1. Introduction

The discovery of dissipative Kerr solitons (DKSs) in optical microresonators attracted the research community due to its distinctive properties such as high repetition rate, low noise and low power consumption [1,2]. Furthermore, it has a compact size, and a source of broadband DKS generation allows for potential applications such as spectroscopy [3,4], telecommunications [5,6], optical clocks [7,8], ranging [9,10] and microwave generation [11,12]. In recent years, rich soliton dynamics have been explored including the dark solitons [13], dark-bright solitons [14], soliton molecules [15], Stokes solitons [16], breather solitons [17–20], breather soliton crystals [21] and perfect soliton crystals [22] in various material platforms.

Multi-mode microresonators platforms have paved the way to access broadband DKS [23–25]. Recently, we have developed the dual-mode microresonator scheme to realize adjacent modes in microcavities for simple access to single and multiple solitons [25]. Careful cavity modeling and dispersion engineering are vital to achieve the desired mode separation. In such microcavities, avoided mode crossings (AMXs) between the modes from the same or different polarization modes are inevitable [26]. Most recently, it has been demonstrated that the presence of AMXs is linked to the tuning of dispersive waves (DWs), this can significantly help to broaden the comb bandwidth [27]. In addition, the AMXs play a crucial role in the variations of the local dispersion and the modulation of the continuous wave (CW) intracavity background, which has led to the study and demonstration of breather solitons [28] and soliton crystals [22,29,30] in microresonators. In particular, defect-free soliton crystals (SCs) are of much interest due to their extraordinary characteristics such as, (i) the SCs pulses are evenly distributed over the circumference of the cavity and each enhanced pulse exhibits characteristics identical to a single soliton. (ii) Unlike a single soliton, the SCs comb lines are separated by an integer number of the free-spectral range (N × FSR) and N² enhanced comb power per line. Hence, the conversion efficiency (CE) of the SCs state is boosted, which is significant for microcomb applications including telecommunications and self-referencing.

In this work, we demonstrate the deterministic generation of an octave spanning (1200-2400 nm) various SCs via AMXs, in a single silicon nitride (Si₃N₄) microresonator. Switching the pump wavelength at relatively low power can deterministically access the desired line spacings. Furthermore, another device with a slightly different coupling gap and mode coupling position can access another set of SCs with the repetition rates of N FSR apart from the AMXs. Moreover, the modes can be effectively engineered to reliably realize the positions of mode crossing. Our approach offers a convenient technique to access SCs with various repetition rates in a single microcavity.

2. Experimental results

To realize versatile SCs, we employed a Si₃N₄ device (D1) in the experiment featuring a radius of 23.3 µm and cross-section of 0.8 × 1.7 µm² (height × width), respectively. The dual-mode device modeling strategy and fabrication details has been reported in our recent work [25]. Figure 1 shows a recorded transverse electric (TE) polarization mode transmission trace (full CW laser range, 1480-1640 nm) of device D1. A Santec multi-port meter (MPM-210), synched with the tunable laser, is employed to record the transmission spectra with an accuracy of 0.1 pm (∼ 12 MHz). From the recorded transmission trace, device D1 can support multi-modes (TE fundamental (TE₀₀) and higher order modes). The calculated intrinsic quality factor (Q_int) for the TE₀₀ pump modes (µ_p) is around one million. For simplicity, we denote the pump wavelengths and mode interaction position with relative mode numbers µ_p and µ_m (with, Δµ = µ_p - µ_m), respectively. All the pump wavelengths are marked with red rectangles and µ_p = 0 is at around 1557 nm. The mode interaction position is marked with a green rectangle and at the µ_m = -7. The zoomed-in image of the AMX position is shown in the right inset of Fig. 1.

 

Fig. 1. Measured transmission trace with TE polarization mode for device D1 at the laser wavelength range between 1480 to 1640 nm. The longitudinal pumped modes (marked with red rectangles) to obtain various SCs are denoted by relative mode numbers (µ_p) far from the AMX position (marked with green rectangle) and denoted by the relative mode number (µ_m). Inset (i) is the calculated D_int for the TE₀₀ mode as a function of relative mode numbers (µ) around the center ∼1557 nm. Inset (ii) is the zoomed-in image of the mode interaction near 1615 nm.

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The left inset of Fig. 1 shows the calculated integrated dispersion of the measured fundamental TE modes. The calculated dispersion profile of the TE mode has an anomalous dispersion i.e. (D2 > 0) with D2 = 32 MHz. The deviation (green marked region) on the dispersion curve at µ_m = -7 is related to the mode interactions.

