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Abstract
As one of major integrated microwave photonics (IMWP) platforms, Si photonics exhibits the intensity-dependent Kerr effect and two-photon absorption (TPA) with associated free carrier effects (FCE). At the commonly used 1.55 µm, TPA losses and the associated FCE would eventually limit the dynamic range of Si photonic links. Resonating structures such as ring resonators (RRs) experience enhanced nonlinear effects due to significant intensity buildup. According to the bandgap characteristics of Si, TPA can be eliminated at and beyond 2.2 µm. In this work, a systemic simulation of straight waveguides and RRs is performed at wavelengths from 1.55 to 2.2 µm where the wavelength-dependent TPA loss is investigated. Moreover, the Kerr effect leads to unwanted change of refractive index, which shifts the RR resonant wavelength at both 1.55 and 2.2 µm, thus needing shift compensation. Compensated RRs operating at 2.2 µm could open a new venue for Si photonics towards IMWP applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Microwave photonics (MWP) combines the fields of radiofrequency (RF) engineering and optoelectronics resulting in a discipline which offers high speed, broadband analog signal processing. It is well acknowledged that MWP technology can provide microwave functionalities that are sophisticated or not even available in the digital RF domain [1]. For example, MWP systems offer ultra-wide-band signal generation, RF signal distribution via fiber, and photonics-based radar systems [2–4]. These functionalities have allowed the MWP field to penetrate into a variety of defense and civil applications, such as 5G systems, wireless, satellite communications, optical signal processing, and medical imaging systems [2]. Bulky MWP systems are not an ideal long-term solution for these applications due to their lack of scalability and high cost and power consumption. Engineering MWP systems on the chip-scale, known as integrated microwave photonics (IMWP) provides the most promising solution. IMWP circuits on the chip scale are able to provide MWP functionality with reduced size-weight-and-power (SWAP) at low cost when manufactured using a foundry, and for these reasons IMWP has attracted intense research activities in recent years.
IMWP systems are currently implemented predominantly in three material platforms: InP, TriPleX technology, and Silicon-on-insulator (SOI). InP enables the monolithic integration of various active and passive photonic components, including lasers, modulators, optical amplifiers, tunable devices, and photodetectors. This monolithic integration enables the construction of highly complex photonic integration circuits (PICs) with hundreds of components contained on a single chip. However, the optical waveguides in this material have relatively high losses that can be an order of magnitude higher when compared to waveguides based on silica or silicon [2], which limits the future scalability of InP-based IMWP circuits. TriPleX technology uses Si3N4 as the waveguiding layer, with SiO2 as the surrounding cladding layers. The result of this technology is extremely low-loss waveguides covering wavelengths from 405 nm up to 2.35 µm [1]. Moreover, TriPleX offers good compatibility with silicon complementary metal-oxide-semiconductor (CMOS) processes [2]. TriPleX also does not suffer the nonlinear losses caused by two-photon absorption (TPA) and the associated effects accompanied with TPA-induced free carriers. However, the critical active components such as lasers, modulators, and detectors can be constructed in this material only through hybrid integration with other materials [1,2].
Si-photonics based on SOI is an exciting and rapidly developing photonic platform that has attracted a lot of research interests. The motivations behind Si-photonics as a platform for IMWP are the excellent compatibility with the mature silicon CMOS fabrication processes—exceeding the CMOS compatibility of the TriPleX system [2]—and the availability of low-loss SOI channel waveguides with high index contrast, which allows the miniaturization and large-scale integration of photonic devices with compact bend radii. The challenges of Si-photonics for IMWP applications are: i) lack of monolithic high-performance light sources. At present, Si-photonics utilizes III-V light sources via hybrid integration; and ii) the presence of two-photon absorption (TPA) which threatens to limit the dynamic range of Si-based IMWP systems when operating at the traditional wavelength of 1.55 µm.
Theoretical study indicates that the bandgap characteristics of Si eliminates TPA at 2.2 µm, where the total energy of two photons is insufficient to span the Si bandgap so that they will not be absorbed [5]. Considering that the SiO2 cladding restricts wavelengths beyond 2.5 µm due to high dioxide absorption, an alternative wavelength window from 2.2 to 2.5 µm may be more suitable for the Si photonics platform. However, there is no satisfactory light source operating in this wavelength range so far.
