Abstract
The process of second-harmonic generation (SHG) in a finite one-dimensional nonlinear medium is analyzed in parallel by the Green-function technique and the Fourier-transform method. Considering the fundamental pump field propagating along a given direction and eliminating back-reflections at the boundaries the terms giving the surface second-harmonic fields in the particular solution of the wave equation are uniquely identified. Using these terms the flow of energy corresponding to the surface second-harmonic fields is analyzed in the vicinity of the boundaries. The formula giving the depth of the nonlinear medium contributing to the surface SHG is obtained. Both approaches for describing the SHG are compared considering complexity and quantization of the interacting fields. In addition, a theoretical model of surface SHG in centrosymmetric media is proposed. The model is built upon assumption that the second-order nonlinearity decays exponentially with distance from the boundary. As an important example, the generation of surface SHG from a thin dielectric nonlinear layer placed on a silicon substrate is analyzed by the proposed model.
Highlights
The process of second-harmonic generation (SHG) is the first ever observed nonlinear optical process[1], soon after the discovery of a laser[2]
As the second-harmonic field is apparently emitted from a thin layer around the interface, we speak about surface SHG, as opposed to the usual volume SHG
We have analyzed the process of second-harmonic generation in a finite 1D nonlinear medium using the Green-function technique
Summary
According to Eq (24) the overall flow of energy from the nonlinear medium in the −z direction is in general nonzero, it vanishes only for specific lengths L (L = 2πn/Δk+bg, n ∈ ) In this case, the surface second-harmonic field generated around the boundary at z1+ perfectly compensates that emitted around the occurs when both surface contributions maximally constructively interfere b[Lou=nd(a1ry+at2zn2−).πA/Δnokt+hbge,rnsp∈ec ifi]c. We apply the transfer-matrix formalism[21,24] to the fundamental field to reveal its forward- and backward-propagating components in all three media This allows us to determine the particular solution for the second-harmonic field of the form of Eqs (26) and (27), i.e. without including back-scattering effects. The second-harmonic field propagating in air is more than twice intense compared to that propagating in the substrate
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