1. Introduction

It is well known that a magneto-optical (MO) enhancement can be realized in magneto-photonic crystals (MPCs) [1, 2, 3, 4, 5], which are very promising for applications in modern photonics. In contrast to the conventional photonic crystals, MPCs possess an additional degree of freedom, spin, so that their optical and MO properties can be tuned with applied magnetic field [6, 7]. In such structures, the time reversal symmetry is broken down and entails nonreciprocity in the system [6, 8, 9]. As a result, they are expected to be a good candidate to provide sufficient signal isolation and suppress parasitic reflections between devices in on-chip large-scale integration. More importantly, one of the advantages of MPCs over the conventional MO devices is that MPCs need much less propagation distance to occupy a large footprint because the weakness of MO effects [10] is overcome by light localization in the vicinity of the defects with a consequent increase in the mean optical-path length of the emitted light [4].

The gyrotropic gratings and the MPCs are different in terms of the geometry for applications, as well as the theoretical model. For example, light generally propagates along the periodic direction of alternating magnetic and nonmagnetic layers in MPCs. For gyrotropic gratings, light is illuminated upon the patterned surface, and more stress is laid on the diffraction efficiency [11, 12, 13], the magnetization reversal process [14, 15] and so on. In fact, one definition includes another, and vice versa [7, 16, 17, 18], because both of them make use of periodic magnetic structures to modulate the MO effects. Antos et al. [19, 20] have observed a MO enhancement experimentally at the first-order diffraction from gyrotropic gratings. However, unlike Kim et al. [16, 18], they did not focus on the significance of the observation. Anyhow, it was supposed that gyrotropic gratings share some of the aforementioned advantages of MPCs. It would be of great significance if the MO enhancement is also able to be obtained through gyrotropic gratings. A huge decrease in thickness compared with the conventional MPCs, from the micrometer level down to the nanometer one, benefits the miniaturization and the integration of MO devices considerably.

In this paper we focus on the demonstration of the geometrical effects on the diffracted MO enhancement utilizing a rigorous theoretical model to achieve trustful and predictable analyses. The influences of the depth of grooves on the diffracted MO enhancement are systematically investigated in order to explore the possibility of miniaturizing and integrating the MO devices. Furthermore, the diffracted MO Faraday spectra are also presented theoretically.

2. Theoretical approach

For the calculation of diffracted MO effects, the rigorous coupled-wave approach (RCWA) implemented as an Airy-like internal-reflection series (AIRS) [21] was employed. Unlike isotropic gratings in Ref. [21], lossy anisotropic gratings with multilayered structures for a polar MO configuration are taken into account, as shown in Fig. 1.

 

Fig. 1. (Color online) Diffracted MO effects from a one-dimensional multilayered gyrotropic grating for the polar magnetization.

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The governing equation is the time-harmonic Maxwell’s equations, for convenience, in which the space coordinates with the free-spacewavenumber k ₀ are transformed to dimensionless ones by setting r̄=k ₀ r:

where H̃=cµ ₀ H. c and µ ₀ represent light velocity in vacuum and magnetic permeability in vacuum, respectively.

The components of the wave vector in periodic media are defined as

where λ, d, and θ_i denote the wavelength of the incident beam, the period of the grating, and the incident angle, respectively.

Using the Floquet theorem and assuming only propagation in the direction of the increasing z coordinate, the propagation eigenmodes of electromagnetic fields are expressed in terms of pseudo-Fourier series:

where k denotes x or y. The permittivity is a periodic function of only one lateral coordinate y with periodicity d:

where [ε_w] and [ε_b] are the permittivity in wires with the width of w and spaces between them, respectively. For the wires made of magnetic materials, [ε_w] is a 3×3 tensor with off-diagonal elements, whose positions are dependent on the direction of magnetization with respect to the incident plane. In the case of polar magnetization, the permittivity is written as below:

The components of the y-dependent permittivity can be expressed in terms of Fourier series:

where α and β represent any of x, y and z. Substituting Eqs. (5), (6) and (9) into Maxwell’s equations, a system composed of ordinary differential equations are obtained as

where [ε_αβ] is a Toeplitz matrix consisting of Fourier coefficients ε _αβ,n and the tangential column vectors are denoted to f _x, f _y, g _x, and g _y composed of f_k and g_k in Eqs. (5) and (6). Here q is a diagonal matrix in the form of

Eventually, matching the boundary conditions at each interface and considering multiple reflections with AIRS, the diffracted MO Kerr rotation can be calculated [22] for the nth order as follows,

where χ ⁽ⁿ⁾ is estimated by the ratio of reflection amplitudes f ^r _y,n/f ^r _x,n. The procedure of modeling diffracted MO Faraday effect is almost identical to the case of the diffracted MO Kerr effect except for replacing the ratio of reflection amplitudes f ^r _y,n/f ^r _x,n with that of transmission amplitudes f ^t _y,n/f _{t x,n}.

