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Abstract
This work shows how phase-shifted time-modulations applied to two or more near-field coupled resonators can be engineered to achieve magnet-free optical isolation in a compact footprint. Because of the strong light-matter interaction afforded by high quality factor resonant cavities, only modest modulation amplitudes are required. A coupled mode theory model is developed which is subject to simulated annealing to optimize device performance. Then it is shown via finite-difference time-domain simulation how the device may be implemented in a one dimensional photonic crystal geometry etched in a silicon ridge waveguide. These devices can be implemented in native silicon with standard electrical contacts thereby maintaining CMOS-compatible fabrication without the need for additional specialized materials. Isolation ratios of over 40 dB with insertion loss less than 1 dB using modulation frequencies under 25 GHz are achievable in this device platform.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Electronic space-time modulation of the dielectric permittivity of nanophotonic devices is an attractive method to break time-reversal symmetry because it does not require magnetic materials or an applied magnetic field, and it can be implemented in a CMOS-compatible silicon photonics platform [1,2]. Breaking time-reversal symmetry is necessary for implementing critical on-chip devices such as optical isolators [3–6] and circulators [7]. In the following, it is shown how near-field coupled microcavity resonators whose materials are subject to space-time modulation via external microwave frequency voltages can provide optical isolation with minimal device footprint and with experimentally realizable modulation amplitudes and frequencies.
This work highlights designs based on two and three near-field coupled resonators. The two-resonator design is similar to a concept previously demonstrated in which a time modulation is applied to two spatially separate regions of a waveguide and phase shifted by $\pi /2$ [8,9]. In that case, closed form design equations are obtainable that relate the device length, modulation amplitude and modulation frequency which makes device optimization straightforward. The drawback is that the phase modulation sections must be connected by waveguides whose lengths are on the order of millimeters [8]. Incorporating 2D materials such as transition metal dichalcogenide monolayers with a larger range of refractive index tunability allows devices to be shrunk to 50$\mu$m [9], but the inhomogeneous material systems may complicate fabrication.
On the other hand, the designs proposed herein rely on the enhanced light-matter interaction afforded by high quality ($Q$) factor microresonators so that the required refractive index variation may be effected in the same silicon material used to construct the passive waveguides. Designs for implementation in 400nm wide ridge waveguides with lengths of 12$\mu$m and 18$\mu$m are shown to exhibit isolation ratios exceeding 40 dB. Because of the complicated energy dynamics of these systems, simple closed form design equations are not forthcoming. Instead, a coupled mode theory (CMT) model is developed [10,11] whose accuracy is verified using full-wave finite-difference time-domain (FDTD) simulations [12]. Device parameters that maximize forward throughput while maintaining a prespecified backward extinction are obtained by subjecting the CMT model to numerical optimization via the simulated annealing method [13,14]. Moving from two to three microresonators generalizes the device concept and provides more design degrees of freedom resulting in better overall device performance via optimization. Subsequently, it is shown how the optimized devices can be implemented in a compact one-dimensional (1D) photonic crystal platform [15,16].
Breaking time-reversal symmetry in this context requires time-modulation of the refractive index in at least two different locations with an appropriate phase shift [17]. Many signal processing functionalities are effected by single-point electrooptic modulation such as intensity and phase modulation [18]. Interesting switching behavior has been demonstrated by time-modulating a single location of a ring resonator [19–22] as well as by modulating two coupled resonators with a $\pi$ phase shift [23]. But none of these device concepts breaks time-reveral symmetry in a way that leads to non-reciprocal wave propagation required for optical isolation. For that purpose, at least two spatially separate modulations with a phase shift other than a multiple of $\pi$ are required.
