1. Introduction

The magneto-optical (MO) effects describe the changes in the polarization of wave reflected (Kerr effect) or transmitted (Faraday effect) from magnetic materials, which are caused by the off-diagonal components of the dielectric tensor. Further, the MO effects can be catagorized by the magnetization direction or the location of off-diagonal elements in the dielectric tensor [1] as the polar, the longitudinal, and the transverse geometry. First, the polar geometry is the case when the magnetization vector is perpendicular to the reflection surface and parallel to the plane of incidence, on the other hand, in the transverse geometry the magnetization is perpendicular to the plane of incidence and parallel to the surface. At last, the magnetization vector in the longitudinal geometry is parallel to both the reflection surface and the plane of incidence. Nowadays, the MO effects are used in many applications, such as MO recording devices which yield a high-density recording at a low cost and whose capacity is still increasing exponentially [2].

Recently, the enhancement ofMO effect from the periodic gyrotropic materials, such as magnetic photonic crystals [3] and magnetic gratings [4, 5, 6], has been observed. In this paper, a rigorous numerical method in the frame of rigorous coupled-wave analysis (RCWA) [7, 8] for the calculation of the diffracted MO effects from gyrotropic or magnetic gratings is presented, and the diffractedMO enhancements of thin permalloy gratings in the polar geometry and of Ni gratings in the longitudinal geometry are studied. A gyrotropic material can be modeled with non-diagonal dielectric tensors in the Maxwell’s equations as aforementioned. In addition, a periodic nature of the grating yields a Bloch type wave solution. Based on these two facts, the field quantities and the dielectric tensor can be expanded in terms of Fourier series. The appropriately truncated expansions (N_max) are substituted into the time harmonic Maxwell’s equations, and the time harmonic Maxwell’s equation can be converted to a system of ordinary differential equations (ODE) with 2(2 N_max+1) unknown Fourier coefficients. Then, the ODE system can be diagonalized with the characteristic variables, and the solution can be written as a product of the initial value and the propagation matrix, which is a diagonal matrix with an exponential function whose argument is the eigenvalues of a coefficient matrix of the ODE system. In turn, using the boundary conditions either between homogeneous medium and grating or between grating and homogeneous layer, the reflection and the transmission matrices can be obtained. However, for a complex dielectric tensor and a deep grating, the convergence is very slow. In order to overcome the convergence issue, Antos et. al. suggested an Airy-like internal-reflection series expansions (AIRS) to consider multiple reflections in the grating region made of isotropic material [9, 10] and anisotropic material in the polar magnetization [11]. In addition, Li’s three Fourier-factorization rules [12] are utilized to overcome the bad convergence resulting from the product of two Fourier expansions of discontinuous function in the longitudinal geometry.

This numerical method can be applied to other applications such as a study on the extraordinary optical transmission and MO effects [13, 14], and combined with micromagnetic simulation tool such as OOMMF [15] for the study of magnetization evolution [16]. However, In [13], a perturbation method is used in order to include off-diagonal elements of the permittivity tensor and coupled mode theory with parabolic approximation, which limits the applicability of the method by the validity of the approximation. In contrast, RCWA implemented with AIRS does not take advantage of any approximation. As a consequence, it does not depend on any approximation theory. Additionally, our method is applied to both longitudinal and polar gratings and compared with the experimental data as oppose to Ref. [13, 14], which only presented the theoretical or numerical results for the polar geometry.

In next two sections, all the mathematical details of RCWA with AIRS for the polar magnetization are presented, and are extended to the longitudinal geometry. In the following section, the diffracted MO enhancements for the permalloy (polar magnetization) and Ni (longitudinal magnetization) gratings are studied with the proposed method and compared with experimental data taken from literatures. At last, paper is concluded with a summary together with the applicability of given method.

