1. Introduction

Magneto-photonic crystals (MPCs) are a special kind of photonic crystals (PCs), constructed of magnetic materials [1–3]. The first report arrived at the end of the 1990s, and at the beginning of the new millennium activity in this area is growing rapidly. One of the attractive points of MPCs is the enhancement of magneto-optic effects seen in one-dimensional MPCs with defects, such as the Faraday and the Kerr effects [1, 4, 5]. These command attention as one of the effective materials for miniaturization of optical isolators and incite studies toward practical use [6]. At the same time, research on two-dimensional (2-D) MPCs has been undertaken recently [7–12], together with expansion of the capability for a new class of devices more suitable for integration. This includes planar nonreciprocal devices specific to 2-D MPCs, such as optical isolators based on line-defect waveguides [13, 14] and circulators based on micro cavities [15], which have been demonstrated theoretically.

Most of this work, however, was based on idealized models of 2-D MPCs of infinite height, i.e. structures periodic in two dimensions and invariant in the third. Naturally, any adequate realization of an MPC structure cannot be of infinite height, and so investigations for three-dimensional (3-D) models are necessary to characterize, in a rigorous way, realistic MPC structures with finite heights. For reciprocal PC structures, in fact, a significant amount of research based on 3-D analysis has been conducted, e.g. shifts of dispersion relations due to confinement of light [16], modes with higher order vertical oscillations [17], out-of-plane scattering losses above the light lines [18], propagation losses due to out-of-plane structural asymmetries [19], etc. In contrast, for MPC structures, few studies based on 3-D analysis are conducted, and these fail to take into account nonreciprocal effects. Although Mondal and Stadler [9] used a 3-D finite-difference time-domain method to characterize optical isolators with MPC waveguides, the nonreciprocal effects of these devices are caused by the ridge waveguides connected to the MPC waveguides. The MPC waveguides operated as attenuators which offer significant losses for only one polarization.

In this paper, we present the first 3-D analysis for nonreciprocal phase shifts in MPC-slab waveguides, including consideration of losses. In particular, we investigate the significance of the waveguide geometry for nonreciprocal phase shifts and losses, to identify several general trends that will become useful for efficient nonreciprocal circuits. Nonreciprocal phase shifts are among the most widely-investigated concepts for the design of integrated nonreciprocal devices [20, 21], along with Faraday rotations and nonreciprocal loss shifts [22, 23]. While the idea of nonreciprocal phase shifts has great advantages (e.g. mode conversion and loss compensation are unnecessary), waveguides with relatively long lengths are required due to the weakness of this effect. For this reason, loss reduction is another important consideration. Our study is based on a 3-D finite element method (FEM). The FEM provides a flexible way to model inhomogeneous and complex structures with arbitrarily anisotropic materials (Application examples in reciprocal PC structures can be found in [24, 25].), and it can be effectively applied to the analysis of MPC waveguides.

The paper is structured as follows. In sec. 2, we first outline how we evaluate nonreciprocal phase shifts and losses in MPC waveguides. In sec. 3, we then discuss the dependence of nonreciprocal phase shifts on the waveguide width and the slab thickness. We also discuss the dependence of losses on the cladding thickness and the hole depth. Finally, the findings are summarized in Section 4.

2. Evaluation method

The following description relates to one period of the MPC waveguide shown in Fig. 1. Modes propagate along the z-axis. The entire waveguide is magnetized parallel to the x-axis. A compensation wall [26–28] divides the MPC waveguide into two equal halves with opposing directions of the magnetization. This is the basis of the nonreciprocal effect. The anisotropic perfectly matched layers (PMLs) [29] are employed to truncate the computational domain in the x- and y-directions. The relative permittivity tensor can be written as

where n is the isotropic refractive index. ξ represents the gyrotropy by the relationship

where Θ_F is the Faraday rotation and k ₀ is the free-space wavenumber. All through this paper, unless stated otherwise, we assume that n and ξ are real numbers (the materials of the waveguides are lossless).

Nonreciprocal phase shifts are given as Δβ = β _fw - β _bw, which is the difference between the propagation constants β _fw and β _bw for transverse-electric (TE)-like modes (with the dominant electric field E_y) propagating in the positive and the negative z-direction. To determine Δβ, we calculate separately the two modes corresponding to each propagation direction: after obtaining the forward propagating mode by setting the directions of the magnetization, i.e. the signs of ξ, as shown in Fig. 1, we calculate the backward propagating mode by reversing these signs.

