1. Introduction

Magneto-optical (MO) effects such as Faraday rotation and nonreciprocal phase-shifts are widely exploited, principally in optical isolators. However, the intrinsic weakness of MO effects hinders miniaturization of such devices. It is known that micro-scale structuring of MO materials, particularly with photonic crystals (PC), is an effective approach to enhancing the response, promising compact, integrated circulators and isolators [1]. It has also been recently shown that magnetic nonreciprocity in plasmon [2] and photonic crystal [3,4] systems may result in true unidirectionality (or one-way propagation) and provide robustness against disorder. This opens up new avenues for magnetic materials in photonics, but the conditions for maximizing nonreciprocality are not well-established. In this work, we reveal the mechanisms of the unidirectionality in low symmetry MO photonic crystals (MPC).

Magnetic nonreciprocity is associated with the breaking of time reversal symmetry due to a static magnetic field or spontaneous magnetization. However, breaking time reversal symmetry is not sufficient to obtain nonreciprocality—the theory of magnetic groups shows that the absence of mirror reflection symmetries is also required [4,5,6]. Previous suggestions to achieve this are the use of birefringent materials [6] or a non-uniform magnetization [4]. Here we propose a new approach based on uniform magnetization, which is much simpler for practical realization. To remove reflection symmetry, we consider ternary photonic crystals composed of three different dielectrics, at least one of which is a transversely-magnetized MO active material. We formulate a transfer matrix approach yielding compact analytical results and elucidate the physical origin of the nonreciprocity. Our approach allows the direct estimation and optimization of the nonreciprocal response of the structures proposed here and for similar structures discussed in Ref. [4].

2. Mathematical treatment

We treat the ternary photonic crystals using a transfer matrix formalism. To formulate the approach, we begin with the problem of the partial reflection of plane waves at the interface between two MO media magnetized transversely (the Cotton-Mouton or Voigt geometry) (Fig. 1). The coefficients relating amplitudes of the incident and scattered waves can be determined using the standard Maxwell boundary conditions. For MO media, these conditions yield relations different from the standard Fresnel equations [7]: with transverse magnetization, the electric field includes a longitudinal component along the wavevector k, and for non-normal incidence, this component contributes to the component of the field tangential to the interface. Straightforward calculations yield the expression:

for a = r (reflection) or t (transmission) as the modified Fresnel coefficients for a p-polarized incident plane wave. The magnetic field amplitudes of the incident and scattered waves are related in the standard way H^a_y = ρ^aH_y. We suppose the dielectric tensor in each medium $\tilde{\epsilon }$_m satisfies ε_m,xx = ε_m,yy =ε_m,zz and ε_m,xz = -ε_m,zx = iΔ_m. Then for the most general case of both materials being MO active, the expressions for ρ ^a ₀ and ρ ^a ₁ are as follows:

 

Fig. 1. Geometry of the problem: (a) ternary MPC and (b) microcavity-type nonreciprocal Fabry-Perot resonator. Magnetization M is along yˆ. Magnetic layers are in green.

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where d_m = ε ² _m - Δ² _m and k_mx(z) is the x(z) component of the wavevector in the medium m. Thus, disregarding effects second order in Δ_m, it can be seen from Eqs. (1) and (2) that the main effect of MO activity is an additional term with an odd dependence on the lateral component of the wavevector k_x (which is the same in every medium) and on Δ_m. As the coefficients ρ ^a ₀ are purely real (we suppose the materials to be nonabsorbing), this dependence manifests mainly as a direction-dependent phase shift ϕ_a = tan^-1(Im(ρ^a)/Re(ρ^a)), which is odd not only with respect to the angle of incidence but also with the magnetization direction (through the sign of Δ_m) [7]. As we show below, this phase shift lies at the root of the nonreciprocity in the dispersion relation of the photonic crystals reported earlier [4].