2. Maxwell’s equation for gyrotropic gratings

First, assume that a gyrotropic grating with period d is placed on the x-y plane and homogeneous along the x-axis and periodic along the y-axis (see Fig. 1). Then, the time harmonic Maxwell’s equations for the gyrotropic grating are specified by a complex non-diagonal dielectric tensor ε. Using the dimensionless spatial coordinate normalized by 2 π/λ, where λ is the wavelength, they can be written as

where H̃=cµ ₀ H. c the speed of light, and µ ₀ the magnetic permeability. ε(ȳ) is a periodic dielectric tensor depending ȳ, namely,

for the polar magnetization, and

 

Fig. 1. Gyrotropic grating with period d and grating depth h, and schematics for Airy like internal reflection series expansions in the grating.

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for the longitudinal magnetization. The grating vector is given by

where q=λ/d. Now, utilizing the periodic nature of gratings, the Bloch theorem can be applied for electric and magnetic fields as follows.

with

θ_i is the incident angle and n ₀ is the refractive index of incidence medium. The dielectric tensor can also be expanded in terms of the Fourier series

where α and β denote x, y or z. In next two subsections, the Bloch wave and the Fourier series are substituted into the Maxwell’s equations and the calculation detail for the polar and the longitudinal geometries is presented.

2.1. Polar magnetization

In case of the polar magnetization, the Maxwell’s equations can be rewritten in six partial differential equations as

First, let us consider Eq. (12). Substituting expansions (8) and (9) into Eq. (12) gives

and taking derivative and rearrangement yield

Then, multiplication of exp(iq_n′ȳ) on both sides and integration over the period give

Similar calculation procedure holds for Eqs. (13) and (14), and give

Now, consider Eq. (15). Substituting Eqs. (8), (9) and (11) into Eq. (15) gives

and taking derivative on the left-hand side of Eq. (23) yields

Again, multiplication of exp(iq_n′ȳ) on both sides of Eq. (24) and integration over the period result in

Now, isolate the derivative term on the left-hand side and set n′=n and n=l for convenience to be

Similar calculation procedure holds for Eqs. (16) and (17), leading to

Now, all six partial differential equations (12)~(17) are converted into a system of ordinary differential equations (20)~(22) and (26)~(28) for coefficients of the expansions and these equations can be rewritten into matrix form as follows.

where (e_j) and (h_j) are vectors consisting of coefficients of the electric and the magnetic fields

[q] is a diagonal matrix with entries composed of q_n and [ε_αβ] where α,β denote x, y or z, which is a Toeplitz matrix consisting of Fourier coefficients of ε_αβ, namely,

Solve Eqs. (31) and (34) for the normal components (h_z) and (e_z) in terms of the tangential components to have

Finally, substituting Eqs. (39) and (40) into Eqs. (29), (30), (32) and (33), and simplification yield four equations only for the tangential components in a matrix form

and Eqs. (39) and (40) for the normal components can be written as a matrix form

Moreover, Eq. (41) can be decomposed into

Taking derivative with respect to z̄ on Eq. (43) and substitution of Eq. (44) into the resultant equation give a second order ordinary differential equation for the tangential components of electric fields only

where f=(e_x,e_y)^t, g=(h_x,h_y)^t,

Equation (45) is a simple system of ordinary differential equation and the coefficient matrix can be diagonalized as

where Λ is a diagonal matrix consisting of the eigenvalues of matrix A, and T is a matrix with the corresponding eigenvectors. With this relation, Eq. (45) can be rewritten as

where f̃ is a characteristic variable defined by f̃=T ^-1 f. Then, a solution of Eq. (50) is

Thus,

where propagation matrix P is defined as

Substituting the solution of electric field into the equation for magnetic field Eq. (46) and simplification yield

where S is a diagonal matrix consisting of -√Λ. Now, applying the boundary conditions between layers (continuity of the tangential components)

where superscript i, r, t, 0 and 1 denote incidence, reflection, transmission, region 0, and region 1, respectively. Then, the reflection and the transmission matrices between region 0 and region 1 can be found as

where

Consider multiple reflections and transmissions as depicted in Fig. 1:

where

and

Then, new reflection and transmission matrices for the polar magnetization can be found to be

where Q=P ¹ R ¹⁰ P ¹ R ¹² and the geometric sum is calculated as

In summary, new reflection and transmission matrices come to be

where

2.2. Longitudinal magnetization

In case of the longitudinal magnetization, the time harmonic Maxwell’s equations can be written as

