Reconstruction of two-dimensional light intensity distribution from the data of photo-count experiments

Abstract

411 analytical method to recover the twodimensional probability density function (PDF) of a randoin light field lroni the PDF of photon counts is presented. Some illustration of tlie inversion procedure are presented with application to tlie problem of light intensity interferometry.Poissoii transform in the optical speckleinterferometry and some other optical processing problems arises when iiiteiisity levels of analyzed field are so low that only few pliotoiis la11 on each pixel of interferogram. In this case the joint distribution of photon counts in two space-time points is given by the equation which is known as Mandel's equation aiid reprcseiits tcTio-dimensional Poisson transform of probability dciisity function (PDF) of iiitensity fluctuations P(I1 ,I2) in observed speckle-structure. As a resultant transform P(n,iii) is determined by two factors (the own field fluctuations which generate the speckle-structure, aiid tlie quantum fluctuations of photodetection), tlie deteriiiination of true radiation iiileiisity fluctualions becomes tlie problem o i real interest. It means the iiiversioni of transform (I), that is, tlie determination or two-dimensional PDF of intensity fluctuations from the photo-count experiment data.Single-dimensional version of similar problem was considered by niaiiy authors [1,2]. Tlie key feature of all these investigations is tlie solution instability which is caused by tlie fact that the problem is typically ill-posed. So, the resultant solutions are unstable and practically do not work even under the extraordinary small level of statistical error. Recently [3] the stable solution based on Padt-approxiiiiants method was proposed, aiid its generalizalioii 011 two-dinieiisioiial distributions was fulfilled [.I.]. This method gives tlie excellent results when tlie average iiuiiiber of photon counts per pixel is extremely small (about one or Icss), aiid Pad& approxiiiiants are based on tlie polynomials of tlie low order. In the iiiterniediate case of average photon counts, tlie nuniber of Pade-approxiinatioii coelficients becomes greater, and tlie problem of tlie solution correctness again takes on its iniportance.In this report we propose another approach to tlie problem of joint two-point iiitensity PDF determination which is based on the orthogonal expansion of tlie solutioin in a series of Poisson transform cigeilfuiictioiis and provides the generalization of correspoiiding one-poinl problem solution [SI. Tlie advantages of this solution are it's high stability, the simplicity of optimal regularization level deterniiiiation, as well as the ability of further extrapolalioli on tlie problems of higher diinensioiialities.Tlie niain idea of supposed method consists in lollowing. Let us define eigeilrunctioiis of oncdiinensioiial Poisson transform by the equation where { p, } aiid { 1; are orthogonal basises in PDF's PQj and P(11) spaces. respectively. It can bc sliowii [ 5 ] , that basises with described properties really exist and that they can be defined by the solutioiis or integral ...