Design Guidelines for a Tunable SOI Based Optical Isolator in a Partially Time-Modulated Ring Resonator

In this paper, we present the design guidelines for a tunable optical isolator in an SOI-based ring resonator with two small time-modulated regions. By considering a physical model, the proper geometrical and modulation parameters are designed, based on a standard CMOS foundry process. The effect of the variation of the key parameters on the performance of the isolator is explained by two counter-acting mechanisms, namely the separation between the resonance frequencies of counter-rotating modes and energy transfer to the side harmonic. We show that there is a trade-off between these parameters to obtain maximum isolation. Consequently, by applying the quadrature phase difference one can obtain the maximum separation between the resonance frequencies and hence the minimum insertion loss, while the maximum isolation is obtained at the modulation phase difference of $-0.78\pi$, which leads to a higher insertion loss. Robustness of the design is investigated through a sensitivity analysis for the fabrication variations in the distance and width of the modulated regions. We demonstrate that there is a trade-off between isolation and insertion loss, and by varying the modulation parameters, we can achieve isolation of 18 (5) dB with 7 (1.8) dB insertion loss.


I. INTRODUCTION
P HOTONIC Integrated Circuit (PIC), which is an effort to integrate multiple high-performance optical elements with different functions on a single chip, has made great progress in recent years, and offers high speed, thermal tuning ability, small footprint, large integration capacity, and compatibility with existing processing flows that allows for high yield, volume manufacturing, and low prices. Recent progress in PICs, enables the integration of various optical devices, such as passive circuits, as well as on-chip lasers, detectors, modulators and fully integrated frequency comb sources. However, on-chip integration of optical non-reciprocal devices has remained as a challenge. Optical nonreciprocal devices, such as isolators and circulators allow light transmission only in one direction while preventing the transmission in the other direction. Hence, they are essential devices in PICs, in order to stabilize lasers and reduce noise by preventing unwanted back-reflection, realizing optical transistors and logic gates for on-chip all-optical information processing systems and facilitating full duplex communications. Traditional approach to achieve optical non-reciprocity is through the magneto-optic effect. In fact, by integrating magneto-optic materials into the waveguides and applying a strong magnetic field, one can achieve different phase shifts for the light traveling in opposite directions [1], [2], [3]. Therefore, by integrating such material in a ring resonator, one can achieve different resonant frequencies for the clockwise (c.w.) and counter-clockwise (c.c.w.) modes [4], [5], [6]. This method typically requires applying high external magnetic fields to the materials and is not compatible with complementary metal-oxidesemiconductor (CMOS) technology. Therefore, the scalability, price and volume manufacturing of this approach still remains a challenge. Furthermore, this method is intrinsically lossy, also as the magneto-optic effect is low at the optical frequencies, the resultant device is usually bulky. Moreover, as such devices usually work in the TM-like mode, mode-converter is required before and after the device, which adds to the design complexity, even further. Another method for achieving non-reciprocity is through the nonlinear effects, such as Kerr nonlinearity [7], [8], [9], [10], [11], [12]. It has been demonstrated in different platforms such as fused silica microrod resonator [7], nonlinear silicon Mach-Zehnder interferometer [8], on-chip silicon nitride ring resonator [9] and silicon ring resonator [10]. Unfortunately this method only works with high-intensity input power, which is not compatible for all of the practical applications, and is accompanied by considerable unwanted harmonics. The third approach utilizes temporal modulation [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]. This approach has attracted considerable interest in recent years, because this is in principle a promising approach to realize optical isolators and circulators for PICs, as it does not require deposition of any new material or applying high power optical signals, and it can be implemented by the conventional foundry process in silicon, hence it has the potential This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0 /  TABLE I  COMPARISON OF THE PROPOSED STRUCTURE WITH EXISTING PROPOSALS FOR INTEGRATED OPTICAL ISOLATOR to become manufactured in high volume. This approach has been demonstrated using various mechanisms and in different structures. Optical nonreciprocal indirect interband transition was demonstrated in a silicon waveguide, by applying a traveling wave modulation to the refractive index, to impart a shift in the momentum and frequency of the optical mode [13]. A similar effect was shown via direct interband transitions [14], [15], where the modulation only imparts the frequency shift. Another scheme is introduced by utilizing tandem phase modulators, where by applying a relative phase delay between them, one can obtain non-reciprocal frequency conversion [16], [17]. In addition to this, spatio-temporal gain-loss modulation in silicon waveguide was studied, where the modulation is applied to the conductivity to impart the intermode transition and hence obtaining non-reciprocal directional amplification [18]. As an attempt for making smaller footprint for the time-modulated devices, optical resonators were utilized to enhance the modulation strength, by confining the optical energy. Photonic transition was also shown in ring resonator [13], in a very similar manner as the one in waveguide. Another variant is the Rabi splitting in ring resonator [19], here the difference is that in the backward direction all the power is dissipated, contrary to the case of photonic transition where the power was transferred to higher frequency, therefore there is no need to an additional filter in the output. In the aforementioned papers, the transition was between different radial modes of the ring resonator. Alternatively, one can work with two degenerate counter-rotating modes and break the degeneracy of them by applying a traveling wave modulation that breaks the time reversal symmetry in a silicon ring resonator [20], [21]. Optical isolators and circulators were also demonstrated in time-modulated photonic crystal cavity array [22], [23], [24], by phase-shifting the modulation signal applied to each cavity, which induces a synthetic linear momentum in the system. It was also demonstrated in a simple macroscopic, fiber-optic resonator that the fundamental time-bandwidth product can be overcome in nonreciprocal time variant resonators [25], [26].
