## Abstract

By introducing nonreciprocal phase shifts into microresonators, we propose new designs for the miniaturization of optical waveguide isolators and circulators. We present detailed design procedures, and numerically demonstrate the operation of these magneto-optical devices. The device sizes can be reduced down to several tens of micrometers. The nonreciprocal function of these devices is due to nonreciprocal resonance shifts. Next, the operation bandwidth can be expanded by increasing the number of resonators (the filter order). This is demonstrated by comparing the characteristics of a single-resonator structure with those of a three-resonator structure. This paper furthermore presents the nonreciprocal characteristics of three-dimensional resonators with finite heights, leading to a guideline for the design of nonreciprocal optical circuits. This involves a demonstration of how the resonators with selected parameters are practical for miniaturized nonreciprocal circuits.

©2007 Optical Society of America

## 1. Introduction

The development of optical fiber communication systems has spurred growing demand for optical isolators, especially to protect semiconductor lasers from unwanted reflected light. At the present time, however, the only readily available isolators are bulky and expensive. There is a need, therefore, for inexpensive and more reliable isolators that can be integrated with other optical devices. To satisfy this demand, numerous designs for optical waveguide isolators have been explored. These waveguide isolators are classified broadly into two categories: configurations relying on nonreciprocal transverse electric/transverse magnetic (TE/TM) mode conversion and configurations relying on nonreciprocal phase shifts [1].

The configurations relying on TE/TM mode conversion have encountered problems due to the phase mismatch between both modes. Recently, alternative attempts have been made to develop heterogeneous integrated isolators, where a bulk Faraday rotator is inserted into a waveguide device. To make such developments possible, it is necessary to miniaturize the Faraday rotators to suppress diffraction loss sufficiently. One-dimensional magneto-photonic crystals (MPCs) [2], i.e. thin-film stacks with magneto-optic materials, represent one attractive option for realizing such miniaturized Faraday rotators. This miniaturization relies on the enhanced Faraday rotations caused by the propagation in defects in MPCs at the resonant frequencies.

On the other hand, the configurations relying on nonreciprocal phase shifts intrinsically require no phase matching, as TE and TM modes are not coupled in the waveguides. An isolator based on this concept, exhibiting a 19 dB extinction ratio, has been successfully demonstrated in a Mach-Zehnder configuration [3]. Furthermore, several groups have begun to study size reduction of the device. They have considered the use of double-layered garnets [4], compensation walls [5], or garnets bound to high-index contrast slabs [6]. It has been estimated that the length of the isolators can be reduced to a minimum of several hundred micrometers [6]. However, the unbreakable limit to further size reductions concerns only transversal waveguide geometries of nonreciprocal phase shifters.

In this paper, we propose nonreciprocal microresonators, and demonstrate the functionability of magneto-optical devices which utilize these resonators. By introducing resonant structures to the configurations that rely on nonreciprocal phase shifts, similar to MPCs for the configurations that rely on the Faraday rotation, we can dramatically reduce the device sizes down to several tens of micrometers. The present configurations, importantly, requires no phase matching and no heterogeneous integration as they are based on nonreciprocal phase shifts.

The remainder of this paper is structured as follows. In Section 2, we present a nonreciprocal microresonator, involving nonreciprocal resonance shifts. We first describe the design procedure and then numerically demonstrate its function as an optical isolator and circulator, using a two-dimensional (2-D) finite-element method for nonreciprocal optical devices [7]. In Section 3, we present parallel-coupled nonreciprocal microresonators. The capability of this structure for expanding the operating bandwidth is distinct from that of the single-resonator structure discussed in Section 2. The parallel-coupled resonator structure still keeps the device size much smaller than that of conventional Mach-Zehnder configurations. Although the discussions in the above sections are for 2-D structures, we also present, in Section 4, characteristics of three-dimensional (3-D) nonreciprocal mircoresonators with finite thickness. We model the 3-D structures using a full-vectorial finite-element method in the cylindrical coordinate system [8]. We investigate the dependence of nonreciprocal phase shifts and radiation losses on the width and the thickness of the microdisk, leading to a guideline for the design of optical isolators, and demonstrate that the 3-D resonators can also function as miniaturized isolators or circulators. Section 5 summarizes important features of the present devices.

