''Sov.Phys.Crystallogr.'' '''9''',418]</ref> (the manuscript received on February 21, 1964). In this paper the electrogyration has been referred to as “electro-optic activity”.
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| 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 <ref name="ifo.lviv">[http://www.ifo.lviv.ua/institute/personal/publ/002.pdf] Vlokh O.G.(1970). "Electrooptical activity of quartz crystals", ''Ukr.Fiz.Zhurn.'''''15'''(5), 758-762.[Blokh O.G. (1970). "Electrooptical activity of quartz crystals", ''Sov.Phys. Ukr.Fiz.Zhurn.'''''15''', 771.]</ref> (the manuscript received on July 7, 1969).<br />
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| 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.<ref name="ifo.lviv" />
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| .<ref>[http://www.jetpletters.ac.ru/ps/712/article_11013.shtml] Vlokh O.G. (1971) "Electrogyration effects in quartz crystals", ''Pis.ZhETF.'' '''13''', 118-121 [Blokh O.G. (1971) "Electrogyration effects in quartz crystals", ''Sov.Phys. Pis.ZhETF.'' '''13''', 81-83.]</ref><ref>[http://www.informaworld.com/smpp/content~content=a752268592~db=all~order=page] Vlokh O.G. (1987), "Electrogyration properties of crystals" ''Ferroelectrics'' '''75''', 119-137.</ref><ref>[http://www.ifo.lviv.ua/journal/UJPO_PDF/2001_2/2001-53-57_new.pdf] Vlokh O.G. (2001) "The historical background of the finding of electrogyration", ''Ukr.J.Phys.Opt.'', '''2'''(2), 53-57</ref>
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| Later in 2003, the gyroelectricity has been extended to gyroelectromagnetic media,<ref name="ingentaconnect.com"/> which account for [[ferromagnetic semiconductors]] and engineered [[metamaterials]], for which gyroelectricity and gyromagnetism ([[Faraday effect]]) may occur at the same time.
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| == Description ==
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| === Electrodynamics relations ===
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| The electric field and the electric displacement vectors of electromagnetic wave propagating in gyrotropic crystals may be written respectively as:
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| <math>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}\, </math>, (1)
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| or
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| <math>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}\, </math>, (2)<br />
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| where <math>B_{ij}^{0}</math> is the optical frequency impermeability [[tensor]], <math>\epsilon_{ij}^{0}</math> the dielectric permittivity [[tensor]], <math>\tilde{g}_{lk}\overline{n}=g_{kl}</math>, <math>\overline{n}</math> the mean refractive index, <math>D_{j}\, </math> - induction, <math>\delta_{ijk}\, </math>, <math>\tilde\delta_{ijk}</math> polar third rank [[tensors]], <math>e_{ijl}\, </math> the unit antisymmetric Levi-Civit pseudo-tensor, <math>k_k\, </math> the [[wave vector]], and <math>g_{lk}\, </math>, <math>\tilde{g}_{lk}</math> the second rank gyration pseudo-tensors. The specific rotation angle of the polarization plane <math>\rho\, </math> caused by the natural [[optical activity]] is defined by the relation:
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| <math>\rho=\frac{\pi}{\lambda n}g_{lk}l_{l}l_{k}=\frac{\pi}{\lambda n}G\, </math>, (3)<br />
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| where <math>n\, </math> is the refractive index, <math>\lambda\, </math> the wavelength, <math>l_{l}\, </math>, <math>l_{k}\, </math> the transformation coefficients between the Cartesian and spherical coordinate systems (<math>l_{1}=\sin \Theta \cos \varphi\, </math>, <math>l_{2}=\sin \Theta \sin \varphi, l_{3}=\cos \Theta</math>), and <math>G\, </math> the pseudo-scalar gyration parameter.