Despite the great deal of research in this field, there have been just a few realistic designs and experimental realizations of an optical isolator with this scheme. In fact applying the time-varying traveling wave modulation to the structure, which is a key parameter in most of the designs, is challenging to implement in practice. Although utilizing the photonic crystal cavity array alleviates the need for the spatio-temporal modulation, it adds additional complexity and more sensitivity to the fabrication imperfections, as it uses coupled photonic crystal defect cavities. Some of the recently proposed structures for integrated optical isolators using temporal modulation are summarized in Table I, and are compared to the isolators realized with magneto-optic effect [1], [6], and nonlinear effects [7]. According to this table, magneto-optical isolators can provide high isolation; however, this is an intrinsically lossy effect and besides the requirement for deposition of the garnet film, it also requires a large footprint and applying high magnetic fields. In the optical isolator realized with Kerr non-linearity, due to the requirement of the high optical confinement, a high-Q resonator, i.e., microrod resonator, is needed which compared to the ring resonator, bears a larger footprint and less compatibility with the standard fabrication processes. Experimental realization of an optical isolator in a silicon waveguide was demonstrated via interband photonic transition in a slotted silicon waveguide, where the traveling wave modulation was sampled with a grid of 166 PN junctions [27], which besides having a large footprint and high insertion loss, is a design with high complexity. An optical isolator based on the indirect interband transition in a silicon nitride ring resonator was demonstrated, with better performance and smaller footprint, however in this structure spatiotemporal modulation is realized using acousto-optic modulation with aluminum nitride piezoelectric actuators, which requires additional step compared to the conventional CMOS fabrication process [28]. Recently an experimental proposal, based on two cascaded silicon microring modulators was presented [29], in which non-reciprocity was obtained by tuning the relative time delay between the RF driving signals and the optical delay between the modulators. Although in this method traveling wave modulation is not required and it provides even better isolation, using two ring resonators requires more space and may add more complexity to the design. On the other hand, it has higher insertion loss, compared to the isolator implemented in silicon nitride [28].
In a previously published paper the idea of the ring resonator with two time-modulated point perturbations for achieving optical non-reciprocity was presented, as a proof of concept [30], where optical isolation is obtained by breaking the degeneracy of the c.w. and c.c.w. modes, through devising the proper phase shift and distance between the modulated points. It is expected that this structure would be simpler to implement compared to the previous proposals, as it does not need the traveling wave modulation applied to the whole ring circumference. However, for practical implementation of such an isolator, we should not only change the design to comply with the design rules of the CMOS foundry process, e.g., for geometrical parameters that are dictated by the foundry process, but also we need to modify our model to account for the phenomena that happen in reality, such as free carrier absorption loss, out-of-plane radiation, bending loss, surface roughness loss, etc. In addition to this, we should consider some restrictions that will be in the characterization stage, i.e., for the modulation parameters. It is crucial to have insight into the effect of variation of each parameter on the performance of the isolator. Not only because in the fabrication process there is always some inevitable imperfection and error, but also because due to adaptation to the fabrication rules, we may need to tune some parameters, compared to the values considered in the initial design. In this paper, such a realistic design guideline based on a CMOS-compatible silicon photonics foundry process is presented. Here we focus on the instructions for such a design by considering the realistic modulation and geometrical parameters. We study the tunability of the device and explain the effect of variation of different parameters, with two main counter-acting mechanisms, namely the separation between the resonance frequencies of counter-rotating modes and energy transfer to the side harmonic. We show that we can enhance isolation, by tuning the modulation phase difference. We examine the sensitivity of the structure to fabrication variations in the angular distance and width of the modulated regions. Our proposed structure has a small footprint and a simple design, i.e., it does not require the deposition of any new material on silicon and can be fabricated on a multi-project wafer with a commercial foundry process. Hence this work can be an important step for facilitating the fabrication and measurement of a time-modulated, compact and CMOS-compatible optical isolator.