## 2. A nonreciprocal microresonator coupled with waveguides

#### 2.1 Design

To design a miniaturized optical isolator, we consider a nonreciprocal microresonator coupled with two straight waveguides, as shown in Fig. 1. The direction of magnetization is parallel to the *x*-direction. A circular domain wall divides the disk resonator into inner and outer domains with opposing directions of magnetization. This is the basis of the nonreciprocal effect for TM modes. We assume a typical magnetic garnet, with the refractive index of *n* = 2.302 and Faraday rotation of |Θ_{F}| ≈ 3000°/cm, as the core material, and air as the cladding material. We assume that there is no variation in the *x*-direction in this example. Later we will discuss, in Section 4, a more realistic 3-D model based on a garnet thin-film. This thin-film can form circular domain walls, so-called magnetic bubbles [9, 10].

As the first stage of our design procedure, we examine the characteristics of the mircroresonator. Figure 2 shows the calculated dependence of the radiation losses per circulation, *γ*, on disk radius *R*
_{D}, for the TM mode at the wavelength of *λ* = 1300 nm. The calculations in this design procedure are based on the finite-element method with the perfectly matched layer (PML) in the cylindrical coordinate system [11]. We assume that the resonator is reciprocal (|Θ_{F}| = 0) in the finite-element calculations. Although the radiation losses increase exponentially as *R*
_{D} decreases, the losses are sufficiently low (*γ* < 10^{-4} dB) even for a very small radius like *R*
_{D} = 2 μm.

Figure 3 shows the mode profiles for different radii, *R*
_{D} = 1, 2, 3, 4, and 5 μm. The modes are tightly confined near the inner edge of the disk. The radiation field is, however, significant for *R*
_{D} = 1 μm.

Figure 4 shows the dependence of Δ*φ*, the nonreciprocal phase shifts per circulation, on *R*
_{D} - *R*
_{B}, the difference of the disk radius and the magnetic bubble radius. We calculate the values of Δ*φ* by substituting the field amplitudes into the following equation, derived from the perturbation theory [12, 13]:

where *R*
_{eff} is the effective disk radius, Δ*β*
_{eff} is the difference of the effective propagation constants, and *ξ* = *nλ*Θ_{F}/*π* is the nondiagonal element of the permittivity tensor. Clearly, the Δ*φ* dependence peaks at the point where the mode profile peaks.

Figures 5, 6, and 7 show the wavelength dispersion of Δ*φ*, *φ*, and *α*, respectively, where *φ* represents the phase shifts per circulation for the reciprocal resonator. In this case, the phase shifts for the anti-clockwise and clockwise directions in the disk can be represented as *φ*
_{a} = *φ* + Δ*φ*/2 and *φ*
_{c} = *φ* - Δ*φ*/2, respectively. Therefore, the resonance wavelength, *λ _{m}*, is represented as

where *m* and *k*
_{0} represents the mode order and the vacuum wave number, respectively. Therefore, the nonreciprocal resonance shift, Δ*λ _{m}*, is represented as

Table 1 summarizes the values of *R*
_{B}, *m*, *λ _{m}*, and Δ

*λ*, as well as the radiation losses at the resonance,

_{m}*γ*, for

_{m}*R*

_{D}= 1, 2, 3, 4, and 5 μm. We select the value of

*m*where

*λ*has the values closest to 1300 nm, for each value of

_{m}*R*

_{D}. We focus on the value of

*R*

_{B}, with the maximum Δ

*φ*value in Fig. 5.

As the second stage of our design procedure we examine the characteristics of the entire structure, which includes two straight waveguides, as an optical isolator. We calculate the transmission from port 1 to 2 or from port 2 to 1, shown in Fig. 1, with the transfer matrix technique [13] using the values of Δ*φ*, *φ*, and *γ* in Figs. 5, 6, and 7. We assume that a single unidirectional mode of a disk resonator is excited and that the coupling is lossless.

As the value of Δ*λ _{m}* is larger for the smaller values of

*R*

_{D}in Table 1, we first focus on the case of

*R*

_{D}= 1 μm. Figure 8 shows the calculated isolation as a function of both

*κ*and

*λ*.

*κ*is the coupling coefficient of the coupling sections. In the range of

*λ*(1345 nm <

*λ*< 1350 nm), the transmission from port 1 to 2,

*T*

_{12}, is greater than the transmission from port 2 to 1,

*T*

_{21}, so the isolation is represented as 10log(

*T*

_{12}/

*T*

_{21}).

