^{1}, Victor A. Malyshev

^{1}and Jasper Knoester

^{1,a)}

### Abstract

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-Blochequations, 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.

This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). Support was also received from NanoNed, a national nanotechnology programme coordinated by the Dutch Ministry of Economic Affairs.

I. INTRODUCTION

II. MODEL AND FORMALISM

A. A single aggregate

B. The Maxwell equation

C. Truncated Maxwell-Blochequations

III. LINEAR REGIME

IV. STEADY-STATE ANALYSIS

A. Bistability equation

B. Phase diagram

C. Spectral distribution of the exciton population

V. TIME-DOMAIN ANALYSIS

A. Hysteresis loop

B. Switching time

VI. DISCUSSION OF DRIVING PARAMETERS

VII. SUMMARY AND CONCLUDING REMARKS

### Key Topics

- Excitons
- 27.0
- Polymers
- 19.0
- Absorption spectra
- 15.0
- Electric dipole moments
- 15.0
- Maxwell equations
- 11.0

## Figures

(a) The joint probability distribution of the transition energy and dimensionless transition dipole moment for -like states on localization segements, obtained for a disorder strength according to Eq. (5). We used chains of length with the monomer transition energy . The sampling was performed over 300 000 disorder realizations. Contour lines correspond to 10% of the peak value of the distribution. (b) The absorption spectrum . (c) The distribution of the transition dipole moment . The solid lines represent the results of calculations, whereas the open circles are fits by a Gaussian.

(a) The joint probability distribution of the transition energy and dimensionless transition dipole moment for -like states on localization segements, obtained for a disorder strength according to Eq. (5). We used chains of length with the monomer transition energy . The sampling was performed over 300 000 disorder realizations. Contour lines correspond to 10% of the peak value of the distribution. (b) The absorption spectrum . (c) The distribution of the transition dipole moment . The solid lines represent the results of calculations, whereas the open circles are fits by a Gaussian.

Examples of the input-output characteristics, demonstrating the occurrence of three-valued solutions to Eq. (12). In simulations, chains of sites and a disorder strength were used, corresponding to a HWHM . (a) The results obtained for different super-radiant constants at the optimal detuning , which corresponds to an incoming field which is resonant with the absorption maximum. The open circles, dotted, and solid curves represent, respectively, the data calculated for (the bistability threshold for ), (below the bistability threshold), and (above the bistability threshold). (b) The results obtained for and various detunings . The dotted and solid curves represent, respectively, the data calculated for and . The open circles show the same data as in panel (a).

Examples of the input-output characteristics, demonstrating the occurrence of three-valued solutions to Eq. (12). In simulations, chains of sites and a disorder strength were used, corresponding to a HWHM . (a) The results obtained for different super-radiant constants at the optimal detuning , which corresponds to an incoming field which is resonant with the absorption maximum. The open circles, dotted, and solid curves represent, respectively, the data calculated for (the bistability threshold for ), (below the bistability threshold), and (above the bistability threshold). (b) The results obtained for and various detunings . The dotted and solid curves represent, respectively, the data calculated for and . The open circles show the same data as in panel (a).

(a) Dependence of the critical superradiant constant on the detuning (solid line) calculated for the disorder strength . The dashed line shows the absorption spectrum (absorption only due to states). The dotted horizontal line indicates calculated for the optimal detuning . (b) Dependence of the switching intensity on the detuning calculated at the corresponding bistability threshold, i.e., with given in the panel (a).

(a) Dependence of the critical superradiant constant on the detuning (solid line) calculated for the disorder strength . The dashed line shows the absorption spectrum (absorption only due to states). The dotted horizontal line indicates calculated for the optimal detuning . (b) Dependence of the switching intensity on the detuning calculated at the corresponding bistability threshold, i.e., with given in the panel (a).

(a) Phase diagram of the bistable optical response of a thin film in the space obtained by solving Eq. (12) for . The open circles represent the numerical data points, whereas the solid line is a guide to the eye. Above (below) the solid line the film behaves in a bistable (stable) fashion. The solid line itself represents the dependence of the critical super-radiant constant , calculated for the optimal detuning , i.e., when the incoming field is tuned to the absorption band maximum. This gives the minimal for each . (b) The same data points as in the panel (a), only replotted as a function of , where is the mean value of the relaxation constant .

(a) Phase diagram of the bistable optical response of a thin film in the space obtained by solving Eq. (12) for . The open circles represent the numerical data points, whereas the solid line is a guide to the eye. Above (below) the solid line the film behaves in a bistable (stable) fashion. The solid line itself represents the dependence of the critical super-radiant constant , calculated for the optimal detuning , i.e., when the incoming field is tuned to the absorption band maximum. This gives the minimal for each . (b) The same data points as in the panel (a), only replotted as a function of , where is the mean value of the relaxation constant .

Population distributions (solid curves), calculated according to Eq. (13) for and , with the optimal detuning indicated by the vertical dashed line. Open circles show the absorption spectrum . Panel (a) represents below the upper switching threshold. The plotted distributions were calculated for the input intensities , and (from bottom to top). Panel (b) shows above the upper switching threshold. In the inset, the dependence of the full width at half maximum (FWHM) of on is plotted in units of the FWHM of the absorption spectrum.

Population distributions (solid curves), calculated according to Eq. (13) for and , with the optimal detuning indicated by the vertical dashed line. Open circles show the absorption spectrum . Panel (a) represents below the upper switching threshold. The plotted distributions were calculated for the input intensities , and (from bottom to top). Panel (b) shows above the upper switching threshold. In the inset, the dependence of the full width at half maximum (FWHM) of on is plotted in units of the FWHM of the absorption spectrum.

An example of the stable optical hysteresis loop of the transmitted intensity (the solid curve with arrows) obtained by numerically solving Eqs. (6a)–(6c) for a linear sweeping up and down of the input field intensity . The sweeping time is . The open circles represent the steady-state solution, Eq. (12). The calculations were performed for the following set of parameters: , , , and .

An example of the stable optical hysteresis loop of the transmitted intensity (the solid curve with arrows) obtained by numerically solving Eqs. (6a)–(6c) for a linear sweeping up and down of the input field intensity . The sweeping time is . The open circles represent the steady-state solution, Eq. (12). The calculations were performed for the following set of parameters: , , , and .

Kinetics of the transmitted field intensity approaching its stationary value (dashed line) after the incident field with intensity is turned on abruptly at . The value exceeds the upper switching threshold . The other parameters were chosen as in Fig. 6.

Kinetics of the transmitted field intensity approaching its stationary value (dashed line) after the incident field with intensity is turned on abruptly at . The value exceeds the upper switching threshold . The other parameters were chosen as in Fig. 6.

Relaxation time as a function of the excess input intensity at the upper switching threshold (indicated by the vertical dotted line). was calculated by turning on abruptly the incoming field at and waiting until the transmitted field intensity approaches its steady-state value (for more details, see the text). The open circles show the numerical results, while the solid line represents a best power-law fit given by Eq. (14). The calculations were performed for the set of parameters of Fig. 6.

Relaxation time as a function of the excess input intensity at the upper switching threshold (indicated by the vertical dotted line). was calculated by turning on abruptly the incoming field at and waiting until the transmitted field intensity approaches its steady-state value (for more details, see the text). The open circles show the numerical results, while the solid line represents a best power-law fit given by Eq. (14). The calculations were performed for the set of parameters of Fig. 6.

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