An optical isolator is typically used to protect an active optical element such as a laser or amplifier from back propagation and back-reflection resulting from other integrated components which can cause instability and device damage [24]. An ideal optical isolator is a device whose scattering matrix is described by
for all frequencies $\omega$[17,25]. The $X$ in the $S_{22}$ position indicates that this parameter will be considered irrelevant in the present context; however, forward reflection may be an important consideration in some applications. Experimentally realizable isolators will fall short of the ideal in ways that depend on their implementation. For example, in the microresonator approach described here, isolation occurs in only a small bandwidth centered at the resonance frequency of the cavity rather than for all frequencies.Additionally, in the present design, there is a tradeoff between minimizing $S_{11}$ and $S_{12}$ to zero and maximizing $S_{21}$ to unity. Since this optimization problem involves multiple objectives, gradient-based optimization approaches are disqualified. Instead, simulated annealing (SA) optimization is used to identify approproate device parameters. Furthermore, the SA optimization is applied to the CMT model rather than to a specific device implementation. This means the optimization routine needs only to solve an algebraic system of equations for each iteration rather than run a full-wave simulation, resulting in orders of magnitude improvement in efficiency [26]. Once optimized device parameters are obtained via CMT SA optimization, a practical device implemented in a properly tuned 1D photonic crystal ridge waveguide is demonstrated via FDTD simulations.
2. Coupled mode theory model for two-resonator system
Figure 1 shows a generic two-resonator configuration consisting of near-field coupled resonators each coupled to a port waveguide. The two resonators are assumed identical with the same resonance frequency $\omega _0$ and intrinsic (absorption and radiation) loss $\gamma _0$. An external microwave-frequency voltage is applied to the material comprising each resonator with a phase shift $\theta _2$ between resonators 1 and 2. The change in refractive index in response to the applied voltage arises from carrier plasma dispersion in silicon lightly doped ($\Delta N \sim 10^{17}$ cm$^{-3}$) to achieve the necessary electrorefraction while minimizing absorption loss [27]. Because this device concept requires only modest changes in the refractive index, it may be implemented in a variety of material systems beyond silicon such as lithium niobate [28–30] and two-dimensional materials on semiconductor substrates [31,32].
Fig. 1. Two generic optical resonators (labeled $v=1,2$) coupled to each other with coupling rate $\kappa$ and each coupled to a waveguide port with coupling rate $\sqrt {d_v/2}$. Input and reflected signals in the ports are denoted by $s_{iv}$ and $s_{rv}$, respectively. The material comprising the two resonators are subject to an applied microwave frequency voltage $V(t)$ which causes a time-dependent perturbation to the resonance frequency $\omega _v(t)$.
The theoretical model used in the optimization algorithm is based on a combination of temporal coupled mode theory [11,33] and first order time-dependent perturbation theory formulated for time-varying electromagnetic systems [10,23]. The model is constructed by expressing the electromagnetic field of a single isolated cavity as $|{\mathbf {F}_v(\mathbf {r})}\rangle \equiv [\mathbf {D}_v(\mathbf {r}) \: \: \mathbf {H}_v(\mathbf {r}) ]^{\textrm {T}}$ where $v = 1,2$ denotes the cavity. Then the total field of the coupled system is expanded in the fields of the separate cavities
Solutions to Eq. (3) take the Floquet form
where $\omega$ is the excitation frequency [34,35]. Inserting Eq. (4) into Eq. (3), the following algebraic system of equations is obtained after matching synchronous frequency terms.3. Device optimization by simulated annealing
Equations (5) and (6) comprise the model subject to SA optimization which involves five design variables associated with the static resonator system: $\omega _0$, $\gamma _0$, $\gamma _{c1}$, $\gamma _{c2}$, $\kappa$; and four design variables associated with the dynamic modulation: $\omega _m$, $\delta \omega _1$, $\delta \omega _2$, $\theta _2$. The cavity resonance frequency $\omega _0$ sets the operating frequency of the device and can be tuned independently of isolator performance, so it was not subject to SA optimization. Preliminary optimization runs showed that the optimal $\gamma _0$ tended to zero, so it was fixed to $\gamma _0 = 2.58\times 10^{9}$ s$^{-1}$ corresponding to zero absorption loss and a resonator $Q$ factor of $2.35\times 10^{5}$ which is within the range of experimental realizability for the photonic crystal nanobeam resonators to be discussed later [16] as well as other cavity types [36–40]. Therefore, the ensuing optimization procedure spans a seven-dimensional parameter space.
