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Molecular Physics

Absorption and emission of light by systems with closely spaced energy levels

J. Michael Schurr

Published Mar 1, 1973• 3 citations
Publisher

Abstract

A complete solution of the time-dependent Schrodinger equation for the overall absorption-emission process is presented for a particular model system possessing an excited state manifold of (nearly) uniformly spaced molecular eigenstates with a lorentzian distribution of transition moment. Results for both narrow and broad band excitation are derived, and it is shown that the fluorescence quantum yield is essentially the same for both broad band and narrow band excitations. It is argued that the lifetime of the incident photon state is important in determining the presence or absence of structure in the absorption spectrum and, consequently, that the absorption spectrum observed depends upon both the optical density of the sample and the intensity of the incident light.

DOI
10.1080/00268977300100531

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A complete solution of the time-dependent Schrodinger equation for the overall absorption-emission process is presented for a particular model system possessing an excited state manifold of (nearly) uniformly spaced molecular eigenstates with a lorentzian distribution of transition moment. Results for both narrow and broad band excitation are derived, and it is shown that the fluorescence quantum yield is essentially the same for both broad band and narrow band excitations. It is argued that the lifetime of the incident photon state is important in determining the presence or absence of structure in the absorption spectrum and, consequently, that the absorption spectrum observed depends upon both the optical density of the sample and the intensity of the incident light.
In this paper we apply the T matrix formalism of scattering theory to derive general expressions for the absorption cross sections, the cross sections for resonance fluorescence and the emission quantum yields from large molecules in the statistical limit. In the simple case of an isolated molecular resonance both the absorption line shape and the photon scattering cross section exhibit a Lorentzian distribution on the photon energy, the emission quantum yields are distributed among the ground state vibronic levels according to their radiative widths and, most important, the emission quantum yields are independent of the photon energy and of the spectral width of the exciting light. We were able to derive general expressions for the resonance scattering from a pair of overlapping resonances, including radiative corrections to infinite order. The absorption cross section does not vanish in the region of destructive interference but assumes a finite value which depends on the radiative widths. A sharp maximum in the partial and in the total emission quantum yields is exhibited in the destructive interference regions. This general scheme was applied to a pair of zero order discrete states, one of which is optically active, which interact with an optically inactive quasicontinuum. The energy dependent quantum yield depends on the total width of the radiatively impotent state and may exhibit a minimum. We have demonstrated that when interference effects are involved the decay characteristics of this system will differ for coherent and for narrow band excitation. The general formalism was utilized to derive approximate relations for the resonance fluorescence cross sections and for the quantum yield in the case of a Fano absorption line shape which are valid away from the interference region. Finally, we have applied the general theoretical scheme to the case of the direct photodissociation spectrum of molecules. We have demonstrated that a finite energy dependent emission quantum yield will be observed when a molecule is optically pumped into a dissociative continuum.
A two-state atom in a perturber gas is irradiated by intense light with an arbitrary spectral profile. The authors derive a general expression for the fluorescence spectrum in terms of the conventional low-intensity absorption profile and the spectral density of the incident light. This result, which contains previous results as special limiting cases, describes the transition from the three-line fluorescence spectrum excited by intense monochromatic light and the power-broadened Lorentzian spectrum produced by broad-band excitation.
