Ls1_unit_9 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 75 IX OPTICAL P ULSE P ROPAGATION THE ELECTROMAGNETIC N ONLINEAR S CHRDINGER EQUATION We

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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 , ϖ ( 29 - v v ∇⋅ v E v r , ϖ ( 29 ( 29 + ϖ 2 c 2 ε 0 ( 29 v v ε ϖ ( 29 v E v r , ϖ ( 29 = -μ 0 ϖ 2 v P NL ( 29 v r , ϖ ( 29 [ IX-1 ] v v v ε ϖ ( 29 v E v r , ϖ ( 29 [ ] =- v v NL ( 29 v r , ϖ ( 29 [ 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 NL ( 29 v r , ϖ ( 29 = ε NL v r , ϖ ( 29 v E v r , ϖ ( 29 [ IX-3 ] where ε NL v r , ϖ ( 29 = 3 4 ε 0 χ xxxx 3 ( 29 v E v r , ϖ ( 29 2 and, thus, Equation [ IX-1 ] simplifies to 2 v E v r , ϖ ( 29 + k 0 2 ε 0 ( 29 ε ϖ ( 29 NL v r , ϖ ( 29 [ ] v E v r , ϖ ( 29 = 0 [ IX-4 ] where k 0 = ϖ 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 , ϖ ( 29 = - ε ϖ ( 29 NL v r , ϖ ( 29 [ ] - 1 v E v r , ϖ ( 29 v ∇ε NL v r , ϖ ( 29 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 ϖ ctr . Thus, we assume a seoaration of variables solution v E v r , ϖ ( 29 = F x , y ( 29 r G z , ϖ-ϖ ctr ( 29 exp - i β ctr z ( 29 [ IX-5 ] where β ctr is a wave or propagation number to be associated with ϖ ctr and, thus, Equation [ IX-4 ] becomes 2 F x 2 + 2 F y 2 + k 2 - β 2 [ ] F r + 2 r z 2 - i 2 β ctr r z + β 2 ctr 2 [ ] r F = 0 [ IX-6] where k = k 0 ε ϖ ( 29 NL v r , ϖ ( 29 [ ] 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 ( 29 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 z 2 - i 2 β ctr r z + β 2 ctr 2 [ ] r = 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 ϖ ( 29 NL v r , ϖ ( 29 [ ] ε 0 = n ϖ ( 29 +∆ n ϖ ( 29 [ ] 2 k 0 2 n 2 ϖ + 2 n ϖ n ϖ k 0 2 [ IX-8a ] β 2 = β+∆β 2 ≈β 2 + 2 β ∆β [ IX-8b ]
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ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 77 R. Victor Jones, March 2, 2000 where n ϖ ( 29 = ε NL v r , ϖ ( 29 2 n ϖ ( 29 = 3 8 χ xxxx 3 ( 29 ϖ n ϖ v E v r , ϖ ( 29 2 . Thus, to first order we need to solve the linear equation 2 F x 2 + 2 F y 2 + n 2 k 0 2 2 [ ] F = 0 . [ IX-9 ] which taken together with appropriate boundary conditions defines the linear eigenvalue problem for propagation in the medium where the functions F are the eigenfunctions and the values β are the eigenfunctions.
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This note was uploaded on 01/31/2011 for the course PHYSICS 108 taught by Professor Staff during the Winter '08 term at UC Davis.

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Ls1_unit_9 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 75 IX OPTICAL P ULSE P ROPAGATION THE ELECTROMAGNETIC N ONLINEAR S CHRDINGER EQUATION We

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