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IEEE Journal of Quantum Electronics

Theory of optical-gain measurements

Peter S. Cross, William G. Oldham

Published May 1, 1975• 66 citations
Publisher

Abstract

The widely used technique for determining optical gain and absorption from single-pass, amplified spontaneous emission (ASE) is analyzed with particular attention given to active-volume saturation and three-dimensional effects. The results for homogeneously broadened media show that the population inversion and hence the gain in the active region is diminished by high levels of spontaneous and ASE. The externally measured gain is further reduced for large output collection apertures that admit incompletely amplified off-axis rays. Criteria are given for the excitation and detection geometries required to ensure accurate (∼10-percent) gain measurement. The simultaneous measurement of the equilibrium absorption coefficient can be made under less restrictive conditions.

DOI
10.1109/jqe.1975.1068592

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The widely used technique for determining optical gain and absorption from single-pass, amplified spontaneous emission (ASE) is analyzed with particular attention given to active-volume saturation and three-dimensional effects. The results for homogeneously broadened media show that the population inversion and hence the gain in the active region is diminished by high levels of spontaneous and ASE. The externally measured gain is further reduced for large output collection apertures that admit incompletely amplified off-axis rays. Criteria are given for the excitation and detection geometries required to ensure accurate (∼10-percent) gain measurement. The simultaneous measurement of the equilibrium absorption coefficient can be made under less restrictive conditions.
Amplified spontaneous emission (ASE) occurs in media with large gain, and affects both the magnitude and the spatial distribution of the inversion. In this work we theoretically study the effect of ASE in three-dimensional, rectangular slab amplifiers, using Monte Carlo type computer simulations. We found that in one-dimensional amplifiers ASE is always larger at the edges so that the inversion has a maxima at the center of the amplifier. However, in two- and three-dimensional amplifiers, the inversion has a minimum at the center of the amplifier for low gain, and a maximum at the center of the amplifier for high gain. Thus, the inversion profile can be changed by increasing the gain from a minimum at the center, through a plateau, to a maximum at the center. A simple analytical theory was developed and agrees with these results.
The theory of amplified spontaneous emission (ASE) and the gain–length product at saturation is revised. Formulas valid for both peak and integrated lines are given. In the latter case the solution for a general Voigt stimulated-emission line shape is presented, whereas in the two limiting cases of Gaussian and Lorentzian line shapes an approximation better than the one obtained by the steepest-descent method is successfully achieved. Various parameters that take into account the ratio of spontaneous to stimulated linewidths, the form factor depending on the linewidths of the inhomogeneous and homogeneous convoluted line shapes, and the population inversion fraction arise from the theory, giving the complete functional dependence for both the ASE photon flux and the gain–length product.
Paraxial Maxwell-Bloch theory is used to describe amplified spontaneous emission (ASE) in strongly inverted media. Counterpropagation of waves coupled through gain saturation and diffraction, and pumping by a laser beam which propagates transverse to the amplifier are included in the treatment. The reactive part of the medium polarization is calculated by population-rate equations in the limit of rapid dephasing of the atomic dipoles. The spontaneous emission noise source needed for the numerical simulation of Maxwell's equations is derived from microscopic theory by coarse-graining, thereby establishing the compatibility of the spherically symmetric microscopic source with paraxial propagation theory. The three coupled Maxwell equations for the counterpropagating ASE waves and the transversely propagating pump wave are solved numerically in time and two spatial dimensions. This simulation is used to calculate the longitudinal profile of the transverse fluorescence, which measures the saturation of the gain, the longitudinal profile of the transmitted pump wave, which measures the saturation of pump absorption, and the transverse coherence function of the radiation emitted from the ends of the amplifier.
A new method for determining the gain in high gain amplifiers is reported. The technique only requires measurements of the relative intensity and the output spectral width of the amplified spontaneous emission from the amplifier as a function of input pumping. The results of an experimental test of this technique using a flashlamp-pumped rhodamine 6G dye amplifier are described. Simple approximate analytical models are used to compare with experiment thereby determining the gain. The results are found to be in good agreement with both more accurate gain measurements and an alternative technique which can be applied over a restricted range of operational parameters.
The study of optical effects in active media is rich with unexpected consequences. For instance, one may imagine that absorption and amplification in a dielectric medium will exhibit some sort of symmetry because both are related to the same imaginary part of the dielectric constant, except for a sign change. It turns out that such a symmetry does not exist [1]. This issue has also been investigated in detail for random or disordered gain media, with varying viewpoints [2]. Traditionally, much of the research on amplifying media has considered the interaction of light within the entire volume of such a medium. Recent interest in metamaterials and other esoteric structures in which plasmons are used to manipulate optical signals has brought attention to the role of surface waves in active dielectrics [3, 4]. From a fundamental perspective, the main difference between active and passive media is that spontaneous emission cannot be avoided in gain media. As a result, a rigorous analysis of gain media demands a quantum-mechanical treatment. Spontaneous emission in a gain medium depends not only on the material properties of that medium but also on the optical modes supported by the structure containing that material [5]. By a clever design of this structure (e.g., photonic crystals or microdisk resonators with a metallic cladding), it is possible to control the local density of optical modes and the spontaneous emission process itself. In Chapter 4, we discuss how to model a gain medium under such conditions by deploying optical Bloch equations.