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Abstract
The energy efficiency of silicon nitride (SiN) microresonator-based frequency combs is investigated using numerical simulations. The formation of stable frequency comb solutions is numerically studied for a resonator with properly engineered dispersion. The required conditions for the pump power and frequency detuning are analyzed to obtain modulation instability (MI-comb), single soliton, multi-soliton, and soliton-crystal. The performance of the micro combs in terms of the power conversion efficiency, the number of generated comb lines, and line spacing required for different applications of the frequency combs are investigated. Our numerical simulations, based on the parameters of the studied microring, show a configurable multi-wavelength source with up to 23% power conversion efficiency, 200 comb-lines with more than -10dBm power, and switchable line spacing from 100 GHz to 3THz is feasible for different regimes of the proposed frequency comb. The efficiency sensitivity to the input pump power is also studied to find the most energy-efficient operation regime for the multi-wavelength source.
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1. Introduction
Kerr frequency combs are intensely studied for numerous applications such as optical logic [1], quantum sensing and timing [2], dimension metrology [3], spectroscopy [4], and microwave photonics [5,6]. Conventionally, mode-locked lasers have been used to generate frequency combs, where an ultra-short pulse is propagating inside the laser cavity. However, new methods for frequency comb generation have been used by introducing chip-scale resonators as the cavity. One of these methods is based on parametric frequency conversion in high-Q micro-resonators, where a continuous wave (CW) pump laser is coupled into the cavity and converted into hundreds of comb lines via parametric four-wave mixing [1]. These methods are characterized by their large mode spacing, compactness, and independence from broadband laser gain media or saturable absorbers.
One of the most interesting features of micro-resonator comb generators is that they can provide ultra-short pulses connected to the formation of dissipative Kerr solitons (DKS) in micro-resonators with anomalous dispersion [7]. DKS forms spontaneously when the CW pump laser is scanned through the effective tuning of the micro-resonator resonance into the red-detuned regime. This laser tuning technique allows reliably generating soliton states, such as multi-soliton state (MS) state [8–15], and soliton-crystal (SC) state [8,16,17]. Other primarily comb states are known as the MI-comb, cnoidal waves, or Turing patterns [13,14,18–21]. Each of these regimes can be useful for a different application. Therefore, a study of the efficiency of the comb in these regimes using experimental parameters is helpful from a system/circuit-level design point of view.
For the optical spectra of formed DKS, the high repetition rate and better power efficiency of the comb lines are important factors. Power conversion quantifies how much power is converted from the CW pump laser into new comb lines [22]. The value of conversion efficiency is essential for specific applications where the comb lines are used as a wavelength source to process electrical signals, such as fiber telecommunications [23], radio over fiber systems [24], and radiofrequency photonic filters [25]. For example, in a photonic delay-line filter, the RF modulation is applied to discrete optical frequency samples, and time-delayed, mode-locked fiber lasers have been used as multi-wavelength sources [26]. However, micro-combs have been recently reported as a proper candidate for microwave photonics because of their high efficiency, small feature size, and low cost [27]. To evaluate the performance of such a source for microwave photonics, one of the main parameters is the number of generated comb lines and the power of each of them. Also, the spacing of lines and the coherency of the spectra are essential factors for many applications. The frequency comb used in a wavelength-division multiplexed system needs spectrums with high conversion efficiency and line uniformity [28,29]. Accordingly, an energy-efficient and configurable comb-source, which enables the formation of different rigorous patterns, is a desirable source that can be used for several applications. The comb conversion efficiency of the DKS for anomalous dispersion and dark pulse for normal dispersion has been discussed earlier [22,30,31]. The conversion efficiency of the Kerr comb has been studied based on analytical solutions for the intracavity field [32–34]. Moreover, the effect of FSR and coupling on the conversion efficiency has been analytically, numerically, and experimentally studied in [33].
Figure 1 represents the schematic of different comb states concerning the optical spectrum. It is shown that different states provide other features in terms of line spacing, the flatness of the spectrum, and conversion efficiency. Targeting application suggests the appropriate form of the comb.
Fig. 1. Schematic of a conventional detuning technique to generate different comb states in a SiN anomalous dispersion ring. Three different states are numerically simulated: MI-comb, single soliton, and multi-soliton. The various features of optical spectrum in terms of line spacing, flatness, and conversion efficiency might be different.
