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Abstract
In this paper we achieve non-reciprocity in a silicon optical ring resonator, by introducing two small time-modulated perturbations into the ring. Isolators are designed using this time-perturbed ring, side-coupled to waveguides. The underlying operation of the time-modulated ring and isolator is analyzed using Temporal Coupled Mode Theory (TCMT). The TCMT is used to find the angular distance, phase difference and thickness of the two time-modulated points on the ring resonator and also to find and justify the optimum values for the modulation frequency and amplitude, which yields maximum isolation in the isolator arrangements. Insight into the major players that determine isolation are also presented, with the aid of TCMT. Our proposed structure is much simpler to implement compared to other ring-based optical isolators, as it does not require spatio-temporal modulation, or large regions with modulation, but only two point perturbations on the ring. All results are obtained using realistic values of modulation and validated using an in-house full-wave solver. We achieve 21 dB isolation and −0.25 dB insertion loss at the telecommunication wavelengths.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Non-reciprocal optical devices such as isolators play a crucial role in nanophotonics, e.g. for protecting sensitive devices such as lasers against reflection of optical signals, realizing optical transistors and logic gates for on-chip all-optical information processing systems and facilitating full duplex communications [1–3]. Any linear time-invariant system having symmetric electric permittivity and magnetic permeability tensors, is reciprocal [4]. Until now the most common approach to realize non-reciprocity is by using magneto-optical effects [5,6]. This method typically requires applying high external magnetic fields to magnetic materials not compatible with today’s CMOS technology. Furthermore, as magneto-optical effects are too weak in optics, the resulting devices will be bulky. Smaller footprints using resonators have been achieved [7–11]. Integrated magneto-optical isolators have also been reported in [10,11], nevertheless a static magnetic field bias is still necessary in all these methods. Combining plasmonics with magneto-optical materials can enhance non-reciprocity, but the resulting structures are too lossy and still need magnetic bias [12]. Another way to obtain non-reciprocity is through non-linear effects [13,14]. Unfortunately this approach requires high intensity signals and is accompanied by considerable unwanted harmonics.
Recently non-reciprocity has been demonstrated by applying a momentum bias to the system via a proper spatio-temporal modulation of the permittivity. This effect has been shown via photonic indirect transitions in a silicon waveguide that simultaneously impart frequency and momentum shift [15]. Since momentum shift is variant under time-reversal, the resulting device will be non-reciprocal. Similar effect has been shown via photonic direct transition [16], in which only frequency shift is imparted. However due to the weak nature of the electro-optic effect (and very small modulation amplitudes), the device length becomes large. Spatio-temporally modulated resonators have been used in order to have smaller footprint [15,17–20]. It was also shown that the fundamental time-bandwidth limit can be overcome in time-varying resonators [21–23].
Recently, designs based on coupled resonators and applying proper phase difference to mimic momentum bias have been proposed in order to achieve smaller footprints, more efficient use of modulation scheme and less restrictions on dispersion engineering [24].
In all previously demonstrated time-modulated ring-resonator based photonic isolators, the mechanism that causes non-reciprocity is by biasing the system with a particular angular momentum that frequency-splits, the two counter rotating degenerate modes of the static ring resonator [17,18]. In these structures, one of the main challenges is the requirement to apply time varying traveling-wave modulation to the whole resonator, or to large regions of it. Practical implementation of this type of modulation is difficult, as it requires a traveling wave modulation. Implementing such modulation requires exciting the ring with a one-way acoustic wave, which is very difficult, as there will be always counter propagating modulating waves in practice. Although [18] provides a scheme based on stepwise modulation implemented with p-n junctions to simplify the structure requirements, practical implementation of this scheme still remains as a challenge since it still requires time varying modulation in the whole ring resonator, to mimic the traveling time-modulation wave, over the entire ring.
