On the efficiency of small-angle Cerenkov second harmonic generation in optical waveguides

It is well known that in the non-depleted pump approximation, the efficiency of a second harmonic generation (SHG) of a guided mode in a non-linear optical waveguide increases quadratically with the interaction length (P-2 proportional to L-2), and linearly (P-2 proportional to L) in the Cerenkov regime. The efficiency of the Cerenkov SHG in the waveguide with a non-linear substrate and linear guiding layer is known to be strongly peaked at a particular pump wavelength and a particular waveguide thickness, with the Cerenkov angle approaching zero. The known theory predicts an infinite efficiency value at the peak, however. In this contribution, a simple integral expression for the SHG efficiency in the Cerenkov regime is derived. For large Cerenkov angles and interaction lengths it yields the expected P-2 proportional to L dependence, while in the limit of small Cerenkov angles the dependence is found to have the form of P proportional to L-3/2, possessing also a finite value at the efficiency peak. The condition determining the accurate position of the efficiency peak in the waveguide thickness pump wavelength plane is given, too.