In this paper, we plan to numerically study and compare the performance of various comb forms for an anomalous dispersion SiN microcavity to provide guidelines for the study and design of the proper comb regime in different applications. The formation of two rigorous forms of Kerr frequency combs is modeled: multi-solitons (MS) and soliton-crystal (SC). The techniques and dynamics needed to generate these comb-line sources will be discussed in section 2. The configurable switching between the comb-lines, the required conditions to obtain stable forms of comb-lines, e.g., MS and SC, and the configurable transitions between different states such as MI-comb and soliton switching, will be explained in section 2. In section 2, the simulation method, as well as the required parameters, are described. Section 3 studies the dynamics of both MS and SC for different laser pump power. Finally, the spectrum of these comb patterns in terms of power conversion efficiency and the number of comb lines are compared in section 4.
2. Model description
DKS formation can be modeled using coupled-mode equations (CMEs), which describe the slow evolution of each comb line of the frequency comb, with each comb mode µ. The results of CMEs are similar to the Lugiato Lefever equation (LLE) when higher order of dispersions than second-order dispersion can be neglected [35]. These equations may be written dimensionless, where the time dependence in the nonlinear terms is removed. Our numerical model is based on these dimensionless CMEs and is modified to consider thermal effects. For this purpose, an additional equation for the normalized temperature variation is solved simultaneously with CMEs. Therefore, the set of equations implemented for numerical simulations can be written as [36]:
To obtain Kerr solitons, the laser tuning technique was developed as an effective method. The CW pump laser is tuned over the microring resonance from the blue-detuned regime to the red-detuned region, known as “forward tuning.” This mechanism results in a triangular trace in the generated comb light over the pump frequency detuning. When the pump enters the red-detuned regime, multiple intracavity solitons can be formed [8]. The generated MS state is accompanied by a step-like trace, where its height corresponds to the number of solitons created inside the micro-resonator. However, single-soliton conditions become rarely accessible with the forward tuning technique [36]. An additional laser tuning towards the blue detuned regime, known as “backward tuning,” provides a way to access single-soliton states starting from arbitrary MS states. This backward tuning method allows for successive extinction of intracavity solitons (soliton switching) down to the single-soliton state [36], which results in a staircase trace. Each switching between MS states occurs with the extinction of one soliton at a time; this successive soliton switching in the backward tuning is attributed to the thermal nonlinearity of micro-resonators [36]. Due to material absorption, the intracavity field thermally shifts the micro-resonator resonance because of the thermal expansion and the thermo-optical effect on the refractive index change.
On the other hand, equally spaced solitons can also form inside the micro-resonator, known as perfect soliton crystal (PSC) [16]. Their frequency combs are characterized by strongly enhanced comb lines separated by multiple free spectral ranges (FSRs). The formation of these structures is linked to the presence of avoided mode crossings (AMXs) [16,37–39], which through localized spectral perturbations of the microring dispersion, change the optical spectrum of a DKS state and lead to the order of DKSs in a crystal-like structure. PSC is the most straightforward and ideal representative of crystal-like patterns, which its behavior is unperturbed by missing or shifted pulses [17]. However, PSC generation strongly depends on the pump power input value.
In order to enable the formation of PSC, the set of coupled-mode equations was perturbated by introducing an additional detuning change for a certain mode number $\xi $, such that the detuning value for that comb line is $\delta {\omega _\xi } = {\omega _\xi } - {\omega _p} = ({{\omega_0} + {D_1}\xi + \mathrm{\Delta }} )- {\omega _p}$ [17], where $\mathrm{\Delta }$ accounts for the mode shifting due to the impact of the spectrally localized avoided modal crossing (AMX) and ${D_1}$ is the FSR in radians per second.
Regarding the intracavity field propagating inside the micro-resonator, the coupled-in power decays due to the absorption and scattering loss. The remaining power is coupled out into the waveguide. Another fraction of the pump power is converted to other comb lines. At the output of the micro-resonator, the power present in the other comb lines, excluding the pump, constitutes the usable comb power. The power conversion efficiency, as a result of this process, is written as follows [22]:
where $P_{pump}^{in}$ is the pump power at the input of bus-waveguide and $P_{other}^{out}$ is the power of the generated comb lines, except the pump line, at the output bus-waveguide.For our numerical simulations, we consider the following parameters, which corresponds to a SiN micro-resonator similar to [36]: $\lambda = 1.553\; \mu m$, ${n_0} = 2.4$, ${n_2} = 2.4x{10^{19}}\; {m^2}/W$, ${V_{eff}} = 1x{10^{15}}\; {m^3}$, $\kappa = 300\; MHz$, ${D_2} = 2.5\; MHz$, $\eta = 0.5$, and the ratio ${n_{2T}}/{n_2} = 10$. We considered a Gaussian noise for the input. Higher order of dispersion, Raman effect, and mode overlaps have been neglected in these simulations. The values for backward- and forward thermal relaxation times (τT) were rescaled small enough to simulate the case similar to [36] effectively. Table 1 shows the parameters and initial conditions considered for the modeling.
