3 in most applications of interest in optics the

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3 In most applications of interest in optics, the spectral amplitudes will have appreciable values only within a spectral width ∆ϖ which is small compared to some mean frequency ϖ . Thus, it often useful to express the analytic signal in terms of a temporal modulation of a mean monochromatic -- viz. E t ( 29 = a ( t )exp i φ t ( 29 - ϖ t [ ] { } . [ VIA-7 ] Using Equation [ VIA-6 ] we see that a ( t )exp i φ t ( 29 [ ] = 2 E ϖ ( 29 exp - i ϖ- ϖ ( 29 t [ ] d ϖ 0 = ˜ E ϖ ( 29 exp - i ϖ t [ ] d ϖ - ϖ [ VIA-8 ] where ˜ E ϖ ( 29 = 2 E ϖ+ ϖ ( 29 : by assumption, ˜ E ϖ ( 29 will be appreciable only near ϖ= ϖ . The following list of transforms are included for reference: a t ( 29 = E r ( 29 t ( 29 [ ] 2 + E i ( 29 t ( 29 [ ] 2 = E t ( 29 E t ( 29 = E t ( 29 [ VIA-9a ] 3 The following identities involving improper functions are invaluable when treating analytic signals -- see Beran and Parrent: 1 2 π exp i x y - y ( 29 ( 29 dx -∞ = δ y - y ( 29 1 2 π exp m ix y - y ( 29 [ ] dx 0 = δ ± y - y ( 29 = 1 2 δ y - y ( 29 m i π Pr y - y
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 4 R. Victor Jones, April 27, 2000 φ t ( 29 = ϖ t + tan - 1 E i ( 29 t ( 29 E r ( 29 t ( 29 = ϖ t + tan - 1 i E t ( 29 - E t ( 29 E t ( 29 - E t ( 29 [ VIA-9b ] E r ( 29 t ( 29 = a ( t ) cos i φ t ( 29 - ϖ t [ ] { } [ VIA-9c ] E i ( 29 t ( 29 = a ( t ) sin i φ t ( 29 - ϖ t [ ] { } . [ VIA-9d ] Using Equations [ VIA-1 ] and [ VIA-6 ], we may easily establish a form of Parseval's formula E r ( 29 t ( 29 [ ] 2 dt -∞ = E i ( 29 t ( 29 [ ] 2 dt -∞ = 1 2 E t ( 29 2 dt -∞ = E ϖ ( 29 E ϖ ( 29 exp - i ϖ+ ϖ ( 29 t [ ] d ϖ -∞ d ϖ -∞ dt -∞ = 2 π E ϖ ( 29 2 d ϖ -∞ = 4 π E ϖ ( 29 2 d ϖ 0 [ VIA-10 ] We have thus far assumed that field is defined for all values of t. In practical terms, the field or, more likely, the observation of the field is defined only within some finite time interval - T t T . In all instances of any consequence, T is much larger than the physically significant times 2 π ∆ϖ and 2 π ϖ so that we may feel some confident in taking the idealization T → ∞ . However, stationary 4 requires that the time averaged energy of the field, which is proportional to 4 A random process is characterized as stationary when all of the moments of the random variables are independent of absolute time .
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE A 5 R. Victor Jones, April 27, 2000 lim T →∞ 1 2 T E r ( 29 t ( 29 [ ] 2 dt - T T , [ VIA-11 ] must be finite and, thus, a stationary field function is not square-integrable ! The problem of analyzing such functions has been a much discussed issue in the mathematics literature. 5 However, as applied physicists, we brush these mathematical niceties under the rug in what follows. We proceed by firmly asserting that optical fields of interest do, indeed, have Fourier transforms and chance the consequences of trucation errors . The cycled-averaged intensity or power spectral density is a crucial element in the analysis of coherence effects and, from Equation [ VIA-6 ], it is proportional E ϖ ( 29 2 = 1 4 π 2 exp i ϖ t - t ( 29 [ ] -∞ E t ( 29 E t ( 29 dt -∞ d t = 1 4 π 2 exp i ϖτ [ ] -∞
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