In this paper we introduce a simple scheme to achieve non-reciprocity and isolation, which consists of a ring resonator with two discrete time-modulation perturbations whose angular positions and temporal phase difference are aptly chosen to maximize isolation without resorting to traveling wave modulation. The time-modulated perturbations cause frequency-splitting of the clockwise (c.w.) and counter clockwise (c.c.w.) modes of the ring resonator, such that they are not degenerate anymore. We utilize Temporal Coupled Mode Theory (TCMT) to model the behavior of this time-varying system. The time-varying ring resonator is then side-coupled to two and one waveguides to yield four-port and two-port isolator behavior, respectively. By designing the geometrical parameters of the structure, as well as the modulation frequency and amplitude, we improve the frequency-splitting between the counter-rotating modes, the isolation and the insertion loss of the entire system. Our designed isolator is magnetic-free, compact, compatible with CMOS technology and significantly easier to implement compared to the previously demonstrated spatio-temporally modulated optical ring resonators [17,18]. Along the way, an in-house multi-frequency Finite Difference Frequency Domain (FDFD) full-wave solver is also developed, and all theory results are validated against the full-wave results.
The paper is organized as follows: in section 2 TCMT is used to analyze our structure, and the derived equations are solved using the harmonic balance method. In section 3, we optimize the modulation parameters to achieve maximum isolation, via TCMT equations derived in section 2, which are much faster than brute-force search using numerical methods. In section 4, we show non-reciprocal transmission in the structure consisting of the designed resonator side coupled to waveguides, and verify the TCMT predictions with full-wave multi-frequency FDFD simulations. At last we find the optimized isolator numerically and verify the optimization in section 3.
2. Theoretical analysis and design
2.1 Modal analysis of time-perturbed ring resonator
The structure under study is a silicon ring resonator with a uniform permittivity of $\varepsilon _s=12$, inner and outer radii of $R_i$ and $R_o$, respectively residing in a background refractive index of 1 (Fig. 1), in which two small discrete regions are additionally time-modulated as a cosine function and with some phase difference:
Fig. 1. (a) Ring resonator with two discrete time-modulated perturbations, (b) time-perturbed ring resonator side coupled to waveguides. Angular position and thickness of the modulated perturbations are $\phi _1,\phi _2$ and $\delta$, respectively.
We now use TCMT to obtain the spectrum of modes in such a resonator. Electromagnetic fields in the time perturbed ring can be expanded in terms of modes of the static ring:
Due to periodicity of the electric permittivity in time and according to Floquet-Bloch’s theorem, one can write the time-varying part of the fields as a sum of temporal Floquet harmonics as:
Taking a finite number for $N$, and then substituting these Bloch waves into Eq. (3), and assuming orthogonality between the $2N+1$ harmonics and applying harmonic balance, we arrive at a set of $2(2N+1)$ phasor domain equations which can be written as an eigenvalue problem in block matrix form as:As stated earlier, the static ring resonator (not time modulated), has two degenerate counter-rotating modes. In the following, we make some simplifying assumptions to find the appropriate values for the geometrical and modulation parameters of the structure under study (see Fig. 1), i.e. the angular distance ($\phi _2-\phi _1$), the modulation phase difference ($\Theta _2-\Theta _1$) and thickness ($\delta$) of the modulated regions. According to Eq. (5), the off-diagonal terms in the coupling coefficients matrix M(t) represent coupling coefficients between the c.w. and c.c.w. modes and thus have the major say in the crosstalk between counter-rotating modes. Therefore, we neglect $\kappa _{s+}$, and look into $\kappa _{m-}$ and $\kappa _{m+}$ for now.