The mode propagation in each direction is described by the vector wave equation:

with

where E and H are the electric and the magnetic field vectors, respectively. [ε]_PML and [μ]_PML are the permittivity and the permeability tensors considering the PML regions, respectively [29]. The electromagnetic fields can be represented as

where β is the propagation constant.

The analysis region is divided into curvilinear tetrahedral edge elements [30]. The advantage of curvilinear elements lies in the fact that they can model curved surfaces with more accuracy and a lesser number of unknowns than rectilinear elements. Cylindrical hollow surfaces in photonic crystals can thus be modeled exactly at low computational cost. The vector field within each element can be expressed as

where {N} is the shape function vector, {ϕ}_e represents the vector of the electromagnetic fields along the edges, and the superscript T denotes transposition. We choose the second-order basis function [31], because higher order basis functions are more suitable than the zeroth- or first-order basis function, taking into account rapidly varying fields due to strong periodic modulation comparable in size to optical wavelengths.

Applying the usual finite element procedure to Eq. (3) with the help of Eqs. (6) and (7), we obtain the following matrix equation:

with

where {ϕ}_i is the field vector on the boundary Γ_i at z = 0 or a, and Σ_e and Σ′_e1 denote the sum over all elements in the whole region Ω and on the boundary Γ_i respectively. In the assembly process of Eq. (8), the presence of the boundary integral on the right-hand side is required only if the target element has edges bordering the boundaries Γ₁ and Γ₂, because PMLs are employed for the other boundaries.

Appling the periodic boundary conditions based on the Floquet theorem for the boundaries Γ₁ and Γ₂, we finally obtain the following eigenvalue equation:

with

where A represents K or M, the subscripts 1 and 2 denote the vector or matrix related to Γ₁ and Γ₂, respectively, and the subscript 0 denotes that for the remaining region.

The method of inverse iteration with shifts of origin can be efficiently applied to solve this eigenvalue equation. The matrix equations to be solved in this iterative procedure are sparse complex unsymmetric linear systems of equations, which can be solved with appropriate software, such as PARDISO library [32] with METIS algorithm [33].

In this formulation, a complex frequency of the eigenvalue is obtained from a predefined propagation constant. To determine nonreciprocal phase shifts, we estimate the real parts of the propagation constants at the frequency to be investigated, by applying the cubic spline interpolation to the dispersion curves calculated by the FEM. To determine wave decays in space, on the other hand, we obtain the imaginary part of the propagation constant by the following equation [34, 35]

where ω is the angular frequency and v_g is the group velocity. Using this value, we estimate the propagation loss, 20|Im(β)|log₁₀ e.

 

Fig. 1. (a) Schematic representation, (b) top cross sectional view, and (c) side cross sectional view of a 2-D MPC-slab waveguide with a compensation wall. The parameters used in sec. 2 are the hole radius r = 0.25a and the Faraday rotation Θ_F ≈ 3000°/cm; the refractive indices of the core layer and the cladding layer are n = 2.3 and 1.45, respectively.

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3. Results and discussion

We consider an MPC-slab waveguide with hexagonally arranged air holes, as shown in Fig. 1. The air holes extend from the upper cladding layer to the lower, through the core layer. To investigate the dependence of nonreciprocal phase shifts as a function of the waveguide width W and the slab thickness h ₂, we begin with an ideal geometry where the thickness of cladding layers is infinite, as is the depth of air holes (h ₁ → ∞ and h ₃ → ∞).

We assume a typical magnetic garnet with n = 2.3 and ξ = ±0.005 (i.e. Θ_F ≈ 3000°/cm) [36] as the core material. The refractive index of the cladding layers are low so that n = 1.45, to ensure a reasonably wide PBG below the light line. The air hole radius is assumed to be r = 0.25a, where a represents the lattice constant. Such small air holes also enable a relatively wide PBG below the light line, because, in PC geometries, band-gaps shift to lower frequency ranges as air holes become smaller. Since this model has a mirror-plane symmetry in x-direction, we restrict our analysis to half of the total region by applying a perfect magnetic conductor (PMC) on the yz-plane at the middle of the core layer.

