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Journal of the Optical Society of America B

Fourth-order interference technique for determining the coherence time of a light beam

Z. Y. Ou, E. C. Gage, Brian E. Magill, L. Mandeļ

Published Jan 1, 1989• 65 citations
Publisher

Abstract

A new experimental procedure for determining the (second-order) coherence time of a light beam is described that is based on a combination of conventional interferometry with an intensity-correlation technique. It permits measurements of coherence times that are several orders of magnitude shorter than the resolving times of the detectors, which ordinarily limit correlation measurements. The validity of the method is demonstrated by an experiment in which the transverse fluorescent light from a dye laser is measured.

DOI
10.1364/josab.6.000100

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A new experimental procedure for determining the (second-order) coherence time of a light beam is described that is based on a combination of conventional interferometry with an intensity-correlation technique. It permits measurements of coherence times that are several orders of magnitude shorter than the resolving times of the detectors, which ordinarily limit correlation measurements. The validity of the method is demonstrated by an experiment in which the transverse fluorescent light from a dye laser is measured.
By using an intensity correlation, or photon coincidence, technique and a Mach-Zehnder interferometer with artificially applied phase fluctuations in one arm, we measured the coherence time of a thermal light beam. The light was derived from the filtered fluorescence of a rhodamine 6G dye solution. The two-photon coincidence rate was measured as a function of path difference and was found to dip 50% when the two arms of the interferometer are balanced. The coherence time of ~100 fs is determined from the measured width of the dip and is consistent with the passband of the filters. The method does not require the stability needed in second-order interference measurements, and its time resolution can be much shorter than that of the photodetectors. The theory of this type of fourth-order interference technique is discussed.
Recently some novel two-photon interference experiments have been performed, where quantities such as the field coherence time and the beat spacing between light beams with different frequencies were measured with high time resolution1. These experiments have overcome the traditional limits on time resolution by combining a Mach-Zehnder interferometer configuration with the two-photon correlation technique. All of these experiments, however, have been performed with cw light sources, and no attempt has yet been made to measure the temporal variation of the field intensity.
Interferometry has been used widely in sensing application. However, the technique is limited by the finite coherence time of the light sources when the interference paths are not balanced. Higher-order interference effects involve intensity correlations between multiple detectors and may have the advantage over the traditional second-order interference effect exhibited in only one detector. We discuss various scenarios with unbalanced delays in different paths in fourth-order interference exhibited in coincidence between two detectors. We find, in some cases, interference effect persists even when the delays are much larger than the coherence time of the sources. We also extend the discussion to nonstationary pulsed fields, which need to consider the pulse shape and require a different treatment. These results will be useful in remote sensing applications.
In this paper we outline a method which allows one, in principle, to measure $〈{E}_{i}({\mathrm{x}}_{1}, t){E}_{j}({\mathrm{x}}_{2}, t){{E}_{k}}^{*}({\mathrm{x}}_{3}, t)\ifmmode\times\else\texttimes\fi{}{{E}_{l}}^{*}({\mathrm{x}}_{4}, t)〉$ (in general, x1{\mathbf{x}}_{1}x1​\ensuremath{\ne}x2{\mathbf{x}}_{2}x2​\ensuremath{\ne}x3{\mathbf{x}}_{3}x3​\ensuremath{\ne}x4{\mathbf{x}}_{4}x4​), for stationary quasimonochromatic light propagating principally in the forward direction. The procedure makes use of the properties of nonlinear dielectrics in a sequence of interference experiments. No correlation of photoelectron currents is used, and as a special case the contracted moment 〈∣Ei(x1,t)∣2∣Ej(x2,t)∣2〉〈{|{E}_{i}({\mathrm{x}}_{1}, t)|}^{2}{|{E}_{j}({\mathrm{x}}_{2}, t)|}^{2}〉〈∣Ei​(x1​,t)∣2∣Ej​(x2​,t)∣2〉 measured by Hanbury Brown and Twiss could be obtained in this manner. It is shown that it should be possible to measure the full fourth-order coherence function in the laboratory for laser light, using presently obtainable intensity levels.
Introduction The investigation of the coherence properties of light sources has traditionally involved the study of correlation functions of different orders of the electric field (Born and Wolf, 1975; Mandel and Wolf, 1965). Field and intensity correlation functions have been measured for laser light (for a review, see Lax and Zwanziger, 1973) and have been compared with predictions obtained from quantum laser theory (Haken, 1970, 1981; Lax, 1969; Louisell, 1973; Sargent, Scully and Lamb, 1974). This was, in fact, one of the major confirmations of the correctness of the basic formalism of quantum laser theory. Most of the experiments that probed the coherence properties of lasers were, however, carried out on a very limited variety of lasing media. Interest was also centered on the regime of operation near threshold, where the light from the laser is extremely weak; hence photoelectron counting and correlation techniques were developed for the measurement of coherence properties of the laser light. While these early experiments were performed with the main intent of testing the basic theory, the large variety of lasers now available, and their ever more demanding applications, make it necessary to develop new methods for the measurement and analysis of laser fluctuations. These techniques should be applicable to the measurement of noise in high power lasers; at the same time, in order to delineate the limits of usage, they must be sensitive enough to quantitatively determine the quantum mechanical sources of noise which may lie buried in external noise that is many orders of magnitude larger.
We present a simple technique for measuring coherence times for stationary light fields using a single detector with tunable time resolution. By measuring the equal-time second-order correlation function at varying instrument response functions it is possible to determine the coherence time and also the shape of the temporal decay without the need to record time-resolved data. The technique is demonstrated for pseudothermal light. Possible applications for dynamic light scattering and photon statistics measurements are discussed.