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Photonics

Dynamic Extended Beer–Lambert Law with Photodissociation and Recombination

Xiaojia Li, Ju Wang, Jun Zhang

Published Sep 30, 2026• 0 citations
Publisher

Abstract

A dynamic extended Beer–Lambert law with photodissociation and recombination is proposed, which is built to describe the interactions between light and molecular gases. The model comprises two spatiotemporally coupled first-order nonlinear partial differential equations. Numerical solutions of these equations systematically reveal the nonlinear transmission properties and steady-state dissociation behavior of the medium from the weak- to intense-light regime. In the weak-light limit, the model naturally reduces to the classic Beer–Lambert law, with the system reaching a steady state characterized by high molecular density and exponential light attenuation. Conversely, in the intense-light regime, low molecular concentration and pronounced photoinduced transparency occurs, and light attenuates linearly instead of exponentially. A method for measuring the recombination rate constant under steady-state conditions is presented, which imposes significantly lower demands on both the light intensity and the temporal resolution of the measurement equipment compared to conventional approaches.

DOI
10.3390/photonics13100930

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A dynamic extended Beer–Lambert law with photodissociation and recombination is proposed, which is built to describe the interactions between light and molecular gases. The model comprises two spatiotemporally coupled first-order nonlinear partial differential equations. Numerical solutions of these equations systematically reveal the nonlinear transmission properties and steady-state dissociation behavior of the medium from the weak- to intense-light regime. In the weak-light limit, the model naturally reduces to the classic Beer–Lambert law, with the system reaching a steady state characterized by high molecular density and exponential light attenuation. Conversely, in the intense-light regime, low molecular concentration and pronounced photoinduced transparency occurs, and light attenuates linearly instead of exponentially. A method for measuring the recombination rate constant under steady-state conditions is presented, which imposes significantly lower demands on both the light intensity and the temporal resolution of the measurement equipment compared to conventional approaches.
The methods of calculating the deviations from Beer-Lambert's law and other related phenomena in solutions illuminated by a ruby laser are analyzed and criticized carefully. These methods include the steady-state and optically thin approximation as compared with the exact solution of a set of coupled differential equations which describe the temporal and spatial changes of the photons and populations of the various energy levels. Critical evaluation of the validity of the approximation is given. The computations are compared with the experimental data of Beer's law deviations, pulse shaping, recovery of the ground state after excitation, and blue and red fluorescence of solutions of cryptocyanine. A set of absorption cross sections and decay rates which fits best all these experiments is deduced.
The traditional Beer law for absorption in an optical medium is precise only at incident power densities lower than a few kW.At higher power densities this law fails because it neglects the processes of stimulated emission and spontaneous emission.In previous models that considering those processes,analytical expression for the absorption law could not be obtained.In order to solving this problem,the two-level energy rate equations is solved analytically and it leads to a generalized absorption law in this paper and then the applied criterion of the generalized Beer law is obtained and analyzed.The generalized absorption law reduces to Beer law at low power densities.By making comparison between the experimental result and the calculated result of the generalized Beer law,it shows that this generalized law is fast and exact because it accounts for absorption,stimulated and spontaneous emissions.
The strong-coupling limit of molecule formation in an atomic Bose-Einstein condensate via two-mode one-color photoassociation or sweep across a Feshbach resonance is examined using a basic nonlinear time-dependent two-state model. For the general class of term-crossing models with constant coupling, a common strategy for attacking the problem is developed based on the reduction of the initial system of semiclassical equations for atom-molecule amplitudes to a third-order nonlinear differential equation for the molecular state probability. This equation provides deriving exact solution for a class of periodic level-crossing models. These models reveal much in common with the Rabi problem. Discussing the strong-coupling limit for the general case of variable detuning, the equation is further truncated to a limit first-order nonlinear equation. Using this equation, the strong nonlinearity regime for the first Nikitin exponential-crossing model is analyzed and accurate asymptotic expressions for the nonlinear transition probability to the molecular state are derived. It is shown that, because of a finite final detuning involved, this model displays essential deviations from the Landau-Zener behavior. In particular, it is shown that in the limit of strong coupling the final conversion probability tends to 1∕61∕61∕6. Thus, in this case the strong interaction limit is not optimal for molecule formation. We have found that if optimal field intensity is applied the molecular probability is increased up to 1∕41∕41∕4 (i.e., the half of the initial atomic population).
The traditional Beer law for absorption in an optical medium is precise only at incident power densities lower than few kW.At higher power densities this law fails because it neglects the processes of stimulated emission and spontaneous emission.In previous models that considered those processes,analytical expression for the absorption law could not obtained.In order to solving this problem,the two-level energy rate equations is solved analytically and it leads to a generalized absorption law in this paper and then the applied criterion of the generalized Beer law is obtained and analyzed.The generalized absorption law reduces to Beer law at low power densities.By making comparison between the experimental result and the calculated result of the generalized Beer law,it shows that this generalized law is fast and exact because it accounts for absorption,stimulated and spontaneous emissions.
In all preceding chapters, we have tried to separate the effect of the light on the atoms from the reverse effect, the modification of the light by the atomic medium. In most cases this allows an explicit solution of the equations of motion and provides stable stationary states. In recent years, another approach to the interaction between light and matter has appeared, which emphasises strong coupling between the two. It uses the interactions as a tool to study general aspects of nonlinear dynamics. This chapter provides an introduction to this field, without trying to give a complete summary. Overview Resonant vapours as optically nonlinear media Mutual interactions So far, we have considered the effect of the laser on the atomic medium separately from the measurement of microscopic dynamics by the polarisation selective detection of transmitted light. In this approximation, the pump laser drives the atoms without suffering significant attenuation or polarisation changes. Conversely, the probe beam, which monitors the optical properties of the atomic medium, changes the microscopic state of the atoms only infinitesimally. This approach guarantees, e.g., that the response of the medium to the probe beam is linear. As stressed before, this assumption of one-sided interactions is always an approximation, since the conservation of energy and angular momentum make it impossible to change either of the two partial systems without compensating changes in the other part.