(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).
(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 .
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 .
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.
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