Based on Eqs. (7) and (8), we can rewrite the coupling coefficients as:
where $\alpha$ and $\beta$ are:Assuming $\Delta \varepsilon _{m1}=\Delta \varepsilon _{m2}=\Delta \varepsilon _{m}$ and the spatial profile of the c.w. and c.c.w. modes to be $\exp (-jl\phi )$ and $\exp (jl\phi )$ (where $l$ is order of the mode), we have:
Now that closed form expressions are obtained for the contribution of positive and negative frequencies in the cross coupling between counter-rotating modes, i.e. $\kappa _{m+}$ and $\kappa _{m-}$, we can find the appropriate values for the geometrical and modulation parameters in Fig. 1, by ensuring that either of the following conditions hold:
It should be noted that by holding either of the above mentioned equations we achieve what applying the angular momentum bias tries to achieve but without resorting to continuous [26] or discrete [18] spatio-temporal modulations. It is rather achieved by two point temporal perturbations whose angular positions and temporal phase difference are aptly chosen. According to Eqs. (15) and (16) the condition in Eq. (21) is best held whenever: Here $\phi _{\alpha }$ and $\phi _{\beta }$ represent phases of $\alpha$ and $\beta$, respectively. From the above equations, the modulation phase difference and the angular distance of the two modulated regions can be found as:We then determine the optimum thickness of the modulated points. Intuitively, we predict smaller perturbations lead to better coupling between the two modes, as the momentum mismatch will be negligible in small lengths. However, for small lengths there will be less energy injected by the perturbation, therefore the modulation effect will be weaker. We speculate that the optimum thickness is determined via a trade-off between the aforementioned factors. In order to maximize coupling between the two modes, we should maximize $\alpha$ and $\beta$ coefficients. Therefore, the optimum thickness is found to be:
We will use these assumptions throughout the paper. It is interesting to note that by using only one modulated point, the above equations clearly demonstrate that the two equations of (3), coalesce into one. Therefore, in this structure, it is not possible to break the degeneracy between c.w. and c.c.w. modes with only one modulated point perturbation.2.2 Time-perturbed ring resonator side-coupled to waveguides
So far we have dealt with the time-perturbed ring resonator on its own, and its governing equations. Our final goal is to devise an isolator, which requires excitation of the ring with waveguides, as shown in Fig. 1(b). Thus an important step is to include the effect of excitation in the TCMT relations. The TCMT for the mode amplitudes with two waveguides side-coupled to the resonator as in Fig. 1(b) will be:
By solving Eqs. (34) and (35), one can find the amplitude and phase of each harmonic and therefore the time perturbation of the two modes, and the reflected and transmitted power at each port. The accuracy of this method is determined by the number of harmonics that are considered. More harmonics are needed for stronger modulations, e.g., higher modulation amplitude $(\Delta \varepsilon _m)$ or if the periodic modulation is not a simple cosine function and has many Fourier components.
3. Optimal isolator design using TCMT
In this section we find the optimum modulation parameters to minimize the overlap between c.w. and c.c.w. modes, since we expect that it leads to maximum isolation. This work generally requires a brute-force optimization with numerical simulations, which is time-consuming. Therefore we find the optimal isolator theoretically by using TCMT equations of section 2.1. For this purpose, by solving Eq. (11) we first study the effect of eigenvalues and then eigenvectors, which represent the Bloch frequencies of the two resultant "supermodes" as well as the spectral content of the c.w. and c.c.w. mode amplitude at each harmonic, respectively. We consider a silicon ring resonator with $\varepsilon _s=12$ and mode-order of $l=11$. In Fig. 2, the eigenvalues (resonant frequencies) of the zeroth order c.c.w. harmonic of the one supermode, as well as the zeroth order c.w. harmonic of the other supermode, is plotted versus the modulation amplitude, for three different modulation frequencies. The difference between these two frequencies i.e., between red and blue curves in Fig. 2, demonstrates breaking of degeneracy between the two supermodes and we call it "frequency-splitting" throughout the paper. According to Fig. 2, for a particular modulation frequency $f_m$, the frequency-splitting between the zeroth order harmonics of the two supermodes increases as the modulation amplitude ($\Delta \varepsilon _{m}$) is increased upto a maximum value of $f_m$. Increasing the modulation beyond the maximum $f_m$ splitting essentially brings us back to a higher temporal Brillioun zone, and thus the reverse trend is observed. An interesting note to be made about Fig. 2 is that for points with frequency-splitting of $f_m$, the two modes will essentially coalesce into one. However an $f_m$ frequency-split is not a good working point, as our purpose is to break the degeneracy. Thus, $0.5f_m$ is the optimum point insofar as the eigenvalues (and eigenvalues only) are concerned. The required modulation amplitude to have frequency-splitting of $0.5f_m$, is $2.719\times 10^{-3}\varepsilon _s$ for $f_{m2} = 6.8$ GHz (which corresponds to the solid curve of Fig. 2). We also note that the frequency-splitting decreases as the modulation frequency is increased at a particular $\Delta \varepsilon _{m}$ (for points in the first temporal Brillioun zone), because the required modulation amplitude to achieve maximum splitting is increased for larger modulation frequencies, and limiting the modulation amplitude will have a more detrimental effect on frequency-splitting.
