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Annalen der Physik

Der optische parametrische Oszillator. II. Amplitudenfluktuationen der Strahlung bei durchlaufender Pumpwelle

W. Brunner, Henryk Paul, A. Bandilla

Published Jan 1, 1971• 7 citations
Publisher

Abstract

Abstract Amplitude fluctuations of the signal wave of an optical parametric oscillator are evaluated for the case of nonresonant pumping by calculating the decrease of a random deviation ε of the signal amplitude from its stationary value during a round trip through the (ring) resonator. From this, a damping term governing the temporal behaviour of ε is deduced which, together with a Langevin force connected with the spontaneous decay of pump photons into both signal and idler photons, determines the amplitude fluctuations of the signal wave. In addition, it is shown that the same results may be obtained from a mode formalism for signal and idler waves taking into account the spatial variation of the pump wave following from Maxwell's equations.

DOI
10.1002/andp.19714820109

Related papers

Der optische parametrische Oszillator. I. Berechnung von Photonenzahl, Schwellenergie und Wirkungsgrad bei resonanter Pumpwelle

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Abstract The optical parametric oscillator is investigated theoretically for the general case of resonant pump, signal and idler waves. Mean photon numbers, threshold energies and conversion rates are calculated explicitly for different types of the waves (standing and running ones), where in the case of both signal and idler being standing waves the appearance of an oppositely running (“reflected”) pump wave is properly taken into account. In addition, from the equations of motion for the photon creation operators approximately valid rate equations are deduced which in certain cases predict a damped oscillating transition (analogous to the relaxation oscillations of a ruby laser) to the stationary state.

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Non-Local Behaviour in the Far-Field of the Optical Parametrical Oscillator Adove Threshold

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SUMMARY Parametric processes in the interaction of electromagnetic field with a medium in an optical cavity are well known to produce states of light with nonclassical properties. The quantum correlations between these states can be employed for a realization of an EinsteinPodolsky-Rosen (EPR) paradox [l], in the original formulation using continuous variables, as proposed [2] and experimentally demonstrated [3] in the case of Nondegenerate Parametric Amplification below threshold. In this paper we study EPR-like behaviour between the signal and idler fields emitted by a degenerate optical parametric oscillator (OPO) above threshold; these fields, generated by conversion of a pump photon of frequency 2w0 in two photons of frequency w. , are spatially separated in the far field, showing non-local quantum correlations. The system is described by the full Hamiltonian for the operators associated with the pump and signal fields; the optical cavity introduces proper damping constants and corresponding vacuum field operators [4,5]. Above the threshold for the generation of the signal field, we consider the fluctuations in the intensity difference and in the phase sum of the signal beams, which are related to their degree of mutual correlation. By linearizing around the stationary state, we calculate the spectral variances of these quantities (i.e. the Fourier transform of the two time correlation functions). The Heisenberg uncertainty principle sets a lower limit to the product of the spectral variances for the fluctuations in the intensity and in the phase of each individual signal field; by using the idler beam to infer results of measurements on the correlated signal beam, we have an EPR paradox if the product of the spectral variances for the fluctuations in the intensity difference and phase sum of the fields, is below the uncertainty limit. This feature is apparent for a wide interval around zero frequency in the far field of the active OPO system, depending on the parameter space; in particular there is maximum correlation between signal beams, and therefore maximum breaking of the uncertainty limit, if the damping constant of the signal field is larger than that of the pump field.

European Quantum Electronics Conference (EQEC)•Jan 1, 1996•0 citations

Theorie der parametrischen Vier‐Photonen‐Wechselwirkung. II., Verstärkung und Oszillation in der Näherung des vorgegebenen Pumpfeldes)

J. Frahm, Reiner Fischer

Abstract The processes of parametric amplification and oscillation in a medium with cubic polarization is investigated in parametric approximation. For the case of amplification the threshold power of the pump wave is calculated. The influence of a finite phase mismatching and of the initial phases on the amplification process is discussed. For larger values of mismatching a periodical change between amplification and depletion may appear. For the cases of a single resonant oscillator (for the signal wave only), a double resonant oscillator (for both signal and idler wave) and a double resonant oscillator with the pump wave reflected at the second mirror the threshold power of the pump wave and the rise time are calculated. Characteristic differences between four‐photon and three‐photon parametric oscillators are discussed. A numerical estimation of the threshold power for some media with nonlinear susceptibilities known from literature leads to the conclusion that the realization of a four‐photon parametric oscillator should be possible. In the appendix we discuss the influence of diffraction, focussing, birefrigence, nonstationarity and depletion of the pump wave on the studied process as well as the conditions under which these effects can be neglected.

