1. Introduction

The concept of parity-time (PT) symmetry in photonics represents an attractive platform to study fundamental physics beyond Hermitian paradigm and it also envisions novel applications enabled by non-Hermitian responses [1–14]. A variety of optical PT-symmetric systems have been explored both theoretically and experimentally, demonstrating two distinct regimes of preserved and spontaneously broken PT-symmetry, separated by exceptional points (EPs) [15,16]. The selective breaking of PT-symmetry has been proposed for various optical functionalities, such as PT-symmetric ring lasers [17,18]. Most of these studies, however, relied on the introduction of gain and loss in the dielectric constant of the materials constituting a PT-symmetric system. The vector nature of electromagnetic waves, however, can enable far richer non-Hermitian responses, which can stem from a special form of permittivity and permeability tensors of the materials, and not just scalar refractive indexes. One of such interesting responses is the circular dichroic responses emerging when certain materials undergo a reduction of time-reversal symmetry [19–27], including magneto-optical materials, such as ferrites, and optically pumped 2D semiconductors [28]. The interplay of such dichroic loss with a chiral structure of the optical near-fields has been recently shown to yield a strong nonreciprocal response. Also other PT-symmetric systems have been explored to achieve extremely asymmetric yet reciprocal responses in waveguides [29], multilayered structures [26] and micro-ring resonators [5]. It is interesting to explore the interplay of two different types of time-reversal symmetry reductions, via introduction of gain and loss, and via magnetization, respectively, which may potentially give rise to even richer non-Hermitian responses.

Photonic devices integrating magneto-optical (MO) materials represent an industry standard for nonreciprocal responses. Motivated by the need for integrated and compact nonreciprocal functionalities, several successful realizations of isolators and circulators based on MO materials have been reported [30–36]. One of the promising implementations is based on ring resonators integrating cerium substituted yttrium iron garnet (Ce:YIG), which exhibits high Q-factor resonances and large isolation [30,31]. The device operation relies on circular birefringence and low-loss in Ce:YIG at optical telecommunication wavelengths. Most of MO materials, however, are quite lossy and their response is often dominated by circular dichroism rather than birefringence. We have previously demonstrated that the dichroic response of Ce:YIG at shorter near-infrared wavelengths ($\lambda \sim 880\; \textrm{nm}$) alone can be used as an alternative to build an optical isolator [37]. The respective platform based on ring resonators and MO materials represents an excellent testbed to explore novel aspects of non-Hermitian responses that may stem from this type of time-reversal symmetry reduction. As a specific example, here we consider the extreme case of vanishing MO birefringence, which naturally takes place for Ce:YIG at $\lambda \sim 880$ nm, when the permittivity (for magnetization along the z-direction) can be expressed as

From this expression, we can immediately notice that, in addition to the conventional imaginary part of the diagonal component of the permittivity tensor, the magnetic nature of the materials gives rise to nonzero values of the off-diagonal components ${\varepsilon _{xy}} ={-} {\varepsilon _{yx}} = \delta $. In what follows, we assume that the real parts of the diagonal terms are equal $\varepsilon {^{\prime}_{xx}} \approx \varepsilon {^{\prime}_{yy}} \approx \varepsilon {^{\prime}_{zz}}$, which is a good approximation for many materials (due to the weak character of the Cotton-Mouton effect), and we also assume that no other loss mechanism is present, yielding the specific form of the permittivity tensor in Eq. (1). This is indeed the case for the dichroic response of Ce:YIG near $\lambda = 880\; \textrm{nm}$, where the absorption is dominated by the Ce³⁺-Fe³⁺ dipole transition [38,39].

2. Non-Hermitian double ring system

The proposed double ring resonator geometry is shown in Fig. 1(a). The TE modes propagating in the ring give rise to circularly polarized evanescent fields in the inner and outer sides of ring. As shown in Fig. 1(a), in the inner side the evanescent fields are left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) for CW and CCW propagation direction of the modes in the ring resonator, respectively [40,41]. Thus, to produce directional losses, the dichroic Ce:YIG film should be placed only on the inner side of the ring, which gives rise to different interactions of the chiral near-fields with CW and CCW modes of the ring resonator with the MO material.