In the actual calculations, the infinite sums of the pseudo-Fourier and Fourier series are replaced with ${\sum }_{n=-n_{max}}^{+n_{max}}$ in Eqs. (3), (4) and (7). Thus, the truncation order n_max is chosen carefully to guarantee a sufficient convergence.

In the analysis, the grating period and the width of gyrotropic rod are 910 and 700 nm, respectively. The thicknesses of Cr₂O₃ capping layer, Ni₈₁Fe₁₉ permalloy layer, and SiO₂ oxide layer on the substrates are 2, 12, and 3 nm, respectively. All the wavelength-dependent optical constants are taken from the literature [23, 24, 25] considering absorption.

3. Results and discussion

Figure 2 presents the theoretical MO spectra, together with the experimental ones of Ref. [19], for the zeroth- and the first-order diffractions. As can be seen in Fig. 2, the overall agreement between the experiment and the theory is good except for some small discrepancies at low photon energies, which is attributed to the free-electron Drude components of the optical constant being sensitive to the effect of the reduced dimensionality in the nanoscale systems [19, 21]. Comparing Fig. 2(a) with Fig. 2(b), the magnitude of Kerr rotation in the zeroth order is one order smaller than that of the first order at most of the energy range. It must, however, be noted that the two Kerr spectra were measured at different incident angles.

 

Fig. 2. (Color online) Theoretical and experimental Kerr rotation spectra in the 0th (a) and the -1st (b) order diffractions for s-polarized incidence. Incident angle was 7° for the 0th order. For the -1st order diffraction, the angle between incident and reflected beams was fixed to 20°. A truncation order of n_max=10 was sufficient enough for the good convergence.

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In order to investigate the MO enhancement in gyrotropic gratings more accurately, the incident angle and wavelength were fixed to 7° and 632.8 nm, that of a He-Ne laser. The diffracted MO enhancement is defined as the magnification M as below:

 

Fig. 3. Simulated Kerr rotation as a function of grating depth. Incident angle was set to 7° for both (a) the 0th and (b) the -1st orders. The truncation order n_max was 50 because of the increase in the grating depth.

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Fig. 4. Magnification M as a function of grating depth.

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where θ ^(-1) _Ks and θ ⁽⁰⁾ _Ks are the Kerr rotation of the first- and the zeroth-order diffraction with s-polarized incidence, respectively. It is possible to observe the MO responses in the zeroth- and the first-order diffraction according to the grating equation [26], for both transmission and reflection, as below:

where θ_D and N represent the diffraction angle and order, respectively. In the following calculations, the effects of the capping and the oxide layers are excluded to investigate the diffracted MO effects from pure gyrotropic gratings. The simulated Kerr rotation is shown in Fig. 3 as a function of the grating depth varying from 10 to 500 nm. The magnitudes of the Kerr rotation in two diffraction orders vary in opposite ways with the increase in the depth even though a fluctuation is found in the depth greater than 100 nm. Obviously, we can suppose that the diffracted MO enhancement through gyrotropic gratings comes out only when the grating is thin compared with the grating period and the width of gyrotropic rod, as presented in Fig. 4, where M monotonously decreases down to zero with depths. It is thought that the exaltation of the internal diffraction-edge effects in thick gratings suppresses the MO effect in high orders. The maximum Kerr rotation is even up to ~36°/µm for the 10 nm grating in the first-order diffraction. The thickness is reduced from the micrometer level down to the nanometer level compared with conventional MPCs to achieve such a huge MO effect. Also in the form of the single layer with the same thickness the estimated Kerr rotation is merely ~5°/µm by the propagation-matrix method [27]. As a result, the realization of the diffracted MO enhancement in thin gyrotropic gratings is of great advantage to the the miniaturization and integration of devices.

In addition to the diffracted MO Kerr effect, the spectra of the diffracted MO Faraday effect is theoretically evaluated for the first time, as shown in Fig. 5. The MO enhancement can also be realized with the Faraday effect as well as the Kerr effect and the magnification is very close to the one illustrated in Ref. [16] experimentally.

 

Fig. 5. Simulated spectra of Faraday rotation in (a) the 0th and (b) the -1st orders. The grating depth was set to 12 nm and the truncation order n_max was equal to 10.

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4. Conclusions

The RCWA implemented as AIRS was utilized to model the diffracted MO effects in one-dimensional gyrotropic gratings. The theoretical and the experimental diffracted MO spectra indicate that there exists a consistency in the zeroth- and the first-order diffractions for a polar magnetization. For the diffracted MO Kerr effect, the calculation illustrates that the maximum of the MO Kerr rotation is even up to ~8 times larger than that of the 10 nm single layer, which is favorable to miniaturize and integrate MO devices. Moreover, the MO enhancement is also demonstrated theoretically in the diffracted MO Faraday effect for the first time. It is believed that the absolute magnitudes and the MO magnification can be improved further by elaborately selecting the grating structures and the gyrotropic materials.