# We assert that component fields radiated by each

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, we assert that component fields radiated by each coherent cell are spherical waves of the form E k P , t ( 29 = E S k , t - P S k c exp - i ϖ t - PS k c PS k . [ VIA-32 ] where E S k , t ( 29 is field radiated by each element of the source and P S k is, obviously, the distance between the source and observation positions. Therefore, we can write E k P 1 , t ( 29 E k P 2 , t ( 29 = E S k , t - P 1 S k c E S k , t - P 2 S k c exp i ϖ P 1 S k - P 2 S k ( 29 c [ ] P 1 S k [ ] P 2 S k [ ] = E S k , t ( 29 E S k , t - P 2 S k - P 1 S k ( 29 c ( 29 exp i ϖ P 1 S k - P 2 S k ( 29 c [ ] P 1 S k [ ] P 2 S k [ ] [ VIA-33 ] In most instances, the path difference will be small compared to the coherence length of the cell, so that, in the limit of many small cells, we obtain the famous van Cittert-Zernike theorem in the form J ( P 1 , P 2 ) = I S ( 29 exp i k P 1 S - P 2 S ( 29 [ ] P 1 S [ ] P 2 S [ ] dA S Source ∫∫ [ VIA-34 ] and the first-order degree of mutual (spatial) coherence is defined as

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 17 R. Victor Jones, April 27, 2000 γ 12 = γ P 1 , P 2 ( 29 = J ( P 1 , P 2 ) I P 1 ( 29 I P 2 ( 29 = 1 I P 1 ( 29 I P 2 ( 29 I S ( 29 exp i k P 1 S - P 2 S ( 29 [ ] P 1 S [ ] P 2 S [ ] dA S Source ∫∫ [ VIA-35a ] where I ( P n ) = I S ( 29 P n S [ ] 2 dA S Source ∫∫ . [ VIA-35b ] For most problems of interest, we may take P n S = R 2 + x n - ξ ( 29 2 + y n - η ( 29 2 R + x n - ξ ( 29 2 + y n - η ( 29 2 2 R [ VIA-36 ] and, hence, we may approximate Equation [ VIA-35a ] as γ 12 2245 I ξ , η ( 29 exp i k x 1 - ξ ( 29 2 - x 2 - ξ ( 29 2 + y 1 - η ( 29 2 - y 2 - η ( 29 2 2 R d ξ d η Source ∫∫ I ξ , η ( 29 d ξ d η Source ∫∫ [ VIA-37a ] Further, we see that in the Fraunhofer or far-field approximation , the mutual coherence is the Fourier transform of the source intensity distribution -- viz.
T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 18 R. Victor Jones, April 27, 2000 γ 12 2245 I ξ , η ( 29 exp i k ξ x 1 - x 2 ( 29 R [ ] exp i k η y 1 - y 2 ( 29 R [ ] d ξ d η Source ∫∫ I ξ , η ( 29 d ξ d η Source ∫∫ ! [ VIA-37b ] -- and, consequently, the measurement of the degree of mutual coherence of a remote source provides a means to measure the angular size of that extended sources -- i.e. stellar objects. We see then that a complete first-order description of the coherence properties of a classical field is embodied in the complex first-order degree of spatial-temporal coherence which is defined as γ 1 (29 ( r r 1 t 1 , r r 2 t 2 ) E r r 1 , t 1 ( 29 E r r 2 , t 2 ( 29 E r r 1 , t 1 ( 29 2 E r r 2 , t 2 ( 29 2 [ VIA-38a ] where E r r 1 , t 1 ( 29 E r r 2 , t 2 ( 29 1 2 T E r r 1 , t 1 + t ( 29 E r r 2 , t 2 + t ( 29 dt - T T . [ VIA-38b ] F REE S PACE P ROPAGATION OF C OHERENCE F UNCTIONS : We derive here an equation which allows us to generalizes the notions implicit in the van Cittert-Zernike theorem by providing us a description of how coherence propagates through space. We start by assuming that the analytic signal representing the field of interest satisfies an wave equation of the form

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 19 R. Victor Jones, April 27, 2000 n 2 E r r n , t n
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