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STUDIA UNIVERSITATIS BABEŞ-BOLYAI, PHYSICA, SPECIAL ISSUE, 2003
MODELING THE PROPAGATION OF FEMTOSECOND LASER PULSES
IN GASEOUS MEDIA
Valer Tosa
National Institute for R&D of Isotopic and Molecular
Technologies, P.O. Box 700, 3400 Cluj-Napoca, Romania
ABSTRACT. A mathematical model for the propagation of femtosecond
laser pulses in gaseous media is presented. The propagation of ultrashort laser
field was solved in the paraxial approximation for a 3D geometry with axial
symmetry. A Fourier transform coupled to a finite difference method was
chosen to solve the propagation equation of the pulsed electromagnetic wave.
The linear and nonlinear dispersion effects as well as the presence of the
electron plasma due to the ionization of the atoms was taken into account.
The atomic rate of ionization was calculated by various models proposed in
the literature and is shown to influence fairly well the field configuration in
the interaction region. The field configuration is found to be in good
agreement with the experimental data of laser produced fluorescence in the
interaction region.
The process of high­order harmonic generation (HHG) by rare­gas atoms in intense
laser fields has become a reliable way to produce high­brightness, tabletop size,
coherent radiation in the vacuum ultraviolet and soft­x­ray ranges (see 1 for a
recent review). The unique properties of the harmonic emission have opened the
way to relevant and new applications in i) atomic and molecular core­level 2--4,
photo­ionization 5, and plasma 6 spectroscopy, ii) x­ray fluorescence analysis 7,
iii) time­resolved solid state physics of surface states 8 and of UV photoemission
spectroscopy 9, iv) nano­ and micro­structured material charac­ terization, such as
porous silicon 10, v) XUV interferometry for the diagnostics of dense plasmas 6,
11, 12.
Tthe propagation of femtosecond (fs) laser pulses in transparent media induces
strong nonlinear behaviours. In particular, in a gaseous medium, where the pulse
induces a considerable ionization plasma, the laser beam departs form its gaussian
form, both in intensity and in phase. The harmonic field, generated by such a
modified laser pulse will essentially depend on the driving field characteristics.
Investigating these characteristics helps one to controle and optimize the harmonic
emission.
NUMERICAL MODEL
Typical calculations of the harmonic field consist of three parts: a) the calculation
of the driving field in the interaction region b) the microscopic response of the
single atom to the driving field, which is then inserted as a source term in the c)
V. TOSA
propagation equation for the harmonic field. It is also important to adopt, within
the model framework, an appropriate model for the ionization of the atoms in
oscillating fields of ultrahigh intensity. In the following we will detail the
description of the physical model adopted for the pulse propagation in ionized
media and we will mention the analytical and numerical methods used for solving
the propagation equations.
In an ionized gas pulse propagation is affected by diffraction, refraction, nonlinear
self-focussing, ionization, and plasma defocusing. The pulse evolution in such
media are described by the wave equation which can be written as:
2 E1(r, z,t ) 
1  2 E1(r, z,t )  2
2 ) E (r, z, t )
 2 (1eff
1
c2
t 2
c
(1)
where E1(r,z,t) is the transverse electric field of the laser, of frequency . Radial
symmetry is assumed, therefore cylindrical coordinates are used throughout. The
effective refractive index of the medium can be written as
eff (r , z, t )  0  2 I (r , z, t ) 
 p2 (r , z, t )
2 2
(2)
The first linear term 0=1+1+i1 accounts for the refraction (1) and absorption
(1) while the second term describes a (3) process depending on laser intensity I,
and known as the optical Kerr effect. Finally the third term contains the plasma
frequency p=(4e2ne/m)1/2 and accounts for the presence of a density ne of
electrons (of mass m and charge e) per unit volume. It is known that if diffraction,
self focussing due to 2, and defocusing due to plasma generation are properly
balanced, a self guided optical beam can be formed and propagated over extended
distances, i.e. many vacuum Rayleigh lenghts. Keeping only the real terms in the
refractive index, the propagation equation for E1(r,z,t) becomes
2 E1(r, z,t ) 
2

