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Annalen der Physik

Theorie der parametrischen Vier‐Photonen‐Wechselwirkung. I. Spontane Streuung

Heinz Steudel, J. Frahm

Published Jan 1, 1972• 4 citations
Publisher

Abstract

Abstract The structure of the phase matching surface in various media is discussed in detail. With respect to collinear scattering in uniaxial crystals simple approximations of the tuning curves near to the degenerated case are found. For a finite pump beam the distribution of the scattered light in k‐space is an interference pattern with its maximum of zeroth order at the phase matching surface. The structure of this pattern is analysed for a GAUSSian beam.

DOI
10.1002/andp.19724830310

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

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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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Abstract The structure of the phase matching surface in various media is discussed in detail. With respect to collinear scattering in uniaxial crystals simple approximations of the tuning curves near to the degenerated case are found. For a finite pump beam the distribution of the scattered light in k‐space is an interference pattern with its maximum of zeroth order at the phase matching surface. The structure of this pattern is analysed for a GAUSSian beam.
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.
A quantum theory is developed for the spontaneous optical parametric process in nonlinear, anisotropic, and dispersive media; the process is treated directly as a scattering problem rather than a problem involving equivalent coupled oscillators. Following the general procedure in quantum field theory, a general expression is obtained for the differential extinction coefficient, which contains all the information on the intensity of the spontaneous emission in the parametric scattering process. The tuning characteristics, beam divergence, and spectral properties of the spontaneous emission for colinear and noncolinear interactions are described in detail and illustrated with numerical examples on such standard nonlinear optical crystals as LiNbO3{\mathrm{O}}_{3}O3​, NH4{\mathrm{H}}_{4}H4​H2{\mathrm{H}}_{2}H2​PO4{\mathrm{O}}_{4}O4​, and KH2{\mathrm{H}}_{2}H2​PO4{\mathrm{O}}_{4}O4​ under various pumping conditions. Explicit formulas for the intensity of the spontaneous emission are also obtained. The spontaneous intensity calculated using the formulas given is shown to be in complete agreement with known experimental data on NH4{\mathrm{H}}_{4}H4​H2{\mathrm{H}}_{2}H2​PO4{\mathrm{O}}_{4}O4​ pumped at 3472 \AA{}.
The transcendental equations of vector phase matching are transformed into a fourth-order polynomial equation that admits an analytical solution. The real roots of this equation provide the optical axis orientations that are useful for efficient down-conversion in nonlinear uniaxial crystals. The production of entangled photon pairs is discussed in both collinear and non-collinear configurations of the spontaneous parametric down-conversion (SPDC) process. Degenerate and non-degenerate cases are also distinguished. As a practical example, SPDC processes of type-I and type-II are studied for beta-barium borate (BBO) crystals. The predictions are in very good agreement with experimental measurements already reported in the literature and include theoretical results of other authors as particular cases. Some properties that seem to be exclusive to BBO crystals are reported; the experimental verification of the latter would allow a better characterization of these crystals.