The paper is organized as follows: In Section II Temporal Coupled Mode Theory (TCMT) is used to model our structure, and the coupling coefficients are obtained for the three-dimensional ring resonator. In Section III, we choose realistic modulation and geometrical parameters based on a CMOS-compatible silicon photonics foundry process. We use Soref equations for determining the realistic modulation parameters and find the optimum point that leads to maximum isolation. Moreover, the effect of the modulation phase on the isolation and insertion loss is evaluated. We show that we can trade off between isolation and insertion loss of the structure and classify the working point of the device into three zones, by changing the modulation parameters. In Section IV sensitivity of the performance to fabrication errors is analyzed.

A. TCMT Formulation for the 3D Structure
The structure under study is a silicon ring resonator that is fabricated in the top silicon layer of a silicon-on-insulator (SOI) substrate and is encapsulated from above by amorphous silica, in which two small discrete regions are additionally timemodulated as a cosine function and with some phase difference ( Fig. 1): (1) Here, Δε(t) is the perturbation in relative permittivity. Δε m and ω m are modulation amplitude and angular frequency. φ i , δ and Θ i represent the position, width and modulation phase of the ith modulated region (i = 1, 2), where we use angular position and width, in radians. Appropriate values for the angular distance (φ 2 − φ 1 ), width (δ) and modulation phase difference (Θ 2 − Θ 1 ) of the two modulated regions, were found with the aid of TCMT [30].
We now use TCMT to obtain the spectrum of modes in such a resonator. In this approach TCMT equations governing this time perturbed ring is obtained by expanding electromagnetic fields in terms of modes of the static ring. Following this approach [31], using perturbation theory, the TCMT relations for the time-varying resonator is obtained: where a(t) is a two row vector of the c.w. and c.c.w. mode amplitudes, and M (t) is the matrix representing the coupling between resonator modes: We suppose that there is no static coupling between the modes. ω 0 is the angular resonance frequency of the static ring resonator. In the above equation, κ s+ represents self-coupling of each mode to itself which leads to the modulation of the mode's resonance frequency, and κ m+ and κ m− represent the contribution of the positive (blue-shift) and negative (red-shift) modulation frequencies of the coupling between c.w. and c.c.w. modes, which are obtained by the overlap integral of the c.w. and c.c.w. modes of the unperturbed ring over the modulation regions, and can be calculated by the following equations [30]: where ε s and l are the relative permittivity of silicon and the resonant mode order of the ring resonator and Γ shows mode confinement in the ring cross section: where e c is the electric field distribution of the c.w. mode of the static ring and ε is the relative permittivity profile. The degeneracy of the two modes is broken via the cross-coupling term, i.e., k m+ and k m− . Hence due to dynamic modulation, there are two supermode spectrums with dominant c.w. and c.c.w. amplitudes, respectively. We define frequency splitting as the distance between the central resonant frequencies of the two supermodes. Therefore, the frequency splitting between the modes increases for highly confined modes. It is also inversely proportional to the mode order (l).
To analyze the optical isolator, we need to include the effect of excitation and decay into the bus waveguide, in the TCMT relations [32]: where K is a matrix representing the coupling coefficients from the incoming waves in waveguide ports (s + , a 2 × 1 vector), to the resonator, and Γ represents the decay rate: The outgoing waves from the waveguide ports are then given by: where, s − is a 2 × 1 vector representing outgoing waves of waveguide ports, and C and D are matrices that represent the direct coupling between waveguide ports and the coupling of the resonator to the waveguide ports, respectively: Equations (7) and (9) are solved using harmonic balance method. According to this method, using Floquet-Bloch's theorem, temporal perturbation of the fields and also the excitation can be expanded as a sum of temporal harmonics. The dynamic equation is then expanded in terms of this series and all terms are grouped by harmonic number. Based on the orthogonality of the basis functions, the coefficients multiplying each should vanish for each harmonic number [33].

