Propagation of quantum fluctuations in single-pass second-harmonic generation for arbitrary interaction length

By solving linearized wave equations of nonlinear polarization for a medium, we derive the input-output relationship for the propagation of the quantum fluctuations in single-pass second-harmonic generation with arbitrary interaction length. Spectra of quadrature-phase squeezing are calculated from the input-output relations. It is found that the intensity fluctuations of both harmonic and fundamental fields are squeezed below the vacuum fluctuation level. For large interaction length, an arbitrary amount of squeezing can be achieved for the fundamental field whereas only 50% squeezing is possible for the harmonic field. The bandwidth of squeezing is determined by the phase-matching condition and can be very broad in this case. The possibility of a quantum-nondemolition measurement in the process is discussed. Some interesting features in quantum fluctuations of the fields are presented.