We present a novel approach to molecular spectroscopy with light pulses of arbitrary strength and duration. The key quantity is the frequency-resolved net energy transfer dE/dω which reveals at which frequencies energy is transferred from the field to the molecule (absorption) or from the molecule to the field (stimulated emission). It is shown that dE/dω can be expressed as the Fourier transform of the cross-correlation function of the molecular polarization P(t) and the time derivative of the applied field, dℰ/dt. In this sense, it is formally equivalent to the absorption cross section under weak-field conditions which, as commonly known, can be represented as the Fourier transform of the autocorrelation function S(t). The time-dependent polarization P(t) is determined by exact integration of the time-dependent Schrödinger equation including the light-matter interaction to all orders. It is shown that under weak-field conditions the expression for dE/dω reduces to the well-known cross section formula in the time-dependent picture of spectroscopy, multiplied by the spectral intensity of the light pulse. Therefore, we consider the expression for the frequency-resolved energy transfer, which is valid for arbitrary electric fields, as the natural extension of the absorption cross section in the weak-field limit. Furthermore, dE/dω is shown to be formally equivalent to the change of the spectral intensity, ΔI(ω), of an optical pulse after transmission through a sample, the latter being derived by solving Maxwell’s equations under well-known approximations. The theory is applied to a simple one-dimensional model with two electronic states and the frequency-resolved energy transfer is investigated as a function of the field strength. For sufficiently strong fields, dE/dω exhibits transitions between essentially all vibrational levels in the ground and all states in the excited electronic manifold. The new expression distinguishes between absorption and emission and that is clearly seen in the spectra.
We develop a simple and general framework for understanding the relationships between absorption and emission in multilevel systems (molecules, semiconductors, condensed matter, etc.) in and out of equilibrium. Conditions under which the absorption and emission spectra display mirror image symmetry are determined. The relationship between the absorption and emission cross sections at frequency \ensuremath{\omega} and temperature T, previously derived by Neporent and McCumber for thermally equilibrated systems, is generalized and a new derivation is presented which extends its range of application to systems out of equilibrium. This derivation requires only microscopic reversibility and that the optically coupled manifold of levels belonging to both initial and final electronic states are each in internal thermal equilibrium at the same temperature T. The initial and final (e.g., ground and excited electronic) states need not be in equilibrium. If more than one excited-electronic-state manifold participates in the dynamics and these excited-electronic-state manifolds are not themselves in equilibrium, a somewhat more complicated relationship between absorption and emission can be obtained, provided that the internal degrees of freedom within the electronic-state manifolds are each in equilibrium at the same temperature.Effects of inhomogeneous broadening are examined. Good agreement is found between the Neporent-McCumber relationship and experimental absorption and emission spectra for the laser dyes rhodamine-6G and 2,5-biphenyloxazol PPO, and the solid-state laser crystal alexandrite. A new relationship between absorption and emission in direct-band-gap semiconductor materials is derived. This relationship allows us to calculate the emission from a direct-band-gap semiconductor provided the dependence of the band-gap energy on excitation level is not large. These relationships hold even when pumping conditions differ for emission and absorption experiments; the dependence of emission on excitation level can thereby be calculated. This permits the gain spectra of semiconductor diode lasers to be determined from their absorption spectra. In the limit that both the absorption and emission are measured for a thermally equilibrated nondegenerate direct-band-gap semiconductor system with identical pumping conditions for the absorption and emission experiments, our new relationship reduces to previous results.
The optical absorption of several substances has been found to depend in a reversible way on the incident light flux [1]-[5] of a high intensity laser. Moreover, it has been observed [6]-[7] that the absorption decreases not only at the laser wavelength but simultaneously within a large band around this wavelength. These effects may be explained by a time-dependant population distribution among the different excited states of absorbing molecules during the strong laser excitation [8]. The following experimental results enable us to determine the different time constants of importance in this phenomenon.
A general theoretical description is derived for the resonance fluorescence of dilute gases and molecular single crystals with special emphasis on optical rotation spectra. The validity of previous theoretical work on dilute gases is confirmed by proving that the damping constants connecting excited states on different molecules are zero. It is shown that the intensities of optical rotation spectra of molecular crystals are much larger than the corresponding quantities for gases. An important theoretical prediction is that the natural line breadth of a resonance fluorescence emission band belonging to an exciton state in a molecular crystal should be larger than the width of the corresponding optical absorption line. The dependence of the intensity of the optical rotation spectrum on the relative orientation of the crystal with respect to the incident beam of light is discussed.