Recently, study of the all-group-IV SiGeSn material system opens a new venue for Si- based light sources. The first optically pumped GeSn laser using the true direct-bandgap GeSn alloy as the gain medium was reported in 2015, followed by the numerous publications worldwide reporting improved device performance [6–11]. In 2020, the first electrically injected GeSn laser was successfully demonstrated [11]. Further development of SiGeSn-based lasers promises to provide monolithic high-performance light sources on Si. It is worth noting that the lasing wavelength covers the 2.0 to 3.0 µm range, and therefore the new operating wavelength window abovementioned could be available if the SiGeSn-based light source were adopted. In such a window the Si TPA can be neglected and the SiGeSn based light sources are available with excellent CMOS compatibility. Due to reduced overall nonlinear optical effects, scaling the SOI IMWP working wavelength up from 1.55 µm to 2.2–2.5 µm could significantly improve the dynamic range of analog signal processing. Note that in the 2.2–3.3 µm region, three photon absorption (3PA) dominates the nonlinear loss [12]. Even with a small number of photons lost, the generated free carriers could lead to significant optical loss [13].
In this work, the effects of high source power upon performance of straight waveguides and ring resonators (RR) in the SOI platform was investigated towards IMWP applications. To the best of our knowledge, such power effects have been generally overlooked in the silicon photonics literature. Guides and RRs can serve as generalized basic building blocks for all passive components for analog signal processing to be performed by IMWP [14]. Waveguides are a critical means of routing light on-chip, which means that TPA-induced propagation losses would impact the overall link gain. RRs function as a fundamental building block for a wide variety of components including microwave filters, delay lines, beamforming networks, ultra-wide band signal generators, and arbitrary waveform generators. RRs require special consideration due to their resonant properties. On-resonance, an RR can achieve standing wave intensities orders-of-magnitude greater than the light input to the component, which enhances the nonlinear characteristics and increases the loss via TPA. Moreover, TPA introduces losses and generates free carriers in the material, which further causes free-carrier absorption (FCA) and free-carrier dispersion (FCD) which modifies the refractive index. In a resonant structure, this refractive index change will unfortunately change the resonance point of the device. An associated nonlinear effect, the Kerr effect, also modifies the refractive index. Therefore, for a complete picture of nonlinear RR behavior both TPA and the Kerr effect should be considered. In practice, the unwanted shift of resonant wavelength is required to be compensated at both 1.55 and 2.2 µm, and operation at 2.2 µm is seen as more desirable as the TPA induced losses are eliminated.
This paper is organized as follows: Section 2 presents the background of third-order nonlinear Kerr-and-TPA effects in Si (plus free carrier effects), followed by section 3 describing the simulation methodology for waveguides and ring resonators. In section 4, the results of the simulations are presented. The results show propagation loss of waveguides with various lengths at wavelength from 1.55 to 2.2 µm as a function of optical input power for both single-mode strip and rib waveguide designs. The RR simulation results show the nonlinear refractive index change which shifts the resonant wavelength. In the case where the resonance wavelength is held constant by using a thermo-optical tuner to compensate the shift, additional simulations show the key RR figures of merit including Q-factor, extinction ratio, and propagation losses. Finally, the relation of the results to IMWP components is discussed.
