The van Cittert–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.