Figure 2 shows the evolution of the SCs, we select the mode at µ_p = -1 (∼ 1556 nm) from the transmission trace of the device D1 (see Fig. 1). To study the soliton crystals existence range and suppress the pump wavelength, we used the Fiber Bragg grating (FBG) filter having a center wavelength of 1566 nm (±1 nm). The comb power trace is plotted in Fig. 2(a). The recorded comb power trace is obtained by filtering the pump and launching an on-chip-power of 255 mW and at the sweep speed of 1 nm/s.

 

Fig. 2. Experimental results of N = 6 SCs in device D1. (a) Measured comb power at an on-chip power of 255 mW. The SCs access region of 0.04 nm i.e, ∼ 5 GHz, is marked with a green rectangle. The red circular dots denote the wavelength stop position to access the three comb states i.e., Primary, MI and SCs microcombs. (b) Measured optical spectra (i) Primary comb (ii) MI comb and (iii) SCs. Insets in states (ii)-(iii) of Fig. 2(b) are the corresponding RF spectra. (c) SCs evolution map and different comb states are marked with white dashed lines (i) Primary comb (ii) MI comb and (iii) SCs with N = 6.

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From Fig. 2(a), an obvious SCs existence step (green marked region) can be observed on the comb power trace with a range of 0.04 nm (∼ 5 GHz). The red circular marked regions (i)-(iii) on the comb power trace corresponds to the comb spectra plotted in Fig. 2(b). At the laser stop wavelengths of 1565.90, 1566.10, and 1566.22 nm, we can deterministically obtain (i) primary comb, (ii) modulation instability (MI) comb and (iii) SCs spectra with the line spacing of N = 6 FSRs apart, respectively. Moreover, the characteristic signature of the SC state is a low noise radio frequency (RF) signal that we confirm with a transition from the high noise level of the MI state (ii) to the low level at the SCs state (iii), as shown in the insets of Fig. 2 (b).

Figure 3 shows the measured spectra of the versatile octave-spanning SCs with various line spacing N and is defined as N = Δµ = µ_p-µ_m, in our device (D1). To experimentally access the SCs, we use the forward tuning technique [1] and a relatively low pump power [22,31]. From Fig. 1, we observe an obvious AMX is at a relative mode number µ_m = -7 and select the pump modes far (N × FSR) from the AMX position to achieve the various SCs. We inject an on-chip power of 255 mW and pump the wavelengths at ∼1541 nm (µ_p= 2), ∼1549 nm (µ_p= 1), ∼1557 nm (µ_p= 0) and ∼1566 nm (µ_p= -1) to obtain the SCs (wavelength range, 1200-2400 nm) with line spacings of N = 7,8,7,6, respectively and depicted in Fig. 3. For the pump modes at µ_p= 1,0,-1 the obtained N = 8,7,6 SCs, i.e., at the difference of one N × FSR while we also achieved N = 7 SCs at the µ_p = 2 mode [30]. All the realized SCs spectra are deterministic and stable as a high resolution of 0.05 nm was used to record the spectra. To monitor the temporal waveforms auto-correlation (AC) measurements were performed. At the output of the waveguide (before AC) fiber polarization controller is used to adjust the dispersion and enhance the pulse. An input beam is split into two beams and overlapped in a nonlinear second harmonic generation (SHG) crystal. The train of AC pulse traces for N = 7,8,7,6 SCs are shown in Fig. 3(b). We measure the pulse periods i.e., inversely proportional to f_rep, of 0.14, 0.125, 0.14 and 0.17 ps for the N = 7,8,7,6 SCs, respectively.

 

Fig. 3. (a) Optical spectra of various octave-spanning SCs obtained in device D1 at the relative pump mode µ_p = 2,1,0,-1 and the mode interaction at µ_m = -7. The symbols circle and cross denote the pump and mode interactions, respectively. (b) Measured auto-correlation traces using APE pulse checker of the obtained different SCs.

