Young, D, Pu, Y. Magnetooptics The electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton CRC Press llc. 2000
Young, D., Pu, Y. “Magnetooptics” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
57 Magnetooptics 57.1 57.2 Classification of Magnetooptic Effects David Young Faraday Rotation or Magnetic Circular Birefringence Rockwell Semiconductor Systems Mouton Effect or Magnetic Linear Birefringence. Ker 57.3 Applications of M Yuan pu Optical Isolator and Circulator. MSw-Based Guided-Wave Applied materials magnetooptic Bragg Cell.Magnetooptic Recording 57.1 Introduction When a magnetic field H is applied to a magnetic medium(crystal), a change in the magnetization M within he medium will occur as described by the constitution relation of the Maxwell equations M=X H where i is the magnetic susceptibility tensor of the medium. The change in magnetization can in turn induce a perturbation in the complex optical permittivity tensor E. This phenomenon is called the magnetooptic effect. Mathematically, the magnetooptic effect can be described by expanding the permittivity tensor as a series in increasing powers of the magnetization [Torfeh et al., 1977] as follows: Soleil 小(M)=ε+ieM+质MLM Here j is the imaginary number. M, M, and M, are the magnetization components along the principal crystal axes X, Y, and Z, respectively. Eo is the permittivity of free space. E, is the relative permittivity of the medium in the paramagnetic state (i.e. M=0), f is the first-order magnetooptic scalar factor, fi i is the second-order magnetooptic tensor factor, 8, is the Kronecker delta, and ejk is the antisymmetric alternate index of the third order. Here we have used Einstein notation of repeated indices and have assumed that the medium is quasi transparent so that e is a Hermitian tensor. Moreover, we have also invoked the Onsager relation in thermo dynamical statistics, i.e., E, (M)=E;(M). The consequences of Hermiticity and Onsager relation are that the real part of the permittivity tensor is an even function of M whereas the imaginary part is an odd function M. For a cubic crystal, such as YIG(yttrium-iron-garnet), the tensor fiu reduces to only three independent terms. In terms of Voigt notation, they are fu f2, and f. In a principal coordinate system, the tensor can be expressed as fu=f12,5x+f4(6+86)+△fδ55k (57.2) c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 57 Magnetooptics 57.1 Introduction 57.2 Classification of Magnetooptic Effects Faraday Rotation or Magnetic Circular Birefringence • CottonMouton Effect or Magnetic Linear Birefringence • Kerr Effects 57.3 Applications of Magnetooptic Effects Optical Isolator and Circulator • MSW-Based Guided-Wave Magnetooptic Bragg Cell • Magnetooptic Recording 57.1 Introduction When a magnetic field H is applied to a magnetic medium (crystal), a change in the magnetization M within the medium will occur as described by the constitution relation of the Maxwell equations M = c ´ ·H where c ´ is the magnetic susceptibility tensor of the medium. The change in magnetization can in turn induce a perturbation in the complex optical permittivity tensor e ´. This phenomenon is called the magnetooptic effect. Mathematically, the magnetooptic effect can be described by expanding the permittivity tensor as a series in increasing powers of the magnetization [Torfeh et al., 1977] as follows: e ´ = e0[eij] (57.1) where eij(M) = erdij + jf1 eijkMk + fijklMkMl Here j is the imaginary number. M1, M2, and M3 are the magnetization components along the principal crystal axes X, Y, and Z, respectively. e0 is the permittivity of free space. er is the relative permittivity of the medium in the paramagnetic state (i.e., M = 0), f1 is the first-order magnetooptic scalar factor, fijk l is the second-order magnetooptic tensor factor, dij is the Kronecker delta, and eijk is the antisymmetric alternate index of the third order. Here we have used Einstein notation of repeated indices and have assumed that the medium is quasitransparent so that e ´ is a Hermitian tensor. Moreover, we have also invoked the Onsager relation in thermodynamical statistics, i.e., eij(M) = eji(–M). The consequences of Hermiticity and Onsager relation are that the real part of the permittivity tensor is an even function of M whereas the imaginary part is an odd function of M. For a cubic crystal, such as YIG (yttrium-iron-garnet), the tensor fijkl reduces to only three independent terms. In terms of Voigt notation, they are f11, f12, and f44. In a principal coordinate system, the tensor can be expressed as fijkl = f12dijdkl + f44 (dildkj + dikdlj) + Df dkldijdjk (57.2) where Df = f11 – f12 – 2f44. David Young Rockwell Semiconductor Systems Yuan Pu Applied Materials
