Interference Stabilization and Possibility of Amplification and Lasing
in Plasma Channel Formed in Gas by Intense Femtosecond Laser
Field
A. V. Bogatskaya
1,2
, E. A. Volkova
1
and A. M. Popov
1,2
1
D. V. Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119991, Russia
2
P. N. Lebedev Physical Institute, RAS, Moscow, 119991, Russia
Keywords: Strong Laser Fields, Multi-photon Ionization, Interference Stabilization, Lasing.
Abstract: The effect of interference stabilization of Rydberg atoms in high-intensity laser field is proposed to create
the plasma channel with population inversion between set of Rydberg states and the ground state for
conversion of the input laser energy into the VUV and XUV frequency band. Furthermore, there is a
possibility to create a population inversion between high-lying Rydberg states which can be used for lasing
and amplification in the IR, mid-IR and sub-terahertz frequency band.
1 INTRODUCTION
Progress in strong-field physics and laser – matter
interaction makes it possible to investigate a number
of new physical processes. In particular, generation
of high order harmonics of the incident laser pulse is
possible as a result of recombination process of the
photoelectron with the parent ion, when the electron
rescatters on it (Agostini and Di Mauro, 2004;
Krausz and Ivanov, 2009). On the other hand,
production of quasi-dc electric current in plasma
arising due to ionization of gases by a few-cycle or
bi-chromatic laser pulses causes the generation of
low-frequency terahertz waves (Kreß, et al, 2006;
Gildenburg and Vvedenskii, 2007; Wu, et al, 2008;
Silaev and Vvedenskii, 2009).
The idea of amplification of radio-frequency
(RF) radiation in ionized gas with population
inversion in the electronic continuum in gases
characterized by a Ramsauer minimum in transport
cross section was proposed long ago (Bekefi, et al,
1961; Bunkin et al, 1972).
In (Bogatskaya and Popov, 2013) it was
mentioned that such an amplification can be simply
realized in heavy rare gases with a Ramsauer
minimum in transport cross section if the gas media
is ionized by UV laser pulse with the duration less
than the characteristic time of relaxation of
photoelectron energy spectrum. This idea was
analyzed in detail in (Bogatskaya et al, 2013;
Bogatskaya et al, 2014).
On the other hand self-focusing and
filamentation of high-intensity laser pulse can lead
to remote lasing action from a filament (Luo, et al,
2003). In (Wang et al, 2013a; Wang et al, 2013b)
lasing action from the Ti-Sa laser femtosecond laser
filament in air was demonstrated experimentally.
Different theoretical analyses of the filament-
initiated nitrogen laser are presented in (Penãno et
al, 2012; Kartashov et al, 2015). Recently,
population inversion in fluorescent fragments of
super-excited molecules inside an air lament and
spontaneous emission from these fragments were
also detected by (Chin and Hu, 2015).
In this paper we would like to pay attention that
interference stabilization (IS) phenomenon predicted
and studied in (Fedorov and Movsesian, 1988;
Fedorov et al, 1996;
Fedorov et al, 2012) or
population trapping in high-lying Rydberg states
(Talebpour et al, 1996; Azarm et al, 2013) in strong
laser fields also results in population inversion
between the set of Rydberg states and low-lying
excited or ground states. It is shown that such a
trapping can be used for lasing and amplification of
electromagnetic radiation in a range from visible to
VUV or XUV frequency band.
Bogatskaya, A., Volkova, E. and Popov, A.
Interference Stabilization and Possibility of Amplification and Lasing in Plasma Channel Formed in Gas by Intense Femtosecond Laser Field.
DOI: 10.5220/0005646802890296
In Proceedings of the 4th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2016), pages 291-298
ISBN: 978-989-758-174-8
Copyright
c
2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
291
2 INTERFERENCE
STABILIZATION,
POPULATION TRAPPING IN
RYDBERG STATES AND IDEA
OF AMPLIFICATION OF
ELECTROMAGNETIC
RADIATION
Interference stabilization (IS) phenomenon consists
in the coherent repopulation of neighboring Rydberg
states via the continuum by Raman Λ - type
transitions, which leads to the destructive
interference of photoionization transition amplitudes
from neighboring Rydberg levels. Theoretical
consideration of this phenomenon was first
discussed in (Fedorov and Movsesian, 1988) for the
case of an atom initially prepared in a Rydberg state.
