Deterministic single soliton generation and compression in microring resonators avoiding the chaotic region

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1 Deterministic single soliton generation and compression in microring resonators avoiding the chaotic region Abstract Jose A. Jaramillo-Villegas,,3,* Xiaoxiao Xue, Pei-Hsun Wang, Daniel E. Leaird, and Andrew M. Weiner, School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 797, USA Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 797, USA 3 Facultad de Ingenierías, Universidad Tecnológica de Pereira, Pereira, Risaralda 3, Colombia * jjv@purdue.edu A path within the parameter space of phase detuning and pump power is demonstrated in order to obtain a single cavity soliton (CS) with certainty in SiN microring resonators in the anomalous dispersion regime. Once the single CS state is reached, it is possible to continue a path to compress it, broadening the corresponding single FSR frequency Kerr comb. This behavior is first obtained by identifying the regions in the parameter space via numerical simulations of the Lugiato-Lefever equation (LLE), and second, defining a path from the stable modulation instability (SMI) region to the stable cavity solitons (SCS) region avoiding the chaotic and unstable regions.. Introduction An optical frequency comb is a light source with a number of highly resolved and nearly equidistant spectral lines. Since its introduction, multiple important applications have been demonstrated in areas such as communications, metrology, spectroscopy, astronomy and optical clocks. Optical frequency combs can be generated using mode-locked lasers or electro-optic modulation of continuous-wave light. Since 7, multiple experiments have reported optical frequency comb generation by means of Kerr nonlinearity (wave mixing processes) in microresonators, which offer potential for highly compact and portable solutions. Such combs are termed Kerr optical frequency combs or simply Kerr combs [-]. The understanding of underlying processes and dynamics in Kerr comb generation is critical to move this technology further to industry and applications. One of the first simulation approaches was introduced by Chembo and Yu in [] using a modal expansion approach. More recently, the Lugiato-Lefever equation (LLE) [3] has been widely adopted [-]. The LLE, eq., is the nonlinear Schrödinger equation (NLSE) with damping, detuning and external pumping, which accurately describes Kerr comb generation. Furthermore, Hansson et al. [] demonstrated the equivalence between the LLE and mode coupling equations models. t R E(t,τ ) t " = $ α iδ + il # $ k β k k! i k! $ % # & + iγ L E ' " τ % & ' E + θ E () in In Eq. E(t,τ) is the complex envelope of the total intracavity field and hereafter simply the field, t is the so called slow time variable, τ is the fast time variable, t R is the round trip time, α is the loss per round trip, δ is the phase detuning, L is the cavity length, β k is the k- order

2 dispersion coefficient, γ is the Kerr coefficient, θ is the coupling coefficient between the waveguide and the microresonator and E in is the pump field (normalized such that P in = E in ). Recently, some authors explored and identified regions corresponding to different types of operation in the phase detuning and pump power (δ,p in ) parameter space in the anomalous dispersion regime [9, ]. The characterized regions are: stable modulation instability (SMI), unstable modulation instability (UMI), stable cavity solitons (SCS), unstable cavity solitons (UCS) and continuous wave (CW). Additionally, Erkintalo and Coen [3] explored the coherence properties of each region.. Single CS generation and compression (a) UMI UCS SCS CW N Peaks (b) U Intra a.u.. δ δ f..3 3 δ f UMI UCS SCS...3 t [µs] (c) Counts N Peaks δ f =. (a) δ (d) N Peaks δ f 3 (e) P in...3 t [ µs] δ Single CS δ f..3 δ f 3 (f) Counts N Peaks δ f =. (d) Fig.. The number of peaks as a function of final value of phase detuning δ f with δ swept as shown in the insets with pump power P in (a) set to a constant value of mw and (d) adjusted through the avoiding chaotic region function (ACRE), Eq., with P in =, both with the same realization of initial noise for all simulations. (b) shows the total intracavity energy as a function of time for the simulation of (a) with δ f =.. Histograms of number of peaks for simulations with different realizations of initial noise with the same pump power and detuning parameters in the case of (c) for the simulation of (a) with δ f =., and (f) for the simulation of (d) with δ f =.. (e) shows a green region in which a single CS is generated when a uniform offset P in = is applied to the ACRE. Single CS generation is a desired goal in much research because it yields a high coherence, single free spectral range (FSR) Kerr comb. It has been suggested that this style of comb can be used as information carriers in optical communications and optical memories. Although the first experimental observation of a single CS was reported in [], the difficulty to control the number of CSs in the SCS region was also emphasized. In particular, in the usual approach in which detuning is swept to a predetermined final value with input power held constant, the number of CSs generated is probabilistic. This statement is corroborated in Fig. (a), which shows the number of CSs produced when the phase detuning is swept linearly from δ = to a final value of δ f over a time interval of.3 µs at a constant P in = mw. These simulations were done initializing the intracavity field in the frequency domain E(ω) with a circularly- symmetric complex Gaussian noise field with a standard deviation σ noise = 9 [W / ] which with normalization P = E is equivalent to a mean power of -5 dbm per cavity mode with uniform

