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Physical Review A

Spatial correlation effects in a laser

Ashish Agarwal, S. Chopra

Published Sep 1, 1996• 2 citations
Publisher

Abstract

Spatial correlation effects are analyzed in a laser near and below its threshold of oscillation by integrating the modifications in the correlation function of intensity fluctuations and related parameters. The pump parameter of the laser is shown to depend on the field spatial coherence which arise due to transverse-mode structure of the laser field implying modifications of the spectrum in the time domain. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.2503

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Spatial correlation effects are analyzed in a laser near and below its threshold of oscillation by integrating the modifications in the correlation function of intensity fluctuations and related parameters. The pump parameter of the laser is shown to depend on the field spatial coherence which arise due to transverse-mode structure of the laser field implying modifications of the spectrum in the time domain. \textcopyright{} 1996 The American Physical Society.
The equations of motion for an inhomogeneously broadened laser operating off resonance in the neighborhood of threshold are solved. It is shown that the effects of detuning cannot be described just by a frequency-dependent laser pump parameter, but that the laser characteristics are modified in other ways also. The relative intensity fluctuations and the correlation time of the laser are calculated as a function of mean intensity and frequency, and it is shown that both are affected by detuning. These theoretical conclusions are then confirmed quantitatively by direct photoelectric counting and two-time correlation measurements of a He:Ne laser.
The form and the decay time of the intensity correlation function of the light from a single-mode He:Ne laser have been measured in the neighborhood of threshold. The experiments were performed at several predetermined values of the pump parametera, ranging froma = -1toa = 4. By combination of the experimental results with the theoretically predicted eigenvalues appearing in the expansion of the correlation function, it was possible also to determine the amplitudes of the first two terms in the expansion over the same range. The experimental results are found to be substantially in agreement with the nonlinear oscillator theory of the laser, without adjustment of any arbitrary scaling parameters.
Expressions for the fluctuations and correlations of the light emitted by a ring laser with counter-rotating traveling wave modes are derived theoretically. It is shown that the competition of the two modes for the excited atomic population leads to negative correlations between their intensity fluctuations, whose magnitude depends on the detuning of the laser cavity from the atomic line center. As a result of the mode competition, the relative intensity fluctuations do not die out in general as the working point of the laser is raised above threshold at line center, and the emitted light does not become fully coherent as in a conventional laser. The two-time amplitude and intensity correlation functions of the light are calculated at line center, and are shown to be expressible in terms of the eigenfunctions and eigenvalues of a certain Schr\"odinger equation, in complete analogy with the theory of the single-mode laser. However, the intensity correlation time increases with pump parameter above threshold, unlike that for the single-mode laser.
It is well known that transverse intensity profiles of lasers may show a nontrivial spatial structure. Usually, depending on the degree of symmetry, the spatial patterns have been associated with modes of a passive resonator or combinations of them which cause oscillations in the laser intensity (mode beating).1 It was recently shown2 that the fluctuations may be due, instead, to the nonlinear coupling between atoms and the field. Here we show that a spontaneous symmetry-breaking process is also at the origin of the formation of complex spatiotemporal dynamics in which oscillations at a single frequency, quasiperiodicity, and chaos may coexist within in the same spatial structure. We compare our experimental results with those obtained from bifurcation theory and numerical integration of the Maxwell–Bloch equations. The transition from a simple (Gaussian) profile to a complex (turbulent) profile is studied by means of the average intensity patterns, frequencies, and phases of the local intensity and cross-correlation functions.
A procedure for the measurement of intensity correlations of the type $〈:\stackrel{^}{I}(\mathrm{x}, t)\stackrel{^}{I}(\mathrm{x}, t+\ensuremath{\tau}):〉$ and $〈:\stackrel{^}{I}(\mathrm{x}, t)\ifmmode\times\else\texttimes\fi{}\stackrel{^}{I}(\mathrm{x}, t+{\ensuremath{\tau}}_{1})\stackrel{^}{I}(\mathrm{x}, t+{\ensuremath{\tau}}_{2}):〉$ with the use of a time-to-amplitude converter is developed. The procedure is used to measure second- and third-order intensity fluctuation correlation functions of the optical field produced by a gas laser operated at various levels near its threshold of oscillation. The results are shown to be in agreement with the predictions of laser theories.
A three-dimensional quantum theory of stimulated Raman scattering is presented which takes into account a general transverse shape of the pump laser. The transverse spatial autocorrelation function and the intensity of the Stokes pulse are derived analytically, by use of the Maxwell-Bloch equations in low-signal-gain conditions and introduction of a many-spatial-modes approximation. We find that the spatial and temporal coherence are coupled. The transverse coherence depends on the pump-laser shape, and the coherence area increases with increasing gain. Time scales for the buildup of stimulated intensity and correlations for the various transverse parts of the Stokes pulse are obtained.