As the first stage of our investigation of this geometry, we examine the relationship of nonreciprocal phase shifts to the waveguide width W. Figures 2(a) and 2(b) show nonreciprocal phase shifts and dispersion curves, respectively, for a constant slab thickness, h ₂ = 0.8a, and for varied waveguide widths, W = 0.5W ₀, 0.6W ₀, 0.7W ₀, and 0.8W ₀ with W ₀ = √3a. While the dispersion curves are shown for the forward propagation, we verified that those for the backward propagation exhibit rather similar tendencies. We calculate the dependence of the nonreciprocal phase shifts on the wave vector, βa/(2π), changing the lattice constant a to set a constant wavelength, λ = 1.3 μm. Clearly, the nonreciprocal phase shifts become larger as the waveguide becomes thinner. This is because magnetic fields are strongly confined in the core region. Figures 3(a), 3(b), 3(c), and 3(d) show cross-sectional distributions of the dominant, x-component of the magnetic fields on the y = 0 plane, along the compensation wall, (shown in the upper figures) and the x = 0 plane (shown in the lower figures) for (a) W = 0.5W ₀ and βa/(2π) = 0.485, (b) W = 0.8W ₀ and βa/(2π) = 0.46, (c) W = 0.5W ₀ and βa/(2π) = 0.495, and (d) W = 0.5W ₀ and βa/(2π) = 0.475, respectively, identified in Fig. 2(b). For quantitative comparison, the normalized magnetic field amplitudes |H_x| at x = 0 and z = 0, respectively, on the y = 0 plane in one period of the waveguide, are also represented in Figs. 3(e) and 3(f). The horizontal dashed lines in Figs. 3(a), 3(b), 3(c), and 3(d) and the vertical dashed and chain lines in Fig. 3(f) indicate the boundaries between the core layer and the cladding layers. Through perturbation theory [28], we recognize that the nonreciprocal phase shifts in magneto-optic rib waveguides with a compensation wall can be evaluated from the integration of the field intensities along the boundaries at which discontinuities of magnetization occur. The main factor contributing to the nonreciprocal phase shifts in the MPC waveguides can be analogized as the field intensity along the compensation wall, because large proportions of fields are confined in the core region (small proportions leak into air holes), as shown in the field distributions on the x = 0 plane (see the lower figures in Figs. 3(a) to 3(d)). While the integration of the fields along the z-axis is similar for every parameter as shown in Fig. 3(e), the integration along the x-axis in the core region for Fig. 3(b), W = 0.8W ₀, is smaller than that for Fig. 3(a), W = 0.5W ₀, due to the thinner core layer, as shown in Fig. 3(f). From this difference, we can understand the difference in nonreciprocal phase shifts in Fig. 2(a). In addition, this result confirms the previously reported tendency, with 2-D analysis, for MPC waveguide couplers in optical isolators to shorten as the waveguide width reduces or the air holes become larger [13]. On the other hand, the fields for Fig. 3(c), βa/(2π) = 0.495, and Fig. 3(d), βa/(2π) = 0.46, are highly similar, although the nonreciprocal phase shifts for W = 0.5W ₀ increase when the mode approaches the light line. In fact, this increment is caused by a different factor; we shall discuss this point in the final stage of this section with Fig. 7.

 

Fig. 2. (a) Nonreciprocal phase shifts and (b) dispersion curves for an MPC waveguide with air holes of infinite depth, a constant slab thickness, h ₂ = 0.8a, and different waveguide widths, W. In the calculation of (a), the lattice constant a is adjusted to set the wavelength as λ = 1.3 μm.

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Fig. 3. (a), (b), (c), and (d) Cross-sectional distributions of the x-component of the magnetic fields on the y = 0 plane (shown in the upper row) and the x = 0 plane (shown in the lower row) for each parameter set identified in Fig. 2. (b). (e) and (f) Normalized magnetic field amplitude |H_x| at x = 0 and z = 0, respectively, on the y = 0 plane.

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As the second step, we examine the relationship of nonreciprocal phase shifts to the slab thickness h ₂. Figures 4(a) and 4(b) show nonreciprocal phase shifts and dispersion curves, respectively, for different slab thickness, h ₂ = 0.6a, 0.8a, and 1.0a. For comparison, Fig. 4 includes the results of 2-D analysis (shown as h ₂ → ∞), modeling the MPC waveguide with a core layer of infinite thickness along the x-direction. The effect of h ₂ on nonreciprocal phase shifts is clearly small. This indicates that the slab with these thicknesses confines magnetic fields enough to saturate the nonreciprocal phase shifts. The nonreciprocal phase shifts would decrease if the slab became thinner. In this case, however, the dispersion curve shifts to a higher frequency range and the band widths of the PBG below the light line reduce and can even vanish. For this reason, we did not consider values smaller than h ₂ = 0.6a. Incidentally, these results of 3-D analysis show values similar to those of 2-D analysis. With regard to the waveguide widths in this analysis, these are adjusted to a constant value, W = 0.56 W ₀, so that the lower frequency edges of the dispersion curves shift close to the lower edges of the PBGs, to widen the low-loss band below the light line [37]. In Figs. 2(a) and 4(a), there are a few deviations; these are thought to be due to variations of the degree of the coupling between TE and TM components and are currently under investigation.