A. Realistic Design Parameters
In this section we consider the parameters of a CMOScompatible silicon photonics foundry process and design the geometrical and modulation parameters of the time-modulated ring resonator based on these realistic parameters. Then we will perform a sensitivity analysis and evaluate the robustness of our design under fabrication errors. The most common method to implement modulation in silicon is through the plasma dispersion effect, in which by applying voltage to a p-i-n junction the concentration of free carriers changes and this leads to a change in refractive index and absorption coefficient of the waveguide, which is described by Soref equations [34]. The cross section of the ring resonator is a rib waveguide with a dimension of 220 nm × 450 nm and a 90-nm slab height, as depicted in Fig. 2(a). A two-dimensional simulation in Ansys Lumerical CHARGE is used to obtain the spatial distribution of charge carriers under different values of the voltage that is applied to the junction through a metal pad on the P++ and N++ regions, as shown in Fig. 2(a). The average carrier density in the center of the waveguide is plotted versus voltage in Fig. 2(b). After obtaining the carrier density for different voltages, the change in the refractive index and absorption is calculated with Soref equations.
To calculate the bending loss of the ring resonator, cross section of the ring resonator (as depicted in Fig. 2(a)) is simulated via Ansys Lumerical MODE, using Finite Difference Eigenmode (FDE) solver, which calculates the modes of the bent waveguide, by solving Maxwell's equations on a cross section of the waveguide. A simulation region of 5.8 μm × 4.5 μm with 400 mesh cells in each direction, is considered.
For choosing the silicon ring radius, two issues should be considered. First it should be large enough to provide working with small bending loss, and second it should be small enough to have acceptable frequency splitting. As an explanation, by decreasing the ring radius, the bending loss, caused by total internal reflection of the light at the bend interface, increases. Therefore the resonance bandwidth is broadened. On the other hand, increasing the ring radius reduces the frequency splitting between c.w. and c.c.w. modes. In fact larger radius is equivalent to larger mode order (l) and according to (5), increasing the mode order reduces the coupling coefficient between c.w. and c.c.w. modes (κ m− or κ m+ ). Therefore, the frequency splitting will decrease. Hence, there is a trade-off between the bending loss and the frequency splitting, for choosing the ring radius. The other two mechanisms of the loss in this ring resonator are free carrier absorption and surface roughness loss. Assuming a change in the refractive index of Δn = 0.004, the absorption caused by the free carriers (α carrier ) is calculated by Soref equations [34]. The absorption coefficient caused by the bending loss (α bend ) is calculated via numerical simulations in Ansys Lumerical MODE. Therefore, the total quality factor (Q T ) of the ring resonator includes the contribution of the quality factor of the ring without electro-optic modulation (Q static ) and quality factor of the ring due to free carrier absorption loss (Q carrier ), and is calculated as: here n g and λ are the group index and the resonance wavelength, respectively. Also α roughness represents the absorption caused by the surface roughness loss. According to Fig. 3, the intrinsic quality factor increases as the ring radius increases. For radii larger than 4 μm, increasing the radius has negligible effect on the quality factor, for α roughness = 2 dB/cm and α roughness = 3 dB/cm. For α roughness = 0.5 dB/cm, as the surface roughness loss becomes comparable to the free carrier absorption loss, by increasing radius, and hence decreasing the modulation region, the free carrier absorption loss decreases, and the quality factor increases.
We consider some points, first we assume working with the 4 μm ring. Second, we suppose the quadrature phase difference between the modulated regions, i.e., Here BW and f m are the intrinsic resonance linewidth and the modulation frequency, respectively.
The distance between the modulated regions is chosen in a way that together with the quadrature phase condition satisfies the condition |κ m+ | >> |κ m− |. The width of the modulated regions is chosen to maximize |κ m+ |. These assumptions leads to maximum frequency splitting [30]. Then we find the optimum point to achieve maximum optical isolation in the next step. For this purpose, we use the scheme used in [30] and consider two parameters to quantify the performance of the isolator, namely frequency splitting and the ratio of the c.w. to the c.c.w. modes for the central harmonics of the two supermodes. These two parameters are plotted versus modulation frequency (f m ) and the voltage amplitude in Fig. 4. Surface roughness absorption coefficient of down to α = 0.5 dB/cm is considered [35]. The operating point can be between the black and red curves in Fig. 4 which represent points where frequency-splitting equals BW and f m − BW , respectively.
