ls2_unit_6

ls2_unit_6 - THE INTERACTION OF RADIATION AND MATTER:...

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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 62 R. Victor Jones, March 16, 2000 62 VI. SEMICLASSICAL LASER THEORY 23 L ASER S ELF -C ONSISTENCY E QUATIONS Lamb's theory of laser operation provides a very powerful means for interpreting and predicting complex time dependent behavior without invoking all the intricacy of a full quantum mechanical theory. It is semiclassical, self-consistent theory in the following sense: statistic summation macroscopic Maxwell theory quantum theory self-consistency Suppose that the field in the laser is represented in the form E z , t ( ) = 1 2 E n t ( ) exp i ω n t + φ n t ( ) ( ) [ ] U n z ( ) n + c . c . [ VI-1 ] where, for example, in a simple a cavity laser U n z ( ) sin k n z = sin n π L z Λ Ν Ξ Π standing wave [ VI-2a ] and in a ring laser U n z ( ) exp i k n z ( ) = exp i n 2 π L z Λ Ν Ξ Π traveling wave [ VI-2b ] 23 An adaptation or interpretation of Lamb's "semiclassical" or "self-consistent" laser theory as first presented in Phys. Rev. 134 , A1429 (1964) and refined in countless other treatments. In these lecture notes we drawn extensively on M. Sargent III, M. O. Scully and W. E. Lamb, Jr., Laser Physics , Addison-Wesley (1974) and P. Meystre and M. Sargent III, Elements of Quantum Optics , Springer-Verlag (1992).
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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 63 R. Victor Jones, March 16, 2000 63 With this representation for the field the induced polarization can be expressed as P z , t ( ) = 1 2 P n t ( ) exp i ω n t + φ n t ( ) ( ) [ ] U n z ( ) n + c.c. [ VI-3 ] where P n t ( ) is the complex, slowly varying component of the polarization of the n th mode. If we take the wave equation in the form 24 −∇ 2 v E o v J t + 1 c 2 2 v E t 2 = μ o 2 v P t 2 [ VI-4 ] where second term is included as a means to account for cavity losses. From Equation [ VI-1 ] assuming that E n , P n , and ˙ φ n are slowly varying functions of time 25 −∇ 2 v E ⇒− 2 E z , t ( ) z 2 = 1 2 E n exp i ω n t + φ n ( ) [ ] 2 U n z 2 n + c . c . = 1 2 c 2 Ω n 2 E n exp i ω n wt + φ n ( ) [ ] U n n + c . c . [ VI-5a ] where Ω n = c k n are the eigenfrequencies of the cold resonator eigenmodes. v E t E z , t ( ) t = 1 2 ˙ E n i E n ω n + ˙ φ n ( ) { } exp i ω n t + φ n ( ) [ ] U n z ( ) n + c . c . [ VI-5b ] 24 In this formulation we are assuming that v v P 0 . 25 The so called slowly-varying amplitude and phase approximation (SVAP) is used extensively in treating problems in laser dynamics. In the SVAP approximation it is assumed that z ln E n ; z ln P n ; z φ n Χ Ψ ± Ϋ ά έ << k n and t ln E n ; t ln P n ; t φ n Χ Ψ ± Ϋ ά έ << Ω n
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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 64 R. Victor Jones, March 16, 2000 64 μ o v J t μ
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ls2_unit_6 - THE INTERACTION OF RADIATION AND MATTER:...

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