1  2 E1(r, z,t ) 2   p

k
 2(1 2 I ) E1(r, z, t )
2
2
2


c
t


(3)
The method adopted to solve this equation is described by Priori et al. and will only
be mentioned briefly here. We write the equation in the moving frame, and, after
performing the paraxial approximation, eliminate the temporal derivative by a
Fourier transform, and obtain the equation:
2 E1(r, z, ) 
2


2i E1(r, z, ) ~  2   p
 F k
2(1 2 I ) E1(r, z,t )
2


c
z
  


(4)
MODELING THE PROPAGATION OF FEMTOSECOND LASER PULSES IN GASEOUS MEDIA
~
where F is the Fourier transform operator acting on the temporal coordinate. We
mention that Priori et al. solved the same equation taking into account only for the
electrons plasma term.
From the numerical point of view, the right hand side of Eq. (4) has both an
implicit (through plasma frequency term and 2I term) and an explicit dependence
on E1. For this reason, Eq. (4) was solved selfconsistently in every z step as
follows. After advancing the solution E(r,z,) by a Crank-Nicolson scheme, we
calculated E(r,z,t) by a back Fourier transform, thus evaluated again the right hand
side term of Eq. (4), firstly in t domain and then in  domain. Crank-Nicolson
scheme was applied again on the same step z and a new E(r,z,) solution was
obtained. The iteration was repeated until the difference between the new and the
old solution was under an imposed threshold.
The energy loss by the pulse during propagation is made up of two contributions:
the photoionization term reprezented by 1 and the energy consumed in ionization
processes. By writing the energy balance for the ionization process we obtain the
immaginary contribution to the refractive index as   (n0 I p w ) / 4I , where n0
is the atomic density, Ip the ionization potential, and w the ionization rate for an
average intensity I. The energy loss was estimated for each (r,z) point after every
successful integration step, and the dumped field was used in the next integration
step. Energy loss by inverse bremstrahlung (collisional absorption) was not
considered here because the working gas pressure is low.
RESULTS AND DISCUSSIONS
We will present in the following two typical results concerning the field
configuration in the interaction region. Plotted in Fig. 1 is the peak intensity of the
pulse after passing through a gaseous medium of constant density. In particular we
have considered here Xe at 0.6 torr. The cell has a total length of 14 cm and is
placed in a loose irradiation geometry produced by a lens with f=5 m. Two
Cell in the diverging beam
Cell in the converging beam
2.0
2
Laser Intensity (10 W/cm )
2
W/cm )
2.0
1.5
z=0
z = 1.4 cm
z = 2.8 cm
z = 4.2 cm
z = 5.6 cm
7 cm < z < 14 cm
1.0
zc = -25 cm
1.5
z=0
z = 1.4 cm
z = 2.8 cm
z = 4.2 cm
z = 5.6 cm
7 cm < z < 14 cm
14
Laser Intensity (10
14
zc = 7 cm
0.5
0.0
1.0
0.5
0.0
0
200
400
r (m)
600
800
0
200
400
600
800
r (m)
Fig. 1. The peak intensity of the laser pulse as a function of z, the distance
from the cell input, and for two cell positions as specified on the graphs.
V. TOSA
positions of the cell with respect to the focus are calculated: one with the cell in the
converging beam, with zc=-25 cm, the other with the input pinhole in the focus, i.e.
zc=7 cm. Here zc represents the distance from the focus to the cell input pinhole. As
one can see, the field configuration is rather different in the two cases: a
convergent beam induces a flat configuration that persists even after 14 cm of
propagation. This is due to the balance between the divergent lens created by the
electron plasma and the convergence of the beam. When the cell is placed in the
focus or after the focus, i.e. in the diverging beam, an initial flat radial distribution
tends to form but is gradually destroyed with increasing z, as seen in Fig. 1.
As distinct from the common self-guiding effect, where the field confinement is
due to the total internal reflection at the guide boundaries, in this case [15] the
effect is obtained due to a strong reflection of the trapped wave from the plasma
boundary that is sharp as compared to the transverse scale. As shown recently [14,
15], these field configurations are particularly favorable for the generation of high
order harmonics, leading to an optimisation of the conversion efficiency.
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