Nevertheless, the phenomenon of IS was also found
to take place for ground states of atoms interacting
with high-intensity laser field (Dubrovskii et al,
1991). In this case the multiphoton excitation via
Freeman resonances (Freeman et al, 1987) appearing
with ac Stark-shifted-levels and IS results in
population trapping in a number of high-lying
(Rydberg) states. Independently, similar idea was
suggested by S L Chin to explain the experimental
results obtained in his group on the strong-field
ionization yields in atoms of rare gases (Talebpour
et al, 1996). We believe that the set of recent
experimental data (Azarm et al, 2013; Nubbemeyer
et al, 2008; Eichmann et al, 2013; Eichmann, et al,
2009) can be interpreted in frames of the discussed
model. Recently the effect of population trapping in
Rydberg states in argon irradiated by the
femtosecond Ti-Sa laser pulse was also observed
experimentally by Fechner et al, 2015.
In this paper we draw attention to the fact that IS
and population trapping in a gas media pumped by
the external source of energy from a physical point
of view is very close to the effect of the population
inversion of gas atoms. This phenomenon reveals a
thought-provoking idea of new methods of
amplification and lasing of electromagnetic radiation
(Bogatskaya and Popov, 2015).
At fig. 1 we present some results of numerical
simulations of ionization and excitation probabilities
versus the peak laser intensity for argon obtained for
the second harmonic (
1.3=
ω
= eV) of the sine-
squared linear-polarized Ti-Sa laser pulse of forty
cycles duration (
53 fs). These data were obtained
by the method of the direct numerical integration of
the TDSE for model single electron atoms with
1E13 1E14 1E15
1E-4
1E-3
0,01
0,1
1
probability
intensity, W/cm
2
excitation
ionization
Figure 1: Probabilities of ionization and excitation of
argon atoms irradiated by the 2nd harmonic of
femtosecond Ti-Sa laser pulse in dependence on laser peak
intensity.
energy spectra similar to that of real Ar atoms
interacting with external laser field. The detailed
information for chosen atomic potential can be
found in (Bogatskaya et al, 2013). One can see the
stabilization phenomenon leading to the substantial
fraction of atoms being trapped in the Rydberg states
and channel closing effect similarly to data obtained
previously (Volkova et al, 2011; Popov et al, 2010;
Fedorov et al, 2012) for the fundamental frequency
of the Ti-Sa laser. The dependence of excitation
probability on the laser intensity reveals a lot of
maxima and minima changing each other. The
maxima observed on the excitation curve are seen to
be separated by approximately the same value of
laser intensity equals to
14
102 ×I
Δ
W/cm
2
, which
exactly corresponds to the condition that the
ponderomotive shift of the ionization threshold
22
0
4
ωε
=
pond
U
reaches the value of the photon
energy
ω
= ,
ω
Δ
==
pond
U
. Confirmation of the
aforesaid can be seen at fig. 2: photoionization peaks
gradually are shifted towards the lower energies as
the pumping intensity goes up. For example, the 6-
photon ionization channel for argon atom is closed
near the intensity value
14
102 × W/cm
2
, next one at
14
104× W/cm
2
and so on (see fig. 2). Furthermore,
the indented structure of the photoelectron spectrum
is related to the coherent repopulation of Rydberg
states via Raman Λ-type transitions. Fig. 3 keeps
more extended information about peak positions and
channel closing over the intensities less than
15
10
W/cm
2
.
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
292
02468101214
0,0
0,2
0,4
0,6
0,8
1,0
spectrum, a.u.
energy, eV
2e14
3e14
4e14
Figure.2: Photoelectron spectra for argon atoms irradiated
by the 2
nd
harmonic of Ti-Sa laser pulse.
0
2x10
14
4x10
14
6x10
14
8x10
14
1x10
15
0,0
3,1
6,2
9,3
12,4
n=6
n=7
n=8
n=9
n=10
peak position, eV
intensity, W/cm
2
Ar
Figure 3: Peak positions in photoelectron spectra for argon
in dependence on laser intensity.
More detailed information about the distribution
of trapped population in dependence on Rydberg
states with different angular momentum and
principal quantum number is performed at fig. 4 for
different laser peak intensities in the case of argon
atoms. One can point out that for the intensity
14
210
W/cm
2
the states with even angular
momentum are dominantly populated (see fig. 4a).
This results from the six-photon resonance between
the ground state and the set of Rydberg states
accompanied by the
Λ
- type transitions
repopulating the Rydberg states with even angular
momentum. As laser intensity grows (
14
104 ×
W/cm
2
), seven-photon ionization channel is found to
be closed and hence we observe seven-photon
excitation which effectively populates states with
odd value of angular momentum (see fig. 4b).
