Modeling third-harmonic generation from layered materials using nonlinear optical matrices

Abstract

A matrix approach is formulated to describe third-harmonic (TH) generation in stacked materials in the small signal limit, in both transmission and reflection geometries. The model takes into account the contribution from the substrate to the total generated TH, interference of fundamental and nonlinear fields inside the stack, the nonlinear signal generation in forward and backward direction, the beam profile of the focused incident beam in the substrate, and the finite spectrum associated with short laser pulses. The model is applied to design stacks of thin films for efficient TH generation.

Characterization and χ^(3) measurements of thin films by third-harmonic microscopy

“…If this is done, the same χ 3 values (normalized to the value for the fused silica substrate) are obtained from transmission and reflection data. We also measured the TH signal in transmission and reflection produced by stacks of HfO 2 and SiO 2 films and found good agreement with the model [13], considering the uncertainties (≈5%) in the thickness of the individual layers.…”

“…A general matrix approach can be used to treat nonlinear optical signal conversion of focused fundamental beams in a stack of layers of different optical properties taking into account multiple reflections of the interacting fields [13]. Here we will use these results for the special case of a single film of thickness much smaller than the Rayleigh range 2d ≪ z 0 , and a substrate of thickness L ≫ z 0 , and assume an incident Gaussian beam of beam waist w 0 .…”

“…The TH field E T in the spatial frequency (ρ) domain is the sum of the field produced in the film, E r , multiplied by the spatial frequency spectrum and the Fresnel propagator, and the field produced in the front surface of the substrate [13]: …”

“…Since 2d ≪ z 0 , we may also neglect the ρ dependence of the phase of fundamental and harmonic fields introduced by the film [13]. The latter implies that for each spectral frequency component ρ the thin-film response is that for a normally incident plane wave.…”

“…Again, because of 2d ≪ z 0 , for each spectral frequency component the nonlinear film response is that for a normally incident plane wave. The field E R ρ is the sum of the TH produced in the film and backward-generated TH in the substrate and transmitted through the film [13]:…”