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Proceedings of the Physical Society

Parametric amplification near resonance in non-linear dispersive media

P N Butcher, R. Loudon, T P McLean

Published Mar 1, 1965• 41 citations
Publisher

Abstract

The behaviour of collinear signal, idler and pump waves in a non-linear medium without inversion symmetry is investigated under small-signal conditions when the frequency ω of either the signal or the idler approaches a transition frequency of the medium. The three waves are coupled together by the first-, second- and third-order susceptibility tensors of the medium. All three tensors exhibit a resonance in the case under consideration. The first-order resonance creates an absorption band and the third-order resonance produces stimulated Raman amplification. It is shown that the strength of the second-order resonance, which enters into the parametric effect, is determined by the product of the strengths of the other two. The growth constant of the signal and idler waves is a maximum when the mismatch of the propagation constants Δ k in the presence of the pump field takes a non-zero value equal to the decay constant at frequency ω times the ratio of the imaginary and real parts of the complex parametric growth constant, a quantity which is independent of the pump field strength. Except in the immediate neighbourhood of resonance, the frequency dependence of the optimum growth constant is controlled by the ratio of the peak Raman growth constant to the off-resonance parametric growth constant. When this ratio is small the optimum growth constant is never significantly larger than the off-resonance parametric value; when it is large the optimum growth constant slightly exceeds the peak Raman growth constant over a frequency range of 10-10 3 linewidths on one side of resonance. In the immediate neighbourhood of the resonance the optimum growth constant exhibits structure which is controlled by the anomalous dispersion and absorption resulting from the first-order resonance.

DOI
10.1088/0370-1328/85/3/316

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We investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, Eω1(x,t) and Eω2(x,t), with central frequencies ω1 and ω2 whose amplitudes are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order Eω1(x,t)Eω2(x,t)* in the field amplitudes. The first RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when ‖ω1−ω2‖ is small or comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T1 times of the system. We solve analytically for the optical response of the system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell–Bloch equations developed here to propagate time-dependent pulses.