2. Nonlinear effects in Si
2.1 Two-photon absorption
The loss introduced into an electromagnetic wave resulting from TPA is proportional to the square of the intensity of the wave [15]. The equation describing the linear and TPA losses is given by:
Equation (1) accounts for both the nonlinear loss due to TPA as well as the linear loss α due primarily to sidewall scattering and free-carrier absorption. Solving Eq. (1) yields:2.2 Free-carrier effects
Two-photon absorption results in the generation of free-carriers in the waveguide material, which in turn causes free-carrier absorption (FCA) and free-carrier dispersion (FCD). FCA further contributes to the absorption loss (but does not generate any carriers) and FCD changes the refractive index of the waveguide material. They both are determined only by the number of free carriers in the material. At 1.55 µm, FCD and FCA are quantified by the following two equations [16,17]:
2.3 Kerr effect
The Kerr effect is a change in the real refractive index of a material in response to the enhanced light intensity. In this work, we consider the optical Kerr effect, in which the refractive index change is induced by the presence of an optical field. The refractive index change due to the optical Kerr effect is described by [18]:
where ${n_0}$ is the refractive index of the material at low power and ${n_2}$ is the Kerr coefficient, which can be experimentally determined. For Si, the Kerr coefficient is 5.516 × 10−14 $\frac{{c{m^2}}}{W}$and 9.153 × 10−14 $\frac{{c{m^2}}}{W}$ at optical wavelengths of 1.55 µm and 2.2 µm, respectively [5]. For the simulations in this work, the light intensity I was determined by multiplying the input power (to the bus waveguide) by the intensity buildup factor of the ring at the input wavelength.3. Simulation methodology
The commercial software Lumerical MODE Solutions was used to simulate nonlinear processes in which the material parameters χ1, χ2, and χ3 can be defined, where χn is the n-th order component of the electric susceptibility of the material, and where χ3 plays a significant role in the Kerr effect. The simulations consist of two parts. First, the impact of TPA on straight waveguides with varying geometry was studied. The second part of simulations investigated all-pass (single-bus) RRs, with selected waveguide geometry, while different coupling factors and RR circumferences were considered. For the RRs, free-carrier and Kerr effects were introduced and their effect on the change of refractive index were studied, which can significantly impact the transfer function characteristics.
3.1 Simulation of waveguides
For the waveguide simulations, various waveguide lengths were studied for a working wavelength from 1.55 to 2.2 µm, where the TPA effect dominates the nonlinear loss. The 3PA effect (dominates at 2.2-3.3 µm) was not included in this study. The waveguide cross-sectional dimensions were chosen to satisfy single-mode operation, which is desirable to avoid multimode signal dispersion. In addition, we considered both the strip and rib waveguides configurations. The rib geometry, exhibiting a relatively larger mode area, is expected to feature reduced TPA since a larger area yields reduced optical intensity at any given input power.
In order to consistently track the impact of TPA on signal, we calculated the 1 dB compression point (P1 dB): the point at which the signal output is compressed/attenuated by 1 dB with respect to a linear relationship. The P1 dB point defines the upper limit of the input power without prohibitive signal compression.
Equation (2) shows that the TPA effect is dependent on the intensity of light in the material. In a waveguide, the intensity of light is distributed according to the cross-sectional mode profile. To apply the TPA equation, simulation of the waveguide structure using RSoft’s FULLWAVE FDTD solver was firstly conducted for both strip and rib waveguides, with dimensions outlined in Table 1. These dimensions were initially designed for operation at 1.55 µm and can be scaled up proportionally to the longer 2.2 µm wavelength. From the FDTD simulation, we exported the spatially-dependent intensity values within the waveguide. Then Eq. (2) was used to determine the TPA losses for a range of input power by importing the obtained intensity distribution. This process was repeated for a variety of waveguide lengths.
Table 1. Summary of simulated waveguides
3.2 Intensity buildup in ring resonators
Both the TPA and Kerr effects are modeled based on the intensity of light in the material. In a resonating structure, high intensity standing waves can be produced, which consequently enhance these nonlinear effects. For RRs, the intensity buildup factor B is dependent on the coupling coefficients t and$\; r$, the round-trip attenuation a, and the round-trip phase shift $\phi .$ The round-trip phase shift is defined as any integer multiple of $2\pi $. The phase shift indicates that the structure is on-resonance at which the intensity buildup is maximized. For single-bus RRs, the expression of the intensity buildup is [19]:
The light intensity in the RR affects the index of refraction via the Kerr effect and TPA-induced free carriers, i.e., the FCD effect. In order to determine the free-carrier concentration, the carrier-generating power absorption was estimated, which provides the carrier-generation rate for any given power. The excess carrier concentration ${D_n}$ was determined from the TPA loss and the carrier lifetime of Si.
3.3 Carrier lifetimes in Si
TPA losses contribute to the generation of free carriers at a rate of one free-electron-hole pair per pair of photons being absorbed. Note that Eq. (1) incorporates both the TPA losses (which generate free carriers) and the linear loss. In this simulation work, both a linear-loss model and the TPA loss model expressed in Eq. (2) were constructed for any input power. Since linear loss is mainly due to scattering by the waveguide sidewalls, it does not contribute to generating free carriers, and so the carrier-generating power loss was taken differently between the TPA model and the linear model. The carrier generation rate was then determined by dividing the carrier-generating power loss by the photon energy at 1.55 µm, and the carrier generation density was calculated by dividing the carrier generation rate by the volume of the RR. Finally, the excess carrier density was found by multiplying the carrier generation density by the overall carrier lifetime.
