1. Introduction

Recent years we have witnessed the transfer of concepts between different subjects of physics if they are governed by similar mathematical operations. Especially, with the similarities between the Schrödinger equation for electrons and the Maxwell’s equations for photons, the optical classic analog of quantum physics [1] has become a well-known area of optics that is still undergoing extensive investigation. Nowadays, by noting that loss is unavoidable in optical systems [2] and gain can be easily accessed, scientists switch their attention to optical non-Hermitian systems [3, 4] that can fully take advantages of the bright side of loss. Among all the non-Hermitian systems the parity-time ($\mathcal{P}\mathcal{T}$) symmetry [5–8] is maybe the most well-known.

The simplest $\mathcal{P}\mathcal{T}$ symmetric system is two parallel coupled waveguides [9–15]. People showed that if the waveguides are identical except for equal amounts of gain and loss, the coupled bulk modes could still possess real wavevectors under a conserved $\mathcal{P}\mathcal{T}$ phase and propagate inside the structure with fixed amplitudes. When the loss/gain rate is greater than the mutual coupling rate κ, the $\mathcal{P}\mathcal{T}$ phase is broken and the wavevectors are complex. Between these two scenarios is a phase transition point termed the exceptional point (EP) [13–16], where the eigenvalues and eigenvectors coalesce. With many attractive properties of $\mathcal{P}\mathcal{T}$ symmetry and EPs, e.g. unidirectional reflectionless [17], polarization filtering [18], unidirectional mode excitation [19], anti-lasing [20], perfect optical absorption [21, 22], enhanced sensitivity [23, 24], and even slow light [25], it is not a surprise that people then discussed how to manipulate the $\mathcal{P}\mathcal{T}$ phase, for example, by tuning the resonant frequency, gain/loss rate, coupling strength, and even by introducing nonlinear optical effects [26–31]. Direction-dependent $\mathcal{P}\mathcal{T}$ phase transition has also been studied by using dynamic modulation scheme [32].

Here we study the influence of magneto-optical (MO) effect on the $\mathcal{P}\mathcal{T}$ phase of two adjacent slab waveguides. Usually people pay their attention to the consequence of modified dispersion from the MO effect, or the topological waveguiding mechanism toward defect-immune and/or backscattering-immune propagation [33–35]. In this article we emphasize another important but easily overlooked phenomenon that could render a nonreciprocal $\mathcal{P}\mathcal{T}$ phase. Although the dispersion of guided bulk mode in a single MO slab waveguide is reciprocal ($\omega \left(-k\right)=\omega \left(k\right)$), the profile of field becomes asymmetric with respect to the middle line of the waveguide due to the MO-induced non-reciprocality of the two boundaries [36]. The coupling strength between adjacent waveguides consequently depends on the direction of the bias field B applied to the waveguides and the propagating direction of wave. The dispersion of guided bulk mode and the associated $\mathcal{P}\mathcal{T}$ phase can be nonreciprocal if the two MO waveguides are properly biased. Such an effect provides us with an interesting way in manipulating the $\mathcal{P}\mathcal{T}$ phase (especially EPs) by simply switching the applied bias fields B or by reversing the propagating direction of wave. We perform theoretical analysis and numerical simulation to prove our prediction. Potential applications and experimental suggestions are discussed.

 

Fig. 1 A schematic of the structure under investigation and the corresponding simple mechanism responsible to the nonreciprocal $\mathcal{P}\mathcal{T}$ phase. (a) Type-I configuration in which the forward propagating fields in the two MO waveguides shift toward each other, and the effective coupling rate increases to κ₊, greater than κ at B = 0. (b) Type-II configuration, that when incident from the backward direction the opposite effect in the field profile takes place, and the coupling rate κ₋ is weaker than κ. Note that the Type-II configuration can be also achieved from the Type-I configuration by simply reversing the bias fields.

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This article is organized as following. In Section 2 we develop rigorous formulas for the guided bulk mode, and briefly discuss the main mechanism of the nonreciprocal $\mathcal{P}\mathcal{T}$ phase. In Section 3 we perform numerical calculation to prove our theory and to provide more information about the influence of MO effect over the $\mathcal{P}\mathcal{T}$ phase. Discussion about the importance of our investigation is made in Section 4.

2. Theory

The configuration under investigation is shown in Fig. 1. It is a simple system made of two geometrically-identical slab waveguides (with a width of b) placed adjacent with each other (with a distance of 2a) in the x − z plane. Without loss of generality we assume that the upper (lower) waveguide WG1 (WG2) provides gain (loss). Medium surrounding these waveguides is air.

