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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Analysis Of Two Spectral Lines By Photon Statistics.

Manuel P. Cagigal, Felipe González, M. Á. Rebolledo

Published Mar 27, 1985• 0 citations

Abstract

Analysis of two spectral lines by photon statistics.M.P.Cagigal, F.Gonzalez and M.A.RebolledoDepartamento de Optica y E.M., Universidad de Santander.Facultad de Ciencias, Avda. de los Castros, 39005 Santander, SpainAbstractThe possibility of analyzing two spectral lines by measuring the second -order factorialmoment,n(2)(T), is studied. The results are compared with the ones obtained with a multicha-nnel analyzer system.IntroductionPhoton statistics is a powerful tool with which fluctuating light beams can be analyzed.So photon statistics has been used to analyze light beams from Rayleigh scattering experi-ments with macromolecules in solution1 -4 excited by laser light. In these experiments the intensity of the resulting light beam fluctuates at random.Photon statistics can be also used to analyze fluctuating light beams where the intensityfluctuates in a quasi- deterministic way as a function of time. This occurs in an experimentwhere the velocity of a fluid is measured by a laser Doppler technique2,5,6. In such an ex-periment a sinusoidal intensity modulated by a Gaussian profile is obtained when a particlepasses through the scattering volume. Since the distance between two consecutive particlesis a random variable, the intensity of the resulting light beam is not completely determinis

DOI: 10.1117/12.943681

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Analysis of two spectral lines by photon statistics.M.P.Cagigal, F.Gonzalez and M.A.RebolledoDepartamento de Optica y E.M., Universidad de Santander.Facultad de Ciencias, Avda. de los Castros, 39005 Santander, SpainAbstractThe possibility of analyzing two spectral lines by measuring the second -order factorialmoment,n(2)(T), is studied. The results are compared with the ones obtained with a multicha-nnel analyzer system.IntroductionPhoton statistics is a powerful tool with which fluctuating light beams can be analyzed.So photon statistics has been used to analyze light beams from Rayleigh scattering experi-ments with macromolecules in solution1 -4 excited by laser light. In these experiments the intensity of the resulting light beam fluctuates at random.Photon statistics can be also used to analyze fluctuating light beams where the intensityfluctuates in a quasi- deterministic way as a function of time. This occurs in an experimentwhere the velocity of a fluid is measured by a laser Doppler technique2,5,6. In such an ex-periment a sinusoidal intensity modulated by a Gaussian profile is obtained when a particlepasses through the scattering volume. Since the distance between two consecutive particlesis a random variable, the intensity of the resulting light beam is not completely determinis

In this paper we study the possibility of analyzing a low intensity light beam whose spectrum has two lines with Lorentzian or Gaussian profiles by measuring n(2)(T) (the normalized second-order factorial moment). To do this the spectrum is scanned periodically by a spectrometer, and the normalized second-order factorial moment of the beam intensity at the exit aperture of the spectrometer is measured. The errors involved in the determination of the width, relative height, and separation between the two lines are calculated and discussed.

The probability distribution

p(n)

for the detection of

n

photons emitted by a single-mode dye laser in a short time at various excitations is measured, together with the distribution of time intervals between pairs of detected photons. The first measurement yields the probability density

\mathcal{p}(I)

and the moments of the light intensity

I

, and the second one the two-time intensity-correlation function. Measurements are also performed at three different wavelengths.

\mathcal{p}(I)

is found to have a two-component structure, and the relative intensity fluctuations $\frac{〈{(\ensuremath{\Delta}I)}^{2}〉}{{〈I〉}^{2}}$ grow, apparently without limit, as the excitation tends to zero, so that no thermal state is reached. A possible explanation in terms of pumping fluctuations is discussed. The correlation function has the form of a sum of three exponential functions, and the amplitudes and decay constants of the three components are derived. The effective decay-time constant exhibits a thirteenth-power-law dependence on frequency of the laser light.

Microscopic scattering processes are discussed for two structures of light incident on thermal particles. One is composed of two collinear light beams originating from different oscillating modes of a laser, and another is composed of two counter propagating beams originating from the same oscillating mode. A comparison of the cumulant expansions for the above two cases has allowed discerning which terms in the scattering structures led to the same statistical properties of the scattered photons. It is shown that, for the scattering of multimode-laser light, the statistical properties of scattered photons are not affected by the noncorrelating random phases of the incident laser fields but are the same as those in the case of single-mode laser light under practical scattering experiments. The factorial moments ${F}_{k}[=〈n(n\ensuremath{-}1)\ensuremath{\cdots}(n\ensuremath{-}k+1)〉]$ up to sixth order of the scattered photons were measured for various scattering situations including the above two cases, and they were found to support the theoretical results.

It is well known that photoelectric measurements yield information about the statistical behavior of fluctuating light beams. The probability of n photoelectrons being ejected, in a fixed time interval, from the photosensitive surface of the photoelectric detector, upon which the light beam is normally incident, is a linear (Poisson) transform of the probability density for the intensity of the beam. With the help of some plausible assumptions, the present analysis provides solutions to the problem of inverting the Poisson transform, thus determining the probability density for the intensity from experimentally obtained photocounting distributions. The effectiveness of the method is demonstrated by an actual inversion of a typical experimental counting distribution. The technique is of particular interest in connection with efforts to understand the statistical behavior of optical fields, especially of laser fields.