The carrier lifetime for Si can be expressed as [20]:
where ${\tau _{SRH}}$, ${\tau _{rad}}$, and ${\tau _{Auger}}$ are the Shockley-Read-Hall (SRH), radiative, and Auger recombination lifetimes, respectively. Because Si is an indirect bandgap material, it is assumed that the radiative lifetime is sufficiently long such that it is negligible when compared with the SRH and Auger recombination mechanisms. Moreover, at lower carrier injection levels, the SRH recombination process is dominant, and was assumed to be 2 × 10−4 seconds [20]. At higher carrier injection levels, the Auger recombination process is dominant. For p-type Si: where ${C_p}$ is the Auger recombination coefficient, ${p_0}$ is the equilibrium hole density, and$\; {D_n}$ is the excess electron density. In this work, ${C_p} = 1.1\; \times \; {10^{ - 30}}\frac{{c{m^6}}}{{s\; }}\; $ was used [20].3.4 Simulation of ring resonators
For the simulation of RRs, the geometry of the RRs mirrored the geometry of the strip waveguides, namely the same mode profile (intensity spatial distribution) was used. This method assumes that the curvature of the waveguide does not significantly distort the distribution of intensity. The cross-sectional geometry for the 1.55 µm operating wavelength was a 220 nm by 450 nm Si-on-SiO2 strip configuration. For the 2.2 µm, the 312 nm by 639 nm cross section was used, determined by scaling up the waveguide geometry proportionally to wavelength. At each wavelength, both the upper and lower cladding layers consisted of SiO2. The RRs were further characterized by their round-trip circumference and their coupling coefficients with respect to the bus waveguide.
Once the intensity distribution of the light within the waveguide was determined, the mesh was exported for processing in Matlab. Since nonlinear effects were not considered in the FDTD simulation, the imported mesh was used as a low-power-limit starting point, where nonlinear effects are negligible. The input power was gradually increased, and the non-linear effect is considered at higher power. Note that it was reported that the mid-infrared frequency comb can be generated under intense pumping power [21]. Such high power is not included in this work.
Figure 1 gives an overview of the simulation process. Since the nonlinear effects considered in this study depend on the intensity of the light, it is necessary to know the distribution of intensity within the waveguide. The intensity distribution was obtained via FDTD simulation, and the results of the simulation were exported as a spatial mesh of intensity values corresponding to Fig. 1(a). The intensity values within this mesh correspond to the fundamental transverse electric (TE) mode, as the waveguide geometries were designed to operate in the single-mode condition. Once the mesh was obtained, the nonlinear effects described by Eqs. (2) and (5) were applied, as described in Fig. 1(b). In the case of the RR the magnitude of these nonlinear effects, enhanced by the intensity buildup factor, simultaneously depends upon and modifies the waveguide loss and the material index (Eq. (6)). Due to this interdependence, the process of applying the equations and updating the material parameters was repeated until a steady-state was reached. The steady-state was reached when there was agreement between the results of the calculation and the updated material parameters. In our simulations, agreement was determined within a tolerance of 10−8%. Figure 1(c) provides a schematic of the RR considered in this study and was defined by the ring circumference and the coupling to the bus waveguide. The coupling was assumed to be lossless obeying the relation of ${t^2} + {r^2} = 1$ [19].
Fig. 1. (a) Schematic showing spatially-distributed light intensity in the waveguide. (b) Flow chart describing the method for simulating RRs. (c) Diagram of RR under study.
The initial optical power input to the input port of the RR was 3.5 × 10−11 mW, which is sufficiently low such that nonlinear effects are negligible. The resonance wavelength influences the intensity buildup factor, which causes nonlinear effects that in-turn influence the resonance wavelength. The change in loss also significantly impacts the buildup factor. For these reasons, the process is repeated until a steady-state condition is met. The steady-state condition was considered reached when the simulated change in the refractive index was less than 10−6%. Once this condition was met, the figures of merit (FoM) of the ring were recorded and the process was repeated with an increased power. The increment used to increase the power was 1.5 mW after each simulation. This was repeated until a power up to 300 mW.
