Nonlinear Diffraction from a Virtual Beam

We observe experimentally a novel type of nonlinear diffraction in the process of two-wave mixing on a nonlinear quadratic grating. We demonstrate that when the nonlinear grating is illuminated simultaneously by two noncollinear beams, a second-harmonic diffraction pattern is generated by a virtual beam propagating along the bisector of the two pump beams. The observed diffraction phenomena is a purely nonlinear effect that has no analogue in linear diffraction.

Figure 1

Schematic of the experiments with (a) one-dimensional and (b) annular periodically poled grating. (c),(d) Experimentally recorded NLD patterns with two fundamental beams, for the corresponding structures.

Figure 2

SH patterns of Raman-Nath and Čerenkov NLD. (a) Single beam without a biprism, (b) with a biprism when both beams cross outside the sample, and (c) with a biprism when the beams cross inside the sample.

Figure 3

Phase-matching diagram for Raman-Nath and Čerenkov NLD when both fundamental beams intersect inside the sample. All wave vectors of the SH beams, ${\overrightarrow{k}}_2$, lie in the $X\mathrm{\text{−}}Z$ plane, while the wave vectors of the two fundamental beams, ${\overrightarrow{k}}_{1A}$, ${\overrightarrow{k}}_{1B}$, are located in the $Y\mathrm{\text{−}}Z$ plane. ${\overrightarrow{k}}_{\mathrm{vb}}$ represents the wave vector of a virtual beam.

Figure 4

Numerical results. (a) Diffraction from a single beam, (b) NLD from two beams crossing outside the sample, (c) NLD from two beams crossing inside the sample, and (d) SH intensity distribution for the virtual-beam NLD.

Figure 5

(a) Microphotograph of the $Z$ surface of the annularly periodically poled structure. (b) Experimental image of NLD with a single fundamental beam and (c) with two beams crossing in the center of the structure. (d) Magnified central part of the pattern (c). (e) Numerically calculated SH emission from the virtual-beam NLD process.