IntroductionExperimental observation of transverse MO effectProperties of transverse MO effectOrigin of transverse MO effectTransverse EllipticityTwo contributions to transverse MO effectMagnetizationdependent optical lossCalculations of transverse MO effect in the case of multilayer structureOptical excitation of spinpolarized electrons utilizing transverse MOPlasmonsGiant Enhancement of Transverse MO effectHistory and Future 
Transverse MagnetoOptical effect
Rigorous solution for transverse MO effect in multilayer structuresA rigorous solution of Maxwell's equations for transverse MO effect in multilayer structures is obtained. A derived scalar dispersion relation (14) describes optical plasmons, waveguide modes and transverse Kerr effect in multilayer structure. It is an example of method, which I have developed for solution of Maxwell's equations in multilayer structure. It is very simple, but a very effective and powerful method. It allows simple quick and rigorous calculations of the transverse MO effect even for a structure with large number of layers. Rigorous calculation means that any assumptions or approximations are not used.I will derive the dispersion relation, which describes the transverse MO effect in cases of waveguide modes and surface plasmons propagating in a multilayer MO slab. Since the permittivity tensor of a MO material has nonzero offdiagonal components, the wave propagation in MO structure is conventionally described by a dispersion relation, which is a combination of (4x4) matrixes. Because of the complexity of this dispersion relation, the approximation of small offdiagonal components is often utilized. However, in the case of the transition metals, the offdiagonal components have the same order of magnitude as the diagonal components and that approximation is not always valid. In the following, without use of any approximations we will derive a scalar dispersion relation, which described the transverse MO effect in a multilayer MO slab. The availability of the simple scalar dispersion relation significantly simplifies analysis and predictions for the transverse MO effect. Let’s us consider a multilayer MO slab, where and tj are the permittivity tensor and the thickness of each jlayer, respectively. The layers of the slab are infinite in the xyplane and the wave propagation direction is along zdirection. Solution of Maxwell's equations for each jlayerSolving Maxwell’s equations for a plane wave we have The solution of (5) and (4) is split per two independent solutions for TE polarized and TM polarized waves. The TE wave does not experience the transverse magnetooptical effect and only TM polarization will be considered below. The solution of first and third equations of (5) is The optical field in jlayer can be described as where Af and Ab are unknowns and c.c. is complex conjugative. Introducing new unknowns the (8) is simplified to where Boundary conditions
Applying boundary conditions at boundary between j and j+1 layers, from (8a) we have where solving (10 a) gives From (10c) Zj can be found. Therefore, a value of Z in jlayer can be found knowing value of Z in (j+1)layer where Solution for multilayerIt is important that the value of Z in the last nlayer is always known.Since there is no back traveling wave Ab=0 when , it leads to Abn=0 and Zn=1. Knowing the value for Z for nlayer, from (11) the value for (n1)layer can be found. Knowing the value for Z for (n1)layer, the value for (n2)layer can be found and so on. Therefore, the general solution describing transverse MO effect in the case of a plain wave propagating in a MO multilayer slab is (14) is main result. The Eqn. (14) can be used to derive reflectivity of MO multilayer and dispersion relation for surface plasmons and waveguide modes. Below we will show how to apply (14) for different structures.
Transverse Kerr effectIn this case we will find the refractivity of a plane wave from a multilayer MO structure. The refractivity is different for two opposite direction of the magnetic field. The reversing of the magnetic field corresponds to the reversing of gama sign . Let us consider a plain wave propagates in nonmagnetic layer j=1 and is reflected by a MO multilayer (Fig.1). alfa is incident angle, so and the optical field in layer j=1 is described as where is intensity of incident light and is intensity of reflected light. The value of kz will be the same in all layers kx will be in the last nlayer optical field will be condition of finite optical when (3), the imaginary part of kx in last layer should satisfy From (9) for nlayer Zn=1 and solution (14) will be from (9) Therefore, the reflectivity will be Also, the light will experience a phase shift during reflection Making simple simplifications as Using (23 b), the (23) is simplified to The (20), (21) and (23c) describe the reflection of the wave from multilayer structure.
Surface plasmons and waveguide modesLet us consider the case of a surface plasmon or a waveguide mode, which propagates in multilayer structure shown in Fig.2 and having effective refractive index kz. kx for each layer can be found from (17). The optical field in the last layer j=n is described by (18) , the imaginary part of kx in this layer should satisfy (19) and Zn=1. The optical field in the first layer j=1 is described as From condition of finite optical field when From (9) and (24) for layer j=1, Z1 equals 1 and calculating Z1 from (14), the dispersion relation for a surface plasmons or waveguide modes in multilayer structure of Fig.2 is The (26) is the dispersion relation describing a waveguiding mode or a surface plasmon propagating in multilayer structure.
Waveguide with singlelayer core and multilayer coverIn the case a multilayer waveguide, the dispersion relation (26) may have several solutions corresponding to each mode number. In the case if only the cover layer of the waveguide is a multilayer and the waveguide core consists of only one layer (See Fig.2), the (25) can be simplified and dispersion relation describing each mode can be found.
where Z1 determined by (20) and m is a mode number.

I will try to answer your questions as soon as possible