The SA optimization algorithm is described in Table 1. The state vector is defined as $\mathbf {s} \equiv [\kappa \:\: \gamma _{c1} \:\: \gamma _{c2} \:\: \omega _m \:\: \delta \omega _1 \:\: \delta \omega _2 \:\: \theta _2]^{\textrm {T}}$. In brief, the algorithm randomly samples parameter space. If a given sample $\mathbf {s}_n$ produces a higher forward throughput $|S_{0,21}(\omega _0)|^{2}$ while maintaining the desired extinctions of $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,12}(\omega _0)|^{2}$, then $\mathbf {s}_n$ is accepted as the new local optimum. This process continues until the algorithm converges to a quasi-global optimum. Convergence is aided by shrinking the neighborhood in which the algorithm samples, which is the cooling schedule in SA. Line 5 shows how the parameter space is randomly sampled from a uniform distribution centered at the most recent local optimal state $\mathbf {s}_{\textrm {opt}}$. The user specifies S11ext and S12ext on line 3 according to desired device specifications.
Table 1. Simulated annealing optimization algorithm. S21max is the maximized scattering element $|S_{0,21}(\omega _0)|^{2}$. S11ext and S12ext are the desired extinction values for $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,12}(\omega _0)|^{2}$. $0 < \lambda _1,\lambda _2 \leq 1$ are the cooling rates.
The traditional SA algorithm allows for probabilistically assigning a new $\mathbf {s}_{\textrm {opt}}$ according to the Boltzmann distribution even if the cost function is not improved. On the other hand, this SA algorithm accepts a new $\mathbf {s}_{\textrm {opt}}$ only if there is improvement in the cost function (increases in $|S_{0,21}(\omega _0)|^{2}$). Hence, this approach is referred to as monotonic SA. Additionally, two different cooling rates are used. $\lambda _1$ is a slower rate (which means it is closer to 1) than $\lambda _2$. This scheme is motivated by a desire to shrink the searchable state space around intermediate optima as the algorithm progresses. Typical values for these coefficients are $\lambda _1 = 0.9999897$ and $\lambda _2 = 0.95$. Figure 2(a) displays the convergence toward the maximum of $|S_{0,21}(\omega _0)|^{2}$ over a span of $10^{5}$ iterations for both traditional SA and monotonic SA. Monotonic SA clearly outperforms traditional SA for this problem.
Fig. 2. (a) Convergence of the optimization algorithm toward the maximum of $|S_{0,21}(\omega _0)|^{2}$. (b) Scattering spectra calculated via CMT for optimized two-resonator isolator when the source is in Port 1. (c) Same as (b) but the source is in Port 2. Device parameters are listed on the first line of Table 2.
Table 2. Typical two-resonator device parameters obtained from SA optimization. $\kappa$, $\gamma _{c1}$ and $\gamma _{c2}$ are in units of $10^{10}$ s$^{-1}$. $f_m$ is in GHz. Additionally, $\omega _0 = 2\pi 193.55$ THz, $\gamma _0 = 2.58\times 10^{9}$ s$^{-1}$ and $f_m$ was restricted to be less than 25 GHz. “ext” refers to the maximum allowed values of $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,12}(\omega _0)|^{2}$. “iso” is defined as iso $=10 \log _{10}(|\frac {S_{0,21}(\omega _0)}{S_{0,12}(\omega _0)}|^{2})$
4. Device results for two-resonator isolator
Figure 2(b) and (c) show the scattering spectra for the zeroth harmonic calculated using CMT with parameters optimized from monotonic SA. Forward throughput reaches $|S_{0,21}(\omega _0)|^{2}=0.491$ while $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,12}(\omega _0)|^{2}$ were held to $<10^{-2}$ producing an isolation ratio of 16.9 dB. The first row of Table 2 summarizes the optimized device parameters corresponding to the spectra depicted in Fig. 2(b) and (c). Table 2 also shows the optimized device parameters when both $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,12}(\omega _0)|^{2}$ are held to less than $10^{-3}$ and $10^{-4}$. Lowering the extinction value results in a lower forward throughput, but the isolation ratio, defined as $10 \log _{10}(|\frac {S_{0,21}(\omega _0)}{S_{0,12}(\omega _0)}|^{2})$, exhibits a net increase from 16.9 dB to 35.6 dB.
In producing the results shown in Table 2, the modulation frequency is held to $f_m < 25$ GHz to ensure experimental realizability [41,42]. The resulting modulation amplitudes are all on the order of $\delta \omega / \omega _0 \sim 10^{-4}$ which is within the range of typical photonic materials such as silicon or lithium niobate. In each case, the modulation phase shift $\theta _2$ between the two cavities converges to a value slightly less than $\pi /2$ suggesting that the underlying time-reversal symmetry breaking mechanism is similar to that exploited in tandem phase shifter isolators previously demonstrated [8,9].
