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Nonlinear Dynamics in Optical Systems

Squeezing Spectrum in Second-Harmonic Generation in the Self-Pulsing Domain

Nicolas Pettiaux, Paul Mandel, Claude Fabre

Published Jan 1, 1990• 0 citations
Publisher

Abstract

We consider the process of second- harmonic generation (SHG) in a cavity that is resonant for both the fundamental and second-harmonic modes. In reduced variables, the system can be written [1] as: where γ is the ratio of the two damping coefficients, Aj=<aj> (j=1 or 2) are the scaled fundamental (j=1) or second-harmonic (j=2) mode and E is the scaled pumping field.

DOI
10.1364/nldos.1990.stsa576

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We consider the process of second- harmonic generation (SHG) in a cavity that is resonant for both the fundamental and second-harmonic modes. In reduced variables, the system can be written [1] as: where γ is the ratio of the two damping coefficients, Aj=<aj> (j=1 or 2) are the scaled fundamental (j=1) or second-harmonic (j=2) mode and E is the scaled pumping field.
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.
Second-harmonic generation (SHG) in doubly resonant optical resonators (resonant for the pump and for the harmonic wave) is a subject rich in outstanding challenges: the demonstration of self-pulsing, in which the intracavity field amplitudes undergo a Hopf bifurcation to a stable limit cycle,1 and chaos; the generation of near-perfect squeezing near but below the self-pulsing threshold; the generation of squeezed radiation in the self-pulsing regime; and the demonstration of quantum nondemolition measurements.
The authors consider the problem of second harmonic generation in an optical cavity which contains both the laser and the nonlinear crystal. They calculate the spectrum of the amplitude fluctuations in the output field, for the mode which is generated by the laser. They focus on the case that the damping constant of this mode is much larger than all the other relaxation rates of the system, and show that in this limit the spectrum is correctly described by the universal normal form for two-photon processes. Finally, they show in general that in the framework of linearised theories the spectrum of the intensity fluctuations in the output field reduces exactly to the spectrum of squeezing of the amplitude quadrature component.
It is known1 that in type-I second-harmonic generation in a doubly resonant cavity where a subharmonic mode is coupled to a harmonio mode through a χ(2) process, self-pulsing may occur for high input fundamental power. Squeezing2 and QND measurements3 are possible near the self-pulsing threshold. For type-II harmonic generation, however, there are two sub-harmonic modes coupled to the harmonic mode through the same χ(2) process but with different arrangements of polarizations. The extra subharmonic mode will make the dynamics of the system totally different.
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.