3 random phase. The number of cavity solitons obtained depends very sensitively on δ f even though the noise initialization was the same for all the simulations of Fig. (a). Additionally, the number of CSs is different when we repeat the simulation with the same power and detuning parameters but different realizations of the initial noise, as shown in the histogram of Fig. (c). In the current work, we show a method not only to obtain a single CS with a high degree of certainty avoiding the chaotic and unstable regions such as UMI and UCS, but also to compress it, moving the system through the SCS to high power values. (a) P in f [mw] P in [mw] P in + - mw 3 δ f. δ f 3 (c) CW SCS UCS SMI UMI P in f[mw] δ δ f 3 P in f (b) P in f [W] δ P in f. δ f δ f (d) 3 5 δ f Fig.. The number of temporal peaks as a function of the final point in (δ,p in ) parameter space at (a) low and (b) high initial Pin. The black curves correspond to ACREs using Eq. with P in of,, and mw in (a) and compression function using Eq. 3 in (b). (c) and (d) show the regions obtained from (a) and (b) respectively. The red triangles show the initial point of the simulations. These simulations were done using the same realization of initial noise. The first step is the characterization of the (δ,p in ) parameter space through numerical simulation of LLE using the split step Fourier method (SSFM). The simulation parameters are: t R = / GHz, β =.7 s m, α =., γ =.9 W m, L = π µm and θ =., corresponding to a Si 3 N microring resonator of µm radius and a loaded quality factor Q of.7. We start the microresonator at the point (δ =,P in = mw) with a welldefined behavior in the SMI region. This point is located in the purple area shown in Fig. (a) where a Turing roll pattern of 3 equally spaced peaks is formed. The intensity and power spectrum obtained at this point are shown in Fig. 3(e). This pattern corresponds to a Kerr comb with a frequency spacing between lines equal to 3 times the microresonator FSR with a high degree of coherence [3]. After ns, we jump in a single step to a point in (δ,p in ) parameter space and keep the system there for. µs. Simulations were repeated using the same realization of initial noise for different final values of detuning and input power up to. and mw, respectively. Fig. (a) shows the number of time domain peaks for each final point. Each point (δ,p in ) is classified based on intensity, spectrum and intracavity energy stability at end of the simulation in order to determine the boundaries of the different operating regions. The result of P in f [W]....