The Journal of Chemical Physics•Dec 1, 1989•20 citations
The behaviour of collinear signal, idler and pump waves in a non-linear medium without inversion symmetry is investigated under small-signal conditions when the frequency ω of either the signal or the idler approaches a transition frequency of the medium. The three waves are coupled together by the first-, second- and third-order susceptibility tensors of the medium. All three tensors exhibit a resonance in the case under consideration. The first-order resonance creates an absorption band and the third-order resonance produces stimulated Raman amplification. It is shown that the strength of the second-order resonance, which enters into the parametric effect, is determined by the product of the strengths of the other two. The growth constant of the signal and idler waves is a maximum when the mismatch of the propagation constants Δ k in the presence of the pump field takes a non-zero value equal to the decay constant at frequency ω times the ratio of the imaginary and real parts of the complex parametric growth constant, a quantity which is independent of the pump field strength. Except in the immediate neighbourhood of resonance, the frequency dependence of the optimum growth constant is controlled by the ratio of the peak Raman growth constant to the off-resonance parametric growth constant. When this ratio is small the optimum growth constant is never significantly larger than the off-resonance parametric value; when it is large the optimum growth constant slightly exceeds the peak Raman growth constant over a frequency range of 10-10 3 linewidths on one side of resonance. In the immediate neighbourhood of the resonance the optimum growth constant exhibits structure which is controlled by the anomalous dispersion and absorption resulting from the first-order resonance.
Abstract We present results of a theoretical analysis of an optical parametric amplification (OPA) in a collinear nearly degenerate four-wave mixing configuration close to the resonance of a two-level transition in χ ′ ′ ′ media. It is performed both for homogeneously and for inhomogeneously broadened spectral line using a standard set of Bloch equations in the semiclassical approximation. Unlike the traditional approach of a coupled propagation of the signal/idler waves, we treat OPA as a coupled propagation of the amplitude and phase modulations of the pump wave with the signal/idler waves as the modulation sidebands. We show that for low frequency off-sets between the pump and the signal/idler waves, OPA process is basically manifested in a unidirectional transformation of the input amplitude modulation of the pump wave into the output phase modulation. The frequency band of the OPA is determined by detuning the pump wave from the resonance transition. The obtained results are applied to evaluate the expected OPA gain for the acetylene-filled micro-structured fiber with the resonance vibration-rotational transitions in the 1500–1550 nm spectral range.
An approximate theory of parametric amplification and stimulated Raman scattering is developed for a pumping field which is incoherent in terms of the transverse coordinates and time. Expressions are obtained for the increments of the average field and the conditions of validity of the obtained expressions are determined. It is shown, in particular, that if the longitudinal length of the correlation between the pump radiation and the signal wave is small in comparison with the length of correlation of this radiation with the idler wave, then the employed approximation describes with sufficient accuracy the transition from the case of δ-correlated pumping (incoherent interaction) to the case when the gain achieved in a distance equal to the correlation length of the pump and idler waves exceeds unity (quasicoherent interaction). The intensity increment is then close to double the increment of the average field or the signal wave. In addition, for the indicated ratio of the correlation lengths, it is relatively easy to calculate the correlation function at a distance from the point of entry into a crystal, which is short but sufficient for an appreciable gain. Analogous conclusions hold also for stimulated Raman scattering in a pumping field with a spectral width exceeding the spontaneous-scattering linewidth.
A quantum-statistical treatment of degenerate (single-mode) parametric amplification is given in which the pump is allowed to have time-dependent amplitude and phase. This relaxes earlier restrictions of a constant pump in exact resonance with the signal. Solutions are obtained for the case where the ratio of the time derivative of the pump phase to the coupling strength between pump and signal is constant. This includes the special case of simple offresonance, where the pump has a constant frequency differing from twice the signal frequency by a given amount (and has constant amplitude). The Wigner distribution for the signal mode is obtained (the PPP distribution does not in general exist as a well-defined function beyond a finite time interval for this process) and is used to determine the time development of the expectation values of various field operators of interest, such as field strengths and fluctuations. A critical point exists in the solution for the photon annihilation operator such that the behavior of the solution and the resulting quantities which it determines is divided into an exponential region and an oscillatory region. The critical point defines, from the expectation value for the photon number, the condition for amplification. For the case of simple off-resonance, this gives the frequency range of amplification. A calculation without using the rotating-wave approximation is given in the Appendix which shows that, to first order, the same range is obtained as when the approximation is used. The effect of damping is given, showing, e.g., how it slows the growth rate of the signal (in the region of amplification) and how it narrows the frequency range of amplification. The effect of a time-dependent pump (and damping) on the breakdown of the PPP distribution is also described.
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.
An analysis is made of generation of the sum frequency ω3 = 2ω1 + ω2 in the case when the pump frequency coincides with that of a two-photon transition 2ω1 = ω21, and the frequency ω3 is tuned near the resonance line of an atom. Under conditions of resonance with the frequency of the generated field, the parametric process has some characteristic features associated with a strong competition between effects involving increases in the nonlinear susceptibility, absorption, and changes in the phase matching. Moreover, in the presence of a fairly high-intensity field due to a neighboring transition at the frequency ω2, a nonlinear Raman process (which changes the characteristics of the medium at the frequency ω3) begins to play an important part. It is found that the highest parametric oscillation powers over a wide spectral range may be achieved by appropriately varying the intensity of the field due to the neighboring transition, the atom concentration, or the interaction length.
We investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, Eω1(x,t) and Eω2(x,t), with central frequencies ω1 and ω2 whose amplitudes are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order Eω1(x,t)Eω2(x,t)* in the field amplitudes. The first RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when ‖ω1−ω2‖ is small or comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T1 times of the system. We solve analytically for the optical response of the system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell–Bloch equations developed here to propagate time-dependent pulses.