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Physical Review A

Amplification of spontaneous emission and coherence in three-dimensional systems

G. Hazak

Published Oct 1, 1990• 2 citations

Abstract

The steady-state wave equation, used in the literature for the description of amplified spontaneous emission (ASE) in a refractive medium with inverted atoms, focuses on a narrow frequency band at the atomic line center. In the present work, the effect of this approximation was studied by comparison to an alternative procedure that accounts for the full line shape. The three-dimensional time-dependent Maxwell-Bloch equations for ASE in a refraction guided system were solved as an initial-value problem by using transverse-mode expansion. The steady-state limit was approached only for quantities that are bilinear in the field (autorcorrelation function, complex degree of coherence, and radiation intensity). It was found that, except for a gain narrowing factor, the inclusion of the full line shape (or equivalently the detailed dynamics of atomic polarization) does not change the theoretical prediction for the amplification of mode intensities. However, it drastically changes the prediction for the behavior of cross correlations of mode amplitudes. In the high-gain limit the steady-state wave equation predicts that these cross correlations, normalized to the intensity of the involved modes, are independent of the system length and of the excitation strength of the modes. The solution of the Maxwell-Bloch equations predicts that the normalized cross correlations are exponentially decreasing functions of the system length and of the difference between the excitation strengths of the modes. It is therefore argued that the inclusion of the detailed atomic line shape is essential to the analysis of spatial coherence in ASE systems.

DOI: 10.1103/physreva.42.4169

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The steady-state wave equation, used in the literature for the description of amplified spontaneous emission (ASE) in a refractive medium with inverted atoms, focuses on a narrow frequency band at the atomic line center. In the present work, the effect of this approximation was studied by comparison to an alternative procedure that accounts for the full line shape. The three-dimensional time-dependent Maxwell-Bloch equations for ASE in a refraction guided system were solved as an initial-value problem by using transverse-mode expansion. The steady-state limit was approached only for quantities that are bilinear in the field (autorcorrelation function, complex degree of coherence, and radiation intensity). It was found that, except for a gain narrowing factor, the inclusion of the full line shape (or equivalently the detailed dynamics of atomic polarization) does not change the theoretical prediction for the amplification of mode intensities. However, it drastically changes the prediction for the behavior of cross correlations of mode amplitudes. In the high-gain limit the steady-state wave equation predicts that these cross correlations, normalized to the intensity of the involved modes, are independent of the system length and of the excitation strength of the modes. The solution of the Maxwell-Bloch equations predicts that the normalized cross correlations are exponentially decreasing functions of the system length and of the difference between the excitation strengths of the modes. It is therefore argued that the inclusion of the detailed atomic line shape is essential to the analysis of spatial coherence in ASE systems.

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.

Competition between amplified spontaneous emission (ASE) and the four-wave-mixing (FWM) process has been observed under conditions of two-photon resonant excitation of the sodium 3d level. The nature of the competition is that the FWM process is able to prevent the occurrence of ASE, even though the gain of the ASE process calculated in the absence of competition effects is much larger than that of FWM. The ASE is suppressed because the fields generated by the FWM process create a new excitation pathway connecting the ground and 3d levels, and under quite general conditions this pathway interferes destructively with that due solely to the applied laser field. These effects are modeled theoretically by solving perturbatively the density-matrix equations of the atomic system, thereby determining the population in the upper level and the nonlinear polarization of the medium. The coupling between the various optical fields due to the nonlinear polarization is described by coupled amplitude equations. The solution to these equations predicts that when the wave-vector mismatch is not too large the fields evolve spatially to reach steady-state values, and that the population excited to the 3d level by the total steady-state optical field is much smaller than that due to the incident laser field alone. We have observed experimentally the suppression of ASE by FWM and have observed that this suppression does not occur when the medium is excited with counterpropagating beams that cannot efficiently excite the FWM process. In addition, we have conducted a series of experiments that shows that the degree of suppression of ASE depends on the intensity and focusing characteristics of the incident laser as expected on the basis of our theoretical model.

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.

We simulate the transition from amplified spontaneous emission (ASE) to lasing in random systems with varying degrees of mode overlap. This is accomplished by solving the stochastic Maxwell-Bloch equations with the finite-difference time-domain method. Below lasing threshold, the continuous emission spectra are narrowed by frequency-dependent amplification. Our simulations reproduce the stochastic emission spikes in the spectra. Well-defined peaks, corresponding to the system resonances, emerge at higher pumping and are narrowed by stimulated emission before lasing takes place. Noise tends to distribute pump energy over many modes, resulting in multimode operation. Well above the lasing threshold, the effects of noise lessen and results become similar to those without noise. By comparing systems of different scattering strength, we find that weaker scattering extends the transition region from ASE to lasing, where the effects of noise are most significant.

The value of the Einstein

B

coefficient for stimulated emission for a collection of atoms interacting with a radiation field is shown to depend upon the velocity distribution of the excited atoms and also upon the variation of field intensity with frequency. An emission line narrowing phenomenon for radiating matter enclosed in a nonresonant cavity with reflecting walls is predicted. It is shown that a loss mechanism must be introduced if significant line narrowing is to result from a single pass of radiation through an amplifying medium. An additional contributing effect to the mode pushing and mode pulling phenomena in a gas laser is demonstrated. A method of setting a gas laser to a predetermined frequency is described.

We present a Hamiltonian-based, 3D theory in paraxial approximation for X-ray amplified spontaneous emission (or superfluorescence) pumped by X-ray free-electron laser. The seed field is included. The ensemble-averaged Heisenberg equations become Maxwell-Bloch equations if factorization of operator products is assumed and are adequate when the stimulated emission is dominant. The spontaneous emission is accounted for by adding a random noise term to the atomic coherence, the magnitude of which is uniquely determined from the fact that the expectation value of the normally-ordered product electric field operators associated with an atom does not factorize. The modified Maxwell-Bloch equation we developed reproduces the results of the previous 1D theory based on correlation functions.