Figure 3 displays isolator performance over a wider range of operating conditions. Figure 3(a) and (b) show the isolation ratio and insertion loss, respectively, for $|S_{0,11}(\omega _0)|^{2}, |S_{0,12}(\omega _0)|^{2}$ extinction values ranging from $3.5 \times 10^{-5}$ to $5 \times 10^{-2}$ and cavity $Q$ factors ranging from $10^{4}$ to $5\times 10^{5}$. The modulation frequency is held to 25 GHz or lower in the SA optimization. The isolation ratio tends to increase as $Q$ factor increases and as extinction decreases. Figure 3(b) shows that insertion loss decreases as $Q$ factor increases, but insertion loss increases as extinction increases which illustrates the tradeoff between isolation ratio and insertion loss. Nevertheless, an important conclusion to draw from Fig. 3(a) and (b) is while higher $Q$ leads to better isolation ratio and lower insertion loss, the two-resonator isolator concept is functional over a wide range of experimentally realizable $Q$ factors with isolation ratios exceeding 30 dB and insertion losses less than 5 dB.
Fig. 3. (a) Isolation, defined as $10 \log _{10}(|\frac {S_{0,21}(\omega _0)}{S_{0,12}(\omega _0)}|^{2})$, as a function of cavity $Q$ factor and extinction of $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,12}(\omega _0)|^{2}$ while the modulation frequency is held to 25 GHz or lower. (b) Same as (a) but for insertion loss defined as $10 \log _{10} (|S_{0,21}|^{2})$. (c) Isolation as a function of modulation frequency $f_m$ and extinction of $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,12}(\omega _0)|^{2}$ while the $Q$ factor is set to $2.35 \times 10^{5}$. (d) Same as (c) but for insertion loss.
Figure 3(c) and (d) also show isolation ratio and insertion loss as a function of $|S_{0,11}(\omega _0)|^{2}$, $|S_{0,12}(\omega _0)|^{2}$ extinction, but $Q$ is held fixed at $2.35 \times 10^{5}$ while $f_m$ is swept from 1 GHz to 50 GHz. The isolation ratio tends to increase as $f_m$ increases and extinction decreases. Figure 3(d) shows that insertion loss decreases as $f_m$ increases, but insertion loss increases as extinction increases which again illustrates the tradeoff between isolation ratio and insertion loss. Again, it is noteworthy that the two-resonator isolator device exhibits isolation ratios exceeding 30 dB and insertion losses less than 5 dB at modulation frequencies from 10 to 50 GHz.
So far, scattering spectra for the two-resonator isolator device have been given in terms of the zeroth order harmonic. The time-modulation at frequency $\omega _m$ causes incident signal energy at frequency $\omega$ to scatter to frequencies $\omega +q\omega _m$. In combination with radiation loss and reflection, this frequency conversion process is a key mechanism in tailoring the scattering spectra. Figure 4 displays the amplitudes of the scattering elements for harmonic orders $|q| = 0, 1, 2, 3$ which illustrates the distribution of incident energy to various harmonic orders. The maximized amplitude of $|S_{0,21}(\omega _0)|^{2}$ along with the suppression of $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,12}(\omega _0)|^{2}$ is clear. However, the amplitudes of the first order harmonics $|S_{\pm 1,11}(\omega _0)|^{2}$ and $|S_{\pm 1,12}(\omega _0)|^{2}$ have amplitudes around 0.1 which violates the desired extinction of $<0.01$ in Port 1. Fortunately, the higher order harmonics are frequency shifted which may render them inconsequential in Port 1, but if needed, a narrowband filter can be placed just before Port 1 to suppress sideband energy from back propagating as shown in [23,43]. Additionally, in Section 7, it is shown how this issue may be mitigated by suppressing the amplitudes of harmonic orders other than $q=0$ in Port 1 using the SA optimization.
Fig. 4. Amplitudes of scattering elements of two-resonator isolator with parameters on first line of Table 2 for harmonic orders $q = -3 \dots 3$.