4. Results and discussions
4.1 Waveguide simulation
The results of the waveguide simulations are summarized in Fig. 2. The mode assumed for both the strips and the ribs is fundamental TE mode, and the rib platform height h is specified in Table 1. Figures 2(a) and 2(b) show the output power vs input power with various lengths for strip and rib waveguides. It is clear that the strip waveguide has significantly higher losses than the rib waveguide. This is due to the rib waveguide distributing the light over a larger area, reducing the average light intensity. The increased area can be quantified using the effective area of the waveguide, as listed in Table 1. For the longer operating wavelength, the TPA effect is reduced. This can be explained by the following: i) the TPA coefficient which determines the absorption parameters of TPA is reduced at longer wavelength [11]; ii) the waveguide geometry used for longer wavelengths was increased proportionally with the wavelength, increasing the effective area of the waveguide.
Fig. 2. Summary of waveguide simulation results for (a) strip and (b) rib waveguides. Insets: cross-section of waveguides.
Figure 3(a) and 3(b) track the P1 dB point of the waveguide with respect to the output power for various waveguide lengths. The P1 dB point is defined to be the upper limit of practical waveguide operation since operation above this point would result in high signal compression. At 1.55 µm, for the short 10 mm strip waveguide, operation up to 170 mW is attainable without prohibitive compression. For the 100 mm strip waveguide, the maximum operation power is only 3.75 mW. While for the rib waveguides, higher power operation is allowed, with the 100 mm waveguide withstanding output power up to 70 mW and the 10 mm waveguide handling power beyond 1 W. As quantified, the shorter 10 mm waveguide is “much more accepting” of high light power than the 100 mm waveguide. Moreover, for both waveguide configurations, at longer wavelength, the maximum power operation was significantly increased, with signal compression totally absent at and beyond 2.2 µm due to the elimination of TPA. Note that for the large-area rib waveguide, the signal compression was relatively small enough such that there is no P1 dB point definable at wavelengths beyond 1.78 µm.
Fig. 3. Summary of P1 dB for (a) strip and (b) rib waveguides. (c) P1 dB as a function of wavelength for 2 to 10 cm waveguide lengths
4.2 Ring resonator
In order to further understand and separate the effects of the TPA and Kerr effects, a comparison of RR behavior at 1.55 µm and 2.2 µm was studied. RRs of different circumferences (100 µm, 20 µm) and different coupling coefficients ($t\; = \; 0.3,\; 0.05$) were investigated. The linear loss of these RRs is assumed to be 1 $\frac{{dB}}{{cm}}$, the same as they are adopted in the strip waveguide simulation. The propagation loss (in dB/cm), the Q-factor of the ring, and the extinction ratio of the transmission spectrum are the focuses. To study the Kerr and FCD effects on resonance shift, the power-dependent resonance wavelength is presented. In addition, the simulations with fixed refractive index were performed, by which the Q-factor and the extinction ratio were studied.
4.2.1 Rings with resonance shift
Figure 4(a) shows the typical transmission spectra for a 100-µm-circumference ring with weak coupling coefficient t = 0.05 operating at 1.55 µm. As input power increases, the resonant wavelength shifts towards longer wavelength. Equations (3) and (5) reveal that the Kerr effect and FCD will shift the effective index oppositely: the Kerr effect increases the index while FCD decreases it. When both FCD and Kerr effect exist, such as operating at 1.55 µm, the resonant wavelength shift is partially cancelled, resulting in a reduced shift of resonance wavelength. While at 2.2 µm this shift is more pronounced due to: i) no FCD effect so the partial cancellation of shift does not occur; ii) the magnitude of the Kerr coefficient ${n_2}$ is almost doubled from ${n_2}$ at 1.55 µm. Direct comparison of performance between 1.55 and 2.2 µm can be seen in Fig. 4(b) for the 100 µm rings. The increased shift at 2.2 µm can be clearly seen. At input power of 50 mW with weak coupling (t = 0.05), the shifts are 0.55 and 0.95 nm for 1.55 and 2.2 µm, respectively. While with strong coupling (t = 0.3), the shifts are 1.92 and 4.23 nm.