Table 1. The considered parameters in modeling the dynamics of the SiN microring.
3. Dynamics of intracavity field formation
This section represents the formation of intracavity fields and corresponding spectra for both MS and PSC in the modeled microring described in the previous section.
3.1 Multi-soliton
To obtain the MS state for the microring with the mentioned parameters, a pump-cavity detuning technique is required in a range of 1-3W. As shown in Fig. 2(a,b), a forward tuning of the laser pump, with a speed of 5 GHz/µs, has been performed for 1W and 3W of pump power. Figure 2(c,d) shows the dynamic evolution of the intracavity field from laser pump, MI-comb (Turing pattern), chaos transition, and finally to an MS state. The formation of 12- and 14 solitons for the input power of 1W and 3W, respectively, is shown. These solitons show an erratic localization along the microcavity. The position of each soliton randomly varies for each simulation run. This behavior is also experimentally observed as both laser sources and EDFAs suffer from different noise sources [36]. As soon as the MS state is formed and the laser is locked, the position of the solitons remains stable. At this point, a relatively slow (0.2 GHz/µs) backward detuning results in a dwindling number of solitons, also known as soliton switching. Figure 2(e,f) demonstrates the effect of backward detuning results in the reduction of solitons down to a single-soliton state. The staircase behavior of the intracavity intensity in a backward detuning represents a step-by-step drop in the number of solitons. If the modulation speed is slow enough, the number of stairs will equal the number of solitons. In contrast, fast backward detuning results in skipping some intermediate steps.
Fig. 2. Average intracavity intensity versus the forward and backward laser tuning for power of 1W (a) and 3W (b). The transient formation of multi-soliton state via forward tuning of the laser for a power of 1W (c) and 3W (d). The number of solitons via backward tuning of the laser for power of 1W (e) and 3W (f). The backward and forward modulation speed of the laser is represented in (a) and (b). The different states of the intracavity field at their corresponding modulation frequency/time is demonstrated (c-f). The continuous dynamics visualization is represented in (see Visualization 1).
3.2 Soliton-crystal
In contrast to the random pattern of MS, with controlled adiabatic pump-laser frequency scans, SC are predictable and, if perfectly crystalized, equally separated along the microcavity. Moreover, they can be obtained at lower pump power than the power required for an MS state. It is shown earlier [17] that there is a threshold for the laser pump power, above which the PSC state loses its uniform shape and turns to an MS state or turns into a defected soliton-crystal [16]. As a result, the required power to obtain the soliton-crystal field in a microcavity is less than the power needed for a multi-soliton state. Concerning our simulations, the power range of 0.3 W to 0.5 W is where the PSC can appear. The dynamic formation of PSC at a pump power of 0.3 W and 0.5 W is shown in Fig. 3(a, c, and e) and Fig. 3(b, d, and f), respectively. Like the MS formation, the evolution of PSC from the laser pump experiences the same states of transition; MI-comb and chaos. As shown in Fig. 3(a,b), the modulation speed of the laser detuning is 5 GHz/µs. However, increasing the pump power and considering similar modulation speed requires much time for laser detuning to obtain the same PSC state. For example, a 15-PSC with the input power of 0.3 W occurs at the modulation time of 0.96 µs (Fig. 3(c)), which is 1.2 µs for the pump power of 0.5 W (Fig. 3(d)). Comparing Fig. 3(a) and Fig. 3(b), more chaos-fluctuations occur with increased pump power, extending the chaos transition to provide an ordered initial condition for the formation of PSC. Thus, larger laser power requires a more extended frequency scan to reach the PSC state. Moreover, increasing the pump power shortens the primary comb state. Consequently, MI-combs turn earlier into chaos regimes before they could experience a power enhancement. As a result, MI-combs at the power of 0.3 W (Fig. 3(c,e)) is almost two times stronger than the ones at a power of 0.5 W (Fig. 3(d,f)).