 

Fig. 1. (a) Schematic of MO ring and gain / loss ring resonators. (b) FEM 2D simulation model. The structural parameters are: width of waveguide $\textrm{w} = 400\; \textrm{nm},{\; \; }$ gap between rings $\textrm{g} = 500\; \textrm{nm},{\; }$ radius of the rings $\; \textrm{R} = 8\; {\mathrm{\mu} \mathrm{m}}$. The permittivity of SiN waveguide ${\varepsilon _{\textrm{SiN}}} = 4,\; $ and for cladding and substrate ${\varepsilon _{\textrm{SiO}2}} = 2.1$.

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As a result of such chiral light-matter interactions, two different complex frequency eigenvalues exist for the two counterpropagating modes, which allows us to have different criteria for a PT-symmetry transition for CW and CCW modes near the Ce³⁺-Fe³⁺ dipole transition resonance frequency, once a second ring with gain/loss is added to the system. The broken PT-symmetry regime and unbroken PT-symmetry regime may thus be achieved for CW and CCW modes independently.

3. Eigenfrequencies of a pair of dichroic non-Hermitian ring resonators

In order to investigate the predicted phenomenon, 2D finite element (FEM) numerical simulations were performed for the MO ring and gain/loss ring system using COMSOL Multiphysics. The geometry of the system is shown in Fig. 1(b). Since Ce:YIG has a relatively large real part of permittivity, the MO film has to be thin enough to ensure that the energy remains predominantly confined within the SiN waveguide. The resonance frequencies of the rings were tuned to one another ${\omega _{MO}} = {\omega _2}$, where ${\omega _{MO}}$ is the resonant frequency of the ring with the MO material, and ${\omega _2}$ is the resonant frequency of the ring resonators with gain/loss. The diagonal term of permittivity in the MO regime was chosen to be the same as the permittivity of cladding and substrate, assuming that the MO film is thin enough, with an effective dichroic loss term $\delta = 0.0075\; $ (Fig. 1. (b)) [37]. The imaginary part of the permittivity in the waveguide of the other ring was then varied from negative to positive (from loss $\varepsilon \prime\prime > 0$ to gain $\varepsilon \prime\prime < 0$) to observe a one-way PT-symmetry transition. The obtained eigenfrequencies are plotted with black dots in Fig. 2. Since the MO ring supports two (CW and CCW) modes with different losses (${\gamma _{\textrm{MO}({\textrm{CCW}} )}} \ne {\gamma _{\textrm{MO}({\textrm{CW}} )}}$), the calculations yield two distinct exceptional points for the two-ring system. This implies that a scenario can emerge for which only one of the modes (CW or CCW) is lossy enough to switch the system into the PT-symmetry broken regime, while the other mode is still in the PT-symmetry preserved regime.

 

Fig. 2. Eigenfrequencies (a) real part (b) imaginary part. The eigenfrequencies obtained by FEM are plotted by black dots. CW mode and CCW mode are shown in bule and red lines, solid and dash lines represent eigenfrequency ${\omega _ + }$ and ${\omega _ - }$, respectively.

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To better understand this phenomenon, we developed a coupled mode theory (CMT) summarized by the system [42]

The eigenvalues for CW and CCW modes can be readily obtained by solving Eqs. (2,3):

To illustrate this asymmetry of the PT-symmetry transition with respect to the two modes, the eigenfrequencies ${\omega _{\textrm{cw}}}$ and ${\omega _{c\textrm{cw}}}$ of the two-ring system are plotted by blue and red solid (dashed) lines in Fig. 2. The analytical model based on CMT shows excellent agreement with FEM results for the values of fitting parameters given in Table 1.

Table 1. Fitting parameters for double ring resonators model based on CMT

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4. Nonreciprocal response of a 2 port-system with side coupled non-Hermitian ring resonators

To understand how the difference in the PT-symmetry transition manifests itself in an open two ring system, we introduce a waveguide placed next to the MO ring resonator (Fig. 3(a)). An additional coupling parameter between the waveguide and the system of coupled rings, ${\kappa _2}$, is correspondingly introduced and the CMT equations are modified accordingly [3,29,41,43]

 

Fig. 3. (a) Schematic of 2-port double ring resonators system. Structural parameters for numerical simulations: gap ${\textrm{g}_1} = 250\; \textrm{nm}$ and ${\textrm{g}_2} = 500\; \textrm{nm}$, width of waveguide $\textrm{w} = 400\; \textrm{nm}$ and radius of rings $\textrm{R} = 8\; {\mathrm{\mu} \mathrm{m}}$. (b) Differential transmission spectra for gain and loss propagation modes with gain (${\gamma _2} ={-} 22\; \textrm{GHz}$) and loss (${\gamma _2} = 39\; \textrm{GHz}$) in loss/gain ring cases. Solid lines are calculated by CMT equation and dots are obtained from FEM simulations.

