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J. Eur. Opt. Society-Rapid Publ. 21, 23( 2025)
OPO resonator was presented in reference [ 37 ], and a twowave model was used to describe the interplay between v( 3) and v( 2) surface effects in a silicon-nitride resonator [ 38 ]. As a general rule, whenever a non-negligible frequency conversion occurs into far different spectral domains, multienvelope mean-field models are more convenient [ 39 ].
The scheme of our OFC generation setup is shown in Figure 1a. A monochromatic continuous-wave( CW) driving field S, around a resonance frequency m 0 = x 0 / 2p, is coupled evanescently in a SHG-Kerr ring resonator from a bus waveguide. Thanks to quadratic nonlinear effects, an important fraction of power is converted to the SH at 2x 0. The intracavity field consists of two nonlinearly coupled waves( A, B), whose spectral broadening is due to the v( 2) + v( 3) interactions. The resulting output OFC owns spectral components around both the carriers x 0 and 2x 0. Figure 1b provides our technological choice for the waveguide cross-section: an AlGaAs core on top of a SiO 2 cladding and a Si wafer.
In the following, we present the two-wave nonlinear dynamics resulting from the v( 2) + v( 3) interplay in AlGaAs waveguides. Since we focus on the relevant case of a pump wavelength in the telecom range, we discuss the crucial problem of the strong normal dispersion in the spectral domain around the SH k SH 0.775 lm. A valid solution is represented by a systematic dispersion engineering approach, which exploits local Bragg mirroring effects [ 40 ] and resonance frequency splittings [ 41 ] in a hybrid PhCmicroring system [ 42 ]. Novel inverse design techniques of waveguides [ 43 ] or microcombs [ 44 ] address this specific topic.
Themanuscriptisorganizedasfollows: inSection 2 we present the design of a doubly resonant AlGaAs microring; in Section 3 we model the resulting SHG-Kerr comb generation dynamics; in Section 4 we propose a purely v( 2) optical feedback and we model the SHG; in Section 5 we discuss fabrication and present preliminary linear measurements; finally, in Section 6 we draw our conclusions and give some perspectives.
2 AlGaAs ring design
The alloy Al 0. 18 Ga 0. 82 As was chosen to both guarantee a high v( 2)( v ð2Þ xyz
238 pm / V at k 1. 55 lm [ 45 ]) and avoid two-photon absorption( TPA) at k 0 = 1.55lm.
The AlGaAs crystal allows to achieve directional quasi phase matching( 4-QPM) between guided modes at x 0 and 2x 0 [ 46, 47 ], and the main design parameters are the waveguide width w, height h and bending radius R. Since optimal dispersion for Kerr-comb excitation around x 0 occurs for waveguide cross sections with( h, w)( 400, 600) nm [ 17, 18 ], we investigate geometries with h = 400 nm and we let w free to vary around its optimum.
2.1 Phase matching and frequency doubling efficiency
We make use of a commercial software( Lumerical, package FDE [ 48 ]) to compute the waveguide eigenmodes and their effective index( n eff). A modal PM condition between fundamental transverse modes( either TE 00 or TM 00) at
Figure 1.( a) Scheme of a passively driven doubly resonant SHG-Kerr ring resonator.( b) Transverse cross-section with height h and width w.
the driving wavelength and a higher-order transverse mode( TE 02) at SH is found. The energy density of the phasematched eigenmodes is reported in Figure 2a, while Figure 2b reports the PM diagram of n eff as a function of the waveguide width. The blue marker corresponds to a type-II PM point, found for w 470 nm. This occurs as the average of the n eff of two pump modes( black dashed line) equals the n eff of the SH mode. The red marker indicates a type-I PM point( found for w 610 nm), which occurs whenever the effective index of the pump mode is equal to that of the SH. We next compute the SHG efficiency [ 25, 26 ]:
j ¼ x 0 0 4
Z
X
v ð2Þ xyz ex;
2x 0 e y x 0; 1e z x 0; 2 þ ey x 0; 2e z x 0; 1
dX;
where e x0; 1, e x0; 2 and e 2x0 are the electric fields at x 0 and 2x 0, respectively, X is the waveguide cross-section, v( 2) is the second order susceptibility tensor, 0 is the electric permittivity, and x, y and z are the spatial coordinates. The symbol * indicates the complex conjugate, and tensor index contraction follows the Einstein summation convention. In equation p( 1) we use the standard normalization e norm ¼ e = ffiffiffiffi
R
N [ 25 ], with N ¼ð1 = 2Þ ðe h Þ^x dX; h the magnetic field, x the propagation direction and the vector product. Since v ð2Þ xyz is a tensor, j is strongly influenced by the orientation of the propagation direction with respect to the crystalline axes. For this reason we define h as the angle, in the( 001) plane, between the propagation direction and the [ 100 ] axis. The SHG generation efficiency | j | 2 is reported in Figure 2c for both type-I and type-II PM conditions. We note that the j assumes comparable values in these two cases. In what follows we will consider the type- I PM case for two main reasons: 1) it results in a better dispersion profile; 2) for the type-II modal PM we are close to a degeneracy point w = h, which is detrimental for broadband applications. Finally, as introduced above, in a ring configuration the condition 4-QPM implies slightly different optimal w. The corresponding waveguide cross-section is thus set as( w, h) =( 630, 400) nm. ð1Þ