Fig. 2. Zeroth order harmonics of two supermodes versus normalized modulation amplitude for $f_{m1}$ (dotted), $f_{m2}$ (solid) and $f_{m3}$ (dash-dotted) where $f_{m1}$>$f_{m2}$>$f_{m3}$. Blue and red represent the c.c.w. and c.w. harmonics, respectively. $f_0$ is the resonance frequency of the static ring resonator.
Next we study the effect of eigenvectors of Eq. (11), that represent the spectrum of the two supermodes as shown in Fig. 3. According to Fig. 3, each supermode consists of both c.w. and c.c.w. mode amplitudes. However, in the first supermode (Fig. 3(a)), the c.c.w. mode is dominant while in the second supermode (Fig. 3(b)) the c.w. components are dominant. There is also a frequency-splitting equal to $0.5f_m$ between the zeroth order harmonics of the two supermodes, as we expected. This is the maximum achievable frequency-splitting between each counter-rotating harmonics, as each supermode consists of a number of equally spaced harmonics with a distance of $f_m$ (see Fig. 3). Considering the effect of eigenvectors, we find out that realizing maximum splitting as already described, is not necessarily the only determining factor for the least overlap of the modes, and we also need to look at the ratio of c.c.w. to the c.w. mode at the central frequency of supermode spectrum (see Fig. 3).
Fig. 3. Spectrum of the modulated ring supermodes for (a) the first supermode and (b) the second supermode. Both supermodes consist of c.c.w. (blue) and c.w. (red) mode amplitudes. It is evident that the first supermode (a) has a dominant c.c.w. spectral content, while (b) has a dominant c.w. spectral content. $f_0$ is the resonance frequency of the static ring resonator.
Therefore, we study the effect of modulation frequency and amplitude on the supermodes by considering two parameters: frequency-splitting between zeroth order harmonics of two supermodes and the ratio of c.c.w. to the c.w. mode at the central frequency of supermode spectrum. These parameters are plotted versus modulation frequency and amplitude in Fig. 4(a) and Fig. 4(b), respectively. It is inferred from Fig. 4(b) that larger modulation frequency leads to the increase in ratio of the counter-rotating modes in the central frequency, which is due to the decrease in amplitude of higher order harmonics. This is because by increasing the modulation frequency, there is larger frequency separation between the central and higher order harmonics and it leads to weaker coupling between them, therefore the ratio of the c.c.w. to the c.w. mode is increased via decreasing the amplitude of the higher order harmonics (according to Fig. 3). Also decreasing the modulation amplitude decreases the coupling coefficients to higher order harmonics in Eqs. (6)–(8), and therefore the ratio of the c.c.w. to the c.w. mode is increased by the same reason. However frequency-splitting is decreased as the modulation frequency (amplitude) increases (decreases), as shown in Fig. 2. Therefore, these two parameters work in the opposite direction. In case of exciting this resonator in a side coupled geometry as in Fig. 1(b), from port 1 and port 2, we speculate that maximum isolation (defined as $\frac {|S_{12}|^2}{|S_{21}|^2}$, where $S_{ij}$ is the transmission to port $i$ for excitation from port $j$) is primarily determined by these two oppositely acting mechanisms. The maximum of isolation occurs at a point where there is sufficient frequency-splitting, while maintaining the maximum ratio of the c.c.w. to the c.w. mode spectral content. Essentially, in these conditions, in order to reach maximum isolation, frequency-splitting should be such that zeroth order harmonics of the two supermodes do not interfere, and this determines the minimum acceptable frequency-splitting which is equal to the bandwidth