Fig. 3. Average intracavity intensity versus the forward laser tuning for power of 0.3 W (a) and 0.5 W (b). The transient formation of soliton-crystal state including 15 equal distanced solitons at the pump power of 0.3 W (c) and 0.5 W (d). The transient formation of soliton-crystal state including 12 equally distanced solitons at the pump power of 0.3 W (e) and 0.5 W (f). The forward modulation speed of the laser is represented in (a) and (b). The different states of the intracavity field at their corresponding modulation frequency/time is demonstrated (c-f). The continuous dynamics visualization is represented in (see Visualization 2).
As discussed in section 2, the various resonant frequency shifts, which lead to localized perturbations of the micro-resonator dispersion, define the order of PSC. Such transition includes the arising of collective solitons and the elimination of destructive ones, which leads to a microcavity of fully occupied enhanced solitons. This phenomenon can be modeled by considering AMXs in the numerical simulation. Besides, it can represent the order of crystallization (number of solitons). The dynamic of 12- and 15-crystallization is shown in Fig. 3(e,f) and Fig. 3(c,d), respectively.
4. Spectrum efficiency analysis
In this section, we discuss the efficiency of the generated patterns for the introduced micro comb states in section 3. The first consideration is the power conversion efficiency, as shown in Eq.(5). Besides, the number of generated comb lines above different power levels and spacing of the combs are reported. A summary of this information for an MS system is illustrated in Fig. 4. The simulation results show that the spectra formed at the MS state provide the highest conversion efficiency (7.57% to 14.19%). As the backward detuning occurs, leading to a staircase reduction of the number of solitons, the conversion efficiency drops to 0.55%. MI-combs can have an efficiency of up to 5%. The number of comb-lines with power more than 5, 0, and -10dBm is calculated. Since the spectra of MI-combs are not a broad pattern in frequency (Fig. 4(a,b)), a few comb-lines are obtained, especially lines with high power.
Fig. 4. Optical spectrum for the comb-states in a multi-soliton approach. (a) and (b) represent the spectrum during a forward detuning at powers of 1 and 3W, respectively. Similarly, backward detuning scenarios are represented in (c) and (d). For all shown states, the power conversion efficiency is illustrated by power percentage of the pump versus the power percentage of the generated comb. The front wall on each plot represents the number of generated comb lines with a power of above 5, 0, and -10 dBm versus the laser modulation frequency. The comb spacing of the MI-combs in (a) and (b) is equal to 25× and 30×FSR and the comb spacing of the states in (c) and (d) are equal to the number of soliton(s) of the state times FSR. The continuous dynamics visualization is represented in (see Visualization 1).
In contrast, the MS spectrum is an enhanced broad pattern with almost 200 comb-lines with a power of more than -10dBm. In a backward detuning by a reduction in the number of solitons, the spectrum became gradually compact and as a consequence, the number of high-power comb-lines reduces. As a result, a single soliton spectrum exhibits no line above the power of 0dBm.
The spacing of the lines in the MI-comb is 25×FSR at the lowest simulated laser pump power (Fig. 4(a)) and 30×FSR at the largest (Fig. 4(b)). For some applications which require a large spectrum zone, MI-combs are a useful source in terms of the large spacing of the comb-lines. This spacing can be increased by applying larger laser pump power. However, the conversion efficiency of the comb pattern drops by increasing the pump power. The simulation results in Fig. 4 offer a high configurable multi-wavelength source that enables switching between states, in which comb-lines above -10dBm can vary from a few up to 200 and the spacing between 100 GHz to 3THz.
Similar features have been investigated for PSC simulations. Figure 5 illustrates the spectra characterization for different comb states while generating the PSC. Contrary to the MS spectrum, in the formation of the PSC, each state presents a higher power conversion efficiency of up to 23% (Fig. 5(a)). The spectra of PSC are similar to MI-combs but with higher side mode suppression and comb-lines with an enhanced power close to the CW background. The spacing of the combs is equal to the order of crystallization times FSR. Which is 15×FSR in Fig. 5(a,b) and 12×FSR in Fig. 5(c,d).
