# 29 1 c 2 2 t n 2 e r r n t n 29 via 39 if we multiply

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( 29 = 1 c 2 2 t n 2 E r r n , t n ( 29 [ VIA-39 ] If we multiply through by E r r 2 , t 2 ( 29 , we see that 1 2 Γ r r 1 t 1 , r r 2 t 2 ( 29 = ∇ 1 2 E r r 1 , t 1 ( 29 E r r 2 , t 2 ( 29 = 1 c 2 2 t 1 2 E r r 1 , t 1 ( 29 E r r 2 , t 2 ( 29 = 1 c 2 2 t 1 2 Γ r r 1 t 1 , r r 2 t 2 ( 29 . [ VIA-40a ] Similarly 2 2 Γ r r 1 t 1 , r r 2 t 2 ( 29 = 1 c 2 2 t 2 2 Γ r r 1 t 1 , r r 2 t 2 ( 29 . [ VIA-40b ] Therefore the coherence function must satisfy the fourth-order equation 1 2 2 2 Γ r r 1 , r r 2 ; τ ( 29 = 1 c 4 2 ∂τ 4 Γ r r 1 , r r 2 ; τ ( 29 [ VIA-41] The implication of Equations [ VIA-40 ] and [ VIA-41 ] is that coherence is a field that propagates through space and may be treat by methods which apply to other field variables. In fact, the content of the van Cittert-Zernike theorem is a description of how the coherence of extended source propagates to remote observation points. This point of view is particularly valuable in the treatment of wave propagation through random media. As a beam propagates through such a medium, the effect of random scattering events scrambles the phase of the beam and the simple notion of a field rapidly loses meaning. However, the evolution of the coherence function provides an important measure of the statistical properties of the random medium.

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 20 R. Victor Jones, April 27, 2000 H IGHER -O RDER C ORRELATION F UNCTIONS -- A CLASSICAL EXPLICATION OF THE FAMOUS H ANBURY B ROWN -T WISS EXPERIMENT : Generalizing Equation [ VIA-38a ], the degree of nth-order spatial-temporal coherence can be defined as 10 γ n ( 29 ( r r 1 , t 1 ; KK ; r r n , t n ; r r n + 1 , t n + 1 ; KK ; r r 2 n , t 2 n ) E r r 1 , t 1 ( 29 KK E r r n , t n ( 29 E r r n + 1 , t n + 1 ( 29 KK E r r 2 n , t 2 n ( 29 E r r 1 , t 1 ( 29 2 KK E r r n , t n ( 29 2 E r r n + 1 , t n + 1 ( 29 2 KK E r r 2 n , t 2 n ( 29 2 [ VIA-42 ] In particular, the degree of second-order temporal coherence is defined as γ 2 ( 29 ( r r ref , τ ) E r r ref , t ( 29 2 E r r ref , t ( 29 2 E r r ref , t ( 29 2 2 = I r r ref , t ( 29 I r r ref , t ( 29 2 I r r ref , t ( 29 2 [ VIA-43 ] We shall see that this function is an important measure of the relative timing of of intensity fluctuations. 11 As a first step, let first find an expression for the average intensity radiated by a collection of independent oscillators -- viz. 10 R. J. Glauber, in Quantum Optics and Electronics, Les Houches, 1964 (edited by C. DeWitt, A. Blandin, and C. Cohen-Tannoudji), p63, Gordon and Breach (1965). 11 By the famous Schwartz inequality a * b 2 a 2 b 2 so that I r r ref , t ( 29 I r r ref , t ( 29 2 I r r ref , t ( 29 2 I r r ref , t ( 29 2 and quite generally γ 2 ( 29 ( r r ref , τ ) 1
T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 21 R. Victor Jones, April 27, 2000 I I t ( 29 = ε 0 c 2 [ ] E t ( 29 2 = ε 0 c 2 [ ] E i t ( 29 i , j E j t ( 29 = ε 0 c 2 [ ] E i t ( 29 2 i [ VIA-44 ] In this expression all of the cross-terms are presumed to vanish, since the radiation from any given atom is statistically uncorrelated with that of any other atom. For the collision- and Doppler-broadening models discussed above, the oscillators differ only in phase and this average can be expressed as I I t ( 29 = N atom ε 0 c 2 [ ] a 2 . [ VIA-45] To obtain the root-mean-square deviation in the cycle average of the intensity, we first calculate I t ( 29 [ ] 2 = ε 0

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