of the resonance. Also this frequency-splitting should not be greater than $f_m-\textrm {BW}$ (where $\textrm {BW}$ is the half-power bandwidth), since otherwise the central harmonic interferes with the first side harmonic. Therefore, acceptable frequency-splitting is determined by the condition: $\textrm {BW}<\textrm {splitting}<f_m-\textrm {BW}$. Therefore the optimum operating point is somewhere between the black and cyan curves in Fig. 4 which represent frequency-splitting of $\textrm {BW}$ and $f_m-\textrm {BW}$, respectively. Maximum isolation occurs at a point in this region, which has the maximum ratio of the c.c.w. to the c.w. mode. We assume that our two point-perturbations have a modulation amplitude no larger than $\Delta \varepsilon _{m}=5\times 10^{-3}\varepsilon _s$, which is the amount that was used in [18]. It should be noted that in [18] spatio-temporal modulation was applied to the entire ring. By taking modulation amplitude to be $\Delta \varepsilon _{m}=5\times 10^{-3}\varepsilon _s$ (dashed yellow line), the optimum operating point occurs at the modulation frequency of $f_m= 71.9$ GHz. One needs to sweep the modulation parameters in the neighborhood of this point to find the exact value of optimum modulation parameters. This will be demonstrated numerically in section 4.2. It is worth noting that our analysis to find the optimum point is valid under the condition that the intrinsic loss of the resonator is high enough to absorb the power transmitted in the reverse direction and causes the transmission ${S_{21}}$ to drop.
Fig. 4. (a) Frequency-splitting between two supermodes and, (b) ratio of c.c.w. to the c.w. mode at the central frequency of supermode spectrum, versus modulation frequency and amplitude. Black and cyan curves represent the splitting of $BW$ and $f_m-BW$, respectively and dashed yellow line indicates modulation amplitude of $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$.
4. Results and discussion
In this section we simulate an optical isolator based on the time-varying ring resonator designed in section 2. For this purpose, first we design the geometrical parameters to achieve a high-Q add-drop filter in telecommunication wavelength of $1.55$ $\mu m$. Then we verify TCMT results by full-wave simulation of the structure via a two dimensional multi-frequency FDFD method. We optimize our design and simulate various scenarios to achieve maximum isolation. We consider a silicon ring resonator with relative permittivity of $\varepsilon _s=12$. The radius of the ring is designed so that it resonates at the telecommunication wavelength of $1.55$ $\mu m$ with order $l=11$. The outer radius of the ring is $R_o=1.094$ $\mu m$. Thanks to the generality of the proposed methodology, one can choose different geometrical parameters and adjust the design to the appropriate ring mode. Waveguide and resonator thickness are $t=0.2$ $\mu m$. As demonstrated earlier, the maximum possible frequency-splitting is $0.5f_m$ that should be equal or greater than the bandwidth of the resonance, for the spectrum of two modes to be completely split. By considering a distance of $0.45$ $\mu m$ between each waveguide and the ring, a bandwidth of $3.4$ GHz is achieved that requires a modulation frequency of $6.8$ GHz.
4.1 Verification using multi-frequency FDFD
The add-drop filter of Fig. 1(b) is simulated using an in-house developed multi-frequency FDFD code to both verify the TCMT equations of section 2 and to demonstrate non-reciprocal transmission. We perform two simulations around each of the split frequencies i.e., $f_0-0.25f_m$ and $f_0+0.25f_m$, one for excitation from port 1 and the other from port 2. Figure 5 shows the transmitted power of each port for these four simulations and for the zeroth order harmonic.
