First-order statistics of displayed speckle patterns in electronic speckle pattern interferometry

The first-order probability density functions of displayed speckle patterns in electronic speckle pattern interferometry are calculated. We show that in specular-reference-beam setups the brightness in the displayed pattern is always (χ)² distributed. For speckle-reference-beam setups the probability density depends on the effective resolution of the recorded speckle pattern. This function is calculated in the case of fully resolved patterns and shown to approach the (χ)² density function as the size of the aperture increases. The statistics obtained with specular-and speckle-reference-beam setups are compared. General expressions are derived for the average value and the standard deviation of the monitor brightness. In specular-reference-beam setups the speckle contrast is found always to be equal to $\sqrt{2}$, and in speckle-reference-beam setups it is found to be in the region from $\sqrt{2}$ to $\sqrt{5}$. In setups in which speckle-reduction techniques have been used the speckle contrast decreases below these numbers. We also calculate this reduced speckle contrast.