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Physical Review A

Beer-Lambert law in the time domain

M. Žitnik, Špela Krušič, K. Bučar, A. Mihelič

Published Jun 29, 2018• 9 citations
Publisher

Abstract

According to the Beer-Lambert law, the spectral density of weak electromagnetic radiation is attenuated exponentially, with the characteristic length depending on atomic density and the frequency-dependent photoabsorption cross section of the target. We discuss an analog of the Beer-Lambert law in the time domain, showing that the temporal profile of the initial light pulse enters explicitly into each term of expansion of the exponential function as a multiconvolution with the corresponding temporal response of an atom. The form invokes a description of pulse propagation in terms of the temporal base functions, which may be precalculated for a chosen initial pulse shape, allowing us to obtain the attenuated profile at a given target depth simply by summing up a sufficient number of weighted base functions. The variant with a 100-fs initial Gaussian pulse is worked out in detail, with the atomic response exhibiting either pure exponential or mixed prompt-exponential decay, characteristic of a photoexcited bound state or a Fano resonance. The method is shown to be valid for an arbitrary temporal profile of the initial electric field and is illustrated for pulses resonantly tuned to the low-lying doubly excited states of the He atom.

DOI
10.1103/physreva.97.063424

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According to the Beer-Lambert law, the spectral density of weak electromagnetic radiation is attenuated exponentially, with the characteristic length depending on atomic density and the frequency-dependent photoabsorption cross section of the target. We discuss an analog of the Beer-Lambert law in the time domain, showing that the temporal profile of the initial light pulse enters explicitly into each term of expansion of the exponential function as a multiconvolution with the corresponding temporal response of an atom. The form invokes a description of pulse propagation in terms of the temporal base functions, which may be precalculated for a chosen initial pulse shape, allowing us to obtain the attenuated profile at a given target depth simply by summing up a sufficient number of weighted base functions. The variant with a 100-fs initial Gaussian pulse is worked out in detail, with the atomic response exhibiting either pure exponential or mixed prompt-exponential decay, characteristic of a photoexcited bound state or a Fano resonance. The method is shown to be valid for an arbitrary temporal profile of the initial electric field and is illustrated for pulses resonantly tuned to the low-lying doubly excited states of the He atom.
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 lossy propagation law (generalization of Lambert-Beer's law for classical radiation loss) for non-classical, dual-mode entangled states is derived from first principles, using an infinite-series of beam splitters to model continuous photon loss. This model is general enough to accommodate stray-photon noise along the propagation, as well as amplitude attenuation. An explicit analytical expression for the density matrix as a function of propagation distance is obtained for completely general input states with bounded photon number in each mode. The result is analyzed numerically for various examples of input states. For N00N state input, the loss of coherence and entanglement is super exponential, as predicted by a number of previous studies. However, for generic input states, where the coefficients are generated randomly, the decay of coherence is very different; in fact no worse than the classical Beer-Lambert law. More surprisingly, there is a plateau at a mid-range interval in propagation distance where the loss is in fact sub-classical, following which it resumes the classical rate. The qualitative behavior of the decay of entanglement for two-mode propagation is also analyzed numerically for ensembles of random states using the behavior of negativity as a function of propagation distance.
The broad white-light continuum generated on the passage of ultrashort pulses of laser light through suitable media has found numerous applications in the physical, chemical, and biological sciences. Many of these are based on use of the continuum to probe the absorption characteristics of transient species, in order to obtain information on their decay kinetics. The processing of data from such experiments is generally based on the implicit assumption that the absorption of white light is subject to the normal Beer-Lambert law, in that the absorption at any particular frequency is linearly proportional to the intensity of the probe light at that frequency. In this paper it is shown that the validity of this underlying assumption should be seriously questioned. At high intensities, the uncertainty principle allows for photon pairs with differing frequencies to provide the energy for the cooperative excitation of molecules in close proximity. This nonlinear absorption, which depends on the frequency autocorrelation function and intensity of the continuum light, results in a rate of absorption which may depart significantly from the normal Beer-Lambert behavior. Hence the interpretation of continuum-probe experiments may need to be reexamined.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
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.
It is shown that homogeneous line-broadening in a diffuse cold atom cloud is proportional to the resonant optical depth of the cloud. Furthermore, it is demonstrated how the strong directionality of the coherent interactions causes the cloud's spectra to depend strongly on its shape, even when the cloud is held at constant densities. These two numerical observations can be predicted analytically by extending the single-photon wave-function model. Lastly, elongating a cloud along the line of laser propagation causes the excitation probability distribution to deviate from the exponential decay predicted by the Beer--Lambert law to the extent where the atoms at the back of the cloud are more excited than the atoms at the front. These calculations are conducted at the low densities relevant to recent experiments.