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Fig. 4. 2D plot of differential transmission (forward minus backward) of 2 ports double ring resonators. (a) FEM (b) CMT. The solid and dash lines represent eigenfrequency with ${\gamma _{\textrm{MO}}} = 3.0\; \textrm{GHz}$ ${\omega _ + }$ and ${\omega _ - }$, respectively.

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Table 2. Fitting parameters for two-port double ring resonators system based on CMT model

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The 2D color plot of the differential transmission is shown in Fig. 4, where a sweep over gain/loss, described by $\varepsilon^{\prime\prime}$ (for the FEM model) and $\; {\gamma _2}$ (for the CMT model), was performed. The eigenfrequencies ${\omega _{({\pm} )}}$ obtained from the CMT are plotted on top of the CMT transmission plots in Fig. 4(b). In the backward mode (${\gamma _{\textrm{MO}}} = 57.5\; \textrm{GHz}$), both EPs are located in the loss regime (see S1 in Supplementary Material) and the incident wave does not strongly couple to the rings due to large losses. On the other hand, each EP is located in both the loss and gain regime for the forward mode (${\gamma _{\textrm{MO}}} = 3.0\; \textrm{GHz}$) because of the low loss in the MO ring, which is relatively comparable to the coupling coefficients ${\kappa _1}$ and ${\kappa _2}$. The balanced loss and gain factors show almost symmetric differential transmission plots in gain and loss in Fig. 4. The transmission at resonance frequency $\omega = {\omega _2}$ (${\Delta _2} = 0$) with ${\gamma _2} = 0$ becomes $T \approx 1$ for both forward and backward propagating modes, yielding no isolation between forward and backward transmissions at the boundary of gain and loss regimes. The forward transmission in both regimes for the second ring exhibit resonance splitting in the strong coupling regime when $4\kappa _1^2 > {[{{\kappa_2} + ({{\gamma_{\textrm{MO}}} - {\gamma_2}} )} ]^2}$. The splitting disappears at the EP (critical coupling) where $4\kappa _1^2 \approx {[{{\kappa_2} + ({{\gamma_{\textrm{MO}}} - {\gamma_2}} )} ]^2}$, leading to large amplification and isolation for forward operation, which can be observed at the resonance wavelength (Fig. 3.(b)). The different PT-symmetry condition for forward and backward operation can be found by adjusting the gap ${\textrm{g}_1}$ between the MO ring and the waveguide.

5. Conclusion

A non-reciprocal MO ring and gain/loss double ring resonators system breaking PT symmetry were theoretically investigated using FEM and analytical calculations based on CMT. A MO film placed in the inner side of one of the rings yields nonreciprocal absorption. We showed that a system with the SiN ring with MO Ce:YIG film at $\lambda = 880\; \textrm{nm}$ may operate in the broken and unbroken PT-symmetry regimes for forward and backward propagation modes. The eigenvalue solutions for the double ring resonators were numerically and analytically obtained, due to different losses in the MO ring for CW and CCW modes, EPs were observed at different frequencies.

We also found that for the system side-coupled to a waveguide, different PT-symmetry conditions for CW and CCW modes manifest in a different transmission in forward and backward directions. Thus, the transmission response becomes non-reciprocal, yielding large unidirectional amplification or unidirectional attenuation exceeding 20 dB in gain and loss regimes, respectively. The proposed double ring system with two complimentary mechanisms of symmetry breaking (time-reversal due to magnetization and PT due to gain and loss) can be applied for one-way micro-ring lasers and isolators with functionality based on new principles. Adding the nonreciprocal features to the PT systems also envisions one-way coherent perfect absorption at absorbing exceptional points [44]. Moreover, recent demonstrations of all-optical time-reversal symmetry breaking [28] envisions all-optical control of such devices.