Here BW and f m are the intrinsic resonance linewidth and the modulation frequency, respectively. As an explanation, due to the temporal harmonic modulation, this structure has two supermode spectrums that are obtained by solving (2). Each supermode consists of c.w. and c.c.w. harmonics that are equally spaced with a distance of f m . Assuming working in the central harmonic of the supermode, the frequency splitting should be such that the spectrum is completely separated from the neighboring counter-rotating spectrums. In fact BW and f m − BW determine the minimum and maximum acceptable frequency splitting, which ensures both the separation between the zeroth order harmonics and the separation between the central harmonic and the first side harmonic. The optimum point which gives maximum isolation, occurs at a point in this region, which has the maximum ratio of the c.w. to the c.c.w. mode. Using this scheme, we choose the optimum working point that gives maximum isolation, while maintaining an insertion loss of -3dB. Forward and backward power transmission spectra for the optimum point are plotted in Fig. 5, assuming surface roughness loss of α = 0.5 dB/cm. The modulation frequency, the voltage amplitude and the intrinsic resonance linewidth are f m = 7 GHz, V = 1.321 V and BW = 950 MHz for this point. These spectra indicate  non-reciprocal transmission in this structure, and an isolation of 10.54 dB and an insertion loss of −3.02 dB is obtained, at the resonance frequency. Isolation (|S 12 | 2 /|S 21 | 2 ) is plotted in Fig. 5(b) versus frequency. According to this figure, the bandwidth at which this device works with the isolation of 3 dB and higher, is 1.03 GHz.
It is worth mentioning that as shown in Fig. 4, the working points on the black curve have approximately the same ratio of c.w. to the c.c.w. mode, this leads to a tunable optical isolator where by varying the voltage and modulation frequency, the optimum working point can be tuned on this curve.

B. Tunability
To demonstrate the tunability of this device, transmission spectrum is plotted for three different values of the modulation voltage and the corresponding change in the carrier density in Fig. 6. According to Fig. 6, by decreasing the voltage to V = 1.233 V (ΔN = 3 × 10 18 ), isolation, insertion loss, frequency splitting and the intrinsic resonance linewidth are 12.61 dB, −3.86 dB, 657 MHz and 789 MHz, respectively, and by decreasing it further to V = 1.121 V (ΔN = 2 × 10 18 ), isolation, insertion loss, frequency splitting and the intrinsic resonance linewidth are 16.09 dB, −5.96 dB, 333 MHz and 573 MHz, respectively. In fact by decreasing the carrier density, the modulation amplitude decreases and according to (5), it will decrease the self-coupling term, i.e., κ s+ , which decreases the power transfer to the side harmonics. Therefore most of the energy remains in the central harmonic and couples back to the waveguide and destructively interferes with the optical wave in the waveguide. This leads to the decrease in the backward transmission, hence the isolation is increased. On the other hand, decreasing the modulation amplitude, decreases the cross-coupling term (κ m+ ), which decreases the frequency splitting. Therefore the forward transmission decreases and hence the insertion loss is increased. Here and in the rest of the paper, when discussing changes in insertion loss, we mean the changes in absolute value of it.
Then we consider the phase difference of the two modulated regions (Θ 2 − Θ 1 as shown in Fig. 1), and its effect on the key parameters, namely, isolation and insertion loss. According to (2)-(4), performance of the isolator is mainly determined by the cross-coupling terms, i.e., |κ m+ | and |κ m− |, as well as the self-coupling term, i.e., |κ s+ |. In fact, if there was no self-coupling, each of the supermodes consists only of one frequency component, i.e., the central harmonic. So each of the supermodes shrinks to one harmonic. However, by increasing the self-coupling term, some power is transferred to the side harmonics. Therefore, smaller portion of the power couples back to the waveguide in the central harmonic to destructively interfere with the optical wave in the waveguide, and to decrease the output. Therefore, by increasing the self-coupling term, the output power at resonance frequency, which is the backward transmission of the isolator, increases, and this results in the decrease of the optical isolation. Next we consider the effect of the cross-coupling terms on the performance. Neglecting the effect of the self-coupling term, considering κ s+ = 0, if κ m+ >> κ m− , according to (2)-(4), the