Fig. 5. (a) Zeroth order harmonic of power transmission of different ports for excitation from port 1 and, (b) from port 2. Solid and dash-dotted lines represent multi-frequency FDFD and TCMT results, respectively. (c) Electric field profile for excitation at frequency $f_0+0.25f_m$ from port 2 and, (d) at $f_0+0.25f_m$ from port 1.
According to Fig. 5(a), for excitation at the central frequency of $f_0-0.25f_m=193.1893$ THz from port 1, the c.c.w. mode of the resonator is excited which results in a drop in transmitted power to port 2, i.e., $|S_{21}|^2$. However for excitation from port 2 at the same frequency (Fig. 5(b)), the c.w. mode is not mainly excited and all of the power (minus some losses) is transmitted to port 1 $(|S_{12}|^2\approx 1)$. The reverse scenario happens for excitation at the central frequency of $f_0+0.25f_m=193.1927$ THz. By excitation from port 2 the c.w. mode is excited (Fig. 5(b)), and for excitation from port 1 the c.c.w. mode is not excited (Fig. 5(a)). As we see from the electric field profile of Fig. 5(c), in the frequency of $f_0+0.25f_m$ c.w. mode is excited and therefore, there is a reduction in the power transmitted to port 1, however according to Fig. 5(d), by exciting from port 1 at the same frequency, c.c.w. mode of the resonator is not excited and therefore, almost all the power is transmitted to port 2. This is because (according to TCMT in section 2.1) the degeneracy of the two counter-rotating modes of the resonator is broken via the special design of the time-modulated points. There now exists a frequency-splitting between the supermodes of the time-perturbed ring resonator, such that one can excite the c.w. (c.c.w.) mode at a frequency without noticeably exciting the other. Our theoretical TCMT results (dash-dotted line) are in excellent agreement with multi-frequency FDFD simulations (solid line).
Next we study the generation of harmonics in the structure. Figure 6(a) and Fig. 6(b) shows the transmitted power to each port for different harmonics (for a total of 5 harmonics), and for excitation from port 1 and port 2, respectively. It is worth noting that for each of the simulations, output powers are calculated for a single frequency input and this scenario is performed for a frequency range equal to the resonance bandwidth around the center resonance frequency. This is equivalent to exciting the structure with a pulse having a bandwidth equals to the linewidth of the resonance. Since the separation $f_m$ is taken twice the resonance bandwidth, no harmonic mixing occurs. According to Fig. 6, for such an excitation, a number of equally distant harmonics are generated, which their amplitudes are reducing for larger harmonic numbers. As can be seen in Fig. 6(a), the side harmonics of $|S_{41}|^2$ and $|S_{21}|^2$ are almost equal. This is because they are both resulting from the symmetric out coupling of the c.c.w. mode energy to the top and bottom waveguides. As we demonstrated in section 2, in this case the c.w. mode is weakly excited via $\kappa _{m+}$ and $\kappa _{m-}$. Therefore, side harmonics of $|S_{31}|^2$ are lower in amplitude, since they represent out coupling of the c.w. mode energy to the waveguide (see Fig. 6(a)). There is a very good agreement between our theoretical TCMT results and those calculated by multi-frequency FDFD simulations, in both the center and side harmonics. Here the in-coupling ($K$) and out-coupling ($D$) coefficients of the resonator are equal. If one aims to overcome the time-bandwidth limit in this time-varying resonator, these two coefficients must be made unequal [23], which may be achievable by time modulating the gap between the ring resonator and waveguide.
Fig. 6. (a) Spectral content of transmission to different ports for excitation at the central frequency of $f_{1}=f_0-0.25f_m$ from port 1 and, (b) for excitation at the central frequency of $f_{2}=f_0+0.25f_m$ from port 2. Dotted and solid lines represent TCMT and multi-frequency FDFD results, respectively.
