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Annalen der Physik

Calculation of Coherence Artefacts in Phase‐Conjugated Signals

B. Wilhelmi

Published Jan 1, 1990• 1 citation
Publisher

Abstract

Abstract The intensity of the phase‐conjugated signal in degenerate fourwave mixing depends on the delay of the signal wave with respect to the pump waves if all waves originate from one and the same primary source. The coherence peak, which appears at zero delay, has been calculated for an ensemble of two‐level absorbers in the rate equation approximation. The height of the coherence peak depends on the strength of the frequency fluctuations of the primary light source and on the relaxation time of the absorber molecules. The width of the coherence peak is given by the correlation time of the frequency fluctuations.

DOI
10.1002/andp.19905020226

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Abstract The intensity of the phase‐conjugated signal in degenerate fourwave mixing depends on the delay of the signal wave with respect to the pump waves if all waves originate from one and the same primary source. The coherence peak, which appears at zero delay, has been calculated for an ensemble of two‐level absorbers in the rate equation approximation. The height of the coherence peak depends on the strength of the frequency fluctuations of the primary light source and on the relaxation time of the absorber molecules. The width of the coherence peak is given by the correlation time of the frequency fluctuations.
Signal intensity as a function of mutual delay time of three incoherent pump beams in the degenerate four-wave mixing (DFWM) process has been calculated with the second-order correlation time of the fields being much shorter than both the dephasing time T 2 and the inverse of the inhomogeneous broadening width δω. The results show that when broadening is purely homogeneous and extremely inhomogeneous, the curve of the DFWM signal intensity as a function of the delay time τ shows no peak shift, while when the inhomogeneous broadening is finite, the peak shift is related to the dephasing time T 2 and the inhomogeneous width δω, and can thus be used to estimate T 2.
The theory of resonant four‐wave mixing using one and two‐photon transitions is described in terms of saturation spectroscopy. Detuning the signal wave from the pump wave in the one‐photon case leads to a narrow‐band reflection spectrum, with width given by 1/T1, where T1 is the lifetime of the population difference. In the corresponding time‐domain picture (the fourth dimension), this produces a reflected pulse at least as long as T1, regardless of the signal. This time‐domain picture also leads to a novel kind of coherent transient spectroscopy. In the two‐photon case, an infinite number of levels are involved in principle, i.e., phase conjugation in infinite dimensions! In this case, two kinds of mechanisms compete in generating conjugate waves leading to two phenomena that do not occur in two‐level media: 1) a double‐peaked reflection spectrum and 2) coupled‐mode oscillation in absorbing media. In addition, due to the dynamic‐Stark shifted line center, the reflection coefficient approaches a constant value at large intensities, unlike the two‐level coefficient which bleaches to zero.
Current experimental studies of degenerate four-wave mixing in saturable absorbers1,2 have shown that the dependence of the reflectivity on the pump intensity exhibits a double maxima. This phenomenon can arise from the fact that the medium approximates a set of four-level quantum systems. We have carried out a detailed analysis of the nonlinear processes of degenerate four-wave mixing in a four-level system. Because resonant saturable absorbers are being considered for systems applications, e.g., optical phase conjugation and wide FOV filters, the results presented are relevant to various optical phase conjugation devices. There are two distinct regimes which are of interest. The first case is when the intramolecular relaxation is very weak or absent. The second is when the intramolecular relaxation is strong. For the weak case the medium behaves like a set of two distinct dipole radiation sources which can generate interference effects.
We have derived an analytic expression for the intensity distribution of two counterpropagating waves of arbitrary input intensity in a saturable absorber. We use this distribution to extend the theory of degenerate four-wave mixing to include the effects of pump wave absorption. In particular, the spatial variation of the nonlinear absorption and coupling constants that appear in the coupled-amplitude equations for the probe and the signal (i.e., conjugate) waves are determined. These coupled-amplitude equations are solved numerically in a noniterative manner, leading to a prediction for the phase-conjugate reflectivity. The nature of the predictions differs qualitatively from those1,2 derived from theories that ignore pumpwave absorption, especially for the case of off-line-center excitation. At large values of the input pump intensities, the phase conjugate reflectivity can be larger than that predicted by theories that ignore pump-wave absorption.
Measurements of the relaxation times T1 and T2 of a two-level system by nonlinear transient spectroscopic methods are discussed, where delayed incident pulses are assumed to have correlation times much shorter than T1 or T2 but varied widths. In particular, a delayed degenerate four-wave mixing (photon echoe) and apump-and-probe type transmission measurement with incoherent long pulses are considered in detail, and a principle of their use for ultrafast spectroscopy is clarified. If the field is steady and T1>>T2, the time resolutionof T2measurement by the delayed four-wave mixlng is determined by the field correlation, therefore by the spectral width of the incident light irrespective of the pulse width. The time resoution of the T1 measurement by the pump-and-probe method is determined by the intensity correlation of the incident light, whereasthe one by the delayed four-wave mixing by a higher order correlation.