Non-reciprocal Transmission of Electromagnetic Waves in Asymmetric Photonic Crystals

1 Introduction

Electromagnetic surface waves (SWs) are a special class of waves excited at the interface between two media of different dielectric parameters. One type of SWs is surface plasmon polaritons (SPPs) [1], [2], which exist at the interface between negative- and positive-permittivity materials. Because of the special and useful properties of SPPs, such as resonance [3], field enhancement [4], and energy localisation [5], [6], SWs have attracted much attention for many applications such as super-resolution imaging [7], spectroscopy [8], imaging [9], and biosensing [10]. However, the SPPs can be excited only by polarised transverse magnetic (TM) mode waves and require specific electromagnetic parameters on both sides of the interface. Another SW type, called the optical Tamm states (OTSs), by analogy with well-known Tamm states for electrons at crystal boundaries [11], [12], [13], has been under the spotlight recently. OTSs, as local optical states, exist at the interface between two photonic crystals with overlapping bandgaps and decay exponentially into the surrounding media. Furthermore, a similar SW wave, called the Tamm plasmon polariton (TPP) wave [14], can exist at the boundary between a metal layer and a Bragg mirror. OTSs and TPPs have been applied in the realisation of optical components such as absorbers [15], filters [16], and bistable switches [17]. Although they have different natures, their properties are very similar.

In contrast to SPPs, OTSs and TTPs can be excited by both transverse electric (TE) and TM polarisation [11]. At the frequencies of the Tamm states, the wave can pass completely through the composite structure by the tunnelling effect of OTSs. When the medium in the photonic crystals is a magneto-optic material (MOM), also known as a magnetophotonic crystal (MPC) [18], [19], the time-reversal symmetry of the OTS is broken and non-reciprocal wave propagation can be observed as a result of the simultaneous violation of reciprocity, time reversal, and all related spatial symmetries in the system. Because of OTSs’ important applications in integrated circulators and isolators, its non-reciprocal properties have also attracted considerable attention [20], [21], [22], [23], [24], [25], [26]. MOMs are divided into electric gyrotropic (EGM) and magnetic gyrotropic (MGM) media. Besides the properties of the Faraday, Cotton–Mouton, and magneto-optic Kerr effects, non-reciprocality of the propagating waves can be switched by the one-way electromagnetic OTSs (or TTPs) in MPCs. In all these works, the non-reciprocal transmission properties of the TM mode stand out.

The non-reciprocality of the propagating waves in photonic crystals is intimately related to their electromagnetic (EM) parameters and decided by the asymmetry of EM gyrotropic parameters. Comparing the gyrotropic parameters of EGM and MGM, aside from the inconspicuous non-reciprocal transmission in the former, its electric gyrotropic permittivities are weaker in external fields than in the latter. Simultaneously, because the magnetic gyrotropic permeability in EGM strongly depends on external magnetic fields, additional non-reciprocal properties can be easily observed and achieved. Furthermore, in photonic crystals, the transmission properties of TE-mode EM fields are different from those of TM-mode EM fields, so investigation of their non-reciprocity in one-dimensional MPCs composed of MPCs has been seldom attempted in previous studies [11].

Therefore, it is advantageous to investigate non-reciprocal OTSs and the transmission properties of TE-mode EM waves in one-dimensional magnetic MPCs. In this article, an asymmetric, complex photonic crystal composed of an anisotropic EGM is constructed, and the non-reciprocal propagation property of the TE-mode waves is analysed. The rest of the article is organized as follows. First, a symmetric MPCs structure is constructed and its reflection coefficient is calculated by the transfer matrix method [20]. Second, the numerical results and discussions associated with our purpose are presented. Finally, we draw conclusions in the last section.

Figure 1:

Schematic diagram of the one-dimensional structure air/(AB)N(B′A)N/air, which is composed of the normal medium A and the magnetic gyrotropic medium B,B′.

2 Physical Model and Computational Method

Figure 1 illustrates a one-dimensional, symmetric MPC, which is composed of two periodic structures (AB)N(B′A)N with alternating anisotropic EGM and isotropic dielectric layers. In this specific model, N is the period number, A is the isotropic dielectric with thickness d_A and relative electromagnetic parameters ϵ_A and μ_A. B and B′ are the anisotropic EGM layers with thickness d_B and relative permittivity ϵ_B. The relative permeability μ¯ of the anisotropic EGM is a matrix, which can be expressed as [26]

where μrg is the relative gyrotropic permeability whose sign depends on the direction of additional magnetic fields. If this direction is along +x-direction, then μrg is positive, and vice versa.

