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Physical Review A

Nonlinear optical susceptibilities of disordered aggregates: A comparison of schemes to account for intermolecular interactions

Jasper Knoester

Published Mar 1, 1993• 77 citations
Publisher

Abstract

The performance of approximate theories for the collective nonlinear optical response of assemblies of interacting molecules is investigated by comparing their predictions to exact calculations. To this end, the nonlinear absorption coefficient of linear molecular aggregates with energetic disorder is studied with the aid of Monte Carlo simulations. It is shown that the well-known local-field approximation (LFA) cannot describe the near-resonance nonlinear absorption spectrum for disorder values up to the nearest-neighbor intermolecular interactions; surprisingly, it is found that the validity of the LFA does not at all improve with growing disorder. By contrast, the ``excitonic two-level system'' approximation, in which each collective one-photon transition (Frenkel exciton) of the assembly is treated as an independent two-level system, turns out to describe the collective nonlinear response of the aggregates in a much better way and, moreover, rapidly improves for growing disorder values. It is shown that both models correctly describe the linear optical response and that, within the framework of the LFA, a close connection exists between the rotating-wave approximation and the Heitler-London approximation.

DOI
10.1103/physreva.47.2083

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The performance of approximate theories for the collective nonlinear optical response of assemblies of interacting molecules is investigated by comparing their predictions to exact calculations. To this end, the nonlinear absorption coefficient of linear molecular aggregates with energetic disorder is studied with the aid of Monte Carlo simulations. It is shown that the well-known local-field approximation (LFA) cannot describe the near-resonance nonlinear absorption spectrum for disorder values up to the nearest-neighbor intermolecular interactions; surprisingly, it is found that the validity of the LFA does not at all improve with growing disorder. By contrast, the ``excitonic two-level system'' approximation, in which each collective one-photon transition (Frenkel exciton) of the assembly is treated as an independent two-level system, turns out to describe the collective nonlinear response of the aggregates in a much better way and, moreover, rapidly improves for growing disorder values. It is shown that both models correctly describe the linear optical response and that, within the framework of the LFA, a close connection exists between the rotating-wave approximation and the Heitler-London approximation.
We perform a theoretical study of the nonlinear optical response of an ultrathin film consisting of oriented linear aggregates. A single aggregate is described by a Frenkel exciton Hamiltonian with uncorrelated on-site disorder. The exciton wave functions and energies are found exactly by numerically diagonalizing the Hamiltonian. The principal restriction we impose is that only the optical transitions between the ground state and optically dominant states of the one-exciton manifold are considered, whereas transitions to other states, including those of higher exciton manifolds, are neglected. The optical dynamics of the system is treated within the framework of truncated optical Maxwell-Bloch equations, in which the electric polarization is calculated by using a joint distribution of the transition frequency and the transition dipole moment of the optically dominant states. This function contains all the statistical information about these two quantities that govern the optical response and is obtained numerically by sampling many disorder realizations. We derive a steady-state equation that establishes a relationship between the output and input intensities of the electric field and show that within a certain range of the parameter space this equation exhibits a three-valued solution for the output field. A time-domain analysis is employed to investigate the stability of different branches of the three-valued solutions and to get insight into switching times. We discuss the possibility to experimentally verify the bistable behavior.
We study the nonlinear optical response of molecular assemblies consisting of coupled two-level molecules with arbitrary geometry including inhomogeneous broadening and exciton dephasing. An analytical expression for the third-order optical susceptibility is derived based on a certain closure of the equation-of-motion hierarchy. An exact expression for the third-order response of small crystalline aggregates with exciton population relaxation is derived, and used to show that for low relaxation rate \ensuremath{\Gamma} the cooperative part of the nonlinear response scales as ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{\ensuremath{-}}1/2}$ in d=1, and up to logarithmic corrections, as ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{\ensuremath{-}}1}$ in d=2 and 3. We show that cooperative enhancement is caused by exciton scattering. This is to be contrasted to other cooperative effects such as superradiance, which are related to the size of molecular coherence domains and therefore scale as ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{\ensuremath{-}}\mathit{d}/2}$. In a numerical study for dipole-dipole interactions in d=1, 2, and 3 dimensions we find that the local-field approximation is valid only in selected cases, and in general fails to reproduce the magnitude and the resonance structure of ${\mathrm{\ensuremath{\chi}}}^{(3)}$.