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| The electro-gyration increment of gyration [[tensor]] occurred under the action of [[electric field]] <math>E_{m}\, </math> or/and <math>E_{n}\, </math> is written as:
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| <math>\Delta g_{lk}=\gamma _{lkm}E_{m}+\beta _{lkmn}E_{m}E_{n}\, </math>, (4)
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| where <math>\gamma _{lkm}\, </math> and <math>\beta _{lkmn}\, </math> 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:
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| <math>\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}</math>. (5)
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| The electrogyration effect may be also induced by spontaneous polarization <math>P_{m}^{s}P_{n}^{s}\, </math> appearing in the course of ferroelectric phase transitions
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| :<ref>[http://www.ifo.lviv.ua/institute/personal/publ/006.pdf] Vlokh O.G., Kutniy I.V., Lazko L.A., and Nesterenko V.Ya. (1971) "Electrogyration of crystals and phase transitions", ''Izv.AN SSSR, ser.fiz.'' '''XXXV''' (9), 1852-1855.</ref>
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| <math>\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}</math>. (6)
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| === Explanation on the basis of symmetry approach ===
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| 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 - <math>\infty 2\, </math> 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, <math>\infty mm\, </math>) 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 <math>E_{m}E_{n} \, </math> or, what is the same, the symmetry of a polar second-rank [[tensor]] (<math>\infty /mmm\, </math>). 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.
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| === Eigenwaves in the presence of electrogyration ===
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| 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 <math>\kappa \, </math> and the azimuth <math>\chi \, </math> are defined respectively by the relations
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| <math>\kappa =\frac{\Delta G}{2\Delta n\overline{n}}\, </math>, (7)<br />
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| <math>\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 )\, </math>, (8)
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| where <math>\alpha \, </math> is the polarization azimuth of the incident light with respect to the principal indicatrix axis, <math>\Delta n\, </math> the linear [[birefringence]], <math>\boldsymbol\Gamma \, </math> the phase retardation, <math>P=\frac{(1-\kappa ^2)^2}{2\kappa (1+\kappa ^2)}\, </math>, and <math>R=\left (\frac{2\kappa }{1+\kappa ^2}\right )^2+\left (\frac{1-\kappa ^2}{1+\kappa ^2}\right )^2\, </math>. In the case of light propagation along optically isotropic directions (i.e., the optic axes), the eigenwave become circularly polarized (<math>\kappa =1\, </math>), 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:
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| <math>2(\alpha -\chi )=\boldsymbol\Gamma \, </math>, (9)
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| or
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| <math>\rho d=\alpha -\frac{\boldsymbol\Gamma}{2}\, </math>, (10)
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| where <math>d\, </math> - is the sample thickness along the direction of light propagation.
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| For the directions of light propagation far from the optic axis, the ellipticity <math>\kappa \, </math> is small and so one can neglect the terms proportional to <math>\kappa ^2\, </math> in Eq.(8). Thus, in order to describe the polarization azimuth at <math>\alpha =0\, </math> and the gyration tensor, simplified relations
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| <math>\tan 2\chi =-2\kappa \sin \boldsymbol\Gamma\, </math>, (11)
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| or
