Nanophotonics 2021; 10(6): 1691–1699 Research article Mingming Nie, Yijun Xie and Shu-Wei Huang* Deterministic generation of parametrically driven dissipative Kerr soliton https://doi.org/10.1515/nanoph-2020-0642 1 Introduction Received December 4, 2020; accepted March 8, 2021; published online March 24, 2021 Recent years have witnessed increasing interest in optical frequency comb (OFC) generation in high-Q quadratic Abstract: We theoretically study the nature of para- nonlinear cavities, benefiting from large χ(2) quadratic metrically driven dissipative Kerr soliton (PD-DKS) in a nonlinearity-induced low threshold, high efficiency, and doubly resonant degenerate micro-optical parametric intrinsic frequency conversion. Either intracavity second- μ χ(2) χ(3) oscillator (DR-D OPO) with the cooperation of and harmonic generation [1–6] or optical parametric oscillator nonlinearities. Lifting the assumption of close-to-zero (OPO) [7–14] can be utilized for the generation of these group velocity mismatch (GVM) that requires extensive quadratic OFCs. In particular, OPO has been demonstrated dispersion engineering, we show that there is a threshold as a versatile and competitive OFC [2, 15, 16] in otherwise μ GVM above which single PD-DKS in DR-D OPO can be difficult-to-access spectral ranges including the mid- generated deterministically. We find that the exact PD-DKS infrared molecular fingerprinting region [17]. Moreover, generation dynamics can be divided into two distinctive intriguing dissipative quadratic soliton dynamics in regimes depending on the phase matching condition. In continuous-wave (cw) pumped micro-OPOs (μOPOs) is both regimes, the perturbative effective third-order recently studied and utilized to enhance the performances nonlinearity resulting from the cascaded quadratic pro- of ultrafast OPOs [9–14]. These prior μOPO studies are cess is responsible for the soliton annihilation and the however limited to the operation regime where pump- deterministic single PD-DKS generation. We also develop signal GVM is close to zero and χ(3) nonlinearity is negli- the experimental design guidelines for accessing such gible [12–14]. deterministic single PD-DKS state. The working principle On the other hand, dissipative Kerr soliton (DKS) can be applied to different material platforms as a formation in fiber-feedback OPO has been recently competitive ultrashort pulse and broadband frequency demonstrated by incorporating the χ(3) nonlinearity of a comb source architecture at the mid-infrared spectral meter-long single-mode fiber to balance the cavity disper- range. sion and the χ(2) parametric gain of a millimeter-long peri- odically poled lithium niobate (PPLN) to compensate for Keywords: frequency combs; nonlinear dynamics; optical the cavity loss [18]. This unique example demonstrates how parametric oscillators; optical solitons; second-order the combination of χ(2) and χ(3) nonlinearities in a singly nonlinear optical processes; self-phase locking. resonant OPO can be utilized to facilitate formation of DKS at signal field that enhances its stability and bandwidth. However, achieving such a clear separation between the χ(2) and χ(3) nonlinearities in chip-scale μOPOs is very chal- lenging if not impossible. Furthermore, most chip- scale μOPOs are doubly resonant where both signal and pump are resonant with the cavity and thus the effect of *Corresponding author: Shu-Wei Huang, Department of Electrical, resonant pump must also be considered in the DKS for- Computer & Energy Engineering, University of Colorado Boulder, mation dynamics. Boulder, CO 80309, USA, E-mail: [email protected]. In this letter, we theoretically study the nature of DKS https://orcid.org/0000-0002-0237-7317 at signal field in a cw driven doubly resonant degen- Mingming Nie and Yijun Xie, Department of Electrical, Computer & erate μOPO (DR-DμOPO) with the cooperation of χ(2) and χ(3) Energy Engineering, University of Colorado Boulder, Boulder, CO 80309, USA, E-mail: [email protected] (M. Nie), nonlinearities. Lifting the assumption of close-to-zero [email protected] (Y. Xie) GVM, we show for the first time that there is a threshold Open Access. © 2021 Mingming Nie et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 1692 M. Nie et al.: Deterministic generation of PD-DKS ″ 2 GVM above which single parametrically driven DKS ∂A αc k ∂ = [ − 1 − i 1 ]A +iκBA∗e−iΔkz +i(γ |A|2 +2γ |B|2)A, (PD-DKS) in DR-DμOPO can be generated deterministically. ∂z 2 2 ∂τ2 1 12 In stark contrast to the previously reported dissipative (1a) quadratic soliton [12], deterministic single PD-DKS can ∂B α ∂ k″ ∂2 stably exist over a wide range of GVM and thus greatly = [ − c2 − Δk′ − i 2 ]B +iκA2 eiΔkz + i(γ |B|2 2 alleviate the need for extensive dispersion