The same derivation strategies can be applied. Substitute all the expansions and organize the unknowns as in case of the polar magnetization. Then, again six equations for the electric and the magnetic fields in matrix form

are obtained with the same notations as the polar geometry. From Eqs. (85) and (88), the equation for the normal components are obtained to be

By substituting back into Eqs. (83), (84), (86) and (87), we have equations for the tangential components in a matrix form

and Eqs. (89) and (90) can be written as a matrix form

Now, apply the Fourier factorization and decompose Eq. (91) in terms of (e_x, h_x)^t and (e_y, h_y)^t as follows.

Taking derivative of Eq. (93) with respect to z̄ and substituting Eq. (94) give a second order ODE system on (e_x,h_x)^t.

where f=(e_x,h_x)^t, g=(e_y,h_y)^t and

 

Fig. 2. Theoretical and experimental Kerr rotation spectra in (a) the 0th and (b) the -1st order diffractions for the s-polarized incidence. Incidence angle was 7° for the 0th order. For the -1st order diffraction, the angle between incident and reflected beams was fixed to be 20°. N_max=10.

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Equations (95) and (96) for the longitudinal geometry are the same as the equations (45) and (46) for the polar geometry with different definition of unknowns f and g. Thus, In the similar manner as for the polar magnetization, the propagation matrix P and the matrix D, which relates the magnetic field and electric field at the interface, can be found and the transmission and reflection matrix can be derived by applying the boundary condition between layers. Finally, the AIRS can be applied and new transmission and reflection matrix can be written as

In the derivation, only three layers consist of homogeneous, grating and homogeneous media are considered. However, multilayer or surface relief grating formulation is a straightforward extension of the matrix product, and described in [9].

3. Numerical results and discussion

In this Section, the reflection [Eqs. (75 and (99)] and the transmission [Eqs. (76) and (100)] matrices are constructed for given material and geometrical parameters of both polar and longitudinal magnetization. Then, the Kerr rotation of the n-th order diffraction

where χ ⁽ⁿ⁾=f^r,ⁿ_y/f^r,ⁿ_x, is calculated for h>0 [17], and the logarithmic enhancement of the n-th order Kerr rotation, defined as

is calculated as a function of grating depth h. All the wavelength-dependent optical constants are taken from Refs. [18, 19, 20].

 

Fig. 3. Relative errors as a function of grating depth.

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3.1. Polar magnetization

For the polar magnetization, consider a grating made of polar magnetized Ni ₈₁Fe₁₉ permalloy gratings with period d=910 nm, width w=700 nm and height h=12 nm on top of 3 nm thick SiO₂ layer on a Si substrate with a 2 nm thick Cr₂O₃ capping layer. The material parameters ε_xx=ε_yy=ε_zz=-6.12-12.01i and ε_xy=-0.16-0.04i are used for permalloy and ε=4.84-0.58i,ε=2.12, and ε=15.06-0.16i are used for Cr ₂O₃, SiO₂, and Si at the wavelength λ=632.8 nm, respectively. In order to verify the method, the experimental results taken from Ref. [10] are compared with the numerical results in Fig 2. In the calculation, N_max=10 was enough to achieve a good convergence because the grating thickness is very small compared to the periodicity and the wavelength. A reasonably good agreement can be seen, and some

 

Fig. 4. Simulated Kerr rotation as a function of grating depth. Incident angle was set to be 7° for (a) the -2nd, (b) the -1st, (c) the 0th, (d) the 1st, (e) the 2nd. and (f) the 3rd orders with a truncation order N_max=50.

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Fig. 5. Logarithmic magnification of (a) the -2nd order A ^(-2), (b) the -1st order A ^(-1), (c) the 1st order A ⁽¹⁾, and (d) the 2nd order A ⁽²⁾ as a function of grating depth.

