Solitons in Nonlocal Media

Appendix C
Analysis of the Index
Perturbation Profile
C.1 FWHM Computation for the Nonlinear Index Per-
turbation
Assuming that the nonlinear index perturbation ∆n is bell-shaped and symmetric
with respect to beam axis (for homogeneous nonlinear media excited by a Gaussian
beam, this hypothesis is certainly verified.) I can take the FWHM ∆n
xi proportional
to x2
i ∆n = x2 i ∆ndxdt/ ∆ndxdt (i = 1, 2; x1 = x, x2 = t), with a coefficient
dependent on the specific ∆n shape. Being ∆n = ∞ m=1 (∆nm/m!) ∆ρm , I get
2
xi ∆n =
∞ m=1 (∆nm/m!) ∆ρm x2 i dxdt
∆ndxdt
∞ m=1
=
(∆nm/m!) x2
i ∆ρm I∆ρm
∆ndxdt
(C.1)
where I∆ρm = ∆ρmdxdt. Equation (C.1) tells me that the width of ∆n is an
average of all the ∆ρm widths, with weight (∆nm/m!)I∆ρm ∞ l=1 (∆nl/l!)I∆ρl . Moreover, if x2
i ∆ρm does
not change with m, I can easily derive x2
i ∆n = x2
i ∆ρ .
Keeping only the first two terms in (C.1) I obtain
2
xi ∆n = x 2 i ∆ρ1 ∆n1I∆ρ
∆n1I∆ρ + ∆n2I ∆ρ 2
112
+ 1 2
xi 2 ∆ρ2 ∆n2I ∆ρ 2
∆n1I∆ρ + ∆n2I ∆ρ 2
(C.2)