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Physical Review A

Two-photon-loss model of intracavity second-harmonic generation

Matthew J. Collett, R. B. Levien

Published May 1, 1991• 29 citations
Publisher

Abstract

The input-output theory, originally developed for a linear coupling between cavity mode a and a bath, is extended to a coupling that is quadratic in the cavity mode operator a. We then apply this to second-harmonic generation, in a cavity that is resonant at the fundamental frequency and has an active laser medium working at the fundamental frequency. We find intensity squeezing of 50% below the shot-noise level in the output light at the second-harmonic frequency, whether or not the cavity is resonant at the second-harmonic frequency. To compare the singly resonant cavity to the doubly resonant cavity, we develop in an appendix a general technique for extracting a single mode from a continuum.

DOI
10.1103/physreva.43.5068

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arXiv:quant-ph/9810020•Oct 7, 1998•0 citations
The input-output theory, originally developed for a linear coupling between cavity mode a and a bath, is extended to a coupling that is quadratic in the cavity mode operator a. We then apply this to second-harmonic generation, in a cavity that is resonant at the fundamental frequency and has an active laser medium working at the fundamental frequency. We find intensity squeezing of 50% below the shot-noise level in the output light at the second-harmonic frequency, whether or not the cavity is resonant at the second-harmonic frequency. To compare the singly resonant cavity to the doubly resonant cavity, we develop in an appendix a general technique for extracting a single mode from a continuum.
The characteristics of squeezed light generated by both passive (occurring in a cavity external to a laser) and active (occurring within a laser cavity) second-harmonic generation (SHG) are investigated and contrasted. Squeezing of both the fundamental and the second harmonic is discussed, and the issue of doubly versus singly resonant behavior is addressed. We examine passive squeezing by explicitly modeling the intrinsically noisy output of the pump laser and coupling it to a passive, multiport, lossy SHG cavity. Low-frequency degradation of the squeezing caused by the laser pump noise is predicted and provides a probable explanation for previous discrepancies between theory and experiment. Active squeezing is quantitatively modeled by a three-level laser model that retains all laser dynamics. Previously disparate predictions are reconciled. For one parameter set two regimes of squeezing are predicted: 50% squeezing at frequencies lower than the laser relaxation oscillation, and near-perfect squeezing at frequencies above. A particular problem of active squeezing is highlighted: The fast dephasing of the laser coherence introduces considerable excess noise likely to mask squeezing in experimental situations. We conclude that, although passive SHG is a practical source of squeezing, active SHG is unlikely to be so in the foreseeable future.
We have generated squeezed states of light by the process of second harmonic conversion within an optical cavity resonant at both fundamental and harmonic frequencies. The experiment consists of a standing-wave cavity containing a crystal of lithium niobate doped with magnesium oxide and driven by the field from a frequency-stabilized Nd:YAG laser (incident power, ~10 mW). The total field reflected from the cavity illuminates a pair of high quantum efficiency photodiodes, whose individual photocurrents are summed. Observations of squeezing are made by analyzing the spectral density of fluctuations for this sum photo-current while the cavity detuning is scanned through resonance. An intensity regulator holds the reflected light intensity constant during the scan by control of the incident light level. Reductions in photocurrent fluctuations of 12% relative to the vacuum-state or shot-noise level are achieved for frequency offsets near 4 MHz. Since significantly higher degrees of squeezing are predicted by our theoretical analysis, we discuss those processes that currently limit the observed noise reduction, including phase noise produced by the intracavity crystal in the absence of nonlinear conversion.
We consider the process of second-harmonic generation (SHG) in a cavity that is resonant for both the fundamental- and second-harmonic modes. It has been previously shown that such a system displays a phase instability leading to self-pulsing and that one gets important intensity squeezing on both the fundamental- and second-harmonic mode for pump intensities close to but below, this instability threshold. Such a squeezing has been recently observed. We have analytically calculated the intensity-noise spectrum of the frequency-doubled field above such an instability threshold, using a semiclassical input-output technique for the treatment of quantum fluctuations.1 Such a method, introduced to deal with fluctuations around a stable solution, has been generalized to the self-pulsing domain. As expected, the instability brings excess noise to the output beams, especially around the zero-frequency, the self-oscillation frequency, and its harmonics. Remarkably, some regions exist in which where the noise increase is small; squeezing still occurs in this frequency range far in the instability domain.
The authors consider a laser with intracavity second-harmonic generation and injected signal at either the second-harmonic or the fundamental frequency. With a strong injected signal at the second-harmonic frequency, near-perfect phase squeezing at non-zero classical intensity is found. A weak injected signal at the second harmonic gives 50% intensity squeezing in the output at the second harmonic, at an adjustable frequency. A moderate injected signal at the fundamental gives 50% intensity squeezing at the second harmonic for a relatively small nonlinearity.
Quantum noise in a model of singly resonant frequency doubling including phase mismatch and driving in the harmonic mode is analyzed. The general formulae about the fixed points and their stability as well as the squeezing spectra calculated linearizing around such points are given. The use of a nonlinear normalization allows to disentangle in the spectra the dynamic response of the system from the contributions of the various noisy inputs. A general ``reference'' model for one-mode systems is developed in which the dynamic aspects of the problem are not contaminated by static contributions from the noisy inputs. The physical insight gained permits the elaboration of general criteria to optimize the noise suppression performance. With respect to the squeezing in the fundamental mode the optimum working point is located near the first turning point of the dispersive bistability induced by cascading of the second order nonlinear response. The nonlinearities induced by conventional crystals appear enough to reach it being the squeezing ultimately limited by the escape efficiency of the cavity. In the case of the harmonic mode both, finite phase mismatch and/or harmonic mode driving allow for an optimum dynamic response of the system something not possible in the standard phase matched Second Harmonic Generation. The squeezing is then limited by the losses in the harmonic mode, allowing for very high degrees of squeezing because of the non-resonant nature of the mode. This opens the possibility of very high performances using artificial materials with resonantly enhanced nonlinearities. It is also shown how it is possible to substantially increase the noise reduction and at the same time to more than double the output power for parameters corresponding to reported experiments.