For simplicity, for a TE mode plane wave Ey(x,z)=ei(kxx+kzz) incident on the stratified medium, the total field in each layer can be written as Ey(x,z)=Ey+(x,z)+Ey−(x,z), where Ey+(x,z) and Ey−(x,z) represent the waves propagating along the positive and negative z-directions, respectively, and their fields are related to those in the neighbouring regions by boundary conditions. According to the transfer matrix method [20], the boundary fields in a region i is connected to those in an adjacent region j by the transfer matrix

where T_ij is the transfer matrix of the adjacent layers i and j, and can be written as

for an anisotropic medium, where Nl=kzl/(ωμ0μ′), Ml=−ikxμrg/(ωμ0μ′), μ′=(μrg2−μr1μr2)/μr2, kzl=ω02c2ϵr(μr1μr2−μrg2)μr2−μr1μr2kx2 is a wave vector of the z-axis in the layer l, and k_x is a transverse wave vector along the x-direction. If the medium is isotropic, then N_l and M_l can be simplified by setting μrg=0 and μr1=μr2. As the TE-mode wave propagates through this multi-layer structure, the incident, reflected, and transmitted electric fields are connected via the total transfer matrix

where P_i is the phase shift factor of one layer from the front boundary to the rear boundary, and defined as

where d_i is the thickness of ith layer. The total transmission and reflection coefficients, t and r, respectively, of the EM field through such a structure is given by

where M₁₁ and M₂₁ are elements of the transfer matrix M.

In (AB)N(B′A)N, the gyroelectric media B and B′ on both sides constitute two asymmetric periodic structures. When two photonic crystals have overlapping bandgaps at certain frequencies, the Block wavevectors are imaginary [11] and the transmission coefficients are equal to 0. That is, the waves cannot transmit in the direction of propagation because of the imaginary propagation constant. But at a certain frequency, the unidirectional OTSs in the surface between two asymmetric photonic crystals can be excited by external fields in the bandgap, and then the incident wave can completely propagate through asymmetric, periodic structures by the tunnelling effect of the OTSs. On the contrary, when the wave at the same frequency is incident from the opposite direction, the OTSs are completely compressed or even removed at the interface, and the incident wave is totally reflected.

3 Numerical Results

In order to demonstrate the non-reciprocity in the designed structure, the transmission properties are numercially calculated when a TE-mode plane wave is incident from the left boundary of the structure. The medium parameters in the numerical simulations are chosen as follows: period number = 8; for medium in layer A, we choose SiO₂, whose relative permittivity ϵr=3.9, relative permeability μr=1, and thickness dA=185.8nm; for the anisotropic EGM medium in the layer B(B′), ferrites subjected to a DC magnetic field are chosen for the +x-direction of layer B and for the –x-direction of layer B′ with thickness dB=99.4nm. The permeability for the ceramic ferrite material is a function of the frequency and the external magnetic field [26]; with increase in the frequency, the relative permittivity approaches a constant, the permeability coefficients μ_{r ⁢ 1} and μ_{r ⁢ 1} approach 1, and the gyro component μrg becomes very weak. In our study, the frequency range used is (1–8) × 10¹⁴ Hz, so we choose ϵr=12, μr1=μr2=1, and μrg=±0.4.

Figure 2:

Transmission spectra of the forward and backward waves, denoted by the solid line and dotted lines for structure (AB)8(B′A)8. There are three pairs of transmission peaks in the three different band gaps for forward and backward waves, namely (1.5–3) × 10¹⁴ Hz, (6–7.5) × 10¹⁴ Hz, and (10.5–12) × 10¹⁴ Hz.

First, a TE-mode wave is incident on the PC structure at the angle θ=30∘ from the boundaries. Figure 2 shows the relationship between the transmission coefficient and the frequency, where the solid and dotted lines correspond to the forward (kx>0) and the backward (kx<0) waves, respectively. Because (AB)N(B′A)N is an asymmetric structure, the OTS can be formed at the interface between the two PCs with overlapping bandgaps, resulting in a transmission peak of a specific frequency. Figure 2 shows that there are three pairs of transmission peaks in the three different bandgaps for forward and backward waves, namely (1.5–3) × 10¹⁴ Hz, (6–7.5) × 10¹⁴ Hz, and (10.5–12) × 10¹⁴ Hz. This is to say, the OTS SWs are excited between the two MPCs with overlapping bandgaps in TE polarisation as in TM polarisation [20]. Furthermore, because of the non-reciprocal gyrotropic electromagnetic parameters in the two PCs, the forward and backward transmission peaks are not coincident at the bandgap, so that the non-reciprocal transmission of electromagnetic waves can be realised. Especially, in the three bandgaps, three pairs of non-reciprocal transmission peaks appear simultaneously, and more non-reciprocal frequencies are available.

Figure 3:

Patterns of the electric fields E_y along the xz-plane for N = 8 and incident angle = 30°: (a) positive incidence and (b) negative incidences for f=7.127GHz; (c) negative incidence and (d) positive incidence for f=6.956GHz.