Unfortunately, the isolation is too small (< 2 dB) to function as an isolator for any value of κ and *λ*. This is due to the low *Q* value of the resonator.

We next examine the characteristics for *R*
_{D} = 2 μm, which exhibits the second largest Δ*λ _{m}* value in Table 1. Figures 9(a) and (b) show the isolation and the insertion loss, respectively. The insertion loss corresponds to the propagation loss for the signal from port 1 to 2. We assume that this propagation loss is equal to the radiation loss in the disk. As shown in Fig. 9(a), when the value of

*κ*increases, the operation bandwidth clearly expands with sufficiently high isolations (e.g. 20 dB or 40 dB). However, at the same time, the insertion loss also increases, as shown in Fig. 9(b). There is, therefore, a tradeoff between the bandwidth and the insertion loss. For example, to suppress the insertion loss below 0.1 dB around the resonance (

*λ*= 1322.04 nm), the value of

*κ*must be less than 0.102. When

*κ*= 0.1, this device functions as an optical isolator, in the 20 dB isolation bandwidth of ~ 0.02 nm.

Similarly, Figs. 10(a) and (b) show the isolation and the insertion loss, respectively, for *R*
_{D} = 3 μm. Aside from the difference of the resonance, the isolation and the insertion loss exhibit similar tendencies to *R*
_{D} = 2 μm, shown in Fig. 9. This structure also functions as an optical isolator. However, the operation bandwidth is narrower than that for *R*
_{D} = 2 μm. This is due to the lower nonreciprocal resonance shift, Δ*λ _{m}*, as shown in Table 1. Importantly, the impact of the insertion loss on the operation bandwidth is smaller than that of Δ

*λ*. Although we do not show the data here, the operation bandwidth becomes increasingly narrower for larger disk radii,

_{m}*R*

_{D}= 4 and 5 μm.

#### 2.2 Finite element simulation

We demonstrate the function of the designed microresonator geometry with two straight waveguides as an optical isolator, by using a finite element method for nonreciprocal optical waveguides [7]. We use the parameters of the disk resonator for *R*
_{D} = 2 μm in Table 1, as this design exhibits the best performance in the comparison in the previous section. The refractive index of the straight waveguides is assumed to be the same as that of the resonator, *n* = 2.302. The waveguide width is set as *W* = 0.4 μm to obtain single-mode operation. Figure 11 shows the dependency of *κ*, the coupling coefficient, on *D*, the gap width between the waveguide and disk resonator. The values of *κ* are calculated using the geometry with a half disk resonator, as shown in the inset of Fig. 11. We evaluate the transmission through the straight waveguide, *T*, with the finite element method, and estimate the values of *κ* from $\kappa =\sqrt{1-T}$. From the figure, we can recognize that the coupling geometry with *D* = 0.31 μm satisfies *κ*= 0.1 (identified in the previous section).

Figure 12(a) shows the normalized transmission characteristics for the fundamental TM mode, evaluated by the finite element method. The red line represents the transmission from port 1 to port 2, *T*
_{12}, the green line represents the transmission from port 2 to port 1, *T*
_{21}. Clearly, the device functions as an optical isolator. *T*
_{21} is minimized around *λ* = 1322.1 nm. At the same time, *T*
_{12} is sufficiently high at this wavelength. However, the band width where the isolation is higher than 20 dB is limited to 0.005 nm, which is narrower than the width predicted in the previous section (≈ 0.02 nm). This is probably due to the asymmetric spectra of the nonreciprocal resonator (asymmetry is clear from the resonance to the shorter wavelengths and to the longer wavelengths).

By modifying the value of *D*, we can obtain wider operation width. Figure 12(b) shows the normalized transmission characteristics for *D* = 0.24 μm. The spectra exhibit patterns similar to those for *D* = 0.31 μm shown in Fig. 12(a). The operation bandwidth with isolations higher than 20 dB, however, expands to 0.018 nm.

## 3. Parallel-coupled nonreciprocal microresonators

#### 3.1 Design

In the previous section, we have demonstrated that a nonreciprocal microresonator coupled with two waveguides can function as an optical isolator. However, the operation bandwidth of this device was limited to only Δ*λ* ≈ 0.02 nm. For a fresh approach to this problem, in this section, we propose parallel-coupled nonreciprocal microresonators, and demonstrate how this configuration is useful for increasing the operation bandwidth.