To continue, we work in the transfer matrix formalism [8]. Using the same boundary conditions that yielded (2), we can derive the interface matrix Mˆ_ij that connects the amplitudes of forward (H ⁺ _y) and backward-going (H ^- _y) waves on either side of the interface between layers i and j as follows:

where F_m = (ε_m k_zm + iΔ_m k_xm)/d_m.

Following the standard procedure [8], we find the total transfer matrix of the elementary cell of the PC in the form Tˆ = Mˆ _N1 Pˆ_N …Pˆ ₃ Mˆ ₂₃ Mˆ ₁₂ Pˆ ₁. Here N is the number of the layers in the elementary cell and Pˆ_m = diag [exp(ik_zma_m),exp(-ik_zma_m)] is the matrix that accounts for the phase shift accumulated by the wave during propagation in the layer m which has thickness a_m [8]. Also k_zm = [k ² _m - k ² _xm)^1/2, where $k_m=\omega /\left[c\sqrt{d_m/{\epsilon }_m}\right]$ is the modulus of the wavevector in the m-th layer. We now see that in contrast to the nonmagnetic case, the total phase shift now consists of two parts: since k_m only depends on Δ_m to second order, the standard propagation phase k_zma_m is only weakly dependent on the magnitude of magnetization, and is independent of its sign. This contribution to the phase is thus reciprocal. However, there is an additional boundary-related phase shift contained in the interface matrix Mˆ_ij which may give rise to a directional dependence of the Bragg diffraction condition and thus to nonreciprocal behavior. Consistent with the magnetic group theory we can show by direct calculation of Tˆ that this only occurs when the elementary cell lacks reflection symmetry. For a two-component cell, i.e. photonic crystals formed from two different dielectrics, this condition can be achieved only with inhomogeneous magnetization of the magnetic component, essentially creating a three-layer structure [4]. However, a magnetization that varies on a sub-micron scale would be difficult to implement in practice. We therefore propose a structure which is formally more complex, but is simpler for fabrication–a ternary PC with a three-layer elementary cell of different dielectrics. As we see below, this provides sufficient asymmetry to achieve nonreciprocality.

2.1. Characterization of nonreciprocality

To consider propagation in the ternary PC, we impose a Bloch condition TˆH̄_y = exp(iK_Ma)H̄_y, where K_M is the Bloch vector and H̄_y = (H ⁺ _y, H ^- _y)^T. This condition is satisfied when det [Tˆ - exp(iK_Ma)Iˆ] = 0, which is just the dispersion relation of the Bloch waves of the MPC. In the general case, this expression is rather difficult to express analytically. However, the terms linear in the lateral component k_x responsible for nonreciprocity may be extracted explicitly. Limiting our consideration to the case of a single MO layer in the elementary cell (Δ₁ = Δ₃ = 0), after tedious but straightforward calculations we obtain the dispersion relation

where K_M and K ₀ are Bloch wave vectors for the magnetic and corresponding non-magnetic structures and second order terms in Δ₂ have been omitted. The first term in Eq. (4) is the reciprocal contribution to the dispersion containing standard nonmagnetic terms due to periodicity [8]. The second explicitly shows the asymmetric contribution to the dispersion relation due to its linear dependence on the lateral wavevector component k_x and Δ₂. Note that reversal of the magnetization direction has the same effect as reversal of the sign of k_x. (Note that a similar approximate form of the dispersion relation as Eq. (4) has been discussed before in the context of birefringence in two-component structures with longitudinal magnetization geometry [9].)