Now, let us assume that the dielectric constants ${\epsilon }_{1,2}$ of WG1 and WG2 can be changed via the MO effect. In the Voigt configuration where the magnetic bias B is perpendicular to the slab waveguide, i.e. in the y direction, the permittivity tensor becomes anisotropic [36]

Furthermore, to make the analysis simple we assume that the gain and loss are induced by proper imaginary parts of $\pm \delta$ in the diagonal elements, that

Paying attention to the guided bulk transverse-magnetic (TM) mode (with a wavevector of k) inside the structure, the field component H_y can be expressed as

From $\nabla \times H=j\omega D$ we can find the expressions of the other two field components E_x and E_z. Since $\nabla \times E=-j\omega {\mu }_{0}H$ should return to Eq. (4), we can prove that

From the boundary continuum conditions on H_y and E_z we can find that the dispersion $\left(\omega ,k\right)$ of the guided bulk mode is governed by an equation of $MH=0$, as

Before resorting to numerical calculation we would like to briefly discuss the mechanism of the non-reciprocal $\mathcal{P}\mathcal{T}$ phase within our interest. It is evident that the introduced MO effect renders a nonzero ${\gamma }_{1,2}$ value, which reverses its sign (${\gamma }_{1,2}\to -{\gamma }_{1,2}$) when B changes its direction. However, Eq. (6) has a quadratic dependence on ${\gamma }_{1,2}$, and in a single slab MO waveguide the two air-MO interfaces are time-reversed partners and switch their roles when $k\to -k$. Consequently, the mode dispersion inside a single MO waveguide is reciprocal [36], that $\omega \left(k\right)=\omega \left(-k\right)$. However, the broken symmetry over the field profile cannot be ignored. As explained in [36], a nonzero γ value would break the degeneracy of the two air-MO interfaces. As a result, the field profile shifts away from the middle of the waveguide and is no longer symmetric or anti-symmetric (see Fig. 1). When the MO waveguide is oppositely biased ($B\to -B$), a mirror inversion is imposed over the field profile and the field is switched to other side of the MO waveguide.

Now, by referring to the two configurations shown in Fig. 1 we can see how the $\mathcal{P}\mathcal{T}$ phase in coupled waveguides is modified by the MO effect. When no magnetic field is applied (${\gamma }_{1,2}=0$), the system can be described by a 2 × 2 non-Hermitian matrix [6, 7]

When the waveguides are oppositely biased, we can firstly consider the Type-I configuration shown in Fig. 1(a). Now the two otherwise independent bulk modes inside the MO waveguides would be pushed together (see right side of Fig. 1(a)), and the effective coupling rate is increased to ${\kappa }_{+}>\kappa$. As a result, to reach the EP a larger value of $g={\kappa }_{+}$ is required.

A totally different scenario is obtained when the wave propagates in the opposite direction (see the Type-II configuration shown in Fig. 1(b)), or when the applied magnetic bias B are reversed. Now the field in each waveguide is switched to the other boundary. In sharp contrast with that of Fig. 1(a), the overlapping of the two bulk modes decreases since the fields are stretched apart, and a smaller effective coupling rate of ${\kappa }_{-}$ is obtained, i.e. ${\kappa }_{-}<\kappa$. The EP is achieved at another frequency where $g={\kappa }_{-}$.

Since these two configurations are time-reversed partners, once the $\mathcal{P}\mathcal{T}$ phases of them are not equal, e.g. EP appears at two different frequencies for forward and backward modes, we can make the conclusion that the system is nonreciprocal. Above analysis also proves that the $\mathcal{P}\mathcal{T}$ phase is reciprocal if the two waveguides are equally biased (${\gamma }_1={\gamma }_2$), which is not discussed here.

 

Fig. 2 Dispersions (k₀ = ω/c,k) of the guided bulk mode when γ₁ = −γ₂ = −1, ε_1,2 = 4±j0.5, a = 0.3 mm, and b = 3 mm. EPs are labeled out by red stars.