Within the context of higher order harmonic generation, it is interesting to compare this isolator concept to optical isolation enacted through engineering a synthetic frequency dimension [44,45]. In both cases, the underlying mechanism involves a space-time modulated coupled resonator system. However, in the synthetic frequency dimension approach, energy is intentionally coupled to nonzero harmonics by using a modulation frequency equal to the free spectral range of a ring resonator. Whereas, here, single-mode resonators are employed, and energy transfer to harmonic frequencies is viewed as detrimental and thus should be suppressed. In comparison to the synthetic frequency dimension approach, the isolator approach described herein maintains its advantage of small device size; it does not require precise alignment of multiple resonance frequencies between resonators; and working exclusively with the fundamental frequency can be advantageous for maintaining high forward throughput.
5. Implementation using photonic crystal nanobeam resonators
In order to implement the two-resonator isolator concept analyzed in the previous section, a device platform capable of tuning each of the parameters listed in Table 2 is required. Due to its compact footprint, unique ability to implement high $Q$ factor resonators and versatile tunability, 1D photonic crystals etched in a ridge waveguide forming nanobeam resonators is chosen [46]. Figure 5(a) depicts the device geometry tuned to implement the two-resonator isolator device. The geometry shown in Fig. 5(a) has $\kappa = 9.30 \times 10^{10}$ s$^{-1}$, $\gamma _0 = 2.58 \times 10^{10}$ s$^{-1}$ ($Q = 2.35 \times 10^{5}$) and was designed symmetrically so that $\gamma _{\textrm {c},1} = \gamma _{\textrm {c},2} = 6.48 \times 10^{10}$ s$^{-1}$ to ease the cavity design and FDTD simulation. The blue regions correspond to silicon with an effective refractive index $n=\sqrt {10}$ used in the two-dimensional FDTD simulations. The white regions denote air.
Fig. 5. (a) Two-resonator isolator implemented in 1D photonic crystal nanobeam platform. (b) Scattering spectra calculated via CMT and via FDTD simulation for optimized two-resonator isolator when the source is in Port 1. (c) Same as (b) but the source is in Port 2. (d) $|H(x,y)|^{2}$ when the isolator is excited in Port 1 by a continuous wave source at frequency $\omega _0$. (e) Same as (d) but with the continuous wave source in Port 2.
Figure 5(a) shows how $\gamma _{\textrm {c},1}$ and $\gamma _{\textrm {c},2}$ may be tuned by the number of air holes etched between the cavities and the port waveguides. Additional fine tuning may be obtained by adjusting the size of the air holes. Holes 1-4 were added between Port 1 and Resonator 1. Holes 2-4 have a radius of $0.3\Lambda$ while hole 1 has a radius of $0.27\Lambda$ where $\Lambda$ is the center-to-center hole spacing as well as the ridge waveguide width. For device operation at $\lambda = 1.55 \mu$m, this parameter is set to $\Lambda = 372.1$ nm. Holes 5, 6, 7 and 8, 9, 10 are tapered symmetrically from the center of Cavity 1. From smallest to largest, their radii are $0.105\Lambda$, $0.15\Lambda$, $0.24\Lambda$. Tapering increases the cavity $Q$ factor by allowing the confined field to spread further into the cladding thereby reducing transverse radiation from the ridge waveguide boundaries. Holes 11-14 were added between Cavities 1 and 2 to adjust $\kappa$. Holes 11-13 have a radius of $0.3\Lambda$, and hole 14 has a radius of $0.18\Lambda$. Holes 1-14 are exactly mirrored for Cavity 2 and Port 2.
