Spatial behavior of light in second-harmonic generation

We investigate the spatial behavior of nonclassical light produced by type I second-harmonic generation in the traveling-wave configuration. An input-output transformation for the system of fundamental and secondharmonic field is derived in the framework of a linearization approach and used to investigate the properties of an optical system, which consists of a crystal with a quadratic nonlinearity pumped at frequency v and enclosed in a two-lens telescopic system. If a faint input image at frequency 2v is injected into this device, for a sufficiently large interaction length, the output displays a pair of symmetric amplified versions of the input image at both fundamental and second-harmonic frequency. The analysis of the quantum fluctuations in the output images shows that under certain conditions, this optical device operates noiselessly with respect to the output at frequency v , whereas the output images at frequency 2v are affected by a slight degradation of the signal-to-noise ratio.