c.w. mode will have a blue-shift equal to the modulation frequency (f m ). Therefore, the frequency splitting is maximized under this condition. When the |κ m+ |/|κ m− | is decreased, the frequency splitting decreases. This will result in the decrease of the forward transmission of the optical isolator and hence the decreased isolation and increased insertion loss. The same interpretation applies for |κ m− |/|κ m+ |, but here the c.w. mode will undergo a red-shift. This can be verified in Fig. 7. Here we consider Θ 1 = 0. According to Fig. 7, at Θ 2 = −π/2 (Θ 2 = π/2), |κ m+ |/|κ m− | (|κ m− |/|κ m+ |) is maximized, therefore in this point the frequency splitting and hence the forward transmission is maximized. Therefore, here the absolute value of the insertion loss is minimized. On the other hand, the minimum of the self-coupling term is at Θ 2 = ±π. As a result, at this point least energy is transferred to the side harmonic, and all the back coupled energy to the waveguide is in the central harmonic and destructively interferes with the power in the waveguide. Hence the backward transmission is minimized. The optimum point for maximum isolation happens somewhere between the Θ 2 = −π/2 and Θ 2 = −π. (or between Θ 2 = π/2 and Θ 2 = π, for the red-shift of the c.w. mode). It means that there is a trade-off between these two factors, for maximum isolation. The phase difference that we already , and due to the symmetry, we consider φ 1 = 0. considered in the previous sections was based on obtaining maximum cross-coupling term. In fact by considering the quadrature phase difference we maximize the frequency splitting and hence minimize the absolute value of insertion loss. However, it is observed (according to Fig. 7(f)) that actually we can achieve better isolation, if we work at Θ 2 = −0.78π (or Θ 2 = 0.78π, for the red-shift of the c.w. mode). At this new Θ 2 , isolation is increased to 14.36 dB and insertion loss is −5.52 dB. At this point, the absolute value of the insertion loss is increased.
Next, we plot the transmission spectrum for and also for the scenarios that Θ 2 − Θ 1 has a deviation equal to ΔΘ from this value. According to Fig. 8, the isolation, insertion loss and frequency splitting will be 7.98 dB, −3.46 dB and 836 MHz, respectively for a 30 percent increase in the phase difference of the two regions. By increasing the variation to 50 percent, the isolation, insertion loss and frequency splitting will be 6.10 dB, −4.44 dB and 660 MHz, respectively. Next, we demonstrate that we can trade off between the optical isolation and insertion loss, by varying the modulation parameters, i.e., Voltage (equivalently ΔN ), f m and Θ 2 . As a consequence, we will classify the working regimes, into three zones. We sweep the modulation frequency and voltage and plot the isolation and insertion loss for different modulation phases, Fig. 8. (a) Forward (S 12 ) and backward (S 21 ) transmission spectra, and b) isolation versus frequency, for Θ 2 = − π 2 and for the scenario that there is a variation of ΔΘ = 0.3( π 2 ) and ΔΘ = 0.5( π 2 ) in the phase difference between the two regions, where ΔΘ = Θ 2 + π 2 .
i.e., Θ 2 = − π 2 and Θ 2 = −0.78π. As shown in Fig. 9, each range of isolation corresponds to a range for insertion loss, and one can achieve higher isolation at the cost of higher insertion loss. Consequently, one can classify the working point into three zones. In the first zone, the isolator works with small insertion loss. Working at this zone can be useful for the applications where very low insertion loss is needed. Therefore, we choose the quadrature modulation phase, i.e., Θ 2 = − π 2 , and a point with moderate isolation of 4.63 dB, but very good insertion loss of −1.76 dB can be achieved at this point. In the second zone, high isolation is needed, but on the other hand higher insertion loss can be tolerated. Hence, we can work with either Θ 2 = − π 2 or Θ 2 = −0.78π. According to Table II and Table III, to achieve the same isolation ratio, one can work with smaller modulation amplitude and obtain better insertion loss, for Θ 2 = − π 2 . On the other hand, one can work at smaller modulation frequency, but achieve higher insertion loss, for Θ 2 = −0.78π. There is also a third zone, between the aforementioned zones, where the device works with moderate isolation of 10.54 dB, and insertion loss of −3.02 dB, for Θ 2 = − π 2 , as obtained in Section III-A.

IV. SENSITIVITY ANALYSIS
In this section robustness of our design to the fabrication variations of parameters such as, angular distance (φ 2 − φ 1 in Fig. 1) and width (δ in Fig. 1) of the modulated regions will be evaluated.

