244 CHINESE OPTICS LETTERS / Vol. 8, No. 2 / February 10, 2010
The role of refocusing in dynamic X-wave formation during
femtosecond laser filamentation in water
Junjie Xu (MMMddd###)
1
, Han Xu (MMM |||)
1
, Hui Xiong (=== )
1
, Ya Cheng (§§§ æææ)
1∗
,
Zhizhan Xu (MMMÐÐÐ)
1
, Weiwei Liu (444)
2
, and See Leang Chin (aaaûûû)
3
1
State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China
2
Key Lab oratory of Opto-Electronic Information Science and Technology, Ministry of Education,
Institute of Modern Optics, Nankai University, Tianjin 300071, China
3
Centre d’Optique, Photonique er Laser (COPL) and D´epartement de Physique, de G´enie Physique et d’Optique,
Universit´e Laval, Qu´ebec, Qu´ebec G1K 7P4, Canada
∗
E-mail: ycheng-45277@hotmail.com
Received April 13, 2009
With the evolution of a laser pulse in water, the formation of a nonlinear X wave during femtosecond fila-
mentation is investigated based on numerical simulations. In particular, we analyze the far-field angularly
resolved spectra obtained for different temporal portions of the ultrashort pulse during its propagation.
Our result shows that the refocusing of ultrashort pulse leads to the formation of dynamic X wave which
essentially manifests itself as conical emission.
OCIS co des: 320.0320, 190.5530, 070.4790.
doi: 10.3788/COL20100802.0244.
Femtosecond filamentation in transparent optical me-
dia has attracted broad attention for more than one
decade
[1−11]
. From a fundamental point of view, fil-
amentation dynamics constitutes a remarkable system
in which many physical effects and properties are ob-
served. It is also particularly attractive for fruitful appli-
cations, such as pulse compression
[12]
, supercontinuum
generation
[13]
, third harmonic generation
[14,15]
and re-
mote sensing
[16]
, etc. The major physics of filamen-
tation was proposed as a dynamic competition among
Kerr-induced self-focusing and plasma-induced defocus-
ing. It has been pointed out that one signature of fila-
mentation is the X-wave generation in normally disper-
sive media
[17]
. Nonlinear X waves (shown as X-shaped
light bullets
[18,19]
in normal dispersive quadratic nonlin-
ear media), which also appear in experiment as conical
emissions, are introduced as the stationary wave packets
propagating without undergoing the natural spreading
due to diffraction or dispersion over long distances, and
exhibiting a conical energy flux with X-shaped intensity
distribution both in the near field (space-time domain)
and in the far field (wave-number-frequency domain). In
fact, the early manifestation of the X-shaped intensity
structures in both space-time domain and wave number-
frequency domain was studied in detail in Refs. [2] and
[4] in three-dimensional (3D) plots from the point of view
of the physics of conical emission, although the word “X-
wave” was not used at that time. Afterwards, inten-
sive experimental and theoretical studies have been per-
formed to study the same problem from the point of view
of X-wave formation in various media
[20−23]
. In most
of these studies, angularly resolved sp ectra of the en-
tire pulses, which are either measured experimentally or
calculated theoretically, frequently show X-shaped struc-
tures. In this letter, we show that by performing an-
gularly resolved spectral analyses on different temporal
portions of an ultrashort pulse during its nonlinear prop-
agation in water, new light can be shed on the dynamics
of X-wave formation or conical emission which has not
yet been revealed so far.
With simulations of a (3+1)-dimensional model, we
study the femtosecond filamentation in water for the pur-
pose of gaining deeper insight into the dynamics of X-
wave formation. The theoretical model is based on an
extended nonlinear Schr¨odinger (NLS) equation govern-
ing the slowly varying envelope of a linearly polarized
laser electric field, coupled with an evolution equation
for the electron density generated by ionization. The
equations can be written as
∂ε
∂z
=
i
2k
∆
⊥
ε −i
k
00
2
∂
2
ε
∂t
2
+ ik
0
n
2
|ε|
2
ε
−
σ
2
(1 + iωτ
c
) ρε −
β
K
2
µ
1 −
ρ
ρ
at
¶
|ε|
2K−2
ε, (1)
∂ρ
∂t
= σ
K
(ρ
at
− ρ) |ε|
2K
+
σ
U
i
ρ |ε|
2
, ρ ¿ ρ
at
, (2)
where t refers to the retarded time variable t = τ −z/v
g
(τ is the real time and v
g
is the group velocity), and z
denotes propagation distance; k = n
0
ω
0
/c and ω
0
are
the wave number and frequency of the carrier wave, re-
spectively. The Laplacian operator ∆
⊥
=
∂
2
∂x
2
+
∂
2
∂y
2
.
Parameters including the linear and nonlinear indices of
the media n
0
and n
2
, the dispersive coefficient k
00
, the
momentum transfer collision time τ
c
, the number of pho-
tons K, the mean ionization potential u
i
, the multipho-
ton absorption cross section β
K
, and the transition rate
σ
K
are the same as those in Ref. [20]. In this model,
as shown in Eq. (1), the linear effects include the trans-
verse diffraction and group-velocity dispersion, and the
1671-7694/2010/020244-04
c
° 2010 Chinese Optics Letters