Thus, we have a variety of emission lines which
could be obtained after the irradiation of the atom by
intense laser pulse. Using transitions between
neighboring Rydberg states it is possible to obtain
quanta from THz to IR frequency range. Cite data
enable us to estimate gain factors for different
0 4 8 12 16
1E-7
1E-6
1E-5
1E-4
1E-3
0,01
population
principal quantum number
s
p
d
f
g
h
i
a
0481216
1E-4
1E-3
0,01
population
principal quantum number
s
p
d
f
g
h
i
b
Figure 4: Population distribution of Rydberg states with
different angular momentum in dependence on principal
quantum number for argon atoms. Laser intensities are
14
102× W/cm
2
(a) and
14
104 × W/cm
2
(b).
types of radiation. For example, the transition
1,,
21
± lnln ( 1,1
211
==Δ>> nnnn ) with the
quanta
1.01
761
3
1
=
n
n
Ω
eV (see fig. 4)
represents the down conversion of the 2
nd
harmonic
of Ti-Sa laser radiation into the far IR frequency
band. Similar, for
54
1
=n , 10
1
n ( 1=Δn ) we
obtain mid IR and THz frequency band respectively.
On the other hand transitions from populated
Rydberg to the low-lying excited states (
3,2
2
=n )
give rise to the emission of visible light.
There is also a possibility to generate high
energy quanta using the transition
snp 1
, which
is available at higher laser intensities when the
ground state is depleted. Data on fig. 5 depict the
difference between population of np and 1s states in
argon-like atom depending on principal quantum
number. Thus, the set of lines from the Layman
series
snp 1
with the quanta 21= Ry
Ω
may
be emitted effectively after the irradiation of atom
by the intense laser radiation of intensities above
14
107 × W/cm
2
.
Interference Stabilization and Possibility of Amplification and Lasing in Plasma Channel Formed in Gas by Intense Femtosecond Laser Field
293
The general expression for the photoabsorption
cross section for transitions
12
1nl n l→± reads
12
2
1
2
nl n l
ph
A
λ
σ
πν
→±
=
Δ
(1)
where
λ
is the transition wavelength,
12
1nl nl
A
→±
is
the Einstein coefficient for the spontaneous emission
and
ν
Δ
is the width of the emitted line
2
1,,
3
3
1,,
2121
3
4
±±
=
lnlnlnln
d
c
A
Ω
(2)
with speed of light
137=c . The matrix element for
the dipole moment in WKB approximation (Delone
et al, 1983) can be estimated as
32 32
112
1
np s
dnn
(
1~l ). Then, for example, for visible radiation one
obtains
.
103
3
1
10
2
1
2
1
3
3
1
2
2
2121
n
d
c
A
n
lnlnlnln
=
±±
Ω
Ω
(3)
The width
ν
Δ
of the Rydberg state arising from the
collisional broadening and for
18
10=N cm
-3
can be
estimated as
6
103
×Δ
ν
(in atomic units). Then
the cross section can be estimated as
1517
1010
÷
ph
σ
cm
2
. If one assumes, that the
population inversion is
15
10ΔN cm
-3
the gain
factor is calculated as
101.0
Δ
×= Ng
ph
σ
cm
-1
.
It means that if
5.0=g
cm
-1
for the propagation
length of 10 cm the intensity of the visible
electromagnetic radiation will be increased 100
times. For longer wavelengths corresponding to IR,
mid IR, THz radiation the gain factor is supposed be
smaller due to lower value of the Einstein coefficient
of the spontaneous emission. Furthermore, for high
values of Rydberg levels it is necessary to consider
that the size of atoms increases as
2
n
: for example,
for
10n = we have gas atoms 100 times larger.
Thus, one should diminish concentration meaning
that the gain factor will decrease as well.
In view of the above argumentations one can
conclude that processes of amplification and lasing
in visible and VUV frequency bands seems to be
more beneficial therefore we are going to dwell
upon them in the next section.
2 4 6 8 10121416
-0,0008
-0,0004
0,0000
0,0004
0,0008
0,0012
0,0016
np-1s
principal quantum number
np-1s (9.5e14)
np-1s (8e14)
np-1s(7e14)
Figure 5: The difference between population of np and 1s
states in argon-like atom in dependence on principal
quantum number for different laser peak intensities.