It is envisioned the space-time modulation results from a microwave frequency voltage applied via electrodes placed near the cavities as shown in Fig. 5(a). However, in the FDTD simulations, the refractive index of the cavity regions was changed explicitly within the algorithm rather than through modeling the electrical properties of the material. The modulation parameters used in the simulations are $f_m = 24.2$ GHz, $\theta _2 = 0.465\pi$, $\Delta n_1 = 8.21\times 10^{-4}$ and $\Delta n_2 = 12.1\times 10^{-4}$. Figure 5(b) and (c) show the spectra centered at $\omega _0$ for each element of the scattering matrix confirming the desired device characteristics obtained from CMT: $|S_{11}(\omega _0)|^{2}, |S_{12}(\omega _0)|^{2} < 10^{-2}$ while maximizing $|S_{21}(\omega _0)|^{2}$ and leaving $|S_{22}(\omega _0)|^{2}$ unconstrained. The spectra correspond to the zeroth harmonic and were obtained using a pulsed excitation (centered at $\omega _0$) and a harmonic extraction technique [35,47,48]. The $[E_x, E_y, H_z]$ polarization was used in the 2D FDTD simulations. Figure 5(d) and (e) depict the scalar magnetic field $|H_z(x,y)|^{2}$ resulting from continuous wave (CW) excitation at $\omega _0$ from Port 1 and Port 2, respectively. Noting the logarithmic color scale, the amplitude in Port 2 of Fig. 5(d) is similar to that of Port 1 illustrating maximized throughput in the forward direction. On the other hand, the amplitude in Port 1 of Fig. 5(e) is reduced by $10^{-2}$ compared to that of Port 2 illustrating suppressed backward throughput.
The strong agreement between the FDTD simulation results and CMT analysis displayed in Fig. 5(b) and (c) is evidence that performing SA optimization on the computationally efficient CMT model will produce the desired results in the less-computatinally-efficient full wave simulations. Another advantage of this design approach is that the specific device implementation of the CMT model is immune to fabrication imperfections so long as the device geometry maintains the key parameters associated with the static structure listed in Table 2. In cases where the static structure parameters deviate from those intended due to fabrication imperfections, the imperfect static device parameters may be re-inputted into the SA optimization procedure to identify new space-time modulation parameters thereby potentially maintaining the isolation effect at the expense of insertion loss.
6. Space-time modulated three-resonator isolator
The space-time modulated resonator concept may be extended to any number of near-field coupled resonators. In this section, a three-resonator topology is considered which is depicted in Fig. 6. The CMT device model is readily extended from two-resonators to three-resonators. In place of Eq. (2), the starting field expansion is $|{\mathbf {F}(\mathbf {r},t)}\rangle = a_{1}(t)|{\mathbf {F}_{1}(\mathbf {r})}\rangle + a_{2}(t)|{\mathbf {F}_{2}(\mathbf {r})}\rangle + a_{3}(t)|{\mathbf {F}_{3}(\mathbf {r})}\rangle$. Then Eq. (5) becomes
Fig. 6. Three generic optical resonators (labeled $v=1,2,3$). Resonators 1 and 2 (2 and 3) are coupled to each other with coupling rate $\kappa _{12}$ ($\kappa _{23}$). Resonators 1 and 3 are each coupled to a waveguide port with coupling rate $\sqrt {d_1/2}$ and $\sqrt {d_3/2}$, respectively. Input and reflected signals in the ports are denoted by $s_{iv}$ and $s_{rv}$, respectively, with $v=1, 3$. $\theta ^{\prime }_3 = \theta _3 - \theta _2$ to align with the CMT model.
Table 3 displays the space-time modulation parameters obtained from SA optimization. Similar to the previous procedure, $|S_{0,31}(\omega _0)|^{2}$ was maximized while restricting $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,13}(\omega _0)|^{2}$ to a desired extinction value. Due to the larger design space, it is expected that larger values of $|S_{31}(\omega _0)|^{2}$ are obtainable [49]. Indeed, Table 3 confirms this hypothesis where insertion losses of 0.781 to 0.906 (-1.1 to -0.43 dB) are seen which are approximately a factor of two better than the two-resonator design. Interestingly, the resulting phase shifts $\theta _2$ and $\theta _3$ are approximately $\pi /2$ and $\pi$ which means the phase shift between each resonator is approximately $\pi /2$ consistent with the results from the two-resonator optimization. It is interesting to observe that while adding more resonators to the system inevitably adds more total loss, the insertion loss of the designed system decreases. This is because the resonators have small intrinsic losses (indicated by their high $Q$ factors) which are negligible compared to the energy lost to nonzero harmonic orders. The increased design freedom of additional resonators allows more control and suppression of interharmonic energy flow while maintaining small intrinsic cavity loss.
