In the doppler model we write the sum of fields

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to that fixed frequency. In the Doppler model, we write the sum of fields emitted by a ensemble of moving radiators as E t ( 29 = E 0 exp - i ϖ i t i [ ] { } i [ VIA-25 ] where the ϖ i 's are the frequencies of individual atoms which are Doppler shifted from the mean radiated frequency ϖ . In this model, the effects of collisions are neglected and, consequently, the first-order correlation function is written as 8 8 Since the dynamics of the individual atoms are uncorrelate, all cross-terms vanish.
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 13 R. Victor Jones, April 27, 2000 E z 1 t 1 ( 29 E z 2 t 2 ( 29 = E 0 2 exp i ϖ i t 1 - z 1 c ( 29 i - ϖ j t 2 - z 2 c ( 29 j [ ] { } i , j = E 0 2 exp - i ϖ i τ [ ] i [ VIA-26 ] where τ= t 2 - z 2 c - t 1 + z 1 c . The sum may be converted to a integral over a Gaussian distribution of Doppler-shifted frequencies and, hence, E z 1 t 1 ( 29 E z 2 t 2 ( 29 = E 0 2 exp - i ϖ i τ [ ] exp - ϖ i - ϖ ( 29 2 2 δ 2 0 d ϖ i = N atom E 0 2 exp - i ϖ τ-δ 2 τ 2 2 [ ] [ VIA-27 ] The corresponding degree of first-order temporal coherence is (see a graph of this function below) γ 1 (29 τ ( 29 = exp - i ϖ τ-δ 2 τ 2 2 [ ] [ VIA-28 ] and the associated normalized power spectral density follows from Equation [ VIA-15 ] as F ϖ ( 29 = 2 πδ 2 [ ] - 12 exp - ϖ- ϖ ( 29 2 2 δ 2 [ VIA-29 ]
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 14 R. Victor Jones, April 27, 2000 D EGREE OF F IRST -O RDER C OHERENCE - A BSOLUTE V ALUE S PATIAL C OHERENCE : T HE VAN C ITTERT -Z ERNIKE In the rudimentary treatment of the Young interference experiment recapitulated above, we assume an ideal point source and, thus, neglect any effects of spatial coherence . In particular, we assume that the field at points P 1 and P 2 due to a given radiator are in time synchronism. However, when we deal with extended sources, we must enlarge our notion of coherence The subject of spatial coherence was first developed by van Cittert and, later, it was extended by Zernike. 9 To define the problem, let us consider the following geometry: 9 See P. H. van Cittert, Physica , 1 , 201 (1934) and F. Zernike, Physica , 5 , 785 (1938).
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 15 R. Victor Jones, April 27, 2000 The task before us is to determine the so called mutual intensity J ( P 1 , P 2 ) due to the extended source which might be observed for the two points located on the observation plane. For simplicity, we assume that extended source is in a plane parallel to the observation plane and that it can be divided into cells of statistically independent radiators. Thus, the total field at any point on the observation plane is given by summing the fields due to each of the cells -- viz. E P , t ( 29 = E k P , t ( 29 k . [ VIA-30 ] The mutual intensity is then defined as J ( P 1 , P 2 ) E P 1 , t ( 29 E P 2 , t ( 29 = E k P 1 , t ( 29 E k P 2 , t ( 29 k + E k P 1 , t ( 29 E k P 2 , t ( 29 k k k . [ VIA-31a ]
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 16 R. Victor Jones, April 27, 2000 Because of the assumed statistical independence of the cells, the cross-terms vanish so that J ( P 1 , P 2 ) = E k P 1 , t ( 29 E k P 2 , t ( 29 k . [ VIA-31b ] It the spirit of the Huygens principle
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