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| <math>g_{kl}=2\chi \Delta n\overline{n}\, </math>. (12)
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| are often used.
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| 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 <math>\boldsymbol\Gamma\, </math> .
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| === Experimental results ===
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| <!-- Image with unknown copyright status removed: [[Image:PGO-FOTO_2.jpg|right|Visualisation of electrogyration effect. Conoscopic patterns of Pb5Ge3O11:0.0049Li% crystals at E<sub>z</sub>=0 (left) and Ez=5kV/cm (right)]] -->
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| 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 quadratic
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| <ref>[http://www.ifo.lviv.ua/institute/personal/publ/007.pdf] Vlokh O.G., Krushel'nitskaya T.D. (1970). "Axial four-rank tensors and quadratic electrogyration", ''Kristallografiya'' '''15'''(3), 587-589 [Vlokh O.G., Krushel'nitskaya T.D. (1970). "Axial four-rank tensors and quadratic electrogyration", ''Sov.Phys.Crystallogr.'', '''15'''(3)]</ref> electrogyrations has been studied in the dielectric (<math>\alpha -\, </math>HIO<sub>3</sub>
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| ,<ref>[http://www.ifo.lviv.ua/institute/personal/publ/008.pdf] Vlokh O.G., Lazko L.A.and Nesterenko V.Ya. (1972). "Revealing of the linear electrogyration effect in <math>\alpha \, </math>HIO<sub>3</sub> crystals", ''Kristallografiya'', '''17'''(6), 1248-1250.[''Sov.Phys.Crystallogr.'','''17'''(6)]</ref> LiIO<sub>3</sub>
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| ,<ref>[http://www.ifo.lviv.ua/institute/personal/publ/009.pdf] Vlokh O.G., Laz'ko L.A., Zheludev I.S. (1975). "Effect of external factors on gyrotropic properties of LiIO<sub>3</sub> crystals", ''Kristallografiya'' '''20'''(3), 654-656 [''Sov.Phys.Crystallogr.'','''20'''(3), 401]</ref> PbMoO<sub>4</sub>,<ref>{{cite journal| last=Vlokh | first=O. G. | coauthors=I. S. Zheludev and I. M. Klimov | year=1975 | trans_title=Optical activity of the centrosymmetric crystals of lead molibdate - PbMoO<sub>4</sub>, induced by electric field (electrogyration) | journal=[[Doklady Akademii Nauk SSSR]] | volume=223 | issue=6 | pages=1391–1393 | url=http://www.ifo.lviv.ua/institute/personal/publ/010.pdf|bibcode = 1975DoSSR.223.1391V| title=Electric field-induced optical activity of centrosymmetrical crystals of lead molybdate PbMoO4 /Electrogyration/| last2=Zheludev| last3=Klimov }}</ref> NaBi(MoO<sub>4</sub>)<sub>2</sub>, Pb<sub>5</sub>SiO<sub>4</sub>(VO<sub>4</sub>)<sub>2</sub>, Pb<sub>5</sub>SeO<sub>4</sub>(VO<sub>4</sub>)<sub>2</sub>, Pb<sub>5</sub>GeO<sub>4</sub>(VO<sub>4</sub>)<sub>2</sub>,<ref>[http://www.sigla.ru/] Vlokh O.G. (1984) ''Spatial dispersion phenomena in parametric crystal optics.'' Lviv: Vyshcha Shkola (in Russian).</ref> alums <ref>[http://www3.interscience.wiley.com/cgi-bin/abstract/112397571/ABSTRACT] Weber H.J. and Haussuhl S. (1974), "Electric-Field-Induced Optical Activity and Circular Dichroism of Cr-Doped KAl(SO<sub>4</sub>)<sub>2</sub> · 12H<sub>2</sub>O " ''Phys. Stat. Sol.(b)'' '''65''', 633-639.</ref><ref>[http://journals.iucr.org/a/issues/1979/01/00/issconts.html] Weber H.J. and Haussuhl S. (1979), "Electrogyration and piezogyration in NaClO<sub>3</sub>" ''Acta Cryst.'' '''A35'''225-232.</ref>
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| <ref>[http://journals.iucr.org/a/issues/1976/05/00/issconts.html] Weber H.J., Haussuhl S. (1976) "Electrogyration effect in alums", ''Acta Cryst.'' '''A32''' 892-895</ref> etc.) semiconductor (AgGaS<sub>2</sub>, CdGa<sub>2</sub>S<sub>4</sub>)
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| ,<ref>[http://www.ifo.lviv.ua/institute/personal/publ/015.pdf] Vlokh O.G., Zarik A.V., Nekrasova I.M. (1983), "On the electrogyration in AgGaS<sub>2</sub> and CdGa<sub>2</sub>S<sub>4</sub> crystals", ''Ukr.Fiz.Zhurn.'', '''28'''(9), 1334-1338.</ref> [[ferroelectric]] (TGS, Rochelle Salt, Pb<sub>5</sub>Ge<sub>3</sub>O<sub>11</sub> and KDP families etc.)
