THE VANCITTERT-ZERNIKE THEOREM IN PULSE ECHO MEASUREMENTS

A classical theorem of statistical optics, the van Cittert-Zernike theorem, is generalized to pulse echo ultrasound. This theorem fully describes the second-order statistics of the spatial fluctuations (the spatial covariance) of the field produced by an incoherent source. As a random scattering medium is insonified, it behaves as an incoherent source. The van Cittert-Zernike theorem can thus predict the spatial covariance of the pressure field backscattered by a random medium. It is shown that this spatial covariance and the incident energy diagram are Fourier pairs. In the case of a focused illumination, the spatial covariance of the backscattered pressure field is proportional to the autocorrelation of the transmitting aperture function. This is independent of frequency and of F/number. Experimental results obtained with a linear array are in good agreement with theoretical expectations. The implications of this theorem in speckle reduction and in focusing in nonhomogenous media are discussed.