In the principal crystal axes [100] coordinate system, the magnetooptic permittivity reduces to the following E where*denotes complex conjugate operation. The elements are given by paramagnetic state E=Elo E. O 00￡ Faraday rotation jfM -ifM, +ifM, -jf,M Cotton-Mouton effect fM+f, M3+f 2M4,, 2MMM,M, 2/4M,M f12M2+f1M2+f12M2 MMM,M, (57.3) 2MM,M, 2M. frMi +fnMS+MMM In order to keep the discussion simple, analytic complexities due to optical absorption of the magnetic mediu have been ignored. Such absorption can give rise to magnetic circular dichroism(MCD)and magnetic linear dichroism(MLD). Interested readers can refer to Hellwege[ 1978] and Arecchi and Schulz-DuBois [1972] for more in-depth discussions on MCD and MLD 57.2 Classification of Magnetooptic Effects Faraday Rotation or Magnetic Circular Birefringence The classic Faraday rotation takes place in a cubic or isotropic transparent medium where the propagation direction of transmitted light is parallel to the direction of applied magnetization within the medium. For example, if the direction of magnetization and the propagation of light is taken as Z, the permittivity tensor becomes(assuming second-order effect is insignificantly small) Er jfM, 0 E≡-ifMe,0 c 2000 by CRC Press LLC
© 2000 by CRC Press LLC In the principal crystal axes [100] coordinate system, the magnetooptic permittivity reduces to the following forms: where * denotes complex conjugate operation. The elements are given by paramagnetic state Faraday rotation Cotton-Mouton effect (57.3) In order to keep the discussion simple, analytic complexities due to optical absorption of the magnetic medium have been ignored. Such absorption can give rise to magnetic circular dichroism (MCD) and magnetic linear dichroism (MLD). Interested readers can refer to Hellwege [1978] and Arecchi and Schulz-DuBois [1972] for more in-depth discussions on MCD and MLD. 57.2 Classification of Magnetooptic Effects Faraday Rotation or Magnetic Circular Birefringence The classic Faraday rotation takes place in a cubic or isotropic transparent medium where the propagation direction of transmitted light is parallel to the direction of applied magnetization within the medium. For example, if the direction of magnetization and the propagation of light is taken as Z, the permittivity tensor becomes (assuming second-order effect is insignificantly small): (57.4) ´ = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ e e e e e e e e e e e 0 11 12 13 12 22 23 13 23 33 * * * ´ = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ e e e e e 0 0 0 0 0 0 0 r r r + + - - + + - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ e0 1 3 1 2 1 3 1 1 1 2 1 1 0 0 0 j f M j f M j f M j f M j f M j f M + + + + + + + È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ e0 11 1 2 12 2 2 12 3 2 44 1 2 44 1 3 44 1 2 12 1 2 11 2 2 12 3 2 44 2 3 44 1 3 44 2 3 12 1 2 12 2 2 11 3 2 2 2 2 2 2 2 f M f M f M f M M f M M f M M f M f M f M f M M f M M f M M f M f M f M ´ @ - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ e e e e e 0 1 3 1 3 0 0 0 0 r r r j f M j f M
The two eigenmodes of light propagation through the magnetooptic medium can be expressed as a right circular polarized(RCP) light wave E1(2)=exp[ j or- 2rn,2 (57.5a) and a left circular polarized(LCP) light wave 2兀t -Expl J where n,2=E, +fMj; o and no are the angular frequency and the wavelength of the incident light, respectively n, and n are the refractive indices of the RCP and LCP modes, respectively. These modes correspond to two counterrotating circularly polarized light waves. The superposition of these two waves produces a linearly polarized wave. The plane of polarization of the resultant wave rotates as one circular wave overtakes the other The rate of rotation is given by 6≡M rad/m B is known as the Faraday rotation(FR)coefficient. When the direction of the magnetization is reversed, the gle of rotation changes its sign. Since two counterrotating