 

Fig. 4. (a) Nonreciprocal phase shifts and (b) dispersion curves for an MPC waveguide with air holes of infinite depth, a constant waveguide width, W = 0.56W ₀, and different slab thicknesses, h ₂. h ₂ → ∞ denotes the results of 2-D calculations.

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Our discussion to this point assumes an ideal structure with infinite thickness of claddings and infinite depth of air holes. Based on these assumptions, the guided modes below the light line exhibit no loss in theory. In actual PC structures, however, the depth of air holes and the thickness of upper cladding must be finite, because these are fabricated by etching techniques in most cases. We therefore discuss, in the following paragraphs, the radiation losses, L_r, caused by finite depth of the air holes and finite thickness of the upper cladding layer, leading to a guideline for design.

In the analysis of the radiation losses L_r due to the finite size of the structure, we assume that the material absorption is negligible, ξ = 0, because the effect of ξ on the losses L_r is so small as to be insignificant. This allows us to restrict our analysis to half of the total region by applying a perfect electric conductor (PEC) on the xz-plane at the middle of the core region (i.e. the y = 0 plane).

The first step of our investigation of this geometry is to examine the relationship of the radiation losses L_r to the thickness of the upper cladding layer h ₁, keeping the depth of the air holes infinite (i.e. h ₃ → ∞). Figures 5(a) and 5(b) show radiation losses and dispersion curves, respectively, for different thicknesses of the upper cladding layer, h ₁ = 0.1a, 0.3a, 0.5a, and h ₁ → ∞. Even a thin cladding layer as with h ₁ = 0.3a can suppress the loss sufficiently, though the losses arise when the thickness is reduced as h₁ = 0.1a. These losses are caused by coupling between TE defect modes and transverse-magnetic (TM) slab modes (with the dominant electric field E_x), which is in turn due to the structural asymmetry of the vertical direction [19].

 

Fig. 5. (a) Losses and (b) dispersion curves for an MPC waveguide with air holes of infinite depth, the slab thickness h ₂ = 0.8a, the waveguide width W = 0.56W ₀, and upper cladding layers of different thicknesses, h ₁.

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The second step is to examine the relationship of the losses to the depth of the air holes. Figures 6(a) and 6(b) show radiation losses and dispersion curves, respectively, for a constant thickness of the upper cladding layer, h ₁ = 0.3a, and for varied depths of the air holes, h ₃ = 2.0a, 3.0a, 4.0a and h ₃ → ∞. Figure 6(b) includes two light lines; one is for the PC cladding (with periodically arranged air holes) and one is for the uniform cladding. In the region below the light line for the uniform cladding, the losses are reasonably low for every value of h ₃. On the other hand, in the region below the light line for the PC cladding and above that for the uniform cladding, the losses increase as the air holes deepen (h ₃ becomes larger). This is compatible with the theory that out-of-plane losses are caused by the coupling of guided waves to radiation modes in claddings, and the coupling is larger for structures with shallower air holes. Figure 6 indicates air holes with the depth of h ₃ = 3.0a are needed to sufficiently suppress the losses for the reasonable range. Even in this case, however, it is difficult to suppress the losses just below the light line. The drawback due to these losses is much larger than the benefit due to the enhancement of nonreciprocal phase shifts for a number of parameters seen in the region just below the light line in Figs. 2(a) and 4(a). Operation well below the light line, therefore, is suitable for nonreciprocal devices with the waveguides.

