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Journal of the Optical Society of America B

Experimental evidence of the validity of the McCumber theory relating emission and absorption for rare-earth glasses

Rodica M. Martin, Richard S. Quimby

Published Aug 31, 2006• 86 citations
Publisher

Abstract

The validity of the theory of McCumber [Phys. Rev.136, A954-A957 (1964)] has been tested by applying it to a number of ground-state transition in various rare-earth-doped glasses. Special attention was given to those aspects of the experimental procedure that can lead to systematic errors, such as reabsorption of fluorescence and baseline subtraction uncertainties in the absorption measurements. To ensure consistency between absorption and fluorescence measurements, we used the same geometry for light collection and measurement. With these experimental procedures properly implemented, we find that in all cases there is excellent agreement between the spectral shape of calculated and measured cross-section spectra at room temperature. This is true even for the thermally coupled (2H9/2,4F5/2) and 4F3/2 levels of Nd, which span an energy range of ∼2000 cm−1, much larger than the typical width of a single Stark level manifold. The results suggest that, at room temperature, the McCumber theory is not restricted to crystalline hosts but remains valid for the broader transitions characteristic of rare-earth-doped glass.

DOI
10.1364/josab.23.001770

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Physical Review A•Aug 1, 1988•37 citations
The validity of the theory of McCumber [Phys. Rev.136, A954-A957 (1964)] has been tested by applying it to a number of ground-state transition in various rare-earth-doped glasses. Special attention was given to those aspects of the experimental procedure that can lead to systematic errors, such as reabsorption of fluorescence and baseline subtraction uncertainties in the absorption measurements. To ensure consistency between absorption and fluorescence measurements, we used the same geometry for light collection and measurement. With these experimental procedures properly implemented, we find that in all cases there is excellent agreement between the spectral shape of calculated and measured cross-section spectra at room temperature. This is true even for the thermally coupled (2H9/2,4F5/2) and 4F3/2 levels of Nd, which span an energy range of ∼2000 cm−1, much larger than the typical width of a single Stark level manifold. The results suggest that, at room temperature, the McCumber theory is not restricted to crystalline hosts but remains valid for the broader transitions characteristic of rare-earth-doped glass.
The relation between absorption and emission spectra in a rare-earth doped solid is explored analytically and through numerical simulations on a model system. It is found that the widely used theory of McCumber [D. E. McCumber, Phys. Rev. 136, A954 (1964)] is strictly valid only when the linewidth of each individual transition between Stark levels is ≪kBT. When this assumption is relaxed, deviations between actual simulated spectra and McCumber-calculated spectra are found for both homogeneous and inhomogeneous broadening. The McCumber transform introduces a characteristic distortion of each component transition, in which the low-energy side of the spectrum is increased, and the high-energy side is decreased. The average degree of distortion for the model system is calculated to be 5% for a homogeneous width of 50 cm−1 at room temperature, but only 1% for a purely inhomogeneous width of 50 cm−1. For both homogeneous and inhomogeneous spectra, the distortion varies with the linewidth Δν as (Δν)3/2. The dependence of the distortion on temperature is different for the two types of line shapes, however: ∼T2 for homogeneous and ∼T−1 for inhomogeneous broadening. For most rare-earth doped glasses and crystals at room temperature, the average degree of distortion is expected to be on the order of 5% or less, although the actual discrepancy in the spectral wings can be much larger. The effects described here are likely to be of importance when using McCumber’s theory to obtain accurate cross section values in the spectral wings of rare-earth transitions.
In this paper, we present a low-temperature study of rare earth doped glasses, which are important for creating compact, efficient gain media. We resolved effective Stark components of erbium doped glasses by analyzing their absorption cross sections at T=293K and T=110K. Using these Stark energy levels and the McCumber theory, we calculated the radiative lifetime of the I/-I/ transition in erbium. Our calculated lifetime agrees with the measured lifetime for this transition better than previously accepted methods.
The reciprocity relation of McCumber which states that the emission and absorption spectra of broad solid-state laser transitions are related by the classical Boltzmann factor is critically examined. It is argued that it should be treated as an exact property of homogeneously broadened amorphous host gain media at room temperature. The contradiction between these conclusions and those of recent literature arguing against the applicability of the McCumber relation for erbium doped silica is resolved. Experimental support for the McCumber relationship in erbium doped silica fiber is provided.
Persistent spectral holeburning is known to occur in rare-earth doped silica-based glasses,1-3 where the mechanism is attributed to light induced changes in the coupling of tunnelling systems (TLS) which are characteristic of the amorphous state. A number of hole parameters such as lifetime, saturation depth and temperature dependence of the width, can provide important information on the density of states and dynamics of the TLS. We present the first data on spectral holeburning in a rare-earth doped heavy metal fluoride glass (HMF, Rennes) and compare with data obtained on a silicate glass (ED-2, Owens-Illinois). For this study we have chosen the well known 4I9/2 ← → 4F3/2 transition of the Nd3+ ion whose absorption peaks around 870nm (HMF) and 880nm (ED-2). Previous holeburning and photon echo measurements have been reported on the 4I9/2 ← → 4G5/2 transition of the Nd3+ of bulk ED-2 glass4 and on the 4I9/2 ← → 4F3/2 transition of the Nd3' transition of a pure silica fiber.3,5
Excited-state absorption spectra of erbium and thulium doped gallium lanthanum sulfide glasses reveal transitions whose final 4f4f4f states do not appear in the ground state absorption spectra due to resonance with the fundamental absorption of the glass host. The results give direct evidence for the model of isolated rare-earth ions with highly shielded 4f4f4f levels showing no noticeable interaction with the host material despite resonance. Only the energy splitting of the 4f4f4f levels is relevant for energy transfer processes between rare-earth ions and the host material, whereas the absolute energetic position is unimportant.
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.