226 J. Eur. Opt. Society-Rapid Publ. 21, 23( 2025) t R
L
t R
L
@ A @ t ¼ �a 1 � id � i b 2; FF @ 2
A
2 @ s 2 þ
ic 1 jAj 2 þ 2ic 12 jBj 2 A þ ijBA þ S;
@ B @ t ¼
�a @
2 � 2id � b 1
@ s � i b 2; SH @ 2 2 @ s 2
þ
ic 2 jBj 2 þ 2ic 21 jAj 2 B þ ij A 2;
B
where p ffiffiffiffiffi A and B are the FF and SH field envelopes in units of W, S is the CW driving field, a1, 2 are the optical losses, t R is the round trip time, L is the total cavity length, d is the phase detuning between the optical source and the closest cavity resonance, Db 1 is the walk-off, c 1, 2 are the self-phase modulation( SPM) coefficients, c 12, 21 are the cross-phase modulation( XPM) coefficients, and j is the SHG contribution defined in equation( 1). We adopt a two-time scales description, where the fast time s governs the intra-cavity field evolution, while the slow time t describes the evolution of the system over time scales larger than t R. Its definition relies on the Ikeda map approximation [ 53 ]:
t R
@ Aðt ¼ nt R; sÞ @ t
¼ A nþ1 ðz ¼ 0; sÞ�A n ðz ¼ 0; sÞ:
The use of the full dispersion profile [ 54 ], computed and discussed in the previous section, is implemented in a well known split-step algorithm [ 49 ] in the Fourier domain. For further details on the numerical implementation, we address the reader to Appendix.
The Kerr SPM and XPM coefficients have been estimated from the transverse mode profiles through [ 55 ]:
c m; n ¼ n 2x c Q m; n; ð6Þ with the overlap integral Q m, n defined as [ 34, 56 ]: Z je t; m j 2 je t; n j 2 dX
X
Q m; n ¼ Z Z: ð7Þ je t; m j 2 dX je t; n j 2 dX
X
Here, e t, 1 and e t, 2 are the profiles of the transverse interacting modes, X the waveguide cross-section, n 2 = 26 10 �18 m 2 W �1 [ 17 ] the nonlinear refractive index of AlGaAs, whose dispersion is neglected for simplicity( i. e. n 2 is considered constant in the whole spectral domain). We computed the Kerr coefficients( c 1, c 2, c 12, c 21) =( 732, 1716, 499, 732) m �1 W �1.
From our previous discussion, the SHG efficiency j has a spatial dependence j = j( h). In the dynamical simulation, this results in a fast time dependence j = j( s), which we
will take into account to properly embed the 4-QPM ring configuration.
We next show the simulations results considering an input power P in = 100 mW and optical losses a 1 = 10dB / cm, a 2 = 20dB / cm.
X ð3Þ
ð4Þ
ð5Þ
3.2 Simulation results
To simulate the OFC generation dynamics from the coupled equations( 3) and( 4), we spectrally sweep a cavity resonance around the driving wavelength k’ 1550 nm. This procedure, which mimics the experimental comb excitation, is shown in Figure 3a. We start the simulation in a blue detuned regime, i. e. by pumping the system at wavelengths k P slightly smaller than a given reference resonance k 0( k P < k 0), then we linearly increase d until we reach a red-detuned regime k P > k 0. At this point k P, and consequently d, are kept constant to verify the stability of the excited OFC regime.
Figure 3b shows the energy stored in the cavity versus the total number of round trips, in the different energy scales associated with waves A( E A) andB( E B). These are computed as the fast-time integral of the intra-cavity intensities | A | 2 and | B | 2:
E A ¼
Z tR = 2
�t R = 2 jAj 2 ds: ð8Þ
These two quantities differ by about three orders of magnitude, being most of the power carried in the spectral domain around the driving wavelength k 0. Nonetheless, we can notice an important energy build-up of the FF wave while d increases and we approach the resonance condition. Above a certain power threshold, a chaotic transition clearly arises and, accordingly, we observe a chaotic nonlinear coupling to the SH domain. Finally, the emergence of a step indicates the likely excitation of a CS regime.
To sustain these statements and fully describe the comb generation, Figure 3 shows four different snapshots of the intra-cavity field( Figs. 3d, 3f, 3h, 3j) and their corresponding spectra( Figs. 3c, 3e, 3g, 3i). In Figures 3c and 3d, the system is in a blue-detuned regime but close to a resonance condition( k P � k 0 = �34 pm). We observe a parametric oscillation which couples efficiently the two spectral domain x 0 ¡ 2x 0. InFigure 3d, we can observe how the two waves have opposite phases. As the detuning increases and bandwidth broadens, the phase drift between the periodic patterns at x 0 and 2x 0 can be read as a signature of the non-zero walk-off which is detrimental to synchronize the dynamical evolution in the two separate spectral domains. Notably, typical SH powers are at a mW level, while the FF wave is three orders of magnitude more powerful( 1 W), and the same conclusion can be drawn for the other reported OFC regimes. Increasing the detuning( k P � k 0 = 86 pm), we can observe, in Figures 3e and 3f, a MI regime and the associated primary comb. The FF wave dominates the resulting dynamics, as the envelope A forms a periodic pattern while B is slaved to it. In the chaotic regime( k P � k 0 = 170 pm, Figs. 3g and 3h), similarly, most of the power is carried by the FF wave and is chaotically coupled to the SH domain. Finally, Figures 3i and 3j show the final excited comb state( for a detuning k P � k 0 = 860 pm). This consists of a single FF cavity soliton which, due to efficient SHG, couples part of the intracavity power around the SH. Thus the resulting dynamics consists of a single CS confined around the FF and coupled to a dispersive wave at SH. The anomalous