Fig. 4. Summary of variable-index RR simulations. (a) Representative power-dependent transmission spectra. (b) Resonant wavelength variation for 100 µm round-trip rings and (c) for 20 µm round-trip rings. Change in resonance at 50 mW of input power is labelled for each RR.
It is noteworthy that the resonance shifting of the strongly coupled rings is much higher than that of the weakly coupled rings. This can be interpreted as: for the weakly coupled rings, the resonance peak width, i.e., the resonance bandwidth, is relatively narrow, which causes high intensity buildup on-resonance. However, slight shifting of the resonance peak leads to a dramatic reduction in the intensity buildup, and therefore a dramatic reduction in the nonlinear effects. While for the strongly coupled ring, the broader resonance bandwidth means that even when the ring is shifted off-resonance, the intensity buildup exceeds that of the weakly coupled rings. Since the intensity buildup for the strongly coupled rings is higher at off-resonance, the nonlinear effects are also more pronounced, leading to an increase in the shifting of the resonance peak. Thus, strongly coupled rings experience more overall resonance shifting than the weakly coupled rings. Study of the quality factor (Q) in the next section reveals that the Q-factor of strongly coupled rings is at least one order-of-magnitude lower than that of weakly coupled rings at any input power (Fig. 5(b)). Lower Q-factor corresponds to broader resonance bandwidth.
Fig. 5. Summary of fixed-index RR simulations. (a) Power-dependent transmission spectra. (b) Q-factor of RR vs. input power. (c) Propagation loss vs. input power (crossover points were labelled at each coupling factor). (d) Extinction Ratio vs. input power. In (b), (c) and (d), solid curves and dashed lines represent operating at 1.55 and 2.2 µm, respectively.
Figure 4(c) shows the shift of resonant wavelength for 20 µm rings. Compared to the 100 µm rings, the shifts are larger under the same input power when operating at both 1.55 and 2.2 µm. This is because the intensity buildup is stronger in smaller rings, giving more pronounced nonlinear effects. Under input power of 50 mW, the shift of 11.99 nm was observed with strong coupling operating at 2.2 µm.
4.2.2 Rings with compensated resonance shift
For a given light source power, the user of the ring applies a steady EO or TO phase shift to the RR of a strength sufficient to compensate for both the real n2 index change and the real index change induced by free carrier refraction. (This is feasible because the temperature rise in Si produces index change with the opposite sign). Having done that, the ring index is “fixed” at the zero-light value. This implies that the rings are on-resonance under any input power. Maintaining resonance in this way leads to greatly increased buildup, and therefore greatly enhanced TPA losses.
Figure 5(a) shows the typical transmission spectra of the compensated 100 µm ring with coupling coefficient t = 0.05 operating at 1.55 µm, indicating that the non-shift of resonant wavelength. For these RRs, the propagation losses show substantial increases as the input power is increased at 1.55 µm. The increased loss within the ring leads to two major effects. First, there is notable reduction in the Q-factor. Weakly coupled rings (with higher Q factors) show the most substantial Q-factor reduction with a 50% - 80% reduction in Q at relatively low input powers, suggesting that high-Q rings in Si at 1.55 µm are not practical for medium to high power using this ring design. For the RRs with stronger coupling, the Q-factor reduction shows less power-dependence, as can be seen in Fig. 5(b) (solid curves). At 2.2 µm, the Q factor stays constant as expected (dashed lines). The second notable effect observed is the variation in the extinction ratio (ER), or notch depth, of the transmission spectrum. The ER depends primarily on the coupling coefficient and the propagation losses of the RR. Increasing the optical power results in increased losses via TPA, which leads to the observed variation in the ER, as shown in Fig. 5(c). This variation could be leveraged in order to modify a filter’s ER in real time by changing the input power without relying on active components in proximity to the RR.
MWP filters based on single-bus ring resonators often operate at or close to the critical coupling condition [22–24]. The critical coupling condition provides highest signal rejection for signals on-resonance with the ring. For example, in Fig. 5(c), for a 100 µm RR with coupling coefficient t = 0.05 operating at 1.55 µm near to the critical coupling condition, the ER of -28 dB was obtained. As the input power increases, TPA and FCA increase the round-trip losses of the ring, causing the ring to rapidly drift away from the critical coupling condition upon experiencing only a few mW of input power. At 10 mW of input power, the ER is decreased to 4.2 dB. Such variation of the extinction ratio would drastically reduce the effectiveness of such a RR used to implement a filter in this way. At 2.2 µm, the TPA and FCA losses could be avoided, allowing for a consistent ER and increased reliability of the filter at a wider range of input powers.