In designing a miniaturized optical isolator with wider bandwidth, we consider parallel-coupled nonreciprocal microdisk resonators, shown in Fig. 13. Such a parallel-coupled scheme can ease the need for fabrication control [15], because the input light does not have to propagate through each and every one of the microdisks before dropping at the output, as in serial coupling configurations. We assume each resonator is the same as in Section 2. Each resonator, composed of a magnetic garnet with *n* = 2.302 and |Θ_{F}| ≈ 3000°/cm, has a circular domain wall. There is no variation in the *x*-direction.

The coupler design presented in [16] begins with the specifications for the free spectral range, *FSR*, and for the 1-dB transmission bandwidth, *B*. The coupling coefficient is expressed as

where the subscript *i* denotes the disk number (*i* = 1, 2, ⋯, *N*) and the *Q* factor is expressed as

with

We select the value of mode order *m* where *λ _{m}* has the values closest to 1300 nm, for each value of

*R*

_{D}, as shown in Table 1. We focus on the parameters for

*R*

_{D}= 2 μm in this example, because this radius is most suitable for nonreciprocal microresonators among the values of

*R*

_{D}in Table 1, as discussed in Section 2. Nonreciprocal resonance shift Δ

*λ*is the highest excepting the case of

_{m}*R*

_{D}= 1 μm, with significant radiation loss

*γ*. The free spectral range is calculated as

_{m}*FSR*= 11.28 THz, for the resonator with

*R*

_{D}= 2 μm.

We first examine the dependence of the isolation and insertion loss on disk separation *L*. We consider the situation where *T*
_{12}, the transmission from port 1 to port 2, corresponds to the through signal and *T*
_{21}, the transmission from port 1 to port 2, corresponds to the isolated signal. In this case, the isolation is determined by the transmission ratio, *T*
_{12}/*T*
_{21}. The insertion loss corresponds to the transmission of the through signal, *T*
_{12}. We calculate these transmissions with the transfer matrix technique [13] using the values of Δ*φ*, *φ*, and *γ* in Figs. 5, 6, and 7. Figures 14(a) and (b) show the calculated isolation and insertion loss, respectively, as a function of both *L* and *λ*, for *B*= 125 GHz and *N* = 3. The values of *κ _{i}* are calculated as

*κ*=

_{1}*κ*= 0.156 and

_{3}*κ*= 0.219, following the above procedure for

_{2}*B*= 125 GHz. There are regions where the isolation is negative (

*T*

_{21}>

*T*

_{12}) in Fig. 14(a). Although we focus on the positive region in this example, we can use the negative region in the same way if we assume that the signal is input at port 2, where

*T*

_{21}corresponds to the through signal. Importantly, Fig. 14(a) shows a pattern of periodic changes depending on

*L*. This is due to the additional resonance through the paths of the half-disks, couplers, and waveguides connecting them, on top of the resonance through each disk [16]. The value of

*L*is one of the keys to achieving wide band operation. This value

*L*must be set so that the isolation is sufficiently high in Fig. 14(a) and, at the same time, the insertion loss is sufficiently low in Fig. 14(b). For example, the operation bandwidth wider than 0.4 nm can be achieved in the range of 5.80 μm <

*L*< 5.83 μm, with both isolation higher than 20 dB and insertion loss lower than 0.1 dB.

We next examine the dependence of the isolation and insertion loss on transmission bandwidth *B*, another key to wide band operation. Figures 15(a) and (b) show the calculated isolation and insertion loss, respectively, as a function of both *B* and *λ*, for *L* = 5.8 μm and *N* = 3. The values of *κ _{i}* are calculated for each value of

*B*, following the above procedure. As shown in Fig. 15(a), when the value of

*B*increases, the bandwidth with sufficiently high isolations (e.g. 20 dB or 40 dB) clearly shifts to longer wavelengths. As shown in Fig. 15(b), on the other hand, the bandwidth with sufficiently low insertion losses (e.g. 0.1 dB) clearly expands without a shift in the center wavelength. We can therefore design an optical isolator with a wider operation bandwidth by adjusting the value of

*B*so that the overlap of the region with sufficiently high isolations in Fig. 15(a) and the region with sufficiently low insertion losses in Fig. 15(b) becomes wider. For example, the operation bandwidth wider than 0.5 nm can be achieved in the range of 120 GHz <

*B*< 130 GHz, with both isolation higher than 20 dB and insertion loss lower than 0.1 dB.

