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J. Eur. Opt. Society-Rapid Publ. 21, 23( 2025)
Figure
4.( a) Scheme of U-shape waveguide design for frequency doubling.( b) Color map of the azimuthal number difference Dm between the TE 00 mode at x 0 and the TM 00 mode at 2x 0.( c) Dm vs the waveguide width for height h = 110 nm. Modal PM and 4-QPM conditions are reported in red and gray dashed line, respectively, while the markers correspond to PM.( d) Fraction of the
power converted from FF to SH. The red line refers to the optimal situation where a change of the waveguide width enables to fulfill the 4-QPM condition in the curved section. The blue dashed line considers a constant width through the whole trajectory.( e) Spectral evolution of I SHG.( f) Asymmetric SHG spectrum of a U-shaped v( 2) waveguide.
condition for PM takes into account the difference between the azimuthal orders, i. e. Dm = 2m FF � m SH =± 2.
From the map of Dm vs. h and w reported in Figure 4b, we can see that typical waveguide sections for phase matching between TE 00( x) andTM 00( 2x) modes are much shallower than those considered above for h = 400 nm. Specifically, we note a more pronounced aspect ratio which tends to a quite-planar waveguide( h w). The resulting guided modes have typically steep dispersion profiles which are hardly manageable in a broadband domain.
Let us now set, for convenience, h = 110 nm. The resulting Dm vs. w diagram is reported in Figure 4c, which shows the 4-QPM and MPM conditions. The intersection with the black dashed line determines the corresponding PM widths for w 0 and w 1. Interestingly, they differ by about hundreds nm, which means that changing the waveguide width in the straight-to-curved section transition may significantly improve the SH conversion.
In order to show that, let us calculate the SHG intensity I SHG under the undepleted-pump approximation. We suppose two straight arms of 100 lm connected by an arc of radius R = 50lm. Figure 4d shows the two cases: I) both the modal PM and 4-QPM conditions are respected in the straight and the curved section, respectively( with phase matched widths w 0 6¼ w 1); II) the modal-PM condition is satisfied in the two straight sections, but the 4-QPM condition is not fulfilled in the arc section and the
widthiskeptconstant( w 0 = w 1). To take into account the dependence of j on Dm, namely j = j( Dm), we calculate the resulting accumulated phase / between the interacting modes TE 00( x) and TM 00( 2x) along the propagation trajectory in equation( 1), i. e. j = j(/( Dm)). In the plot we show the fraction of converted power I SHG / I 0, beingI 0 the driving field intensity, as a function of the propagation length x. We can see that the parabolic trend of the SH intensity buildup is preserved in the straight arms for the two cases, since the modal PM condition is fulfilled. In the curved section, the typical QPM trend occurs for case I [ 61 ], while I SHG / I 0 goes down for case II( phase mismatch).
Interestingly, in the ideal case where PM is satisfied all along a U-shaped lossless waveguide, the net power conversion is 16 % for a total propagation length of just 300 lm. It is worth mentioning that, in the seek of maximizing the SHG efficiency, it is crucial to consider realistic pump depletion and optical losses, here neglected for simplicity. While this will be dealt with in future work, for the moment we may clearly state that AlGaAs may lead to an efficient x 0? 2x 0 on-chipconversioninanintegrated circuit with minimal propagation length.
4.1 SHG spectral symmetry breaking
Besides the monochromatic frequency conversion x 0? 2x 0, if we aim to integrate the v( 2) U-shaped waveguide in a compact v( 2) + v( 3) device, it is important to