Delay lines are another important component for many MWP signal processing operations, such as beamforming. RRs have been effectively utilized to design compact delay lines using stronger coupling than RRs used in filter designs [25–27]. In this work, the propagation loss in rings cross-coupling coefficients of $t\; = \; 0.3$. and$\; 0.05$ were investigated, as shown in Fig. 5(d). The increased nonlinear losses as the input power increases were observed. Since the TPA losses are enhanced by the buildup factor B (Eq. (6)), it is expected that the RRs with higher buildup factors will have greater TPA losses. Comparing the 20 µm and the 100 µm RRs, the shorter length of the 20 µm leads to lower round-trip losses and therefore a higher buildup factor and larger TPA losses are obtained (per unit length). Furthermore, a comparison between the weakly coupled ($t\; = \; 0.05$) and strongly coupled ($t\; = \; 0.3$) rings reveals that at lower input power, loss remains relatively lower, and the weakly coupled rings exhibit a stronger intensity buildup and thus a higher TPA loss. As the input power increases, the propagation loss increases, the round-trip attenuation a becomes dominant the buildup factor B. As the strongly coupled ring features higher attenuation a, its propagation loss is high than that of weakly coupled ring at higher input power. The crossover points were extracted as input powers of 36 mW and 184 mW for 100 µm and 20 µm rings, respectively. For the purposes of delay lines, these increased losses will directly translate to a reduction in the signal of interest. The magnitude of this signal reduction is dependent on the RR circumference. At 50 mW of optical power, the 20 µm and 100 µm RR experience 1 dB and 3.6 dB signal reduction, respectively. This signal reduction is accumulated per-ring, and therefore for the designs using a series of cascaded rings, such as in the side-coupled integrated spaced sequence of resonators (SCISSOR) [26,27], the overall loss would be added up according to the number of RRs in the device.
5. Conclusions
The impact of nonlinear effects in Si was in-depth studied in this work. The results of the waveguide simulations show the TPA impacting a signal travelling down strip and rib waveguides of varying lengths at a wavelength of operation within 1.55 to 2.2 µm range. Significant improvements were seen in the rib waveguide over the strip waveguide due to the increased waveguide cross-sectional area lowering the overall intensity. Operation at longer wavelengths also shows improvement in loss reduction due to a combination of the increased effective waveguide area and the lower overall impact of TPA in Si at longer wavelengths. Elimination of the TPA effect was observed at 2.2 µm.
The simulations of RRs give a direct comparison of nonlinear effects in Si between 1.55 µm and 2.2 µm. At both wavelengths, there is a notable resonance-shifting due to the Kerr and FCD effects as input power is increased. Compensation of the index via thermo-optic or electrooptic effects may be required to ensure proper device function. For the fixed-index simulations, at 1.55 µm, TPA increases the losses within the ring which causes significant reduction in Q-factor, particularly for higher Q rings. In addition, the ER of the transmission spectrum varies with the input power due to the increased losses. At 2.2 µm, TPA is eliminated, which leads to a consistent Q-factor and an ER. It is worth noting that the resonance shifting experienced at 2.2 µm is greatly increased due to the elimination of FCD and an almost doubled n2 coefficient. However, since refractive index compensation is required for both 1.55 and 2.2 µm operation, and since 2.2 µm operation features eliminated TPA loss, a 2.2–2.5 µm optical window for SOI can be considered to offer improved device performance.
In general, the dynamic range of IMWP circuits are limited due to the nonlinearities in the modulator. As modulators become more linear, the spurious signals will be reduced and allow these circuits to perform at higher power, in which the nonlinear effects studied in this work become more pronounced. Therefore, it is important to consider what impact TPA could have on IMWP components toward future device design.
Funding
Air Force Office of Scientific Research (FA9550-18-1-0361, FA9550-21-1-0347).
Acknowledgments
Dr. Wei Du appreciates support from Provost's Research & Scholarship Fund at Wilkes University.
Disclosures
The authors declare no conflicts 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 authors upon reasonable request.
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