The nonreciprocity of the dispersion in Eq. (4) is more apparent if we note the small value of the parameter Δ₂, and consider the no nreciprocal contribution δ ≪ K ₀ to the dispersion as a small correction to its reciprocal part K ₀. Substituting K_M = K ₀ + δ into (4) and expanding the cosine function on the left hand side of Eq. (4) gives

It is evident that Eq. (5) is the term of first order in k_x and Δ₂ responsible for the nonreciprocity because simultaneous reversal of the wavevector k and K ₀ results in a different sign of δ and therefore alters the dispersion relation for forward and backward propagation. As expected, the nonreciprocity vanishes in the symmetric configuration ε ₁ = ε ₃ confirming the necessity of breaking the reflection symmetry mentioned earlier. From Eq. (5), it is also obvious that the nonreciprocity reaches maximal values near the band edges where the sine in the denominator tends to zero. Figure 2 shows an exact calculation of the photonic band structure (Fig. 2(a)) using the transfer matrix technique, and the difference between eigenvectors K_M for opposite propagation directions (Fig. 2(b)), and confirms our expectations. The nonreciprocity reaches its maximum at the band edges. It has a complex resonant dependence on the transverse wavevector k_x due to the presence of interference in the layers, which is reflected in the sine functions in the numerator of Eqs. (4) and (5). Thus the nonreciprocity is extremely sensitive to the structure parameters and optimization will be required for every particular application. For clarity, Fig. 3 shows slices through the energy surfaces in Fig. 2 for the case of θ_i = 30° incidence. Parameters of the structure used in all our calculation were chosen to maximize the nonreciprocity with the short-wavelength band edge lying near optical communication wavelengths for θ_i = 30°. Figure 3(b) clearly shows that at this particular wavelength, the difference in the Bloch wavevectors for opposite propagation directions reaches its maximum. Note that results obtained with use of the approximate expression Eq. (5) (shown by the thick red line) are in excellent agreement with the exact result (shown by the thin blue line). In fact this approximation fails only in a very narrow (≈2 nm) wavelength range in close proximity to the band edges where it diverges (see insets to Fig. 3(b)).

 

Fig. 2. Photonic band structure (a) and its nonreciprocity (b) in 1D MPC with three-layer elementary cell formed from layers of SiO₂ (a ₁=220 nm, ε ₁=2.1), Ce:BIG (a ₂=130 nm, ε ₂=6.25, Δ₂=0.06) and Ta₂O₅ (a ₃=110 nm, ε ₃=4.7), respectively.

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At this point, it is instructive to reconsider the origin of the nonreciprocity, namely, asymmetric scattering and the additional phase shift due to magnetization described above. As mentioned earlier, for transverse magnetization, the MO activity does not induce circular birefringence and so the correction to the dynamical propagation phase is a second order effect in Δ₂, and therefore reciprocal. Thus the overall nonreciprocity is purely a boundary effect associated with the layer interfaces, and the thickness of the MO and other layers is only important in so far as it provides a resonance and so enables multiple reflections. In contrast, for enhanced Faraday rotation in periodic multilayers with longitudinal magnetization [1,10], the enhancement is entirely associated with the increased optical path length within the structure. From this picture of boundary-related phase shifts, it is also evident why the nonreciprocity disappears for symmetric configurations—the phase shift at two opposite boundaries cancels [see Eqs. (2)] and the Bragg condition is identical to that of the nonmagnetic system (up to second order terms in Δ₂, which are reciprocal).

As was already mentioned, the results given above show considerable nonreciprocity for the structure with optimized parameters. However, the nonreciprocity could certainly be improved further. At first glance, optimization of the structure seems straight-forward because one simply has to make the correction due to the nonreciprocal term in Eq. (4) maximal. For a given angle of incidence (i.e. fixed k_x) this is achieved when modulus of all the sine functions in Eqs. (4) and (5) is simultaneously equal to unity. This condition is satisfied when the thickness of each layer satisfies the condition $a_m=\left(\frac{1}{2}+n\right)\pi /k_{\mathrm{zm}}\left(n=\mathrm{1,2,3}...\right)$. However, any variation in the thickness of layers of course affect the band structure. Therefore, as the nonreciprocity is maximal at band edges, simultaneously with the conditions imposed on the thickness of layers one should guarantee that a band edge appears at the frequency of particular interest. An optimal structure must therefore satisfy sin(K(a ₁,a ₂,a ₃)(a ₁ +a ₂ +a ₃)) = 0, as well as the conditions on the thickness of the individual layers. The complete optimization task is therefore quite involved. To concentrate on the physical picture, we will not consider the optimization procedure any further, limiting our consideration to the particular structure introduced above (see Fig. 2).