A. Angular Distance Variation
We choose the quadrature phase difference (Θ 2 − Θ 1 = − π 2 ), which minimizes the absolute value of insertion loss, and proceed to investigate the effect of the fabrication errors on the performance of the isolator. First we consider fabrication error in the distance between the two points, i.e., φ 2 − φ 1 as shown in Fig. 1. According to (5), the condition Θ 2 − Θ 1 = −π/2 together with 2lφ 2 = 2lφ 1 + 2nπ + π 2 leads to the condition |κ m+ | >> |κ m− |, hence the frequency splitting is maximized and the c.w. spectrum will undergo a blue-shift. Without loss of generality, and due to the symmetry, we consider φ 1 = 0. For the case that φ 2 is deviated from the ideal value φ i = (2lφ 1 +2nπ+ π 2 ) 2 l , |κ m+ |/|κ m− | is decreased, hence the frequency splitting and therefore, the forward transmission decreases. Therefore, it will degrade both the isolation and insertion loss. However the selfcoupling coefficient |κ s+ | and therefore the amplitude of the side harmonic is independent from the φ 2 , hence, as explained earlier the backward transmission will not change. These predictions can be verified in Fig. 10. According to (5), if the variation of φ 2 is equal to π 2 l , |κ m+ | and |κ m− | and hence the split c.w. and c.c.w. modes are swapped. Therefore according to Fig. 10, frequency splitting is a periodic function of φ 2 , with the period π 2 l . Also, as the forward transmission is determined by the frequency splitting, and since the backward transmission does not change by varying φ 2 , insertion loss and isolation are also periodic functions, with the same period. Another interesting point, is that at the points where |κ m+ | = |κ m− |, frequency splitting is minimized, since according to (2)-(4), in this condition c.w. and c.c.w. modes have the same frequency shift.
Next, the transmission spectra are plotted for the scenario that there are some fabrication imperfections in the distance between the two points. The isolation, insertion loss, and frequency splitting will be 7.17 dB, −5.97 dB, and 510 MHz, respectively for a Δφ = 0.0125 rad (50 nm shifting of the second region along the circumference of the ring due to fabrication imperfections). Here is the ideal value of φ 2 . By decreasing this value to Δφ = 0.0075 rad (equivalently 30 nm shift), the isolation, insertion loss, and frequency splitting will be 9.52 dB, −3.81 dB, and 781 MHz, respectively. According to Fig. 11, here the backward transmission does not change, and the degradation of the isolation is originated from the decreased frequency splitting and hence the decreased forward transmission, caused by the deviation from the aforementioned condition, i.e., 2lφ 2 = 2lφ 1 + 2nπ + π 2 . To obtain better insight of the effect of these parameters, Θ 2 and Δφ are swept together. As shown in Fig. 12, frequency splitting is maximized in the points where Θ 2 = ± π 2 and 2lφ 2 = 2lφ 1 + 2nπ + π 2 . Therefore these points represent the zone where the absolute value of the insertion loss is minimized. As shown in Fig. 12(a) and (f), at the points where 2lφ 2 = 2lφ 1 + 2nπ + π, cross-coupling term has equal redshift and blue-shift contributions, i.e., |κ m+ | = |κ m− |. Consequently, isolation will be zero in this region. At Θ 2 = ±π and Θ 2 = 0, frequency splitting is zero, hence the isolation will also be zero in this zone. As Δφ moves away from Δφ = − π 4 l , the zone of maximum isolation moves to the smaller |Θ 2 |.  Therefore, if there is some variation in the distance between the two regions, it can be compensated, to some extent, by tuning the modulation phase difference accordingly.

B. Width Variation
Other important parameter is the width of the modulated regions (δ). According to (5), the cross-coupling term is a sinusoidal function of δ, and is maximized at δ i = π 2 l . As the width of the modulated regions is deviated from this value, the frequency splitting which is proportional to the cross-coupling term, is reduced, as plotted in Fig. 13. On the other hand, by reducing the width of the modulated regions, the free carrier absorption loss decreases and hence the linewidth reduces. Therefore, the absolute value of insertion loss is minimized at a width between δ i and the smallest modulation width. Moreover, according to (5), by increasing the modulation width, the self-coupling term, which represents the modulation of the resonance frequency, increases. This broadens the supermode spectrum and increases the amplitude of the side harmonic, as shown in Fig. 13. Under this condition, some of the energy of the resonating ring couples to the side harmonics, and smaller portion of the energy is coupled back to the waveguide in the central frequency,  therefore the transmission, which had to be extinct due to the destructive interference with the wave in the waveguide, is not dropped completely. Consequently, the backward transmission increases, which leads to further decrease in the isolation. Next, transmission spectrum is plotted for the scenario that there is a 50 nm error in the width of the modulated regions, compared to the ideal value, i.e., δ i . According to Fig. 14, transmission spectrum is broadened by increasing δ, due to the increase in the free carrier absorption loss, and the isolation, insertion loss, frequency splitting and the intrinsic resonance linewidth will be 5.54 dB, −5.05 dB, 741 MHz and 1.21 GHz, respectively for a 50 nm change in the width of the modulated regions, i.e., (δ − δ i )R. Results of this section can be useful both in the design and measurement stages, as we investigated the effect of each parameter on the key performance metrics, i.e., isolation, insertion loss and frequency splitting.