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| <ref>[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JAPIAU000049000002000809000001&idtype=cvips&gifs=Yes] 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<sub>2</sub>PO<sub>4</sub>
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| ", ''J.Appl. Phys.'' '''49''', 809-815.</ref>
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| <ref>[http://www.informaworld.com/smpp/content~content=a755716926~db=all~order=page] Kobayashi J., Uesu Y. and Sorimachi H. (1978), "Optical activity of some non-enantiomorphous ferroelectrics", ''Ferroelectrics''. '''21''', 345-346.</ref>
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| <ref>[http://prola.aps.org/abstract/PRL/v42/i21/p1427_1] Uesu Y., Sorimachi H. and Kobayashi J. (1979), "Electrogyration of a Nonenantiomorphic Crystal, Ferroelectric KH<sub>2</sub>PO<sub>4</sub> " ''Phys. Rev. Lett.'' '''42''', 1427-1430.</ref>
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| ,<ref>[http://www3.interscience.wiley.com/cgi-bin/abstract/112443012/ABSTRACT] Vlokh O.G., Lazgko L.A., Shopa Y.I. (1981), "Electrooptic and Electrogyration Properties of the Solid Solutions on the Basis of Lead Germanate", ''Phys.Stat.Sol. (a)'' '''65''': 371-378.</ref> as well as the [[photorefractive]] (Bi<sub>12</sub>SiO<sub>20</sub>, Bi<sub>12</sub>GeO<sub>20</sub>, Bi<sub>12</sub>TiO<sub>20</sub>) materials
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| <ref>[http://www.ifo.lviv.ua/institute/personal/publ/020.pdf] Vlokh O.G., Zarik A.V. (1977), "The effect of electric field on the polarization of light in the Bi<sub>12</sub>SiO<sub>20</sub>, Bi<sub>12</sub>GeO<sub>20</sub>, NaBrO<sub>3</sub> crystals", ''Ukr.Fiz.Zhurn.'' '''22'''(6), 1027-1031.</ref>
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| <ref>[http://dx.doi.org/10.1063/1.1828585] Deliolanis N.C., Kourmoulis I.M., Asimellis G., Apostolidis A.G., Vanidhis E.D., and Vainos N.A. (2005), "Direct measurement of the dispersion of electrogyration coefficient of photorefractive Bi<sub>12</sub>GeO<sub>20</sub>", ''J. Appl. Phys.'' '''97''', 023531.</ref>
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| .<ref>[http://dx.doi.org/10.1007/s00340-006-2437-1] Deliolanis N.C, Vanidhis E.D, and Vainos N.A. (2006), "Dispersion of electogyration in sillenite crystals", ''Appl. Phys. B'' '''85'''(4), 591-596.</ref> 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
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| <ref>[http://www.jetpletters.ac.ru/ps/1451/article_22078.shtml] Akhmanov S.A., Zhdanov B.V., Zheludev N.I., Kovrigin N.I., Kuznetsov V.I. (1979). "Nonlinear optical activity in crystals", ''Pis.ZhETF''. '''29''', 294-298.</ref>
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| .<ref>[http://www.jetpletters.ac.ru/ps/1413/article_21509.shtml] Zheludev N.I., Karasev V.Yu., Kostov Z.M. Nunuparov M.S.(1986) "Giant exciton resonance in nonlinear optical activity", ''Pis.ZhETF'', '''43'''(12), 578-581.</ref> The influence of electro-gyration on the [[photorefraction]] storage has been investigated in
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| ,<ref>[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVF-46JGY09-2BR&_user=10&_coverDate=04%2F01%2F1990&_rdoc=6&_fmt=summary&_orig=browse&_srch=doc-info(%23toc%235533%231990%23999239998%23333412%23FLP%23display%23Volume)&_cdi=5533&_sort=d&_docanchor=&view=c&_ct=19&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=27d23e12b85d69af10cee771d30389e5] Brodin M.S., Volkov V.I., Kukhtarev N.V. and Privalko A.V. (1990), "Nanosecond electrogyration selfdiffraction in Bi12TiO20 (BTO) crystal", ''Optics Communications'', '''76'''(1), 21-24.</ref><ref>[http://www.turpion.org/php/paper.phtml?journal_id=qe&paper_id=5220] Kukhtarev N.V., Dovgalenko G.E. (1986) "Self-diffraction electrogyration and electroellipticity in centrosymmetric crystals", ''Sov.J. Quantum Electron.'', , 16 (1), 113-114.</ref> 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 <ref>[http://www3.interscience.wiley.com/cgi-bin/abstract/112454661/ABSTRACT] Vlokh R.O. (1991). "Nonlinear medium polarization with account of gradient invariants.", ''Phys. Stat.Sol (b)'', '''168''', k47-K50.</ref> is taken into account. In that sense, the electrogyration effect represents the first of the gradient nonlinear optical phenomena ever revealed.
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| == See also ==
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| *[[Piezo-gyration]]
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| *[[Faraday effect]]
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| == References ==
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| {{reflist}}
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| {{DEFAULTSORT:Electro-Gyration}}
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| [[Category:Optics]]
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| [[Category:Electromagnetism]]
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