Table 3. Typical three-resonator device parameters obtained from SA optimization. $\omega _0 = 2\pi 193.55$ THz, $\gamma _0 = 2.58\times 10^{9}$ s$^{-1}$ and $f_m$ was set to 24.2 GHz. “ext” refers to the maximum allowed values of $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,13}(\omega _0)|^{2}$. “iso” is defined as iso $=10 \log _{10}(|\frac {S_{0,21}(\omega _0)}{S_{0,12}(\omega _0)}|^{2})$. The static parameters ($\kappa _{12}$, $\kappa _{23}$, $\gamma _{c1}$, $\gamma _{c3}$) for each row are given in Table 4 in Appendix A.
Figure 7(a) displays the three-resonator isolator implemented in the same 1D photonic crystal platform discussed previously. As before, a device topology with mirror symmetry along $x$ is used to ease implementation and simulation (device parameters are given in Table 5 in Appendix A). Figure 7(b) and (c) show the zeroth harmonic order scattering spectra for Ports 1 and 3 illustrating good agreement between the CMT model and FDTD simulations. This device has a forward throughput of 0.929 while propagation to Port 1 was suppressed to 0.005 resulting in an isolation ratio of 22.7 dB with -0.3 dB insertion loss. Noting the logarithmic color scale in Fig. 7(d), the amplitude in Port 3 is similar to that of Port 1 illustrating maximized throughput in the forward direction. On the other hand, the amplitude in Port 1 of Fig. 7(e) is reduced by more than $5\times 10^{-3}$ compared to that of Port 3 illustrating suppressed backward throughput.
Fig. 7. (a) Three-resonator isolator implemented in 1D photonic crystal nanobeam platform. (b) Scattering spectra calculated via CMT and via FDTD simulation for optimized three-resonator isolator when the source is in Port 1. (c) Same as (b) but the source is in Port 3. (d) $|H(x,y)|^{2}$ when the isolator is excited in Port 1 by a continuous wave source at frequency $\omega _0$. (e) Same as (d) but with the continuous wave source in Port 3.
7. Increasing isolator bandwidth
The bandwidth of the space-time modulated near-field coupled resonator isolator concept is limited by two mechanisms. The first mechanism results from the time modulation itself: for an optical carrier centered at $\omega _0$, the modulated signal is limited to the spectral region $\omega _0 \pm \omega _m/2$ to avoid interference with the signal spectra centered at the $q=\pm 1$ harmonics thereby limiting the total signal bandwidth to $\omega _m$. This bandwidth limitation applies to both resonant and non-resonant time-modulated device topologies. Fortunately, this limitation can be easily mitigated using the SA optimization approach by including the suppression of scattering coefficients corresponding to higher order harmonics in the optimization procedure. Figure 8 displays the scattering coefficients up to $|q| = 3$ for a three-resonator isolator optimized to maximize $|S_{0,31}(\omega _0)|^{2}$ while suppressing all harmonics of $|S_{q,11}(\omega _0)|^{2}$ and $|S_{q,13}(\omega _0)|^{2}$ to less than $10^{-2}$ (optimized device parameters are provided in Table 6 in Appendix A). Comparing to Table 3, the added restriction of suppressing all harmonics of $|S_{q,11}(\omega _0)|^{2}$ and $|S_{q,13}(\omega _0)|^{2}$ instead of only $q=0$ results in a modest reduction in the forward throughput from 0.906 to 0.805. In addition to eliminating the inter-harmonic bandwidth limitation, suppressing back reflection and back propagation at all harmonics provides complete isolation at Port 1 and eliminates the need for an additional narrowband filter. This is an important advantage over tandem phase modulator versions of this concept previously reported [8,9].
Fig. 8. Amplitudes of scattering elements of a three-resonator isolator for harmonic numbers $q = -3 \dots 3$. Corresponding device parameters are provided on the first line of Table 6 in Appendix A.
