ls1_unit_9

ls1_unit_9 - ON CLASSICAL ELECTROMAGNETIC FIELDS IX. PAGE...

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ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 75 R. Victor Jones, March 2, 2000 IX. OPTICAL PULSE PROPAGATION T HE E LECTROMAGNETIC N ONLINEAR S CHRÖDINGER E QUATION : We begin our discussion of optical pulse propagation 35 with a derivation of the nonlinear Schrödinger (NLS) equation. To that end, we recall Equations [ VII-23 ] and [ VII-23 ] from the early lecture set entitled Nonlinear Optics I -- i.e. 2 v E v r , ω ( ) v v v E v r , ω ( ) ( ) + ω 2 c 2 ε 0 ( ) v v ε ω ( ) v E v r , ω ( ) = μ 0 ω 2 v P NL ( ) v r , ω ( ) [ IX-1 ] v v v ε ω ( ) v E v r , ω ( ) [ ] = v v P NL ( ) v r , ω ( ) [ IX-2 ] In this treatment we will confine our attention to wave propagation in uniform, isotropic optical materials -- viz., glass fibers. For such materials, we can write v P NL ( ) v r , ω ( ) = ε NL v r , ω ( ) v E v r , ω ( ) [ IX-3 ] where ε NL v r , ω ( ) = 3 4 ε 0 χ xxxx 3 ( ) v E v r , ω ( ) 2 and, thus, Equation [ IX-1 ] simplifies to 2 v E v r , ω ( ) + k 0 2 ε 0 ( ) ε ω ( ) + ε NL v r , ω ( ) [ ] v E v r , ω ( ) = 0 [ IX-4 ] where k 0 = ω c . 36 To proceed, postulate that this nonlinear Helmholtz equation can be treated by separation of variables methods. In particular, we are looking for a time-localized solution (a 35 An excellent reference on this subject is Govind P. Agrawal’s Nonlinear Fiber Optics , Academic Press (1989) ISBN 0-12-045140-9. 36 In this simplification, we have taken v v E v r , ω ( ) = − ε ω ( ) −ε NL v r , ω ( ) [ ] 1 v E v r , ω ( ) v ε NL v r , ω ( ) 0.
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ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 76 R. Victor Jones, March 2, 2000 pulse) with a relatively narrow frequency spectrum (or “group” of frequencies centered on a frequency ω ctr . Thus, we assume a seoaration of variables solution v E v r , ω ( ) = F x , y ( ) r G z , ω−ω ctr ( ) exp i β ctr z ( ) [ IX-5 ] where β ctr is a wave or propagation number to be associated with ω ctr and, thus, Equation [ IX-4 ] becomes 2 F x 2 + 2 F y 2 + k 2 β 2 [ ] F Χ Ψ Ω Ϋ ά έ r G + 2 r G z 2 i 2 β ctr r G z + β 2 −β ctr 2 [ ] r G Χ Ψ Ω Ϋ ά έ F = 0 [ IX-6] where k = k 0 ε ω ( ) + ε NL v r , ω ( ) [ ] ε 0 . In the linear problem β 2 would be the “separation constant,” but in this case we will need a bit more elaboration. Nevertheless, we shall assume that we can find a set of functions F x , y ( ) and values β 2 that satisfy the equation 2 F x 2 + 2 F y 2 + k 2 β 2 [ ] F = 0 [ IX-7a ] so that 2 r G z 2 i 2 β ctr r G z + β 2 −β ctr 2 [ ] r G = 0 [ IX-7b ] To use perturbation theory, we first reduce Equation [ IX-7a ] to a solvable linear problem by writing k 2 = k 0 2 ε ω ( ) + ε NL v r , ω ( ) [ ] ε 0 = n ω ( ) + Δ n ω ( ) [ ] 2 k 0 2 n 2 ω ( ) + 2 n ω ( ) Δ n ω ( ) [ ] k 0 2 [ IX-8a ] β 2 = β + Δβ [ ] 2 ≈β 2 + 2 β Δβ [ IX-8b ]
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This note was uploaded on 02/29/2012 for the course PHYSICS 216 taught by Professor Staff during the Fall '11 term at BU.

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ls1_unit_9 - ON CLASSICAL ELECTROMAGNETIC FIELDS IX. PAGE...

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