4 this classification is shown in Fig. (c). A similar process was performed to explore behavior over a larger range of parameter space (final values of detuning and input power up to.5 and W, respectively), starting at initial point (δ =,P in =. W). The results of the simulations in this range and the classification are shown in Figs. (b) and (d), respectively. (a). ω [Prad/s]... (b) τ [ps] (c) 3 (d) t [µs] U Intra a.u. P in [mw] δ [db] a. u. (e) t =.µs (f) t =.µs 9 3 (g) 5 t =3.µs 5 E (t, τ ) a.u. τ [ps] Ẽ(t, ω) [db] ω [Prad/s] Fig. 3. Generation and compression processes of a single CS through ACRE and compression function using Eq. 3 respectively, from top to bottom: (a) spectrum, (b) intensity, (c) total intracavity energy, (d)(blue) pump power and (d)(green) phase detuning vs. slow time t. Intensity (right) and spectrum (left) at (e) t = µs, steady state initial point, (g) t =. µs, after generation and (h) t = 3. µs, after compression of a single CS. Because instantaneously jumping to a single final state in detuning-power parameter space as in the simulations of Figs. (a) and (b) is not physically realizable, we next define a path that can be followed smoothly between our starting point (δ =,P in = mw) and a target point chosen to fall within the single CS region (δ =.7,P in = mw). In particular, we select a trajectory that goes around the chaotic and unstable regions. We call such a trajectory an avoiding chaotic region function (ACRE). In this study we consider an ACRE given by the following: P in =. exp( 9 δ ) +. exp(.9 δ ) + ΔP in () The ACRE was defined using a curve-fitting tool and takes the form of a -term exponential function, Eq.. Additionally, we introduced a power offset parameter P in which allows us to describe a family of ACRE functions, portrayed by the black curves in Fig. (a). To test the performance of the ACRE we run simulations in three stages. In the first stage we dwell at the selected initial point (δ =,P in = mw) for. µs to produce a steady state Kerr comb in the SMI regime. Next, we sweep the detuning linearly from δ to a final value δ f over a.3 µs interval (7 round trip times) while varying the input power according to the ACRE function, eq.. We then continue the simulation at fixed detuning and input power for an additional µs. Fig. (d) shows the number of temporal peaks versus the final value δ f for fixed realization of initial noise. The insets illustrate the variation of δ and P in in time. The light green area depicts a

5 range of δ f values (from.59 to.9) for which a single CS is always obtained. In another case we keep δ f fixed but repeat the simulation for different realizations of initial noise. As shown in the histogram of Fig. (f), a single CS is obtained every time. Fig. (e) is similar to Fig. (d), in that we keep the noise initialization fixed and vary δ f, except that now we use different values of P in from -5 to 5 mw to the ACRE function. The light green shaded area again shows the cases where we generate a single CS. This demonstrates that the ACRE function to repeatable generation of a single CS is nonunique and can be robust over a finite range of operation. When microresonator operating mode is crossing the UMI region, the total intracavity energy U intra exhibits noisy behavior, as shown in Fig. (b) and in ref. [], respectively. Additionally, when the operating mode crosses the UCS regime, U intra undergoes periodic oscillations. Fig. 3(c) shows U intra along an ACRE trajectory leading to repeatable single CS generation and subsequent compression (see below). Neither noisy nor oscillatory behavior is seen, providing evidence that the system remains in SMI or SCS regions for all simulation time and should consequently be highly coherent [3]. Our final goal is to demonstrate compression behavior when we move the system from relatively low to relatively high detuning and power under a trajectory that remains within the SCS region. Here we simulate the comb evolution starting from (δ =.7,P in = mw), the end of a previous ACRE trajectory, to (δ =.,P in = W) using a trajectory given by P in =. exp(.53 δ ) (3) Again, this function is not unique, and a wide range of end points δ and P in lead to similar results, provided that the trajectory remains within the SCS region. Our simulation shows that the cavity soliton is compressed from 73 fs to 9 fs full width at half maximum (FWHM) pulse width; the bandwidth ω/π goes from 3.9 to. THz. Fig. 3(g) shows the final state of the intensity and spectrum after the compression process. Although we are not taking into account high order dispersion terms and other nonlinear effects (e.g. Raman) that will be present in real systems, we believe these results make clear that once a single CS state is reached compression should be possible while maintaining stability through appropriate coordinated increases in δ and P in. 3. Conclusion In this work, we have presented a novel method to generate a single CS in a highly deterministic way using a coordinated tuning of pump frequency and power through a function, which avoids the chaotic and unstable regions. The simulation results presented here could help accelerate progress towards establishing microring resonators as highly coherent and stable single FSR Kerr comb sources for practical applications. Furthermore, we have also demonstrated a way to compress the single CS by further coordinated variation of the pump parameters in the region where CSs are stable. Acknowledgments This work was supported in part by the National Science Foundation under grant ECCS-, by the Air Force Office of Scientific Research under grant FA , and by the DARPA PULSE program through grant W3P- 3-- from AMRDEC. JAJ acknowledges support by Colciencias Colombia through the Francisco Jose de Caldas Conv. 59 scholarship and Fulbright Colombia. JAJ also acknowledge fruitful discussions with Victor Torres-Company and Evgenii Narimanov.

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