The second bandwidth limitation results from the narrow passband of a high $Q$ factor resonator. For a waveguide-loaded $Q$ factor of $10^{4}$ (which is typical for the devices analyzed herein), the transmission bandwidth is approximately 20 GHz. But this overestimates the usable isolation bandwidth as the extinction in Port 1 is enforced in the optimization procedure at only the resonance frequency. However, as before, the flexibility afforded by the optimization procedure can be deployed to widen the isolation bandwidth. As an example, Fig. 9 displays the zeroth order scattering spectra resulting from a three-resonator isolator design in which $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,13}(\omega _0)|^{2}$ are extinguished to $<10^{-2}$ in the range $\omega _0 \pm 1.01 \omega _m/2$ while maximizing the average of $|S_{0,31}(\omega _0)|^{2}$ in this same range (optimized device parameters are provided on the second line of Table 6 in Appendix A). Both the passband and stopband spectra display flattened responses, with an average forward throughput of 0.309 and a maximum throughput of 0.61. A passband slightly beyond $\omega _0 \pm \frac {\omega _m}{2}$ is targeted as it requires simultaneous suppression of the $q=\pm 1$ harmonics at $\omega _0 \pm 1.01\frac {\omega _m}{2}$ and at $\omega _0 \pm 0.99\frac {\omega _m}{2}$ to avoid harmonic distortion. It becomes increasingly difficult to find solutions to Eq. (7) that meet the design specifications as the bandwidth is widened further. However, adding a fourth (or more) resonator to the system will increase the dimensionality of the parameter space with the potential for additional increase in the isolation bandwidth and lower insertion loss.
Fig. 9. (a) Scattering spectra calculated via CMT for three-resonator isolator when the source is in Port 1. $|S_{0,11}(\omega _0)|^{2}$ and $|S_{0,13}(\omega _0)|^{2}$ are constrained to be less than $10^{-2}$ in the frequency range $f_0 \pm 1.01\frac {f_m}{2}$. (b) Same as (a) but the source is in Port 3. Device parameters are listed on the second line of Table 6 in Appendix A.
8. Conclusion
This work presents a versatile design concept for compact magnet-free CMOS-compatible on-chip isolators. The nonreciprocal wave propagation required for isolation is achieved by tailored space-time modulation of near-field coupled reonators. By exploiting the enhanced light-matter interaction of high $Q$ factor resonators, existing electrorefractive effects in lightly doped silicon or native lithium niobate are sufficient to implement the required dynamic modulation with experimentally realizable modulation frequencies. The electrodynamics of the device are accurately described using a coupled mode theory model which is used in the simulated annealing optimization of device performance. Isolation ratios over 40 dB are attainable with less than 1 dB insertion loss. The versatility of this design along with its optimization approach is illustrated by suppressing sideband frequencies and by enlarging the isolation bandwidth. Using three coupled resonators instead of two increased the design parameter space which loosened constraints on the cost function. For example, forward throughput increased from 49.1% to 90.9% when using three resonators instead of two. By adding additional resonators, it is predicted that further enhancement of bandwidth and forward throughput is possible while maintaining one of the smallest isolator footprints currently reported in the literature.
Appendix A: Device parameters used in coupled mode theory models
Table 4 provides the static device parameters that accompany the dynamic space-time parameters shown in Table 3.
Table 4. Three-resonator static device parameters obtained from SA optimization. The four lines correspond to the four lines displayed in Table 3. All parameters are in units of $10^{10}$ s$^{-1}$.
Table 5 provides the device parameters used in the results displayed in Fig. 7.
Table 5. Device parameters used in the results displayed in Fig. 7. $\kappa _{12} = \kappa _{23}$ and $\gamma _{c1} = \gamma _{c2}$ are in units of $10^{10}$ s$^{-1}$. Modulation frequency is set to $f_m = 24.2$ GHz. Additionally, $\omega _0 = 2\pi 193.55$ THz and $\gamma _0 = 2.27\times 10^{9}$ s$^{-1}$.
Table 6 provides the device parameters that produce the CMT results displayed in Figs. 8 and9.
Table 6. First line: device parameters corresponding to results depicted in Fig. 8. Second line: device parameters corresponding to results depicted in Fig. 9. $\kappa _{12}$, $\kappa _{23}$, $\gamma _{c1}$ and $\gamma _{c2}$ are in units of $10^{10}$ s$^{-1}$. Modulation frequency is set to $f_m = 24.2$ GHz. Additionally, $\omega _0 = 2\pi 193.55$ THz, $\gamma _0 = 2.58\times 10^{9}$ s$^{-1}$.
Acknowledgments
This work was supported in part through computational resources and services provided by the Institute for Cyber-Enabled Research at Michigan State University.
Disclosures
The author declares no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
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