To further show the non-reciprocal transmission in the structure, the field patterns of Ez of the non-reciprocal transmission peaks in the frequency range (6–7.5) × 10¹⁴ Hz are shown in Figure 3, with the same physical conditions as in Figure 2. Figure 3a and b shows the field patterns Ez of f = 7.127 GHz for the forward (kx>0) and backward (kx<0) waves when counter-propagating TE plane waves are incident on the left and right boundaries. From the figures, it is seen that there are strong field distributions in MPCs due to the OTS SWs, which is consistent with those excited by the TM mode waves at the interface between the two MPCs. The OTSs were precisely strong, and the MPCs looked like transparent media through which the fields could completely pass. On the contrary, for backward incidence, the transmitted waves were almost suppressed, OTSs could not be excited at the interface, and the fields could not transmit through the MPCs. Therefore, stationary waves in the incident region were formed by the interaction between the incident waves and the reflected waves. For the frequency f = 6.956 GHz, similar results are shown in Figure 3c and d: the backward propagating wave can completely pass through the MPCs but the forward incident wave is reflected. Thus, for TE-mode waves, the non-reciprocal propagating properties can also be realised by the MPCs.

Figure 4:

Transmission spectra of the designed model with N = 8 at different incident angles: (a) incident angle = π / 7, (b) incident angle = π / 6, (c) incident angle = π / 4, and (d) incident angle = π / 3.

In order to further investigate the non-reciprocal properties of the structure, we calculated the transmission coefficients for different incident angles (as shown in Fig. 4), with the other conditions being the same as in Figure 2. When the incident angle was increased from π / 7 to π / 3, the non-reciprocal transmission peaks still appeared and the transmission maximum almost remained unchanged for the forward and backward waves. Therefore, the non-reciprocal properties could still be achieved. As the incident angle was increased, the transmission coefficients shifted to high frequencies, and the non-reciprocal frequencies of the forward and backward waves broadened. That is to say, the non-reciprocity of the structure got enhanced with increasing angle.

Figure 5:

Transmission spectra of designed model with N = 8 at different gyrotropic permeability coefficients: (a) μxz=−μzx=0.4i, (b) μxz=−μzx=0.3i, (c) μxz=−μzx=0.2i, and (d) μxz=−μzx=0.1i.

The non-reciprocal transmission in the structure was achieved by the asymmetric gyrotropic permeability in the two PCs, whose different values affected the transmission. However, the gyrotropic property of an MGM (such as ferrite) is a function of the frequency and tends to be very weak at high frequencies. Figure 5 shows the relationship between the transmission coefficient and the frequency as the gyrotropic permeability decreases, with the incident angle being π / 6 and all other conditions remaining the same as in Figure 2. As illustrated, the non-reciprocal transmission remained the same, and the transmission peak remained constant, when the gyrotropic permeability coefficients were changed from 0.4 to 0.1. Moreover, with decreasing gyrotropic permeability, the peaks of the non-reciprocal frequencies simultaneously shifted from high to low values (red shift), while the width of the non-reciprocal frequencies of the forward and backward waves became smaller. These results indicate that the non-reciprocal properties are closely related to the gyrotropic parameters, which became weaker as a result of decreasing gyrotropic permeability.

Figure 6:

Relationship between transmission coefficient and frequency for different layer numbers N: incident angle θ = 30°; N range 6–9.

Furthermore, the period number of the two MPCs also influenced the non-reciprocal properties. Figure 6 shows the transmission coefficients with different MPC period numbers, with all other conditions remaining the same as in Figure 2. Note that when the layer number decreased from 9 to 6, a pair of non-reciprocal transmission peaks still appeared, but the transmission properties changed. From the figure, it can be seen that the larger the layer number, the narrower the transmission spectrum. On the other hand, as the layer number decreased, the non-reciprocal transmission peaks shifted slightly to high frequencies (blue shift), while the transmission spectrum broadened.

4 Conclusion

In this article, we investigated OTCs excited by TE-mode waves in a two-MPC structure. Because of their non-reciprocity, non-reciprocal transmission properties appeared in the asymmetrical structure. Three pairs of non-reciprocal transmission peaks occurred in the three bandgaps over the frequency range 1×1014Hz to 8×1014Hz. As the incident angle increased, these peaks apparently red-shifted, and the width of the non-reciprocal frequencies increased. The gyrotropic permeability, another fundamental determinant of the non-reciprocal transmission, was also examined. As such, with decreasing gyrotropic permeability, the non-reciprocal transmission peaks appeared blue-shifted; at the same time, the non-reciprocal properties weakened as a result of the decrease in the width of non-reciprocal frequencies. As the non-reciprocal properties can be achieved by TE-mode waves, such design has further underpinned SWs; further applications also include integrated circulators and isolators.