V. CONCLUSION
In this paper, design guidelines for a tunable optical isolator in a silicon ring resonator with two time-modulated parts has been presented. The design is based on a CMOS compatible silicon photonics foundry process. It has been demonstrated that there is a trade-off between the resonance bandwidth and frequency splitting for choosing the ring radius. Isolation, insertion loss and frequency splitting of 10.54 dB, −3.02 dB and 950 MHz have been achieved for a modulation frequency and voltage amplitude of f m = 7 GHz and V=1.321 V, respectively, and for Θ 2 − Θ 1 = − π 2 . The bandwidth at which this device works with the isolation of 3 dB and higher, is 1.03 GHz. It has been demonstrated that the two main parameters that affect the performance, are the amplitude of the cross-coupling and self-coupling terms in TCMT equations, that determine the frequency splitting and energy transfer to the side harmonics, respectively. In fact, frequency splitting determines the forward transmission and amplitude of the side harmonics affects the backward transmission of the isolator. We have shown that by applying the 90 degrees phase difference between the modulated points, one can obtain the maximum frequency splitting and hence the lowest insertion loss, while the maximum isolation of 14.36 dB occurs for Θ 2 = −0.78π, which results in a higher insertion loss of −5.52 dB. In other words there is a trade-off between the maximum cross-coupling and the minimum selfcoupling terms, to obtain the maximum isolation. It has been shown that the isolation, insertion loss, and frequency splitting will be 7.98 dB, −3.46 dB, and 836 MHz, respectively for a 30 percent increase in the phase difference of the two regions, i.e., ΔΘ, and they will be 6.10 dB, −4.44 dB, and 660 MHz, respectively, by increasing the variation to 50 percent. Tunability of the device under variations of the change in the carrier density has been demonstrated and explained by similar reasoning. In fact by decreasing the ΔN to 3 × 10 18 , isolation, insertion loss, frequency splitting and the intrinsic resonance linewidth are 12.61 dB, −3.86 dB, 657 MHz, and 789 MHz, respectively, and by decreasing it further to ΔN = 2 × 10 18 , isolation, insertion loss, frequency splitting, and the intrinsic resonance linewidth are 16.09 dB, −5.96 dB, 333 MHz, and 573 MHz, respectively. The sensitivity of the device performance to the imperfections has been analyzed. It has been shown that variation in the distance of the regions degrades the performance by changing the frequency splitting and hence the forward transmission and for a 30 nm shifting of the second region along the circumference of the ring, i.e., (Δφ)R, the isolation, insertion loss, and frequency splitting will be 9.52 dB, −3.81 dB and 781 MHz, respectively. By increasing this value to 50 nm, the isolation, insertion loss and frequency splitting will be 7.17 dB, −5.97 dB, and 510 MHz, respectively.
It has been observed that for Θ 2 = 0 and Θ 2 = ±π and also at the points where 2lφ 2 = 2lφ 1 + 2nπ + π, cross-coupling term has equal red-shift and blue-shift contributions, i.e., |κ m+ | = |κ m− |. Consequently, isolation will be zero in this region. Isolation, insertion loss, frequency splitting and the intrinsic resonance linewidth will be 5.54 dB, −5.05 dB, 741 MHz and 1.21 GHz, respectively for a 50 nm change in the width of the modulated regions, i.e., (δ − δ i )R. Moreover, based on the analysis, there is a trade-off between insertion loss and isolation, by tuning the modulation parameters, and it has been shown that an insertion loss of only −1.8 dB can be obtained for an isolation of 5 dB whereas an isolation of up to 18 dB can be achieved if an insertion loss of −7 dB can be tolerated. Our proposed structure is simple to implement and has a small footprint, as it uses only one ring resonator with two small, modulated regions. Moreover, it is completely compatible with the conventional CMOS fabrication processes. We believe that this work can pave the way toward the fabrication of a non-magnetic, CMOS-compatible, and compact optical isolator.