Analysis of the technique of dark speckles for detection of exo-planets

The present paper gives a formal analysis of the dark speckle technique, in particular concerning the effect due to the finite pixel size. Probability laws for the high light level case are established first. Photon counting laws are then obtained using the Poisson-Mandel transform. The computation makes use of the characteristic function of the light intensity. Formal expressions for the general case and for the particular case of a fully developed speckle pattern are given. The application of the computation to the dark speckle technique is immediate: the probability of detecting '0' photoelectron in a pixel can be written as a particular value of the high light level characteristic function. Several results are obtained. For a given fixed overall integration time, there is an optimum number of images to take that maximizes the signal-to-noise ratio (SNR). The loss of contrast induced by use of extended pixels is shown to depend, besides on a geometrical effect, on the average number of photons per pixel. For a given overall observation time, an optimum number of frames (or exposure time) may be defined. The problem of oversampling indicated by the previous analyses of others might be partly overcome by choosing an adequate exposure time. If the number of frames cannot be increased as needed, a better SNR may be obtained by clipping the images at the level of several photoelectrons. A comparison of the SNR of the dark speckle technique with that of a long time exposure experiment is discussed.