circular polarized optical waves are used to explain FR. the effect also known as optical magnetic circular birefringence(MCB ). Furthermore, since the senses of polarization rotation of forward traveling and backward traveling light waves are opposite, FR is a nonreciprocal optical effect. Optical devices such as optical isolators and optical circulators use the Faraday effect to achieve their nonreciprocal functions For ferromagnetic and ferrimagnetic media, the FR is charac terized under a magnetically saturated condition, i.e., M,= Ms the saturation magnetization of the medium. For paramagnetic or diamagnetic materials, the magnetization is proportional to the external applied magnetic field Ho. Therefore, the FR is proportional to the external field or AF= VHo where V=Xof/o ve, )is called the Verdet constant and %o is the magnetic susceptibility of free space Cotton. Mouton effect or magnetic linear birefringence When transmitted light is propagating perpendicular to the magnetization direction, the first-order isotropic R effect will vanish and the second-order anisotropic Cotton-Mouton(CM)effect will dominate. For example, if the direction of magnetization is along the Z axis and the light wave is propagating along the X axis, the E,+fu,M 0 0 0E,+f12M30 Er+fMi c 2000 by CRC Press LLC
© 2000 by CRC Press LLC The two eigenmodes of light propagation through the magnetooptic medium can be expressed as a right circular polarized (RCP) light wave (57.5a) and a left circular polarized (LCP) light wave (57.5b) where n± 2 @ er ± f1M3; w and l0 are the angular frequency and the wavelength of the incident light, respectively. n+ and n– are the refractive indices of the RCP and LCP modes, respectively. These modes correspond to two counterrotating circularly polarized light waves. The superposition of these two waves produces a linearly polarized wave. The plane of polarization of the resultant wave rotates as one circular wave overtakes the other. The rate of rotation is given by (57.6) qF is known as the Faraday rotation (FR) coefficient. When the direction of the magnetization is reversed, the angle of rotation changes its sign. Since two counterrotating circular polarized optical waves are used to explain FR, the effect is thus also known as optical magnetic circular birefringence (MCB). Furthermore, since the senses of polarization rotation of forward traveling and backward traveling light waves are opposite, FR is a nonreciprocal optical effect. Optical devices such as optical isolators and optical circulators use the Faraday effect to achieve their nonreciprocal functions. For ferromagnetic and ferrimagnetic media, the FR is characterized under a magnetically saturated condition, i.e., M3 = MS, the saturation magnetization of the medium. For paramagnetic or diamagnetic materials, the magnetization is proportional to the external applied magnetic field H0. Therefore, the FR is proportional to the external field or qF = VH0 where V = c0f1p/(l0 ) is called the Verdet constant and c0 is the magnetic susceptibility of free space. Cotton-Mouton Effect or Magnetic Linear Birefringence When transmitted light is propagating perpendicular to the magnetization direction, the first-order isotropic FR effect will vanish and the second-order anisotropic Cotton-Mouton (CM) effect will dominate. For example, if the direction of magnetization is along the Z axis and the light wave is propagating along the X axis, the permittivity tensor becomes (57.7) ˜ E Z j exp j t n 1 Z 0 1 0 2 ( ) = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ - Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ + w p l ˜ E Z j exp j t n 2 Z 0 1 0 2 ( ) = - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ - Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ - w p l q p l e l e F r r f M f M @ = 1 3 0 1 3 0 1 8 rad/m degree/cm . er ´ = + + + È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ e e e e e 0 12 3 2 12 3 2 11 3 2 0 0 0 0 0 0 r r r f M f M f M