 

Fig. 3. Photonic band structure of nonmagnetic (a) and nonreciprocity of magnetic (b) ternary 1D PCs, respectively. The structure parameters are the same as in Fig. 2(a) and the angle of incidence is 30°. Thin blue lines correspond to the exact result and the thick red line is obtained from the approximate expression Eq. (5). Insets in (b) show the nonreciprocity in the proximity of the band edges.

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Fig. 4. Transmittance of nonmagnetic (a) and nonreciprocal transmittance of magnetic (b) ternary 1D PCs, respectively. Figure (a) shows the whole bandgap region, while (b) shows the transmittance in the proximity of the short-wavelength band edge. The structure parameters are the same as in Fig. 2(a) and the angle of incidence is 30°.

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3. Designing one-way structures

With the analysis developed above, we are also able to predict narrow bands of unidirectional transmission in finite structures similar to that observed in previous work [4] (see Fig. 4(b)) having nonreciprocal band structure shown in Fig. 3. Figure 4 shows the transmittance of a ternary MPC of 50 cells (150 layers) and confirms that in the proximity of the band edge at λ ≈ 1545 nm, there is a narrow frequency range where the transmittance in the forward direction and backward direction is significantly different—the structure is transparent in the forward direction and almost opaque in the backwards direction.

Unfortunately, the phase shift produced at the boundaries of the MO layers is small, and according to our and previous [4] calculations, stacks of order one hundred or more layers would be required to produce effective unidirectionality ∣ T(k⃗) - T(-k⃗) ∣ ≈ 1. Constructing such large stacks would be very difficult. Structures fabricated to date have been limited to around ten layers, since in order to preserve strong magnetic effects, an annealing step is required after each garnet layer is deposited, and this ultimately induces serious cracking [1, 11,12]. Thus, fabrication of crystals with sufficient nonreciprocity by this strategy seems unfeasible.

 

Fig. 5. Transmittance of nonmagnetic (a) and nonreciprocal transmittance of magnetic (b) microcavities, respectively. Figure (a) shows the whole bandgap region, while (b) shows the proximity of the resonance only. The structure consists of a single Ce:BIG layer (a_M=155 nm, ε_M=6.25, Δ_M=0.06) sandwiched between two Bragg mirrors formed from a sequence often SiO₂ (a ₁=267 nm, ε ₁=2.1) and Ta₂O₅ (a ₂=179 nm, ε ₂=4.7) layers. The asymmetry is achieved through bounding the Ce:BIG layer with different layers of the Bragg mirrors, i.e. SiO₂ and Ta₂O₅ for left and right boundaries, respectively. The angle of incidence is 30°.

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However, with the origin of the nonreciprocity in transversely-magnetized crystals clarified, we can now propose an alternative system which would be simpler for fabrication and more compact. Instead of a MPC, we propose to obtain strong nonreciprocity using an asymmetric magnetic Fabry-Perot structure: a single MO layer bounded by two non-identical non-magnetic Bragg mirrors (see schematic in Fig. 1(b)). Note that similar structures with longitudinal magnetization were proposed before for the increase of Faraday rotation [1,10–12], but we mention again that the nature of the transverse magnetization effect we consider here is quite different. For example, in the symmetric configuration proposed before [11,12], i.e. with a MO layer bound symmetrically by identical mirrors, the phase shifts at the two boundaries of the MO layer cancel each other and the effect of the nonreciprocity disappears. However, if the structure is asymmetric, the phase shifts after reflection at two opposite boundaries are different, and even if the shifts are of different sign, the net shift acquired by the wave after multiple reflections in the cavity does not vanish. Thus, a Fabry-Perot resonance of sufficient quality factor can give rise to a very large value of the total phase shift accumulated in the microcavity. This additional shift obviously alters the conventional Fabry-Perot condition in a similar way that the Bragg condition was modified in MPCs. As the phase shift is nonreciprocal, and in fact has the same absolute value but different sign for forward and backward propagation, this results in nonreciprocal resonant transmittance T(k⃗) ≠ T(-k⃗), which manifests as a frequency gap between transmission peaks for waves propagating in opposite directions (Fig. 5). It can be easily shown that the frequency gap δω ∝ k_xΔ_M, and that the differential transmittance ∣T(k⃗) - T(-k⃗)∣ ≈ 1 if the quality factor of the resonance is sufficient to satisfy the condition δλ < δω, where δλ is bandwidth of the resonance. More precisely, the resonance condition is exactly the same as for the case of a reciprocal microcavity