Finally, we analyze the nonreciprocal phase shifts in an MPC waveguide with air holes of a finite depth and an upper cladding layer of a finite thickness (i.e. the structure precisely as depicted in Fig. 1), using the parameters h ₁ = 0.3a and h ₃ = 3a, which have been proven to be reasonable for loss reduction in the above examination. Figures 7(a) and 7(b) show nonreciprocal phase shifts and dispersion curves, respectively, for the waveguide width W = 0.56W ₀ and the slab thickness h ₂ = 0.8a. For comparison, Fig. 7 contains the above calculated results for the structure with air holes of infinite depth, like those plotted in Fig. 4 for h ₂ = 0.8a. Well below the light line, the nonreciprocal phase shifts for the structure with the finite air holes are lower than those for the structure with the infinite air holes, although the difference is as small as 0.1 dB/mm. This results from the loss of the in-plane mirror symmetry in the structure with the finite air holes; this asymmetric geometry can cause higher polarization mixing than that in the symmetric geometry with infinite air holes. Nonreciprocal phase shifts due to vertical compensation walls occur only for TE modes. Therefore, the phase shifts are dependent on the degree that these modes deviate from a pure TE state; the mode in this asymmetric waveguide is largely TE-like, thereby still supporting large nonreciprocal phase shifts, close to those in the symmetric waveguide. On the other hand, the phase shifts increase near the light line at different rates in both structures. These increments occur because the TE-like guided modes couple to radiation modes just above and roughly parallel to the light line, and the frequencies at which this coupling yields large group-velocity dispersion are different, depending on the propagation directions; this mechanism is represented conceptually in the inset of Fig. 7(b). The wave vectors, βa/(2π), at which the nonreciprocal phase shifts increase are therefore different according to the geometry, because of the difference in intersections of the dispersion curves for the guided modes and those for the radiation modes.

 

Fig. 6. (a) Losses and (b) dispersion curves for an MPC waveguide with the upper cladding thickness h ₁ = 0.3a, the slab thickness h ₂ = 0.8a, the waveguide width W = 0.56W ₀, and air holes of different depths.

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Fig. 7. (a) Nonreciprocal phase shifts, figures of merit, and (b) dispersion curves for an MPC waveguide with the waveguide width W = 0.56W ₀ and the slab thickness h ₂ = 0.8a. The inset in (b) is a conceptual diagram representing the mechanism of the increments in nonreciprocal phase shifts near the light line.

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In our discussion of the nonreciprocal efficiency, we calculate a figure of merit, which we defined as nonreciprocal phase shifts divided by propagation losses, FOM = Δβ/L _p, for the structure with the finite air holes, as shown in Fig. 7(a). The figure of merit decreases near the light line. This indicates that the increments of the losses, due to the coupling into radiation modes as mentioned above, are larger than the enhancement of the nonreciprocal phase shifts. We propose here that the propagation losses L _p are caused, not only by the radiation losses analyzed above, L _p = L _r, but also by material absorption, L _t = L _r + L _m, where the material absorption L _m is assumed to be 0.05 dB/mm [38]. In this calculation, we assume that the material absorption do not affect nonreciprocal phase shifts. To confirm the validity of this assumption, that these terms are negligible for nonreciprocal phase shifts, we have modeled this material absorption by adding isotropic or anisotropic imaginary terms in the permittivity tensor. It follows, of course, that lower material absorption directly yield higher figures of merit. In real manufactured structures, however, scattering losses due to structural disorders would also affect the nonreciprocal efficiency. Much work has indeed been done on the impact of imperfections on the performance of reciprocal PC structures [39–44], and similar work will be important for nonreciprocal MPC structures.

4. Summary

We have analyzed nonreciprocal guided-wave properties of 2-D MPC-slab waveguides, using the 3-D FEM. A vertical compensation wall at the waveguide center induces nonreciprocal phase shifts for TE-like modes. The waveguides with reduced width can enhance nonreciprocal phase shifts, while the slab thickness (the waveguide height) has little effect on the nonreciprocal phase shifts in typical thicknesses of PC slabs. The guided modes exhibit low losses when the air holes are sufficiently deep and there is an upper cladding layer, which can be relatively thin. 2-D simulations express similarities in nonreciprocal phase shifts, and can provide helpful approximations, although the frequencies of the guided modes shift.

For optimizing a nonreciprocal optical device using MPC waveguides, it will be important to consider various issues, on top of the nonreciprocal efficiency discussed here, such as bandwidth, coupling efficiency, and scattering losses from structural disorders. Although these issues are beyond the scope of this paper, we believe that the general trends of nonreciprocal efficiency, demonstrated here, will be a crucial factor in determining the future direction for the realization of integrated nonreciprocal circuits.