Page 462 - High Power Laser Handbook

430 Fi b er L a s er s Intr oduction to Optical Fiber Lasers 431

The nonlinear length L is defined as the effective fiber length
NL
for which φ NL = 1:
1
L NL = (15.21)
P γ 0

–5
The nonlinear coefficient γ in silica fibers typically varies from 1 × 10
–1
to 0.1 (mW) . For a gaussian pulse, the maximum induced chirp is
17
given in Agrawal as
δω = . 052 ν φ (15.22)
∆
max sNL
We obtain P *L /A = 1.92 × 10 MW/m for SPM. For example,
7
eff
cr
eff
for L = 1 m and A = 1000 µm, we have P = 19.2 kW for SPM; the
eff
eff
cr
critical power is independent of pulse width for SPM.
Nonlinear Self-Focusing Although SPM can be viewed as the work
of the Kerr nonlinearity in the temporal domain, nonlinear self-
focusing arises from the Kerr nonlinearity operating in the spatial
domain. For the fundamental mode of an optical fiber, the refrac-
tive index distribution of the waveguide confines the mode. In a
bulk medium, there is no waveguide, and a propagating mode nat-
urally diffracts. Self-focusing by Kerr nonlinearity reduces diffrac-
tion and eventually leads to a balance between the effects of
nonlinear self-focusing and diffraction. Self-focusing collapses the
beam and can produce optical damage when the effect of nonlinear
self-focusing overcomes diffraction. A larger beam size equally
reduces the nonlinearity coefficient and diffraction, leading to a
critical power P , not intensity, which determines the onset of non-
cr
linear self-focusing.
In a waveguide, the effect of diffraction is already balanced by
waveguiding. The presence of the Kerr nonlinearity below the critical
power P leads to a nonlinear guided stationary mode with a smaller
cr
mode size than the waveguide’s original fundamental mode. Self-
focusing takes place above P 21
cr
186
.
P = (15.23)
cr
kn n
00 2
where k = 2π/λ is the vacuum wave number and n is the fundamen-
0
0
tal mode index. The critical power for nonlinear self-focusing is esti-
mated to be around 4–6 MW in optical fibers. The normalized spot
sizes of nonlinear-guided stationary modes versus normalized power
are plotted in Fig. 15.15a for various V values. The evolution of the
22
normalized spot size along an optical fiber at various normalized
powers is illustrated in Fig. 15.15b. 22