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Annalen der Physik

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

W. Brunner, Henryk Paul, A. Bandilla

Published Jan 1, 1971• 8 citations
Publisher

Abstract

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.

DOI
10.1002/andp.19714820108

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Theorie der parametrischen Vier‐Photonen‐Wechselwirkung. IV. Der optische parametrische Oszillator mit resonanter Pumpwelle.

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

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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.

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Theorie der parametrischen Vier‐Photonen‐Wechselwirkung. III. Verstärkung und Oszillation bei Berücksichtigung der Rückwirkung auf die Pumpwelle

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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.

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Ulrike Herzog, Matthias Scholz, Oliver J. Benson

We present a quantum-theoretical treatment of biphoton generation in single-resonant type-II parametric down-conversion. The nonlinear medium is continuously pumped and is placed inside a cavity which is resonant for the signal field, but nonresonant for the idler deflected by an intracavity polarizing beam splitter. The intensity of the classical pump is assumed to be sufficiently low in order to yield a biphoton production rate that is small compared to the cavity loss rate. Explicit expressions are derived for the rate of biphoton generation and for the biphoton wave function. The output spectra of the signal and idler field are determined, as well as the second-order signal-idler cross-correlation function which is shown to be asymmetric with respect to the time delay. Due to frequency entanglement in the signal-idler photon pair, the idler spectrum is found to reveal the longitudinal mode structure of the cavity, even though the idler field is not resonant.

arXiv:0709.2829•Physical Review A•Feb 28, 2008•39 citations
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.
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 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.
Abstract The processes of parametric amplification and oscillation in a medium with cubic polarization are investigated theoretically taking into consideration the depletion of the pump wave. We suppose that phase‐matching is realized exactly (a possible nonlinear detuning of the exact phase‐matching should be compensated by a suitable subsequent adjustment of the frequencies). A consequence of linear losses is the appearance of a threshold for the parametric amplification. There exist a maximum length, a minimum length and an optimum length of the nonlinear medium for the process of parametric amplification. The threshold intensity, the optimum length of the nonlinear medium and the optimum reflection coefficient for outcoupling are determined for two arrangements of the double resonant parametric four‐photon oscillator (ring resonator, FABRY‐PEROT resonator). It is shown that for the above mentioned arrangements of the single resonant and the double resonant four‐photon oscillator the efficiency is a monotonous function of the threshold excess. On the contrary in case of the three‐photon oscillator there exists an optimum value of the threshold excess.
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.
We present a quantum-theoretical treatment of biphoton generation in single-resonant type-II parametric down-conversion. The nonlinear medium is continuously pumped and is placed inside a cavity which is resonant for the signal field, but nonresonant for the idler deflected by an intracavity polarizing beam splitter. The intensity of the classical pump is assumed to be sufficiently low in order to yield a biphoton production rate that is small compared to the cavity loss rate. Explicit expressions are derived for the rate of biphoton generation and for the biphoton wave function. The output spectra of the signal and idler field are determined, as well as the second-order signal-idler cross-correlation function which is shown to be asymmetric with respect to the time delay. Due to frequency entanglement in the signal-idler photon pair, the idler spectrum is found to reveal the longitudinal mode structure of the cavity, even though the idler field is not resonant.