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Abstract
High index doped silica glass exhibiting low loss property and CMOS compatibility is a promising material in nonlinear optics. In this work, mode-locked soliton crystals (SCs) are demonstrated in a high-Q (>106) micro-ring resonator (MRR) made in this platform. The asymmetric spectra of SCs are numerically investigated and interpreted as the combined impact of Raman self-frequency shift (RSFS) and the wavelength-dependent loss. By precisely comparing the experimental and simulated spectra based on the perturbed Lugiato-Lefever equation (LLE), the Raman shock time is inferred to be at the range of 2.5 fs to 2.7 fs for this material.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Benefiting from the breakthrough in theory and fabrication processing over the last two decades [1,2], optical micro-cavities with Q-factor as high as 1011 have been demonstrated [3]. The photon-storage time in such cavities corresponds to hundreds of microseconds, which induces ultra-high optical intensity [4] due to constructive interference inside the cavities and stimulates intriguing nonlinear effects in such system. For the centrosymmetric materials such as Si, Si3N4, and SiO2, the 3rd order nonlinear effect including self-phase modulation (SPM), cross-phase modulation (XPM) and four wave mixing (FWM) [5], are introduced due to the cavity-power-driven field-enhancement effect [6]. These nonlinear effects have been exploited in chirality [7,8], Raman laser [9] and frequency conversion [10]. Recently, researchers’ interests have been attracted by another iconic nonlinear phenomena in high-Q micro-cavity, that is, the on-chip soliton micro-comb [11–17], which has been successfully utilized in microwave generation [18], ranging measurement [19,20], dual-comb spectroscopy [21,22], and coherent communication [23,24] with unprecedented precision. Theoretically, dissipative cavity solitons (DCSs) rely on the double balance between anomalous cavity dispersion and Kerr nonlinearity on the one hand, the cavity loss and parametric gain on the other. Meanwhile, the high peak power of such solitons will stimulate high order nonlinear effect, such as Raman effect, which has been demonstrated in former DCS generation experiment [25,26], where DCSs experience the so-called Raman self-frequency shift (RSFS), resulting in asymmetric power-spectra distribution about the pump. Phenomenally, the maximum of spectra is red-shifted away from the pump by a certain amount, which is mainly dependent on the pulse-width, group velocity dispersion (GVD) and material-associated Raman shock time [25].
In this paper, the RSFS effect on soliton crystals (SCs) is numerically and experimentally investigated. The SCs are generated in a high-Q micro-ring resonator (MRR) fabricated on a high-index doped silica glass platform, using thermal-controlled method [27], the spectra exhibiting obvious asymmetric characteristic are obtained. Such asymmetry is caused by the combined contribution from wavelength-dependent loss and RSFS. In order to precisely determine the Raman shock time for this material, the loss term is measured and taken into account in our theoretical model based on the perturbed Lugiato-Lefever equation (LLE). By comparing couples of experimental and simulated optical spectra, Raman shock time between 2.5 fs to 2.7 fs is inferred for this material. This result contributes to the understanding of Kerr frequency combs in the solitons regimes and is relevant to improve the accuracy of numerical simulations.
2. Devices
Figure 1(a) shows the schematic of the add-drop type MRR made of high index doped silica glass (n=1.6) using SiO2 as cladding. The MRR is fabricated by CMOS-compatible process, the core films are deposited using plasma enhanced chemical vapor deposition (PECVD), and then the device patterns are printed in photoresist using an in-line stepper and reactive ion etching to produce exceptional low sidewall roughness on the core layer [28–31]. The radius of the MRR is 592.1µm , corresponding to the free spectral range (FSR) of ~49 GHz, and the gap between the MRR and the bus waveguide is 1µm. Figure 1(b) is the scanning electron microscope (SEM) image of the cross section of the MRR, the ring and the bus waveguide are both 2µm ×3 µm. A mode transformer (MT) structure is added to in-out port of the MRR and the coupling loss is ~2.5 dB per facet at 1550 nm. Figure 1(c) shows the measured transmission loss of the waveguide and the calculated fundamental transverse magnetic (TM) optical field which is expected to be the SC-forming mode is shown in Fig. 1(d).
Fig. 1 (a) Schematic illustration of add-drop type MRR. (b) SEM image of the cross section of the MRR. (c) Measured loss parameter versus wavelength. (d) Calculated fundamental TM mode used for generating SC.