4.2 Optimized isolator
The amount of isolation, or the ratio between the amplitudes of $|S_{12}|^2$ and $|S_{21}|^2$ in Fig. 5, was only about 2 at the central harmonic, as we had only applied the condition of maximum frequency-splitting. Here we seek to increase this ratio such that one way transmission is achieved at the central harmonic, and we arrive at a very good optical isolator. The TCMT equations along with the multi-frequency FDFD simulations allow us to arrive at optimum isolation levels. As discussed in section 3 and Fig. 4, we expect that maximum isolation occurs at a point around $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$ and $f_m = 71.9$ GHz. Therefore, we sweep the modulation frequency for a fixed modulation amplitude of $5\times 10^{-3}\varepsilon _s$. For each point, structure is driven at $f_0-\textrm {splitting}/2$ and then the frequency $f_m$ is swept. Figure 7(a) shows $|S_{12}|^2$ and $|S_{21}|^2$ and Fig. 7(b) shows isolation ($\frac {|S_{12}|^2}{|S_{21}|^2}$) for different modulation frequencies in the structure with two waveguides. Maximum isolation is determined by the two factors discussed in section 3 (frequency-splitting and the ratio of c.c.w. to the c.w. mode) and occurs at $f_m=74.2$ GHz. In fact these two mechanisms work in the opposite direction, therefore maximum isolation is achieved at an optimum modulation frequency and amplitude around a point that maximizes the ratio of the c.c.w. to the c.w. mode in Fig. 4(b). It is worth to mention that, depending on the application, one may choose a point in Fig. 7 with less isolation, but better insertion loss.
Fig. 7. (a) $|S_{21}|^2$ (blue) and $|S_{12}|^2$ (red) and, (b) Isolation versus modulation frequency in the structure with two waveguides for $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$. Maximum isolation occurs at $f_m=74.2$ GHz.
In order to achieve an even better performing isolator, we optimize structure of Fig. 1(b), where only coupled to one waveguide, which is a two port device. We perform the same optimization by sweeping $f_m$ and driving structure at $f_0-\textrm {splitting}/2$. According to Fig. 8(c), large isolation of 123 ($20\log _{10}\frac {|S_{12}|}{|S_{21}|}=21$ dB) is achieved at the optimum point ($f_m = 19.5$ GHz , $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$), while obtaining an insertion loss ($|S_{12}|^2$) of about 0.944 ($20\log _{10}|S_{12}|=-0.25$ dB) (see Fig. 8(a)). It is worth noting that large isolation at the optimum point here is achieved at the cost of smaller frequency-splitting i.e., $0.2303f_m = 4.49$ GHz. By removing one of the waveguides, quality factor is increased, therefore the required modulation frequency and amplitude to achieve a desired isolation is reduced, consequently the wave in the resonator undergoes a smaller phase shift and it leads to the expected destructive interference between the out-coupled energy of the resonator and the incoming power from the input port of waveguide. As a result, the power of port 2 ($|S_{21}|^2$) drops and isolation increases. As shown in Fig. 8(a), the power transmission in the reverse direction i.e., $|S_{21}|^2$ increases for modulation frequencies greater than $19.5$ GHz, it means intrinsic loss of the resonator is not high enough to absorb the power and it does not satisfy the critical coupling condition. Therefore we do not use TCMT analysis of section 3, but directly sweep $f_m$ in the multi-frequency FDFD simulation for finding the optimum point. In other words, our analytical optimization works in the critically-coupled condition for one-waveguide structure. If we were to consider higher intrinsic loss for the resonator, via adding the material loss of silicon caused by injecting the carriers ($\varepsilon _s = 12-2.052\times 10^{-5}j$), we see that the condition of critical coupling is achieved. Then we can use TCMT for optimization. Optimum modulation frequency predicted by TCMT is about $f_m = 80$ GHz. Optimum modulation frequency obtained with multi-frequency FDFD simulations occurs at $f_m = 76$ GHz, as shown in Fig. 8(d). According to Fig. 8(b), $|S_{21}|^2\approx 0$ and critical coupling condition is satisfied for this point. Since bandwidth of the resonance differs for different modulation frequencies and is slightly different from the bandwidth of the static resonator, the optimum