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Results in Physics

Analyzing coherent and incoherent optical susceptibilities in four-wave mixing: stochasticity, symmetry, and intramolecular coupling

José Luis Paz, Cristhian Dávila, Marcos Loroño, Lenin A. González-Paz, Edgar Márquez, José R. Mora, Ysaías J. Alvarado

Published May 18, 2024• 2 citations
Publisher

Abstract

The nonlinear optical susceptibilities are analyzed in their optical propagation length considering the incoherent-coherent contributions with the dependence on the relaxation times. In addition, some symmetry properties of the Four-Wave mixing (FWM) signal concerning the frequency detuning exchange are used in the study, without restricting ourselves to the maxima of population oscillations. We analyze the behaviors of these optical responses in a vibronic coupling scheme using a molecule model consisting of two coupled harmonic curves of electronic energies with displaced minima in position and nuclear energies. The solvent effect in our model is treated through the natural Bohr frequency shift to a time-dependent function, with explicit manifestations in its comparison as if the upper state were broadened. Our results for both FWM signal propagation and nonlinear optical susceptibilities are sensitive to the intramolecular coupling parameters, solvent stochasticity, and perturbation order for the treatment of the weak probe beam. The consideration of a second-order probe beam in FWM would not represent a significant contribution to the predicted results, since its percentage contribution is negligible within the model development framework; however, this contribution becomes important when increasing the ratio between longitudinal and transverse relaxation times.

DOI
10.1016/j.rinp.2024.107765

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We study the nonlinear absorptive and dispersive optical properties of molecular systems immersed in a thermal reservoir interacting with a four-wave mixing (FWM) signal. Residual spin-orbit Hamiltonians are considered in order to take into account the internal structure of the molecule. As system parameters in the dissipation processes, transverse and longitudinal relaxation times are considered for stochastic solute–solvent interaction processes. The intramolecular coupling effects on the optical responses are studied using a molecule model consisting of two coupled harmonic curves of electronic energies with displaced minima in nuclear energies and positions. In this study, the complete frequency space is considered through the pump–probe detuning, without restricting the derivations to only maximums of population oscillations. This approach opens the possibility of studying the behavior of optical responses, which is very useful in experimental design. Our results indicate the sensitivity of the optical responses to parameters of the molecular structure as well as to those derived from the photonic process of FWM signal generation.
We consider the Four-Wave mixing (FWM) signal and the effects of the pump beam on the nonlinear optical responses of a two-level system where intramolecular coupling is modelled by two coupled harmonic electronic potentials, displaced both in their equilibrium minimum position and energy. We investigate the modifications introduced in the FWM signal, considering a simple representation for the vibrational structure and including molecular electronic states with nonzero permanent dipole moments. The critical quantities for this analysis are the transition and permanent dipole moments, which depend strongly on the intramolecular coupling. Our results show how both absorption and dispersion are affected by the vibronic coupling.
The nonlinear optical properties are studied theoretically of a molecular system immersed in a thermal bath and interacting with a four-wave mixing signal. A methodology based on cumulant expansions to obtain the average in the Fourier components associated with the coherences and populations, calculated by the optical stochastic Bloch equations, is employed. The system properties are analyzed as a function of the frequency detuning of the high-intensity pump beam and parameterized by the frequency detuning of the probe beam, both with respect to the Bohr frequency of the system. The longitudinal and transversal relaxation times and the bath effects, modeled as white noise, are also included as parameters in the calculation. Different cases, with degenerate and non-degenerate optical frequencies are examined and compared, resulting in differences related to the maximum in the population pulsation effects.
A strong pump-power dependence of the four-wave mixing (FWM) signal for an aqueous solution of Malachite green is reported. The characteristics of the pump-power dependence of the nonlinear signal are reproduced by a theoretical model based on the coupling between pump-probe, considering signal fields and propagation effects. The effect of the intramolecular coupling on the nonlinear intensity of the FWM signal is studied using a model molecule consisting of two-coupled harmonic curves of electronic energies with minima displaced in energy and nuclear positions. Two-vibrational states are considered while including non-adiabatic effects for the two-state model. Moreover, the coupling among the field components, as well as the propagation effects, are studied by considering a constant pump-intensity. Our calculation scheme, considering both the intramolecular coupling effects in the description of the molecular structure and the effects produced by the propagation of the FWM signal along the optical length, allows the exponential dependence of the latter, as the intensity of the pumping beam increases. Our treatments do not require the inclusion of other non-resonant processes outside the RWA approximation, due to the consideration of an adiabatic basis.
A study has been made of the effect of relaxation times on the four-wave mixing (FWM) signal response in frequency space for a two-level system where intramolecular coupling is explicitly taken into account within a vibronic model. The model is a generalization of the conventional two-level approach in that it includes a simplified description of the molecular structure and the explicit consideration of permanent dipole moments. Significant changes in both the intensity and shape of the peaks of the FWM spectrum are observed, due mainly to changes in the ratio between the longitudinal and transversal relaxation times, T 1/T 2.
This work studies the spatial propagation of the Four-Wave Mixing (FWM) signal of a molecular system (malachite green) with two electronic levels immersed in a thermal bath, under a perturbative treatment at all order in the pump field and to first order in the probe and signal fields.Using the Optical Stochastic Bloch Equations OSBE, we present three approximations, two of them with analytical solutions and the third with a numerical approach, where the effects of the variation of pump intensity through the path optical are considered.We compare these results with the analogous in absence of the thermal reservoir.The stochastic effects induced by the solvent, due to the experimental conditions as relaxation times, chemical concentration of the solution and optical frequencies of the fields, diminishes the intensity responses compared with the same in absence of the bath.