Fig. 5. Optical spectrum for the comb-states in soliton-crystal approach. (a) and (b) represent the spectrum during forward detuning for a 15-soliton-crystal at powers of 0.3 and 0.5 W, respectively. Similarly, for a 12-soliton-crystal the spectrum is represented in (c) and (d). For all shown states, the power conversion efficiency is illustrated by power of the pump versus the power percentage of the generated comb. The front wall on each plot represents the number of generated comb lines with a power of above 5, 0, and -10 dBm versus the laser modulation frequency. The comb spacing of the MI-combs in (a) and (c) are equal to 19×FSR and in (b) and (d) is 21×FSR. At soliton-crystal, the spacing of the lines is related to the order of crystallization which is 12× and 15×FSR. The continuous dynamics visualization is represented in (see Visualization 2).
In the formation of PSC, the calculated number of comb lines above the reported power is less than half of the ones reported for the spectra of MS dynamics. Because the shaped states in the formation of PSC, e.g., MI-comb and chaos, are much more coherent and less broad than the formation of MS. The number of comb-lines above -10dBm in 15-PSC is below 70 (Fig. 5(a,b)) and as the order of crystallization reduces, the number of lines drops significantly. For instance, in the 12-PSC state, the spectrum can hardly produce 10 lines (Fig. 5(c,d)). As shown in Fig. 5, the MI-combs also provide higher comb conversion efficiency (9.7% to 16%). However, the comb spacing is less than those in the MS case, which is 19×FSR and 21×FSR at a pump power of 0.3W and 0.5W, respectively.
To understand the impact of laser pump power, we have simulated the same microring by considering different pump powers. The summarized results in Fig. 6 illustrate this effect on comb power conversion efficiency and the number of comb lines with a power of more than 0dBm. It is shown for MS and PSC states that the conversion efficiency decreases as the pump power increases (Fig. 6(a,b)). The conversion efficiency of the MI-comb, however, shows approximately an oscillatory response to the pump power, particularly at lower power and among the formation of PSC (Fig. 6(d)). Therefore, an increase in pump power does not necessarily lead to obtaining higher comb power efficiency and there is a trade-off in choosing the power which has the highest conversion efficiency. To compare the numerical results with the analytical approach such as in [34], we recalculate the conversion efficiency using the analytical solution for the intracavity field. The power efficiency when the pumped line is excluded can be calculated analytically as:
For a single soliton state, the results show the same trend of conversion efficiency versus pump power sweep; however, the level of conversion efficiency in the analytical approach is 0.8% higher than our numerical results. This gap might be due to the thermal effect considered in the numerical method and/or the initial noise defined for the pump field.
Fig. 6. The power conversion efficiency (solid lines) and number of lines with a power of more than 0dBm (triangle) versus pump power in multi-soliton (a), soliton-crystal (b) MI-comb for the formation of multi-soliton (c), and MI-comb for the formation of soliton-crystal dynamics (d) states.
The number of comb lines with a power of more than 0dBm can go up to 50 for an MS state. In contrast, this value is much less for MI-comb and PSC (Table 2). This is due to the coherency and broadened of the optical spectrum of these different comb states.
Table 2. A summary of obtained line spacing, conversion efficiency, and the number of lines for different studied comb states. The results are based on a microring with parameters detailed in Table 1.
5. Conclusion
For a SiN microring resonator with anomalous dispersion waveguide, the formation of various Kerr comb states i.e., MI-comb, multi-soliton (MS), and perfect soliton-crystal (PSC) have been simulated and compared to provide guidelines for the designers to select the proper regime for their practical applications. The critical conditions for pump power and wavelength tuning in different regimes have been studied. Three main features of the spectrum of the micro combs: power conversion efficiency, number of generated comb lines, and the spacing of the comb lines are investigated to find practical values for the operation of the combs. The studied microring in this work can be used as a configurable multi-wavelength source with up to 23% power conversion efficiency in the PSC regime, 200 comb-lines with more than -10dBm power in MS regime, and switchable line spacing from 100 GHz to 3THz in MI-comb regimes. Although, achieving higher conversion efficiency is possible by optimized engineering of the microring. It has been shown that a higher pump power does not necessarily provide a comb pattern with higher conversion efficiency. The trade-off between the parameters which need to be optimized to achieve higher energy-efficient systems is investigated.
Funding
Deutsche Forschungsgemeinschaft (JA 2401/11).
Acknowledgment
This work is supported (in part) by the German Research Foundation (DFG) within the project: Silicon-on-Insulator based Integrated Optical Frequency Combs.
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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