#### 3.2 Finite element simulation

In this section, we demonstrate the function of the three-disk geometry as an optical isolator, using the finite element method [7]. The disk separation is set as *L* = 5.8 μm, in the designed range. The gap widths between the waveguides and disk resonators are determined as *D*
_{1} = *D*
_{3} = 0.157 μm and *D*
_{2} = 0.06 μm to obtain the calculated values of *κ _{i}* using the calculated dependency of

*κ*on

*D*shown in Fig. 11. Figure 16 shows the normalized transmission characteristics for the fundamental TM mode, identified with the finite element method. The red line represents the transmission from port 1 to port 2,

*T*

_{12}, the green line represents the transmission from port 2 to port 1,

*T*

_{21}, and the blue line represents the transmission from port 2 to port 4,

*T*

_{24}. Clearly, the device functions as an optical isolator or circulator.

*T*

_{21}is less than -21 dB around

*λ*= 1322.4 nm. At the same time,

*T*

_{12}is sufficiently high, greater than -0.1 dB, at this band. The bandwidth where the isolation is higher than 20 dB and the insertion loss is lower than 0.1 dB is Δ

*λ*≈ 0.4 nm; the width approximates 20 times greater than the structure discussed in Section 2.

## 4. Characteristics of three-dimentional nonreciprocal resonators

In Sections 2 and 3, we have proposed and demonstrated optical isolators based on nonreciprocal microresonators. We assumed, however, that there was no variation in the *x*-direction (the models are two-dimensional). Naturally, any adequate realization of a resonator structure cannot have infinite height, and so investigations of 3-D models are necessary to characterize, in a rigorous way, realistic resonator structures with finite heights. In this section, we present the full-vector analysis for nonreciprocal resonance shifts in 3-D nonreciprocal microdisk resonators, including consideration of radiation losses. In particular, we investigate the significance of the resonator geometry for nonreciprocal phase shifts and losses, to demonstrate how the 3-D resonator is practical for miniaturized nonreciprocal circuits.

We consider the nonrecirprocal microdisk resonator shown in Fig. 17. We assume a typical magnetic garnet with *n* = 2.302 and |Θ_{F}| ≈ 3000°/cm as the core material. The refractive index of the cladding layers are low so that *n* = 1.45 (SiO_{2}), to reduce radiation losses of the microresonator. This structure (garnet on SiO_{2}) can be realized using techniques such as epitaxial liftoff [18], wafer bonding [19], and a method of thin film growth with thermal treatment [20]. We assume that a magnetic bubble can form at the center of the disk. This yields the nonreciprocal effect for the TE-like mode. The bubble radius is controlled by the film thickness and the external magnetic field [10]. In addition, significant absorption of the garnet film clearly can damage the characteristics of the microdisk. Fortunately, it has been reported that liquid-phase-epitaxy films with ion compensation processes exhibit absorption losses as low as 0.5 dB/cm [3], which do not significantly affect the device operation. We therefore do not involve material absorption in our discussion here.

We examine the characteristics of the 3-D mircroresonator with the full-vectorial finite-element method in the cylindrical coordinate system [8]. The computational domain is in the cross-section of the resonator, as represented by the dashed lined in Fig. 17(a) and Fig. 17(b). This domain is divided into more than 20000 linear tangential and quadratic normal (LT/QN) elements [21]. Radiation losses can be identified from the imaginary part of the complex propagation constants of the leaky modes. To deal properly with these leaky modes, the solution region must be truncated with an appropriate absorbing boundary condition. One of the most flexible and efficient absorbing boundary conditions is the PML [11]. We select, therefore, the parameters for the PML: *ρ*
_{max} = 5 μm, *x*
_{min} = -4 μm, *x*
_{max} = 4 μm, tan *δ*= 5, and *d*
_{PML} = 1 μm, where *d*
_{PML} is the thickness of the PML and *δ* is the loss angle on the exterior boundary of the PML. Each PML region is divided into 15 layers of elements.