Next, the major parameters of the MRR are calculated and analyzed based on the aforementioned parameters. Typically, the deviation of cavity resonance from equidistant spacing can be expressed as [32,33]:
where ωμ and ω0 are resonant and pump angular frequency, μ is the relative mode order. D1/(2π) corresponds to the FSR, D2 = -c/n0D12β2 illustrates the variation between adjacent FSRs, β2 is GVD value, c and n0 are light velocity in vacuum and linear refractive index of this material, respectively. D3 is third order dispersion (TOD) parameter and Dk (k>3) is higher order dispersion (HOD) coefficients. For our device, D2/(2π) is 175 kHz, the Dint curve shown in Fig. 2(a) exhibiting perfect parabolic feature suggests that TOD and HOD can be neglected in this work. Mode interaction between SC-forming mode and interaction-expected mode 2nd transverse electric (TE) mode (90 × FSR) is also observed from the calculations, which is roughly consistent with previous experimental demonstrations (102 × FSR) [27] and the difference between calculation and experiment is expected to be attributed to thermal effects. Another essential parameter used in subsequent simulation is the nonlinear coefficient: γ = ω0n2/(cAeff), where n2 = 1.15×10-19 m2/W [28] is the nonlinear refractive index, Aeff = [∫∫|E(x,y)|2dxdy]2/[∫∫|E(x,y)|4dxdy] is effective mode area with E(x,y) of mode field distribution. The effective mode area, loaded Q, FSR and group index are simulated and presented in Fig. 2(c) and Fig. 2(d), respectively.Fig. 2 Calculated major parameters of the MRR. (a) Integrated dispersion curve at pump wavelength (1556.3 nm in experiment), exhibiting perfect parabolic characteristic. The intersection curve belonging to 2nd TE mode induces mode interaction. Insets: calculated optical field of SC-forming and interaction mode. (b) GVD and D2 curves. (c) Effective mode area and loaded Q value of the MRR. (d) FSR and group refractive group index.
3. Experiments and results
The experimental set-up is shown in Fig. 3. The pump laser is an external cavity diode laser (ECDL) with the linewidth of ~100 kHz at a fixed wavelength of 1556.3nm and boosted by a commercial erbium doped fiber amplifier (EDFA). A fiber polarization controller (FPC) controls the polarization states inside the resonator. The monolithic MRR is pigtailed with a standard fiber array and packaged in a 14-pins butterfly package [27]. The optical spectrum analyzer (OSA) and power meter directly monitors the spectrum and energy inside the MRR. A high precision thermoelectric cooler (TEC) is used to tune the resonance wavelength using the thermo-optic effect.
Fig. 3 Experiment set-up of the SC. ECDL: external cavity diode laser. EDFA: erbium doped fiber amplifier. FPC: fiber polarization controller. MRR: micro-ring resonator. TEC: thermos-electric cooler. OSA: optical spectrum analyzer. PM: power meter.
With proper polarization state and pump power (~31.7 dBm in the experiment), primary comb (Turing pattern), chaotic state comb (modulation instability comb) and mode-locked SCs are generated in sequence, as shown in Fig. 4. The case of SC with one vacancy [27] is presented in Fig. 4(c), which presents asymmetric spectrum about the pump from the sech2 fit line shown in orange. Apparently, the amount of shift could be measured by the difference of center frequency of SC spectrum and the pump frequency. However, for most of SC spectra, it is not precise to determine the central wavelength because the defects caused by special distributions of soliton ensembles keeps the spectrum bumpy and uneven. Fortunately, the power difference between symmetric “primary like” comb line with dominant power can be taken as a reference to speculate RSFS effects. Next, the Raman shock time will be analyzed through comparing the theoretical and experimental spectra.
