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.
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.
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.
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.
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.
We have applied the theory of McCumber to the (4)4I(13/2) ? (4)I(15/2) transition of Er(3+) -doped glasses and confirmed its ability to provide both spectra and accurate values of cross sections. For an Al/P-silica fiber, the exact McCumber treatment is within 3% of the measured emission cross section, whereas an Einstein analysis is off by >50%. An approximate treatment that eliminates the McCumber theory's requirement of detailed electronic structure has been developed and found to usually provide more accurate values than an Einstein analysis.
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.
Rapid development of optical fiber amplifiers, and more especially of the erbium-doped fiber amplifiers (EDFA) during the last decade, has benefited from extensive work on the configuration of the system itself: determination of pump wavelength, pumping configuration, improvement of noise performance, etc. However, less effort have been dedicated to study the interaction that influences the spectral profile of the gain bandwidth and particularly the relation between the glass composition and the spectroscopy of erbium. Here, we report a systematic determination of key parameters responsible for the global profile of the gain bandwidth: stark splitting, homogeneous broadening, and inhomogeneous broadening. The correlation between these parameters and the glass structure from one part and the behavior of the EDFA in amplification regime from another part is discussed. The quasiregular crystal-field splitting of tellurite and fluoride glasses is found to correlate well with the flatness of the gain profile. Moreover, the low-temperature homogeneous bandwidths extrapolated to room temperature can be correlated to the saturation spectral hole widths measured in the amplification regime.