Predicting the filamentation of high-power beams and pulses without numerical integration:
A nonlinear geometrical optics method
Nir Gavish
*
and Gadi Fibich
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Luat T. Vuong and Alexander L. Gaeta
School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA
Received 26 June 2008; published 7 October 2008
We present an analytic method for predicting the initial self-focusing dynamics of high-power beams and
pulses. Using this method we study the filamentation pattern of a variety of input profiles, without solving
partial differential equations. In particular, this method shows that in the anomalous regime, high-power pulses
with temporal super-Gaussian profiles will split in time into two shorter pulses, and spherically symmetric
pulses with super-Gaussian spatiotemporal profiles will evolve into a spatiotemporal shell.
DOI: 10.1103/PhysRevA.78.043807 PACS numbers: 42.65.Sf
I. INTRODUCTION
Nonlinear wave collapse is a phenomenon intrinsic to
many areas of physics. In nonlinear optics, the intensity-
dependent refractive index leads to self-focusing where a
beam with input power greater than the critical power P
cr
may undergo collapse 1.
In self-focusing experiments, it is observed that when the
input beam power exceeds several critical powers, the trans-
verse spatial beam profile breaks up into several filaments, a
phenomenon known as multiple filamentation 2. For many
years, the only explanation for multiple filamentation, due to
Bespalov and Talanov 3, has been that multiple filamenta-
tion is initiated by random noise in the input beam profile.
Subsequently, it turned out that the modulation instability
analysis of Bespalov and Talanov is only valid for very high
input powers where P = O100 P
cr
4. At lower powers
e.g., P = O10 P
cr
, noisy beams initially collapse as a
single filament 5. After collapse is arrested due to higher-
order effects, the beams undergo a few focusing-defocusing
cycles, at which point they typically break into multiple fila-
ments, whose locations are dependent on noise.
Since noise is by definition random, the multiple filamen-
tation patterns of noise-induced multiple filamentation are
different from shot to shot. This constitutes a serious draw-
back in applications, in which precise localization is crucial.
Therefore a method for predicting and controlling multiple
filamentation is desired.
The first model for deterministic multiple filamentation
was suggested by Fibich and Ilan 6,5, who showed theo-
retically that the deterministic breakup of cylindrical symme-
try by a linear-polarization state can lead to deterministic
multiple filamentation patterns. However, multiple filamen-
tation due to vectorial effects has not been observed in ex-
periments 7. The reason for this is that in order for vectorial
effects to lead to multiple filamentation, the beam radius
should self-focus down to approximately two wavelengths.
In experiments, however, self-focusing is arrested prior to
this stage, due to temporal or higher-order effects such as
plasma. In subsequent research, deterministic multiple fila-
mentation patterns i.e., control of the number and location
of the filaments is achieved through a deterministic breakup
of the cylindrical symmetry of the input pulse by inducing
astigmatism 8,9 or ellipticity 10, applying amplitude or
phase masks 9,11, propagating the beam through grids and
slits 12–14, and also by shaping super-Gaussian input
beams 15. Each of the above methods induces a character-
istic filamentation pattern by breaking the cylindrical sym-
metry of the input pulse. For example, filaments induced by
ellipticity are distributed along the ellipse axes, whereas in
the collapse of super-Gaussian input beams the filaments are
distributed around a circle. These deterministic filamentation
patterns are generally deduced via qualitative arguments,
however, and not from a quantitative theory that is capable of
predicting the number and locations of the filaments.
In 15, we presented a nonlinear geometrical optics
NGO method for predicting the initial self-focusing dy-
namics of high-power P P
cr
input beams. The idea of the
method is as follows. In this high-power regime, diffraction
can be initially neglected and beam propagation is dictated
by self-phase-modulation SPM. Therefore the phase can be
calculated analytically and used to solve the eikonal equation
for the rays trajectories analytically. This method explained
why high-power Gaussian input beams collapse to a single
filament, while super-Gaussian input beams evolve into a
ring that subsequently breaks into a ring of filaments. These
analytic results were in agreement with experimental mul-
tiple filamentation patterns produced by propagation in water
15.
In this study, we further improve the NGO method by
calculating the beam amplitude along the rays. This allows
us to predict the intensity distribution of the beam along its
propagation, which is much more informative than the rays
distribution. The numerical calculation of the rays trajecto-
ries and of the amplitude evolution involves solving linear
ordinary differential equations along each ray. Because it is
not necessary to solve the nonlinear Schrödinger equation
NLSE or any other PDE, the computational costs are mini-
mal. We show that the results of the NGO method are in
*
nirgvsh@tau.ac.il; URL: www.tau.ac.il/nirgvsh
PHYSICAL REVIEW A 78, 043807 2008
1050-2947/2008/784/04380716 ©2008 The American Physical Society 043807-1