Numerical simulations of self-focusing of ultrafast laser pulses
Gadi Fibich*
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Weiqing Ren
Courant Institute of Mathematical Science, New York University, New York, New York 10012
Xiao-Ping Wang
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Received 14 November 2002; published 7 May 2003
Simulation of nonlinear propagation of intense ultrafast laser pulses is a hard problem, because of the steep
spatial gradients and the temporal shocks that form during the propagation. In this study we adapt the iterative
grid distribution method of Ren and Wang J. Comput. Phys. 159, 246 2000 to solve the two-dimensional
nonlinear Schro ¨dinger equation with normal time dispersion, space-time focusing, and self-steepening. Our
simulations show that, after the asymmetric temporal pulse splitting, the rear peak self-focuses faster than the
front one. As a result, the collapse of the rear peak is arrested before that of the front peak. Unlike what has
sometimes been conjectured, however, collapse of the two peaks is not arrested through multiple splittings, but
rather through temporal dispersion.
DOI: 10.1103/PhysRevE.67.056603 PACS numbers: 42.25.Bs, 42.65.Sf, 42.65.Jx
I. INTRODUCTION
The nonlinear Schro
¨
dinger equation NLS
i
z
z , x , y +
+| |
2
=0 1
is the model equation for the propagation of cw continuous
wave laser beams in Kerr media. Here, is the electric field
envelope, z is the axial distance in the direction of beam
propagation, x and y are the coordinates in the transverse
plane, and
=
xx
+
yy
is the diffraction term. It is well
known that when the power, or L
2
norm, of the input beam is
sufficiently high, solutions of Eq. 1 can self-focus and be-
come singular in a finite distance z 1,2. Because of the
infinitely large gradients that exist at the singularity, standard
numerical methods break down after the solution undergoes
moderate focusing 3. Therefore, as part of the research ef-
fort during the 1980s to find the blowup rate of the NLS,
McLaughlin et al. developed the numerical method of dy-
namical rescaling 4, which can resolve the solution near
the singularity at extremely high amplitudes. This method
exploits the known self-similar structure of the collapsing
part of the solution near the singularity, which relates the
shrinking transverse width of the solution to the increase in
its norm. Therefore, the solution is computed on a fixed com-
putational grid, which in physical space corresponds to a grid
that shrinks uniformly toward the singularity. The focusing
rate of the grid points is determined dynamically from some
norm of the solution ( | | |
2
dxdy , max
x,y
||, etc.. Be-
cause the focusing rate of the grid points can be chosen to be
the same as the physical focusing rate, in the transformed
variables the solution remains smooth and can thus be solved
using ‘‘standard’’ methods.
The method of dynamic rescaling works extremely well
for solutions of the NLS with radially symmetric initial con-
ditions, in which case focusing by 10
10
or more can easily be
realized see, e.g., Fig. 3.5 in 1. Although the method of
dynamic rescaling has been extended to NLS’s with noniso-
tropic initial conditions 5 and to perturbed NLS’s e.g.,
NLS’s with normal time dispersion 6, in such cases dy-
namic rescaling is considerably less efficient, because the
solution does not focus uniformly and/or it is not clear how
to extract the physical focusing rate from the solution. The
iterative grid redistribution IGR method, developed by Ren
and Wang, overcomes these difficulties by allowing the grid
points to move independently rather than uniformly accord-
ing to a general variational minimization principle. This
method has been showed to be highly effective for solving
partial differential equations PDE’s with singular behavior
such as the NLS 1 and the Keller-Segal equations with
multiple blowup points 7. As we shall see, however, apply-
ing the IGR method to nonstationary NLS models that de-
scribe the propagation of ultrashort pulses turns out to be
considerably more demanding.
A. Simulations of ultrashort pulses
The NLS 1 does not include temporal effects. These
effects become important in the case of ultrashort laser
pulses, whose propagation can be modeled by the dimension-
less nonstationary NLS 8
i
z
z , x , y , t +
+| |
2
+
1
zz
+i
2
| |
2
t
-
t
-
3
tt
=0, 2
where now is also a function of time t. The dimensionless
parameters are given by *Electronic address: fibich@math.tau.ac.il
PHYSICAL REVIEW E 67, 056603 2003
1063-651X/2003/675/0566039/$20.00 ©2003 The American Physical Society 67 056603-1