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discrepancies at low photon energies result from the inaccurate material parameters obtained from the free-electron Drude model. The relative errors for the 0th order,

where Nⁱ _max=10, 20, 30, 40, 50 and 60 for i=1,2, …, 6, respectively, are calculated to determine the truncation order N_max in Fig. 3. The relative errors are bounded by about 2% when the N_max is set to be 30 or more in a range of grating depth in our interest. Figure 4 displays the -2nd, the -1st, the 0th, the 1st, the 2nd and the 3rd diffraction order Kerr rotations, calculated as a function of grating depth h for a fixed wavelength of 632.8 nm with the s-polarized incidence. Figuare 5 shows A ^(-2), A ^(-1), A ⁽¹⁾ and A ⁽²⁾, that is, the enhancements of the -2nd, the -1st, the 1st and the 2nd order Kerr rotation, respectively. Clearly, the MO enhancements of the 1st and the -1st order Kerr rotation, compare to the 0th order, in a range of grating depth less than about 50 nm can be observed from Figs. 5(b) and 5(c), respectively. Especially, for h=12 nm, which is used in Ref. 2, the 1st order and the -1st order MO Kerr rotations are enhanced 4 and 5 times, respectively.

 

Fig. 6. Absolute errors as a function of grating depth.

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Fig. 7. Kerr rotations of the -3rd, the 0th, the 1st, and the 9th orders from the numerical calculation (black triangular dots) and the experiments (red square dots).

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Fig. 8. (a) The 0th, (b) the ± 1st, (c) the ± 2nd, and (d) the ± 3rd order diffracted Kerr rotations and (e) MO enhancement A ⁽ⁿ⁾ as a function of grating depth h for Ni grating.

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3.2. Longitudinal magnetization

Consider a Ni grating in the longitudinal magnetization. The material parameters are as follows; ε_xx=ε_yy=ε_zz=-13.2-16.5i and ε_xz=-0.24-0.02i on a Si substrate with d=20 µm, w=4 µm, and h=20 nm at an incident angle θ_i=45° of the s-polarized light with a wavelength λ=635 nm. The schematics of the grating is the same as Fig. 1, and the magnetization vector M⃗ in the grating region is parallel to the surface and directed along the y-axis. Again, the convergence was checked using the absolute errors $\mid {\theta }_K^{\left(n\right),N_{max}^{i+1}}-{\theta }_K^{\left(n\right),N_{max}^i}\mid$ for the 0th order Kerr rotation as a function of grating depth in Fig. 6. It should be noted that the relative error could not be calculated because θ ⁽ⁿ⁾ _K appears to be zero at some grating depths. The truncation order N_max=100 is selected for all the calculations in order to ensure, at least, an accuracy of 10^-5. Figure 7 shows the experimental data from Ref. [21] and the simulation results. For the comparison, the saturated values for the 0th and the 1st order diffraction and the maximum values of the -3rd and the 9th order are taken from the hysteresis loop (owing to an unsaturated range of magnetic field). A reasonably good agreement between theory and experiment can be observed. The discrepancies between theoretical calculation and experiments for the -3rd and the 9th orders are believed to result from the unsaturated experimental data. Again, the Kerr rotation enhancement is calculated, and Figs. 8(a) through 8(d) depict the Kerr rotations of various orders (the ±2nd, the ±1st and the 0th order) for the s-polarization. Figure 8(e) shows the enhancements of the ±1st and the ±2nd diffraction order Kerr rotations as a function of grating depth. Siginificant enhancements of all the orders can be observed around 50 and 500 nm, where the 0th order Kerr rotation becomes very close to zero or changes the sign.

4. Conclusions

A detail of the RCWA with AIRS and Fourier factorization for the polar and the longitudinal magnetization is presented, and the enhancements of diffracted MO Kerr effect for the high-order diffracted beams from thin polar and longitudinally magnetized gratings are numerically shown. The experimental results on permalloy and Ni gratings also support the diffracted MO enhancements and are coincident very well with the numerically calculated values. The MO enhancements of the thin grating can be beneficial for the miniaturization and/or the integration of optical devices, and robust numerical technique can assist us in designing the structures and selecting materials for the MO devices with appropriate properties.