ls3_unit6A

ls3_unit6A - THE INTERACTION OF RADIATION AND MATTER:...

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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE A 1 R. Victor Jones, April 27, 2000 VIA. ON CLASSICAL OPTICAL INTERFERENCE, COHERENCE AND FLUCTUATIONS Light from a real physical source is never, strictly speaking, monochromatic nor does it emanate from a single point in space. Both the amplitude and the phase of the wave field generated by a real source undergoes irregular fluctuations. Within a time interval of the order of the putative coherence time the amplitude and phase of the radiation remain relatively stable and, thus, it may be treated as monochromatic. For time intervals long compared to the coherence time, the effects of fluctuations become manifest. In particular, interference phenomena, the hallmark of wave behavior, are observed only when the interfering beams traverse paths that differ in length by an amount small compared to the coherence length -- i.e. the product of the coherence time and the velocity of light. In this first section we review the classical analysis and interpretation of interference effects in terms of the notions of partial coherence . COMPLEX REPRESENTATION OF POLYCHROMATIC FIELDS: Let us first carefully define the mathematical framework of the subject. We assume that the observable optical field may be represented by a Fourier integral transform 1 E r ( ) t ( ) = E ω ( ) exp i ω t [ ] d ω −∞ [ VIA-1 ] Since of necessity E r ( ) t ( ) must be real 1 For this assumption to be valid, it is sufficient that the function E t ( ) be square-integrable -- i.e. E r ( ) t ( ) [ ] 2 dt −∞ < .
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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE A 2 R. Victor Jones, April 27, 2000 E −ω ( ) = E ω ( ) [ VIA-2 ] and, consequently, Equation [ VIA-1 ] may be re-expressed in terms of an integral over positive frequencies -- viz. E r ( ) t ( ) = E ω ( ) exp i ω t [ ] + c . c . { } d ω 0 . [ VIA-3 ] Further, if we write E ω ( ) = 1 2 a ω ( ) exp i φ ω ( ) [ ] , [ VIA-4 ] where a ω ( ) and φ ω ( ) are both real, then the observable optical field may be expressed in the form E r ( ) t ( ) = a ω ( ) cos i φ ω ( ) −ω t [ ] { } d ω 0 [ VIA-5 ] which is a strict generalization of the complex or phasor representation of real monochromatic fields. To complete the generalization, we follow Dennis Gabor 2 and introduce the so called complex analytic signal E t ( ) = a ω ( ) exp i φ ω ( ) −ω t [ ] { } d ω 0 = 2 E ω ( ) exp i ω t [ ] d ω 0 [ VIA-6 ] which is to be associated with the real observable field -- viz. E r ( ) t ( ) = R e E t ( ) [ ] . As we shall see, this rather careful, or should we say pedantic, introduction of the analytic signal 2 D. Gabor, J. Inst. Elec. Engrs. (London), 93 , 2529 (1946).
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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE A 3 R. Victor Jones, April 27, 2000 greatly facilitates and validates the use of complex quantities in handling cycle-averaged properties -- e.g. energy and intensity --
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ls3_unit6A - THE INTERACTION OF RADIATION AND MATTER:...

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