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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Antiphased states in intracavity second harmonic generation

Jingyi Wang, Paul Mandel, Thomas Erneux

Published Apr 1, 1995• 24 citations

Abstract

We analyse the problem of intracavity second harmonic generation to determine properties of the antiphased states that have been observed in this system. We study the time-periodic solutions by means of a perturbation expansion in powers of a small parameter which is the ratio of the cavity lifetime to the population inversion relaxation time. We prove that each mode intensity is characterized by two oscillation frequencies though the total intensity oscillates with only one of these frequencies. In first approximation, we find that the bifurcation from the steady state to the periodic state remains vertical. We resolve this singularity by a complementary numerical analysis. We show that there are at least four distinct types of antiphase dynamics, including partial clustering of the oscillating modes into subsets of antiphasing states. Finally, we analyse the scaling exponent of the oscillation amplitudes near the bifurcation point and determine that the generic 1/2 exponent is the rule for only two of four types of antiphasing. The exponents of the other two cases depend on the mode number and partition.

DOI: 10.1088/1355-5111/7/2/007

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We analyse the problem of intracavity second harmonic generation to determine properties of the antiphased states that have been observed in this system. We study the time-periodic solutions by means of a perturbation expansion in powers of a small parameter which is the ratio of the cavity lifetime to the population inversion relaxation time. We prove that each mode intensity is characterized by two oscillation frequencies though the total intensity oscillates with only one of these frequencies. In first approximation, we find that the bifurcation from the steady state to the periodic state remains vertical. We resolve this singularity by a complementary numerical analysis. We show that there are at least four distinct types of antiphase dynamics, including partial clustering of the oscillating modes into subsets of antiphasing states. Finally, we analyse the scaling exponent of the oscillation amplitudes near the bifurcation point and determine that the generic 1/2 exponent is the rule for only two of four types of antiphasing. The exponents of the other two cases depend on the mode number and partition.

We consider second-harmonic generation by a non-linear crystal placed in an empty resonant cavity and pumped by an external coherent field. It was recently predicted that the stationary state loses its stability through a Hopf bifurcation when the input intensity reaches a critical value. A new amplitude self-modulated state then arises. We construct explicitly this new time-periodic solution and prove its stability in the vicinity of the transition point. As the input intensity is further increased, numerical investigations indicate a new transition to another time-periodic solution; this transition involves an hysteresis cycle. The two periodic solutions differ by the number of maxima in one period: at low input intensity, there is one maximum per period, whereas at high input intensity there are two.

For the process of intracavity second-harmonic generation in the region of unstable dynamics, the joint fluctuations of numbers of photons and phases of the fundamental mode and second harmonic are studied in positive P representation. It is shown that above the bifurcation point of the optical system the distribution functions of the phases of the fundamental mode and second harmonic have a two-component structure. The quantum trajectories of the phases of these modes are not localized in the components of their states, which demonstrates the presence of quantum-mechanical interference between these components. In both components of each state the same values of numbers of photons are realized. It is shown that in the case of coherent initial states of both modes the joint distribution functions of the numbers of photons and phases are in the oscillatory regime. With time, the modes pass from a two-component delocalized state to a one-component one and then back to the two-component state. The time of oscillation of the distribution function is of the order of the photon's lifetime in the cavity volume. Self-matching of phases of interacting modes is investigated above the bifurcation point.

We investigate quantum-mechanical counterpart of a classical instability in a phase space by the numerical method of quantum trajectories with moving basis. As an application the model of coupled two oscillators driven by a monochromatic force in the presence of dissipation (intracavity second harmonic generation) is analyzed. The system of interest is characterized by two bifurcations leading to ranges of instability: the Hopf bifurcation which connects a steady state dynamics of the oscillatory modes to a self-pulsing temporal dynamics and the bifurcation of the period-doubling. The both two regimes are analyzed on the framework of the semiclassical phase trajectories and the Wigner functions of the oscillatory modes in phase space.

Quantum fluctuation and quantum entanglement of the pump fields reflected from an optical cavity for type-II second-harmonic generation are theoretically analyzed. The correlation spectra of quadrature components between the reflected subharmonic fields are interpreted in terms of pump parameter, intracavity losses, and normalized frequency. High correlation of both amplitude and phase quadratures can be accessed in a triple resonant cavity before the pitchfork bifurcation occurs. The two reflected subharmonic fields are in an entangled state with quantum correlation of phase quadratures and anticorrelation of amplitude quadratures. The proposed system can be exploited as a source for generating entangled states of continuous variables.