engineering. In ∂z 2 ∂τ 2 ∂τ2 addition, by simplifying the coupled-wave equation into a + γ |A|2)B, 2 21 (1b) single mean-field equation, we clearly identify the different roles of χ(2) and χ(3) nonlinearities in the signal PD-DKS and the boundary conditions: √̅̅̅̅̅ formation. With above-threshold GVM, material Kerr −iδ1 Am+1(0, τ) = 1 − θ1Am(L, τ)e , (2a) nonlinearity (MKN) will dominate the properties of PD-DKS √̅̅̅̅̅ √̅̅ −iδ2 while effective third-order nonlinearity becomes soliton Bm+1(0, τ) = 1 − θ2Bm(L, τ)e + θ2Bin, (2b) perturbation that is responsible for the soliton annihilation A fi B fi and the deterministic single PD-DKS generation. The exact where is the signal eld envelope, is the pump eld B α PD-DKS generation dynamics can be divided into two envelope, in is the cw pump, c1,2 are the propagation Δk′ k″ distinctive regimes depending on the phase matching losses per unit length, is the GVM, and 1,2 are the group condition. With large phase mismatch, deterministic single velocity dispersion (GVD) coefficients. Without loss of PD-DKS can be obtained with a lower GVM threshold but at generality, normal GVD at the pump wavelength and the cost of higher pump power. Our theoretical analysis anomalous GVD at the signal wavelength are chosen in this provides the basis for comprehensive understanding of work. Higher-order dispersion√̅ and nonlinearity√̅̅̅̅̅̅̅̅ are κ = ω d /(A c3 n2 n ϵ ) deterministic single PD-DKS generation recently observed neglected for simplicity. 2 0 eff eff 1 2 0 is in an aluminum nitride (AlN) DR-DμOPO [19]. Finally, we the normalized second-order nonlinear coupling coeffi- have developed the experimental design guidelines for cient, where deff is the effective second-order nonlinear accessing such deterministic single PD-DKS state, consid- coefficient, Aeff is the effective mode area, c is the speed of ering various parameters such as GVM, effective second- light, ε0 is the vacuum permittivity, and n1,2 are the order nonlinear coefficient and phase mismatch. refractive indices. Δk is the wave-vector mismatch, γ1,2 are self-phase modulation (SPM) coefficients and γ12 and γ21 are cross-phase modulation (XPM) coefficients. L is the θ 2 Theoretical analysis and nonlinear medium length, 1,2 are the coupler transmission coefficients and δ1,2 are the signal-resonance and pump- numerical results resonance phase detuning [4]. By solving the coupled-wave equations via the The field evolution of a cw-pumped DR-DμOPO with both standard split-step Fourier method (see Section S1, χ(2) and χ(3) nonlinearities (Figure 1) obeys the coupled Supplementary material), Gaussian-pulse-seeded genera- equations in the retarded time frame of the signal [4]: tion of DKS pulse profile and optical spectrum at signal Figure 1: Schematic of the deterministic single PD-DKS generation in a DR-DμOPO with both quadratic and cubic non- linearities. FWM: four-wave mixing. M. Nie et al.: Deterministic generation of PD-DKS 1693 field are obtained as shown in Figure 2 (red solid lines). Due pump field Bin and the signal field A, ξ = ΔkL/2 is the wave- to the large GVM, the pump is only quasi-cw without any vector mismatch parameter. Equation (3) is the para- clear pulse profile (red dashed lines in Figure 2). Impor- metrically driven nonlinear Schrödinger equation tantly, DKS at signal field can only exist when MKNs (SPM (PDNLSE) [12–14, 20] with a perturbation term represent- and XPM terms) in Eq. (1) are included in the numerical ing effective third-order nonlinearity J(τ)=F−1[̂J(Ω)], − simulation, revealing the key role of MKN in the PD-DKS ̂ ′ ″ 2 1 where J(Ω)=(α2 + iδ2 − iΔk LΩ − ik LΩ /2) and Ω is the generation (see Section S2, Supplementary material). We 2 offset angular frequency with respect to the signal reso- further find that the balance between signal SPM and nance frequency. As shown in Figure 2 with magenta signal GVD dominates the properties of PD-DKS (blue solid lines, the validity of PDNLSE is verified and it provides a lines in Figure 2). It can be understood as the large GVM computationally efficient way to study the deterministic renders XPM ineffective and SPM by the weak quasi-cw PD-DKS generation in this section. pump is negligible. Our numerical simulation confirms that Figure 3(a) and (b) shows the histogram of 100 inde- the PD-DKS is parametrically driven through the χ(2) OPO pendent intra-cavity average power traces with and process while its anomalous GVD is balanced by the MKN without the last term in Eq. (3), respectively. The pump χ(3) SPM process. The exact PD-DKS generation dynamics phase detuning δ2 (δ2 =2δ1, Section S4, Supplementary can be divided into two distinctive regimes depending on material) is tuned linearly from blue to red side in 180 ns the phase matching condition.