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Next, employing another device denoted as D2, using the AMX method we realized near octave-spanning SCs with various comb line spacings. The device design and dimension parameters are similar to our device D1 but with a relatively large coupling gap between the straight waveguide and microring. Figure 4(a) shows the typical transmission trace of the TE polarization mode for device D2. The coupling of TE₀₀ and TE₁₀ modes is observed at mode µ_m = -8 and indicated with the green rectangles. The pump modes at µ_p = 0,-2,-3 are marked with red rectangles. The realized SCs at the particular pump mode and various line spacings are shown in Fig. 4(b). At a relatively low on-chip power of 175 mW, we obtained SCs with N = 4 at the pump mode of µ_p = 0 (∼1548 nm). At µ_p = 0 the achieved N = 4 SCs may be due to a possible AMX at lower wavelength at the difference of integer number multiple N × FSR and is subject to future study. Similarly, at an on-chip power of 275 mW, we select the pump modes of µ_p = -2 (∼1565 nm) and µ_p = -3 (∼1573 nm) to achieve N = 6,5 SCs which are at the difference of one N × FSR. The temporal waveforms of N = 4,5,6 SCs are measured with the AC and plotted in Fig. 4 (c). We can calculate the pulse periods of 0.25, 0.17 and 0.2 ps from the AC measurements for the N = 4,6,5 SCs, respectively.

 

Fig. 4. (a) Measured transmission spectrum of D2 at the TE polarization mode in the laser wavelength range 1480-1640 nm. The pump modes are marked with red rectangles far from the mode interaction position (marked with a green rectangle). Inset is the calculated integrated dispersion for the TE₀₀ mode as a function of relative mode numbers µ at the around µ_p = 0, i.e., 1548 nm. (b) Optical spectra of different SCs in device D2. The symbols circle and cross denote the pump and mode interaction, respectively. (c) Measured corresponding AC traces.

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We further experimentally observed various types of soliton crystals with broadband spectra (1200-2400 nm) compared to the previously reported [29]. The ordering of the soliton crystals with the background wave led to a rich variety of soliton crystals, including irregular inter-soliton spacings and Frenkel defects arising from the soliton shift [29]. The measured optical spectra of these soliton crystals are plotted in Fig. 5(a)-(c). Figure 5(a) and (b) shows the soliton crystals with irregular inter-soliton spacing and line spacing of 6 and 8 FSRs, respectively. Lastly in Fig. 5(c), from the recorded spectra we observe the effects of both Frenkel defects and irregular inter-soliton spacing.

 

Fig. 5. (a) Measured experimental optical spectra of variety of soliton crystals. (a) and (b) Soliton crystals with irregular inter-solitons distribution. The separation between each supermodes are 6 and 8 FSRs. (c) Soliton crystals with an effect of Frenkel defects and irregular inter-soliton distribution.

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A higher conversion efficiency (CE) is critical and a key merit in the miniaturization and on-chip integration of the Kerr microcombs [32–34]. Recently, immense efforts have been made on the microcavity design and pumping schemes to improve the overall performance [35,36]. Therefore, we investigate the conversion efficiency (CE) of the obtained various SCs. To obtain the CE we measure the ratio between the input pump power to the power converted into the comb lines at the output excluding the pump i.e., P_comblines/P_pump [37]. In the case of device D1, the CE of the various SCs has striking performance of above 50%. Specifically, the SCs N = 7, 8 at the pump modes µ_p = 2, 1 can achieve CEs of 58% and 55% at an error of 4%. Whereas for the N = 7,6 SCs at µ_p = 0,-1 the CEs are calculated to be 52% and 52%. Furthermore, in the device D2 with the achieved SCs N = 4,6,5 at the relative pump modes µ_p = 0,-2,-3 the CEs are 28%, 25% and 24% with an error of 3%, respectively. Table 1 lists the employed devices and generated various SCs and their CEs, respectively. It is important to note that the comparatively low performance of the D2 resonator can possibly be attributed to the relatively large waveguide coupling gap and poor performance of the external coupling power. The achieved broadband SCs in this work are in the range of an octave-span and with the performance of above 50% CEs makes it a potential candidate for diverse industrial and commercial applications.

3. Conclusion

In conclusion, we have demonstrated broadband soliton crystals with tunable line spacing via AMX in Si₃N₄ microresonators. The AMX-induced SCs were experimentally demonstrated in two devices, thus confirming the deterministic and reliable approach. Our devices enabled an on-demand generation of the SCs. This is a significant addition in simplifying the complexity of achieving soliton crystals on demand [30,38], but requires a rigorous design and engineering of the AMXs position. In addition, we also tested the performance of the devices by measuring the CEs. The performance of device D1 with more than 50% efficiency and octave-spanning (1200-2400 nm) stable spectra are ideal for applications such as data communication [6,39] and self-referencing [40].

Table 1. Summary of the dimensions of the employed devices D1, D2 and the calculated CE of the obtained diverse SCs, with the selected pump wavelengths at the relate mode numbers µ_p and the mode interaction positions µ_m, respectively.

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