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3. Simulation

To prove the existence of nonreciprocal $\mathcal{P}\mathcal{T}$ phase based on the mechanism discussed in the former section, we calculate the dispersion relation ($k_0=\omega /c$, k) of guided bulk modes by using Eq. (7). In this section we assume that all the parameters (including ε_m and δ) in the permittivity tensor are constants and do not vary with the angular frequency ω. It violates the causality principle [37], but makes it easier to observe and analysis the nonreciprocal features in our proof-of-concept numerical investigation. Parameters utilized in our calculation are b = 3

mm, $a=0.3$ mm, ${\epsilon }_m=4$, $\delta =0.5$, and ${\gamma }_1=-{\gamma }_2=-1$. The dispersion (k₀, k) is found by firstly calculating the determinant of the 8 × 8 matrix M in Eq. (7) versus k₀ and k, and then projecting the colored parametric surface of

 

Fig. 3 Variation of Ω versus k₀ when k = 0.785 mm⁻¹ (blue line) and k = −0.785 mm⁻¹ (red line), respectively. In each case two eigenmodes characterized by dips can be observed.

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Fig. 4 Distributions of H_y (dark) and phase (red, in the unit of π) in the structure at the four eigenmodes of Fig. 3.

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Figure 2 shows the mode dispersions of the two configurations. We can see in both cases a loop of dispersion similar to that in a $\mathcal{P}\mathcal{T}$ symmetric system [25] is obtained in the low frequency regime (before EPs). It implies that the system is $\mathcal{P}\mathcal{T}$ symmetric there. When the magnitude of the wavevector k increases, each dispersion curve coalesces at an EP (red star), across which the $\mathcal{P}\mathcal{T}$ phase becomes broken. The most important character of Fig. 2 is that the dispersion loop of the Type-II configuration is smaller and narrower than that of the Type-I configuration, and the EP has a smaller k value. Figure 2 verifies that the $\mathcal{P}\mathcal{T}$ phase of this system is nonreciprocal.

To provide more information about the nonreciprocal $\mathcal{P}\mathcal{T}$ phase, we plot the curves of Ω versus k₀ at $k=\pm 0.785$ mm${ }^{-1}$ in Fig. 3. In each curve we can observe two dips between $k_0=0.50$ mm${ }^{-1}$ and $k_0=0.61$ mm${ }^{-1}$. Each dip represents a guided bulk mode. It is very clear that the guided resonances are sensitive to the propagating direction of wave, or the magnetic bias.

Figure 4 shows the distributions of field component H_y and the phase of these guided bulk modes. We can see the magnitude of H_y is symmetric with respect to the middle of the structure. However, the phase is not a constant and varies smoothly in the two waveguides. It is in sharp contrast with the scenarios of Hermitian system (δ = 0), in which the phase difference between the two waveguides is either zero or π (not shown here). Furthermore, the distribution of field is evidently sensitive to the sign of k, especially the ones shown in Figs. 4(a) and 4(c). This result proves the mechanism we proposed above.

 

Fig. 5 Variation of Ω versus the distance a and wavevector k at a fixed angular frequency of ω = 2πc/λ₀ with λ₀ = 8 mm. EPs are labeled out by red stars.

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Above calculation is based on the assumption that all the parameters are constants and dispersionless. However, such an assumption violates the causality principle [37], which permits the satisfaction of ${\epsilon }_1={\epsilon }_2^{*}$ only at isolate frequencies. The dispersion in ε_m and γ due to the cyclotronic motion of effective charges [36] should also be taken into account in experiments. Now we propose to fix the angular frequency $\omega =c{k}_0$ and the waveguide thickness b, and then change the distance 2a to check the nonreciprocal $\mathcal{P}\mathcal{T}$ phase. This approach obeys the limitations set by the causality principle [37], and can be utilized as a possible guideline in verifying our proposed mechanism in future experiments.

Assuming that ${\epsilon }_m=4$, $\delta =0.1$, and $\left|{\gamma }_{1,2}\right|=1$ can be realized at $k_0=0.785$ mm${ }^{-1}$ (${\lambda }_0=8$ mm), we analyze the different $\mathcal{P}\mathcal{T}$ phases in the two configurations. The variation of Ω versus a is shown in Fig. 5. We can see the EPs of the two configurations have different a values, which split the whole figure into three different regions. When $a<0.655$mm (Region-1) the $\mathcal{P}\mathcal{T}$ phases are conserved. When $a>0.824$ mm (Region-3) the phases are broken. Between them (Region-2) the $\mathcal{P}\mathcal{T}$ phase is broken (conserved) in the Type-II (Type-I) configuration. Region-2 might possess novel applications because here the system can be switched between broken and conserved $\mathcal{P}\mathcal{T}$ phases by simply reversing the propagating direction of wave, or by reversing the bias fields.

 

Fig. 6 Variation of Ω versus k when a = 0.655 mm. Inset shows the distributions of field and phase for the EP in the Type-II configuration. Here δ = 0.1 is assumed.