We generalize our recent work on the optical bistability of thin films of molecular aggregates [J. A. Klugkist et al., J. Chem. Phys. 127, 164705 (2007)] by accounting for the optical transitions from the one-exciton manifold to the two-exciton manifold as well as the exciton-exciton annihilation of the two-exciton states via a high-lying molecular vibronic term. We also include the relaxation from the vibronic level back to both the one-exciton manifold and the ground state. By selecting the dominant optical transitions between the ground state, the one-exciton manifold, and the two-exciton manifold, we reduce the problem to four levels, enabling us to describe the nonlinear optical response of the film. The one- and two-exciton states are obtained by diagonalizing a Frenkel Hamiltonian with an uncorrelated on-site (diagonal) disorder. The optical dynamics is described by means of the density matrix equations coupled to the electromagnetic field in the film. We show that the one- to two-exciton transitions followed by a fast exciton-exciton annihilation promote the occurrence of bistability and reduce the switching intensity. We provide estimates of pertinent parameters for actual materials and conclude that the effect can be realized.
We investigate the third-order nonlinear-optical susceptibility ${\ensuremath{\chi}}^{(3)}$ in molecular aggregates and its scaling with aggregate size. The aggregate is modeled as a collection of N-interacting, homogeneously broadened two-level systems, where the interaction includes both the static dipole-dipole coupling and superradiant coupling. Our expression for ${\ensuremath{\chi}}^{(3)}$, derived using Liouville-space Green-function techniques, contains contributions from excitons as well as biexcitons. The scaling of ${\ensuremath{\chi}}^{(3)}$ with aggregate size and its dependence on the laser detuning and homogeneous and inhomogeneous dephasing rates are analyzed. ${\ensuremath{\chi}}^{(3)}$ contains terms that scale as \ensuremath{\sim}N2{N}^{2}N2 and \ensuremath{\sim}N(N-1). However, under off-resonant conditions these terms interfere destructively, resulting in a linear \ensuremath{\sim}N scaling, which is precisely what is expected for N monomers. Detailed calculations of phase-conjugate degenerate four-wave mixing and pump-probe spectroscopy are performed. In the first case we predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton line shape. Pump-probe spectroscopy shows a series of biexciton absorption lines, some of which are superradiantly broadened.
The stationary nonlinear reflection and the time resolved four wave mixing signal from a molecular monolayer are calculated using Green function techniques. Cooperative resonant nonlinear response found in small aggregates suggests the existence of coherence size of order of optical wavelength. A new peak in the nonlinear reflection spectrum is predicted, which is missed by the local field approximation. For an infinite two dimensional molecular monolayer with transition dipole moments in the lattice plane, the momentum-dependent two exciton decay rate is found to be larger than the sum of the single exciton radiative decay rates, as predicted by the local field approximation.
A theoretical study of transmittivity of an ultrathin film (of thickness smaller than an optical wavelength) consisting of oriented linear aggregates is carried out. We show that within a certain domain of the incident field amplitude, optical transitions of a single disordered aggregate, the eigenstates of which are represented by the Frenkel exciton states, can be treated in the framework of a two-level approximation. This allows us to claim that an ensemble of linear aggregates is analogous to that comprised of inhomogeneously broadened two-level systems. Based on this analogy, we show that the transmittivity of the film can manifest a bistable behavior originating from the coupling of the aggregates via the emission (far-zone) field. This mechanism of bistability differs from that occurring in a single perfect aggregate. In the latter case, bistability results from the population-dependent resonance frequency shift of an individual aggregate, originating from the coupling of the molecules via the dipole–dipole (near-zone) field. Estimates of the driving parameters show that real candidates, to realize the type of behavior we found, are such well-studied objects as J-aggregates of polymethine dyes or thin films of thiophene oligomers.