The eigenmodes are two linearly polarized light waves polarized along and perpendicular to the magnetization E1(x)=0exor~2兀 (57.8a) E(x)=1Jexplilot-2T m (57.8b) 0 with ny=E,+fnM3and n2=E,+fmM; ny and n, are the refractive indices of the parallel and perpendicular linearly polarized modes, respectively. The difference in phase velocities between these two waves gives rise to a magnetic linear birefringence(MLB) of light which is also known as the CM or Voigt effect. In this case, the nw-ny. The phase shift or retardation can be found by the following expressiong. y depends on the difference light transmitted through the crystal has elliptic polarization. The degree of ellipticit π(f1-f2)M2 rad/m 1.8(f1-f12)M3 legree/cm 入 Since the sense of this phase shift is unchanged when the direction of light propagation is reversed, the CM effect is a reciprocal effect. Kerr Effects Kerr effects occur when a light beam is reflected from a magnetooptic medium. There are three distinct of Kerr effects, namely, polar, longitudinal (or meridional), and transverse(or equatorial). Figure 57.1 shows the configurations of these Kerr effects. A reflectivity tensor relation between the incident light and the reflected light can be used to describe the phenomena as follows E (57.10) r21r22 where r: is the reflectance matrix. En and Ev are, respectively, the perpendicular(TE)and parallel (TM)electric field components of the incident light waves(with respect to the plane of incidence). E, and Em are, respectively, the perpendicular and parallel electric field components of the reflected light waves The diagonal elements Tu and r can be calculated by Fresnel reflection coefficients and Snells law. The off diagonal elements Tiz and T2 can be derived from the magnetooptic permittivity tensor, the applied magneti zation and Maxwell equations with the use of appropriate boundary conditions [Arecchi and Schulz-DuBois 1972]. It is important to note that all the elements of the reflectance matrix ri, are dependent on the angle incidence between the incident light and the magnetooptic film surface c 2000 by CRC Press LLC
© 2000 by CRC Press LLC The eigenmodes are two linearly polarized light waves polarized along and perpendicular to the magnetization direction: (57.8a) (57.8b) with n// 2 = er + f11M3 2 and n^ 2 = er + f12M3 2 ; n// and n^ are the refractive indices of the parallel and perpendicular linearly polarized modes, respectively. The difference in phase velocities between these two waves gives rise to a magnetic linear birefringence (MLB) of light which is also known as the CM or Voigt effect. In this case, the light transmitted through the crystal has elliptic polarization. The degree of ellipticity depends on the difference n// – n^. The phase shift or retardation can be found by the following expression: (57.9) Since the sense of this phase shift is unchanged when the direction of light propagation is reversed, the CM effect is a reciprocal effect. Kerr Effects Kerr effects occur when a light beam is reflected from a magnetooptic medium. There are three distinct types of Kerr effects, namely, polar, longitudinal (or meridional), and transverse (or equatorial). Figure 57.1 shows the configurations of these Kerr effects. A reflectivity tensor relation between the incident light and the reflected light can be used to describe the phenomena as follows: (57.10) where rij is the reflectance matrix. Ei^ and Ei// are, respectively, the perpendicular (TE) and parallel (TM) electric field components of the incident light waves (with respect to the plane of incidence). Er^ and Er// are,respectively, the perpendicular and parallel electric field components of the reflected light waves. The diagonal elements r11 and r22 can be calculated by Fresnel reflection coefficients and Snell’s law. The offdiagonal elements r12 and r21 can be derived from the magnetooptic permittivity tensor, the applied magnetization and Maxwell equations with the use of appropriate boundary conditions [Arecchi and Schulz-DuBois, 1972]. It is important to note that all the elements of the reflectance matrix rij are dependent on the angle of incidence between the incident light and the magnetooptic film surface. ˜ E x // exp // ( ) = j t n x È Î Í Í Í ˘ ˚ ˙ ˙ ˙ - Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ 0 0 1 2 0 w p l ˜ E x exp j t n x ^ ^ = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ - Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ ( ) 0 1 0 2 0 w p l y p l e l e cm r r f f M f f M @ - - ( ) . ( ) 11 12 3 2 0 11 12 3 2 0 1 8 rad/m degree/cm or E E r r r r E E r r i i ^ ^ È Î Í Í ˘ ˚ ˙ ˙ = È Î Í Í ˘ ˚ ˙ ˙ È Î Í Í ˘ ˚ ˙ // //˙ 11 12 21 22