J. Eur. Opt. Society-Rapid Publ. 21, 23( 2025) 225
Figure
2.( a) Energy density of the phase-matched guided modes at x 0( TE 00 and TM 00) and 2x 0( TE 02, SH).( b) n eff vs. w diagram of type-I( red marker) and type-II( blue marker) modal PM.( c) SHG efficiency as a function of the angle h between the propagation direction and the [ 100 ] axis.( d) Effective index n eff,( e) group velocity b 1, and( f) GVD b 2 for the TE 00, FF( blue dashed line), TM 00, FF( blue pointed line) and TE 02, SH modes( red dashed line). In( d, e, f) the colored arrows indicate the corresponding x-axis.
2.2 Chromatic dispersion
Chromatic dispersion plays a pivotal role for broadband frequency generation, as the propagation constant has a different spectral dependence for each mode. It is convenient to expand it in Taylor series around a carrier frequency x 0 [ 49 ]:
bðxÞ ¼n eff ðxÞ x c ¼ X ½x � x 0 Š k b k; ð2Þ k! k
where b k ¼ d k b = dx k x¼x 0
; b �1
1 is the group velocity and b 2 the group velocity dispersion( GVD). In a mean-field approximation, which has been largely proven to accurately describe the physics of OFC generation [ 50 ], the crucial quest for spectral broadening is that of anomalous dispersion, i. e. b 2 < 0. This is meant to compensate and balance nonlinear effects, directly responsible for the formation of novel spectral components. Moreover, a small | b 2 | is typically preferable to minimize the OPO power threshold. In the ideal case, the target is b 2 < 0 and | b 2 | 0 in a large spectral domain.
When SHG effects are non negligible, a substantial part of the driving field might be converted into the SH spectral domain. Under the doubly resonant condition, the energy carried by the SH wave remains confined in the optical cavity. In such situations, a double-envelope model provides accurate and physically consistent descriptions [ 39 ]. This considers two mean-field equations coupled by Kerr and SHG terms, describing the field evolution around the fundamental frequency( FF) and the SH components, respectively, with different propagation constants b FF and b SH. In a doubly resonant SHG-Kerr cavity, besides the usual requirement of anomalous GVD for both b 2, FF and b 2, SH around FF and SH, also the corresponding group velocities b 1, FF and b 1, SH must take on prescribed values. In particular, the group velocity mismatch Db 1 = b 1, SH – b 1, FF must be moderately small, or ideally zero [ 32 ]. To model realistic situations, it is crucial to incorporate a detailed representation of dispersive effects into the dynamics. For this reason, we compute the full dispersion profiles for the phase-matched modes, and Figures 2d – 2f reports n eff, b 1 and b 2 around x 0 and 2x 0. It is apparent that we have anomalous dispersion at x 0 but not at 2x 0. In the next section, we will see how this is detrimental for CS generation.
It is worth to mention that, beyond the propagation constant, other parameters exhibit a non-zero chromatic dispersion. For instance, variations in the waveguide mode profiles over a broad spectral domain could slightly affect the resulting dynamics [ 51 ]. In this work, we neglect these effects for simplicity, and we ascribe all the dispersive effects of the underlying dynamics solely to b( x).
3 OFC dynamics
3.1 The model
Let us consider the system of two mean-field equations passively driven and damped, coupled by means of nonlinear Kerr and SHG terms [ 32, 33, 38, 52 ]: