Van Cittert-Zernike theorem

Thus the mutual intensity at the observer is the Fourier transform of the source. This is a special case of the van Cittert-Zernike theorem. The mutual intensity is translation invariant or homogeneous, i.e., it depends only on the separation of Pi and P2. The intensity at the observer is simply / = J. Measuring the mutual intensity will give Fourier components of the object. 

The coherence area om is measured at the entrance slit, because this is where the object is imagined to exist. From Fig. 1, this slit is distance R from the spectral source, which is incoherent and of area w2. Then the Van Cittert-Zernike theorem (Born and Wolf, 1959) gives... 

Figure 1.14. Diagram illustrating the van Cittert-Zernike theorem. An extended source a illuminates a diffracting object in the xj -plane, a distance R away.

Same considerations are valid for spatial coherence. As the advanced wave passes through a narrow aperture, it gains some degree of transverse coherence according to the van Cittert-Zernike theorem. Because the nonlinear interaction is restricted to the area where the pump field is present, the resulting signal (DFG) field is also partially coherent provided the pump beam diameter is smaller than the coherence width of the advanced wave in the plane of the crystal [Aichele 2002],... 

The Van-Cittert-Zernike theorem describes the relation between the complex visibility of an object and its brightness distribution on the plane of the sky. It states that for sources in the far field the normalised value of the spatial coherence function (or complex visibility) is equal to the Fourier transform of the normalised sky brightness distribution. If a source with a brightness distribution 7(0), where 0 = 6y)... 

Going back to Eq.2.34, the complex visibility or spatial coherence function was defined as the mutual coherence function when r = 0. According to the van-Cittert-Zernike theorem, the normalised spatial coherence funetion is the Fourier transform of the normalised sky brightness distribution (Eq.2.35). The temporal coherence function is defined as the mutual coherence function for b = 0, and according to the Weiner-Khinchin theorem, the normalised value of the temporal coherence function is equal to the Eourier transform of the normalised spectral energy distribution of the source, this is... 

Given the considerable spatial extent of most celestial objects, it might seem surprising that measurements of the electric field coherence at the Earth could yield any information about their structures. Indeed, most radio sources emit incoherent radiation At the source F = 0 unless r, = rj and f, = tj. Figure 1 illustrates how coherence arises in a distant radiation field. The two point sources Pi and P2 are incoherent. However, at the distant points Qi and Q2, the electric field contains contributions from both Pi and P2 and is therefore partially coherent. The generalization of this idea leads to the van Cittert-Zernike theorem. A simple version of this theorem produces a Fourier transform relation between the sky brightness B(x, y) and the measured coherence function... 

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