Figure 18 shows the calculated dependence of the radiation losses per circulation, *γ*, on disk radius *R*
_{D}, at the wavelength of *λ* = 1300 nm, for different disk thickness, *H* = 300 nm, 400 nm, and 500 nm. Clearly, the radiation losses decrease as the values of *H* increase and the values of *R*
_{D} increase. For example, the disk geometry with *H* = 500 nm and *R*
_{D} = 4 μm exhibits *γ* < 10^{-4} dB. We have shown in Sections 2 and 3 that this small loss does not significantly damage the isolation operation.

Figures 19(a), (b), and (c) show the dependence of Δ*φ*, the nonreciprocal phase shifts per circulation, on *R*
_{D} - *R*
_{B}, the difference of the disk radius and the magnetic bubble radius, for *H* = 300 nm, 400 nm, and 500 nm, respectively. The values of Δ*φ* for *H* = 300 nm shown in Fig. 19(a) are smaller than those for *H* = 400 nm shown in (b), especially for smaller values of *R*
_{D}. On the other hand, there is no significant difference between Δ*φ* values for *H* = 400 nm and 500 nm, shown in Figs. 19(b) and (c) respectively. The thickness of *H* = 400 nm is sufficient for the nonreciprocal phase shifts.

Finally we demonstrate the characteristics of an optical isolator based on the 3-D nonreciprocal resonator. This resonator can support several transverse higher-order modes depending on the structural parameters (*H* and *R*
_{D}). Yet the free-spectral range of these modes is very large, because the radius of the resonator is as small as several micrometers (e.g. *FSR* > 5 THz for *H* = 500 nm and *R*
_{D} = 4 μm). With the selection of adequate parameters, therefore, the higher-order modes are confined to outside the operation band, and these modes do not affect the device operation. We consider only the fundamental TE-mode as the mode impacting the device operation. We focus on the parallel coupled resonator configuration, presented in Section 3, as this configuration allows a wider operation bandwidth. Figure 20 shows the calculated transmission spectra of the parallel coupled 3-D resonators, for *H* = 500 nm and *R*
_{D} = 4 μm. The coupling coefficients *κ _{i}* are calculated as

*κ*=

_{1}*κ*= 0.2025 and

_{3}*κ*= 0.2832, following the above procedure for

_{2}*B*= 50 GHz. The disk separation is assumed to be

*L*= 10.08 μm. The calculation is based on the transfer matrix technique, using the values of Δ

*φ*,

*φ*, and

*γ*identified with the vectorial finite-element method.

*T*

_{12}corresponds to the through signal,

*T*

_{21}corresponds to the isolated signal, and, when the device works as an optical circulator,

*T*

_{24}corresponds to the drop signal. This 3-D structure clearly functions as an optical isolator or circulator.

*T*

_{21}is less than -22 dB around

*λ*= 1295.7 nm. At the same time,

*T*

_{12}is sufficiently high, greater than -0.1 dB, at this band. The bandwidth where the isolation is higher than 20 dB and the insertion loss is lower than 0.1 dB approximates 0.4 nm. This width is comparable to that of the 2-D structure discussed in Section 3. In addition, in terms of the device size, smaller than 10 μm × 40 μm, the 3-D model maintains the advantage due to the introduction of resonators.

## 5. Summary

We have proposed a form of nonreciprocal microresonators, which can dramatically reduce the size of optical waveguide isolators and circulators. We present design procedures, and numerically demonstrate the operations for two configurations of these devices: an isolator consisting of a single nonreciprocal resonator and an isolator consisting of parallel-coupled nonreciprocal resonators. The size of magneto-optical devices based on these configurations can be reduced down to several tens of micrometers. The configuration based on parallel-coupled nonreciprocal resonators is capable of expanding the operating bandwidth of nonreciprocal optical devices derived from microresonators.

We have characterized nonreciprocal phase shifts and radiation losses in 3-D magneto-optical resonators with finite heights. We modeled the 3-D structure with the full-vectorial finite-element method in the cylindrical coordinate system, which is efficient and accurate in the analysis of microdisk resonators. We have investigated the dependence of nonreciprocal phase shifts and radiation losses on the width and the thickness of the microdisk, leading to a guideline for the design of optical isolators. We have therefore demonstrated how the resonators with effective parameters are practical for miniaturized nonreciprocal circuits.

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