modulation frequencies predicted in section 3 differs slightly from that derived by numerical simulations. However, as TCMT optimization is much faster, it enables us to find the neighborhood of the optimum point. To find the exact value of the optimum point one should completely consider this effect and also the effect of waveguides by numerically simulating the structure in the neighborhood of the optimum point predicted by TCMT. Figure 9(a) shows power transmission to port 2, for excitation from port 1 ($|S_{21}|^2$) and power transmission to port 1, for excitation from port 2 ($|S_{12}|^2$) in the optimum point of Fig. 8(c), i.e., $f_m=19.5$ GHz and $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$ and Fig. 9(b) shows $|S_{21}|^2$ and $|S_{12}|^2$ in the optimum point of Fig. 8(d), i.e., $f_m=76$ GHz and $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$. As can be seen in Fig. 9, bandwidth is slightly increased in the lossy structure to about 1.13 GHz. Figure 9(a) shows isolation of 123($=21$ dB) and insertion loss of 0.944($=-0.25$ dB) for the structure with real electrical permittivity of $\varepsilon _s = 12$ and Fig. 9(b) shows isolation of 110($=20$ dB) and insertion loss of 0.834($=-0.78$ dB) for the structure with electrical permittivity of $\varepsilon _s = 12-2.052\times 10^{-5}j$. Here, the 10 dB isolation bandwidth (defined as the bandwidth in which the device works with the isolation of greater than or equal to 10 dB) is about 338 MHz. It is worth mentioning that in comparison to the state-of-the-art schemes employing integrated magneto-optic materials, e.g., [11], our designed isolator is much more compact, provides much better insertion loss and achieves the same isolation level. This is while our proposal does not require deposition of magneto-optic layers as in [11], which reduces complexity and makes it compatible with CMOS processes.
Fig. 8. (a) $|S_{21}|^2$ (blue) and $|S_{12}|^2$ (red) and, (c) isolation versus modulation frequency in one-waveguide structure for $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$. Maximum isolation occurs at $f_m=19.5$ GHz. (b) and (d) the same parameters for resonator with higher intrinsic loss, with $\varepsilon _s = 12-2.052\times 10^{-5}j$. Maximum isolation occurs at $f_m=76$ GHz.
Fig. 9. Two-port transmission response of the isolator, $|S_{21}|^2$ (blue) and $|S_{12}|^2$ (red) are shown for (a) the optimum point of Fig. 8(c) ($f_m=19.5$ GHz and $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$), and (b) the optimum point of Fig. 8(d) ($f_m=76$ GHz and $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$).
5. Conclusions
In this paper we demonstrate that by adding two small time-modulated perturbations to a ring resonator, one can disrupt the degeneracy of the two counter-traveling eigenmodes of the static ring, and achieve frequency-splitting of the two resultant supermodes. We analyze the structure using TCMT and find the optimum modulation parameters theoretically, which is further confirmed by numerical simulations. We demonstrate non-reciprocal transmission using this time-perturbed resonator when coupled to one or two waveguides arranged as either a two-port isolator or a four-port non-reciprocal channel drop filter, respectively. Almost full isolation is achieved with our simple design, which does not require spatio-temporal modulation of the entire resonator as utilized in other methods. Results of full-wave multi-frequency FDFD simulations are in excellent agreement with our TCMT formulation, validating the theory and the isolation levels achieved. The intricate mechanisms at play that largely determine the isolation are found and justified via the TCMT model, which are namely the frequency-splitting of the supermode frequencies (eigenvalues) and the dominance of c.w. or c.c.w. waves in their frequency contents (eigenvectors). Our designed silicon isolator is magnetic-free, compact, compatible with CMOS technology and significantly easier to implement compared to the previously demonstrated spatio-temporally modulated optical ring resonators. We achieved isolation of 21 dB and insertion loss of −0.25 dB with realistic parameters for the modulation.
Funding
Iran National Science Foundation (97008712).
Disclosures
The authors declare no conflicts of interest.
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