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Noise Calculations for Optical Parametric Oscillators

David R. White, W. H. Louisell

A general approach for describing complex quantum-mechanical systems in terms of $c$-number variables is discussed on the basis of a formalism developed by Lax. We apply this formalism to the particular problem of noise fluctuations in parametric oscillators. From a microscopic Hamiltonian describing light interacting with optically active atoms in a cavity with loss, we arrive at an equation for the motion in terms of $c$-number variables. We then solve this system for the noise properties of the signal mode near threshold and above, when the pump and idler modes may be treated adiabatically, and compare our work with the results of Graham, who treated the signal and idler symmetrically.

Physical Review A•May 1, 1970•16 citations
Abstract Amplitude fluctuations of the signal wave of an optical parametric oscillator are evaluated for the case of nonresonant pumping by calculating the decrease of a random deviation ε of the signal amplitude from its stationary value during a round trip through the (ring) resonator. From this, a damping term governing the temporal behaviour of ε is deduced which, together with a Langevin force connected with the spontaneous decay of pump photons into both signal and idler photons, determines the amplitude fluctuations of the signal wave. In addition, it is shown that the same results may be obtained from a mode formalism for signal and idler waves taking into account the spatial variation of the pump wave following from Maxwell's equations.
Abstract The optical parametric oscillator is investigated theoretically for the general case of resonant pump, signal and idler waves. Mean photon numbers, threshold energies and conversion rates are calculated explicitly for different types of the waves (standing and running ones), where in the case of both signal and idler being standing waves the appearance of an oppositely running (“reflected”) pump wave is properly taken into account. In addition, from the equations of motion for the photon creation operators approximately valid rate equations are deduced which in certain cases predict a damped oscillating transition (analogous to the relaxation oscillations of a ruby laser) to the stationary state.
SUMMARY Parametric processes in the interaction of electromagnetic field with a medium in an optical cavity are well known to produce states of light with nonclassical properties. The quantum correlations between these states can be employed for a realization of an EinsteinPodolsky-Rosen (EPR) paradox [l], in the original formulation using continuous variables, as proposed [2] and experimentally demonstrated [3] in the case of Nondegenerate Parametric Amplification below threshold. In this paper we study EPR-like behaviour between the signal and idler fields emitted by a degenerate optical parametric oscillator (OPO) above threshold; these fields, generated by conversion of a pump photon of frequency 2w0 in two photons of frequency w. , are spatially separated in the far field, showing non-local quantum correlations. The system is described by the full Hamiltonian for the operators associated with the pump and signal fields; the optical cavity introduces proper damping constants and corresponding vacuum field operators [4,5]. Above the threshold for the generation of the signal field, we consider the fluctuations in the intensity difference and in the phase sum of the signal beams, which are related to their degree of mutual correlation. By linearizing around the stationary state, we calculate the spectral variances of these quantities (i.e. the Fourier transform of the two time correlation functions). The Heisenberg uncertainty principle sets a lower limit to the product of the spectral variances for the fluctuations in the intensity and in the phase of each individual signal field; by using the idler beam to infer results of measurements on the correlated signal beam, we have an EPR paradox if the product of the spectral variances for the fluctuations in the intensity difference and phase sum of the fields, is below the uncertainty limit. This feature is apparent for a wide interval around zero frequency in the far field of the active OPO system, depending on the parameter space; in particular there is maximum correlation between signal beams, and therefore maximum breaking of the uncertainty limit, if the damping constant of the signal field is larger than that of the pump field.
Abstract The processes of parametric amplification and oscillation in a medium with cubic polarization is investigated in parametric approximation. For the case of amplification the threshold power of the pump wave is calculated. The influence of a finite phase mismatching and of the initial phases on the amplification process is discussed. For larger values of mismatching a periodical change between amplification and depletion may appear. For the cases of a single resonant oscillator (for the signal wave only), a double resonant oscillator (for both signal and idler wave) and a double resonant oscillator with the pump wave reflected at the second mirror the threshold power of the pump wave and the rise time are calculated. Characteristic differences between four‐photon and three‐photon parametric oscillators are discussed. A numerical estimation of the threshold power for some media with nonlinear susceptibilities known from literature leads to the conclusion that the realization of a four‐photon parametric oscillator should be possible. In the appendix we discuss the influence of diffraction, focussing, birefrigence, nonstationarity and depletion of the pump wave on the studied process as well as the conditions under which these effects can be neglected.
A general approach for describing complex quantum-mechanical systems in terms of ccc-number variables is discussed on the basis of a formalism developed by Lax. We apply this formalism to the particular problem of noise fluctuations in parametric oscillators. From a microscopic Hamiltonian describing light interacting with optically active atoms in a cavity with loss, we arrive at an equation for the motion in terms of ccc-number variables. We then solve this system for the noise properties of the signal mode near threshold and above, when the pump and idler modes may be treated adiabatically, and compare our work with the results of Graham, who treated the signal and idler symmetrically.