where δϕ_m = tan^-1(Im(ρ^m)/Re(ρ^m)) is the nonreciprocal phase shift due to reflection at the interface between magnetic microcavity and the first (m = 1) or second (m = 2) nonmagnetic bounding layers (see Eqs. 2), and ϕ ₀ = ϕ_D + ϕ ₁ + ϕ ₂ is the reciprocal phase which includes the dynamical phase ϕ_D = 2k_Mzd_M and two boundary phases ϕ_m associated with reflection from the Bragg mirrors. Thus the nonreciprocal response of the structure can be optimized through: (i) an increase in the number of mirror layers which will result in narrower resonances reducing overlap of the forward and backward resonances (at the expense of the transmission bandwidth of course); or (ii) increase in the nonreciprocal phases by increasing the angle of incidence or choosing appropriate characteristics of the materials (refractive indexes and MO parameter) which will result in increase of the nonreciprocal phase shifts (according to Eqs. 2) and stronger separation of the forward and backward resonances, again reducing their overlap.

The thicknesses of the layers appear to be not as critical for the microcavity type structure as for the ternary PC structure. The thicknesses of the layers in the Bragg mirrors should be chosen to provide photonic bandgap center at a particular frequency of interest, while the thickness of the central layer is determined by the Fabry-Perot condition Eq. (6). Thus optimization of the microcavity design is a significantly simpler task than optimization of the ternary MPC structure discussed in the previous section. In addition, our calculations show that as compared to the MPC, for the microcavity structure, a significantly smaller number of layers is required to obtain equivalent nonreciprocity. The example in Fig. 5 contains 41 layers (20 layers in every mirror plus the MO defect layer) compared to the 150 layers (50 ternary elementary cells) for the MPC in Fig. 4.

4. Conclusion

We propose the basic components described in this work as building blocks for a new class of nonreciprocal devices. Nonreciprocal MPCs of both ternary and microcavity types may be combined with waveguides to generate a new class of isolators and circulators that are compact, but simpler to fabricate than traditional 2D MO PC approaches [13,14]. Since a single magnetization is applied throughout the structure, a change in the isolation functionality could be obtained by a change in the direction of the applied magnetization. This would be extremely awkward in previously proposed systems with layers of different magnetizations. In addition, a significant advantage of MO devices based on structures of this kind is that they eliminate the polarizers that are required in the Faraday (longitudinal) configurations. At the same time, the Fabry-Perot structures proposed here may be superior to the conventional Faraday configuration, because of the complete transmission at the peak of the resonance. Nonreciprocal devices based on Faraday rotation show an intrinsic trade-off between the shift in the polarization angle and the transmission. At the peak of the Faraday rotation, the transmission of a Faraday device [11,12] is never equal to 100 percent.

Our results therefore demonstrate that low-symmetry magnetic photonic crystals are excellent candidates for realization of various compact nonreciprocal photonic devices. Paradoxically perhaps, we have shown that by reducing symmetry, we may actually simplify fabrication challenges, and bring dynamically-tunable MO photonic crystal devices closer to reality.