Derivation of the generalized NLS equation

APPENDIX C

Derivation of the generalized NLS equation The starting point is the wave equation (2.3.1) in Section 2.3. We convert it to the frequency domain and use the form of PNL (r, t) given in Eq. (2.1.9) to obtain ˜ + n2 (r, ω) ∇ 2E

ω2 ˜ ω2 ˜ E = − PNL (r, ω), c2 0 c 2

(C.1)

In the absence of nonlinear polarization, the solution of this equation was obtained in Section 2.2 in terms of fiber modes. If we consider a single-mode fiber and assume that the optical pulse maintains its state of polarization inside the fiber, the nonlinear effects can be included by writing the solution of Eq. (C.1) in the form E˜ (r, ω) = xF ˆ (x, y, ω)B(z, ω) exp[iβ(ω)z],

(C.2)

where β(ω) is the propagation constant of the mode with the profile F (x, y, ω) obtained by solving ∇T2 F + (n2 k20 − β 2 )F = 0.

(C.3)

The amplitude B(z, ω) varies with z because of PNL (r, t) in Eq. (C.1). We substitute the solution in Eq. (C.2) into Eq. (C.1), make use of Eq. (C.3), and neglect the second derivative of B with respect to z because of its slowly varying nature. The result is 2iβ

∂B ω2 F (x, y, ω)eiβ z = − 2 xˆ · P˜ NL (r, ω). ∂z 0 c

(C.4)

Multiplying the preceding equation with F ∗ (x, y, ω) and integrating over the transverse plane, we obtain ∂B iωe−iβ z = ∂z 20 c n¯ (ω0 )



F ∗ (x, y, ω)xˆ · P˜ NL (r, ω) dx dy,

(C.5) 

where we assumed that the mode profile is normalized such that F ∗ F dx dy = 1 and used the relation β = n¯ ω/c. We also replaced n¯ (ω) with its value at ω0 . We need to convert Eq. (C.2) to the time domain by using E(r, t) =

1 2π



∞ −∞

E˜ (r, ω) exp(−iωt) dω,

(C.6)

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Derivation of the generalized NLS equation

where we use a scalar field in view of its fixed polarization. If we assume that the mode profile does not change much over the pulse bandwidth, we can replace F (x, y, ω) with its value at the carrier frequency ω0 of the pulse to obtain 

E(r, t) = F (x, y, ω0 )

B(z, ω)ei(β z−ωt) dω.

(C.7)

We write it as E(r, t) = F (x, y, ω0 )A(z, t)ei(β0 z−ω0 t) , where the slowly varying function A(z, t) is defined as 

A(z, t) =

B(z, ω)ei(β z−ωt) dω,

(C.8)

with β = β − β0 and ω = ω − ω0 . We use this equation to calculate the partial derivative of A(z, t) in the form ∂A = ∂z

   ∂B + iβ B ei(β z−ωt) dω. ∂z

(C.9)

We substitute Eq. (C.5) into Eq. (C.9) and expand β(ω) in a Taylor series around ω0 as β(ω) = β0 + β1 ω + 12 β2 (ω)2 + 16 β3 (ω)3 + · · · ,

(C.10)

where βn is the nth-order dispersion parameter. Noting that ω is replaced with i ∂∂t in the time domain, we obtain the following time-domain equation:  ∞ m m  ∂A i ∂ A ie−iβ0 z  ∂ −i = + i F ∗ (x, y, ω0 )xˆ · PNL (r, t) dx dy. ω 0 m ∂z m ! ∂ t 2  c n ¯ ∂ t 0 m=1

(C.11)

The right-side of this equation includes all the nonlinear effects. Note the time derivative resulting from the presence of ω = ω0 + ω in Eq. (C.5). It is responsible for the phenomenon of self-steepening. The final step is to use an appropriate form of PNL (r, t). In the case of pulses polarized linearly along the x direction, we use Eq. (B.12) with E(r, t) = F (x, y)A(z, t)eiβ0 z to obtain NL

P

(r, t) = xF ˆ (x, y)A(z, t)e

iβ0 z 30

4

 (3)

χxxxx

R(t − t )|F (x, y)|2 |A(z, t )|2 dt .

(C.12)

Using this form in Eq. (C.11), the generalized NLS equation becomes     ∞ ∞ m m  ∂A i ∂ A 1 ∂   2  −i = i γ (ω ) 1 + i A ( z , t ) R ( t )| A ( z , t − t )| dt . (C.13) 0 ∂z m! ∂ tm ω 0 ∂t 0 m=1

Derivation of the generalized NLS equation

where the nonlinear parameter γ (ω0 ) is defined as γ (ω0 ) =

3ω0 (3) ω0 n2 χ = , 8c n¯ Aeff xxxx cAeff



(C.14)

with the effective area Aeff = ( |F (x, y)|4 dx dy)−1 . This definition is identical to that in Eq. (2.3.31) once we recall that F (x, y) is normalized here such that  |F (x, y)|2 dx dy = 1.

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