4. Model and simulation
The theoretical model to research on the frequency-shift is based on the damped, driven LLE equation augmented by mode-crossing and Raman terms [34,35]:
where E(t,τ) is the complex slowly varying amplitude of the intra-cavity field with t and τ correspond to slow and fast time evolution of the field, respectively. tR and α are the round-trip time and intra-cavity losses shown in Fig. 1(c). κ corresponds to power transmission coefficient of the input coupler. δ is the cavity detuning between pump and corresponding resonance. Ein is the external pump term and L is the total cavity length. R(τ) is the nonlinear response of high index doped silica, including Kerr and Raman response [5]. Since the soliton pulse in our experiment is measured to be ~ 120 fs [27] (simulated under the perfect SC case), which is larger than 100 fs [5], thus the Taylor expansion and necessary normalized condition [25] can be applied to simplify the last term in Eq. (2) as:where the first term is Kerr effect, and the second term containing Raman shock time τR causes the RSFS. Mode-crossing effect is included by adding a phase shift Φω = a/(ω-ωl) to the perturbed modes [36], where a reflects the strength of the interaction and ωl is the interaction position which locates at around 1517 nm (102 × FSR away from the pump) [27], so the interaction is imposed on the 102nd comb line away from the pump wavelength. Numerical simulation is realized via split-step Fourier algorithm [37] and fourth-order Runge-Kutta method [38]. The wavelength-dependent loss effects are taken into account within the Fourier term during simulation.In order to stably simulate SCs of our experiment, Gaussian pulse distributed in specific positions determined by experimental spectra is chosen to be the initial condition [39]. During the simulation, there are two critical variables, δ and τR. After extensive calculations, δ between 0.015 and 0.0188, τR ranging from 2.3 fs to 3.1 fs are effective and consistent with the experimental spectra. Values beyond these ranges are proved to be either unstable or inconsistent with the experiments, for example, Figs. 5(a) and 5(b), Figs. 5(c) and 5(d) show the unstable (δ = 0.018, τR = 2.55 fs) and stable crystal (δ = 0.013, τR = 2.55 fs) evolution in time and frequency domain with aforementioned parameters, respectively.
Fig. 5 SC evolution within different condition. Unstable state in (a) time domain and (b) frequency domain with δ = 0.013 and τR = 2.55 fs. Stable state in (c-d) shows robust evolution with δ = 0.018 and τR = 2.55 fs.
Figure 6 shows the simulation results of 3 different SCs. For each spectrum, different values of τR and δ are used to numerically calculate the perturbed LLE, then the power difference between mode + 34 and −34 (first pair of symmetric “prime-comb” modes about the pump) are accumulated. The right column of Fig. 6 presents the power difference variation with different combinations of τR and δ, the horizontal dashed lines are their corresponding experimental power difference values. The simulated curve is not effective unless it has intersection points with the dashed line. Note that τR and δ beyond the calculation ranges are unstable in the simulation. Figure 6(c) shows that when 2.5 fs < τR < 3.1 fs, there are intersections between simulated and experimental spectra. While the range of τR is 2.3 fs to 2.5 fs for Fig. 6(f) and 2.3 fs to 2.9 fs for Fig. 6(i). According to these results, 2.5 fs < τR <2.7 fs is effective for all the three cases. In further simulation, τR = 2.55 fs is selected for Figs. 6(b), 6(e) and 6(h), with varying δ values between the range mentioned above, all the simulated spectra are in good agreement with experiments.
Fig. 6 Experimental (first column) and Simulated spectra (the middle column) with Raman interaction, τR = 2.55 fs. (a)-(c) Experimental and simulated SC spectra with “one-gap-peak” feature, experimental power difference is 3.4 dB, δ = 0.0182. (d-f) Spectra present “two-peaks”, power difference is 3.11 dB, δ = 0.015. (g-i) Spectra with “Stack” feature with power difference of 3.31 dB and δ of 0.0170. Horizontal dashed lines in (c), (f) and (i) are the corresponding power difference from experiments.
5. Conclusion
In conclusion, the RSFS of SCs in a high-Q MRR made of high index doped silica is demonstrated. It is precisely researched that the strongly wavelength-dependent loss and RSFS jointly contribute to the asymmetric spectra about the pump. Raman shock time at the range of 2.5 fs to 2.7 fs is inferred by comparing the experiments and the simulations. This work not only improves the modeling accuracy for this material, but also gives reasonable prediction for RSFS on single-soliton that will be researched in the near future.
Funding
National Natural Science Foundation of China (Grant No. 61635013; 61675231; 61475188; and 61705257) and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB24030600).
Acknowledgments
The authors thank T. T. Yu, L. Wang and P. X. Chen for help in the device design.
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