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Physical Review A

Effects of spectral diffusion on frequency-domain four-wave-mixing spectroscopy

Hailin Wang, Duncan G. Steel

Published Apr 1, 1991• 21 citations
Publisher

Abstract

We report the results of a theoretical study of line shapes obtained in frequency-domain four-wave mixing in systems inhomogeneously broadened due to random fields in the presence of spectral diffusion. The results are based on a solution of the modified optical Bloch equations solved perturbatively to third order in the applied optical fields, assuming a strong redistribution model for the spectral diffusion. The results show that the redistribution of excitation can be directly observed in the line-shape function and can also lead to the presence of interference effects in the line shape.

DOI
10.1103/physreva.43.3823

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Physical Review A•Sep 1, 1997•21 citations
We report the results of a theoretical study of line shapes obtained in frequency-domain four-wave mixing in systems inhomogeneously broadened due to random fields in the presence of spectral diffusion. The results are based on a solution of the modified optical Bloch equations solved perturbatively to third order in the applied optical fields, assuming a strong redistribution model for the spectral diffusion. The results show that the redistribution of excitation can be directly observed in the line-shape function and can also lead to the presence of interference effects in the line shape.
Effects of spectral diffusion on the third-order nonlinear susceptibility for two- and three-level quantum systems, immersed in a thermal reservoir, are evaluated with a four-wave mixing (FWM) signal through the density matrix formalism. For this, inhomogeneously broadened two- and three-level quantum systems are used. In these models, the distribution of natural frequencies follows a Lorentzian function. The absorptive and dispersive nonlinear optical responses are determined for the considered quantum models. The results presented in this work show potential applications in the development of optical switches and provide a potential method to measure spectral diffusion relaxation times in three-level quantum systems.
The effect of an intense pump field and the chemical concentration of the solution on the forward four-wave mixing signal of molecular systems with an inhomogeneous distribution of resonance frequencies, has been studied. A perturbative treatment to all orders in the pump and to first order in the probe and signal beams, including the propagation of the fields as the variable to be studied, was used. The present results are compared to local regimes of propagation and to the case of a homogeneously distributed spectral line. The behaviour of the non-linear signal, characterized by tridimensional intensity surfaces, is explained in terms of saturation and/or absorption processes taking place under the present study conditions.
We have incorporated spectral diffusion in a system of three quantum states immersed in a thermal bath for the study of optical responses in four-wave mixing spectroscopy, using the density matrix formalism and the development of quantum relaxation theory. The primary objective of this work is to establish a generalized theoretical framework that evaluates how internal relaxation within all three quantum states—including a non-resonant third state—modifies the macroscopic nonlinear polarization of the medium. By introducing specific scaling parameters, we aim to isolate and quantify the influence of spectral diffusion in the third state, a feature typically omitted in existing literature. The results found allow giving explanations about the parametric amplification, as well as the possibility of generation of sensors in regions close to the degeneracy of optical frequencies. In our study we have been able to recognize previously published results for the optical susceptibility responses of absorption and scattering. Our model, unlike those already presented in this topic, assumes all states of the quantum system with internal relaxations. We found that the inclusion of spectral diffusion in the third state significantly impacts the system’s kinetic behavior, specifically eliminating “hole burning” effects in dispersion and parametric amplification in nonlinear absorption. These findings suggest that interband and intraband kinetic constants must be considered simultaneously to accurately model the ultra-fast relaxation times in complex molecular systems like organic dyes. The development of the problem is subject to rotating wave considerations in the selection of the Fourier component terms associated with both the coherences induced in the system and those coming from the canonical populations of the states. In addition, it involves the development of perturbation theory in the amplitudes of the incident electromagnetic fields associated with this nonlinear FWM type technique. Our treatments have been performed for frequency differences in the absorption range of organic dyes, without any restriction in the cases of near-zero tunings. Our proposal for the third state with spectral diffusion effects completely eliminates both the “hole burning” created in the dispersion and the parametric amplification generated in nonlinear absorption, when compared with results with the state outside resonance treated without spectral diffusion effects within the band. This highlights the importance of interband kinetic constants in comparison with those within the band.
The effects of finite-bandwidth excitation are incorporated into the theoretical treatment of degenerate four-wave mixing (DFWM) spectroscopy by employing a general description of applied electromagnetic fields, thereby permitting the transient nature of typical laser sources to be taken explicitly into account. A computationally efficient perturbative (weak-field) formalism is developed by exploiting exponentially-decaying wave forms as a flexible basis for expanding the temporal envelope functions of incident quasimonochromatic pulses. This approach has the distinct advantage of yielding time-domain response integrals that can be evaluated analytically since they are almost as simple in form as their monochromatic counterparts. The resulting frequency-domain expressions for the induced DFWM signal polarization reflect the pronounced influence of finite-bandwidth excitation without needing to cobble such behavior into phenomenological rates for population decay and coherence dissipation. Spectral line shapes are examined for both stationary and nonstationary ensembles of isolated (gas-phase) target molecules, with the latter analyses demonstrating the systematic degradation of spatial discrimination introduced by nonmonochromatic implementations of the sub-Doppler (phase-conjugate) four-wave mixing scheme. In keeping with recent experimental findings, the precipitous drop in signal intensity accompanying the action of collisional dephasing processes is predicted to be tempered greatly by the use of short-duration excitation pulses.
The spectral line shape of the four-wave mixing (FWM) signal, induced by broad bandwidth laser fields, is derived in the case of a nonresonant interaction. The laser fields are assumed to have a Lorentzian spectrum arising from Markovian stochastic fluctuations described by the phase-diffusion, chaotic field, and Gaussian-amplitude models. The usual ``phase conjugating'' geometry for FWM is considered where the nonresonant interaction is determined only by the statistics of, and the correlations between, the pump and probe fields. For each one of the four possible correlation states between the three input fields and for each of the three fluctuating field models, the line shape of the FWM signal is calculated and shown to be particularly sensitive to the field statistics.