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We check the distributions of field H_y and its phase at EPs of Fig. 5. A standard result is shown in Fig. 6 for $a=0.655$ mm. Now an EP is obtained in the Type-II configuration, while in the Type-I configuration the $\mathcal{P}\mathcal{T}$ phase is conserved because two dips can be observed. From the inset we can see the magnitude of H_y is still symmetric distributed. However, in sharp contrast with these shown in Fig. 4, the phase in the two waveguides is jumped by exactly $\pi /2$. This is a standard property of EP that reveals its chirality feature [16].

4. Discussion

Above analysis and numerical simulation prove that nonreciprocal $\mathcal{P}\mathcal{T}$ phase can be obtained in two MO waveguides with properly bias fields. As for potential applications, the nonreciprocal $\mathcal{P}\mathcal{T}$ phase demonstrated here enables us to extend the state-of-the-art advances of $\mathcal{P}\mathcal{T}$ symmetry [6, 7]. For example, it might enable us to control the switch of field between two adjacent waveguides.

Using the parameters of Fig. 5 we perform full-wave numerical simulation by using COMSOL Multiphysics 5.4. A y-directional magnetic current is placed in WG1 to excite the field with a wavelength of ${\lambda }_0=8$ mm, and the simulation is performed in the frequency domain. Figure 7 shows the results at $a=0.655$ mm where the Type-II configuration reaches EP while the Type-I configuration holds a conserved $\mathcal{P}\mathcal{T}$ phase. We can see in the Type-I configuration the field switches periodically between the two MO waveguides due to the beating of the two $\mathcal{P}\mathcal{T}$ symmetric eigenmodes (see Fig. 7(a)). However, in the Type-II configuration the field is very strong (note that the colormap in Fig. 7(b) is 10 times greater than that in Fig. 7(a)) and is distributed uniformly in the two waveguides. Figure 7(b) demonstrates an standard evidence of EP where only a single chiral solution exists [16], and highlights the presence of slow light [25] because a zero group velocity permits the extreme concentration of field power. Similar effect of efficiently excitation of field at EPs can be found from [38], and the enhanced spontaneous emission by utilizing the high density of states at EPs has been discussed in [39, 40]. Figure 7 indeed proves that thefield dynamics in coupled MO waveguides can be manipulated, for example, by dynamically tuning the bias fields.

 

Fig. 7 COMSOL simulation of the field inside the structure with λ₀ = 8 mm and a = 0.655 mm. (a) For a forward propagation the $\mathcal{P}\mathcal{T}$ phase is conserved and the field switches periodically between the two adjacent MO waveguides. (b) For a backward propagation the field is distributed uniformly in the two waveguides because an EP is reached.

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Our investigation presented here is about the coupled modes in two parallel waveguides. It is also of great interest to discuss how similar idea performs in multi-waveguide systems, especially the nonreciprocal feature and the emergence of high-order EPs [16, 23, 41, 42].

Experimental realization of the proposed scheme can refer to literatures of $\mathcal{P}\mathcal{T}$ symmetry and MO effect [33–35, 43]. We believe that the experiments prefer the microwave region where the metal has a low loss and the cyclotronic frequency of electrons is comparable to that of electromagnetic field [36, 43]. The low frequency nature of microwave also enables us to easily control the geometric parameters of the structure especially the distance a. Note that since the ratio of off-diagonal element γ to the diagonal element ε is critical in determining the sensitivity of MO performance [36], we can also utilize the concept of zero-index medium [44–46] to enhance the ratio and make the experimental observation more pronounced. Before finishing this article we also notice that Thomas etc. al. [47] have proposed a user-friendly framework of lumped circuits to study the giant nonreciprocity near EPs in $\mathcal{P}\mathcal{T}$ symmetric gyrotropic structures in the microwave domain. The feasibility in utilizing this approach to verify our theory is an interesting question to be answered.

5. Conclusion

In summary, we show that the $\mathcal{P}\mathcal{T}$ phase in coupled MO waveguides is sensitive to the directions of external applied biased fields. Although the bulk mode of a single MO slab waveguide is reciprocal, the coupling between two adjacent MO waveguides leads to two pairs of distinct modes propagating differently along opposite directions. When $\mathcal{P}\mathcal{T}$ symmetry is introduced, the guided bulk modes can be modified to achieve EPs that exists asymmetrically for forward and backward propagations, which is a by-product of non-reciprocality. To prove that the nonreciprocal $\mathcal{P}\mathcal{T}$ phase is closely related to the asymmetric field profile induced by the MO effect, we develop proper formula and provide numerical simulations. Potential applications and experimental suggestions are discussed.