3 RATE EQUATIONS AND
PROCESSES OF GENERATION
AND AMPLIFICATION OF
VISIBLE AND VUV PULSES IN
PLASMA CHANNEL
To study the phenomena of generation and
amplification of electromagnetic pulses in a plasma
channel with population inversion the approach of
rate equations was applied (Kartashov et al, 2015).
Based on this approach the following system for the
forward generation (amplification) can be written:
N
l
d
zIg
z
zI
s
Δ
+=
2
22
),()(
),(
τ
ω
ττ
τ
=
,
(4)
ion
ph
N
N
zI
d
Nd
τω
τ
σ
τ
Δ
Δ=
Δ
=
),(
.
(5)
Here
)(
τ
g is the gain factor, czt =
τ
is the
retarded time,
s
τ
and
ion
τ
is the time of
spontaneous decay and of ionization of Rydberg
levels by electron impact respectively,
NΔ is
population inversion for the certain transition. The
factor
ld 2 (d is the plasma channel diameter, and l
is the plasma channel length) means the part of
spontaneous emission that lies in the direction of
ionizing pulse propagation and can be used for
amplification.
It is possible to provide estimation for the
intensities of generated (amplified) pulse when we
are able to neglect its impact on the population
inversion decay:
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
294
210
W/cm10~
phion
I
στ
ω
=
<<
.
(6)
This estimation was performed in the assumption
that the electronic density is ~10
17
cm
-3
, and that the
electron energy in continuum is significantly larger
than the binding energy of Rydberg atoms. Provided
the condition (6) is accomplished at propagation
distance L one obtains the following expression for
the population inversion decay:
()
ion
NN
τ
τ
Δ=Δ
exp
0
.
(7)
Formulas (4) and (7) give us the following
dependence of the generated signal intensity:
()
()
()
[]
.1expexp
22
,
0
2
Δ
×
=
zN
l
d
zI
ionph
phs
ττσ
στ
ω
τ
=
(8)
For the Rydberg states with principal quantum
number 106
=
n
ion
τ
can be estimated as 100 fs.
Fig. 6 and 7 show the spatio-temporal
development of generation of the VUV and visible
radiation respectively in a plasma channel with
population inversion. The data obtained for the
visible radiation correspond to the laser intensity
14
104 ×
W/cm
2
while for VUV radiation generation
the laser intensity was chosen significantly higher (
14
105.9 ×
W/cm
2
) in order to reach the regime of
the depletion of the ground state and higher values
of population inversion. The quantity of population
inversion for transition
sp 17
(VUV radiation)
was taken from fig. 5 and from fig. 4 for transitions
snp 2
(visible light). For VUV radiation we
observe significant lasing process based on the
spontaneous decay of the Rydberg state, in particular
generated intensity could reach
5
10
W/cm
2
at a
distance of 5 cm. However in the case of visible
radiation mainly due to the less value of generated
frequency it is possible to attain generation much
less than for the VUV (see fig. 7).
The last question we would like to discuss in this
section is the process of amplification of a seed
pulse in such a plasma channel. We assume the input
pulse (
0=z
) to be characterized by the sine-squared
envelope
)2(sin),0(
2
p
tzI
τπτ
==
(9)
012345
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
I
, W/cm
2
z,
cm
τ
=
t-z/c
=0
τ
=30 fs
τ
=100 fs
a
0
2x10
-14
4x10
-14
6x10
-14
8x10
-14
1x10
-13
10
1
10
2
10
3
10
4
10
5
z =
1 cm
z =
2 cm
z =
5 cm
I
, W/cm
2
t-z/c,
s
b
Figure 6: Spatial dependence of generated VUV pulse at
different instants of retarded time (a), temporal
dependence of generated VUV pulse at different
propagation distances (b) and two-dimensional
distribution of propagated pulse (c).
with
20=
p
τ
fs. Then general solution of (4) can be
written in a form
()
[]
zN
zI
ionph
p
ττσ
τπττ
Δ
×=
expexp
)2(sin),(
0
2
.
(10)
I(z,t-z/c), W/cm
2
z, cm t-z/c, s
c
Interference Stabilization and Possibility of Amplification and Lasing in Plasma Channel Formed in Gas by Intense Femtosecond Laser Field
295
0 1020304050
0
50
100
150
200
250
I,
W/cm
2
z
, cm
τ =
t
z/c=
0
τ
= 30 fs
τ
= 20 fs
a
0
2x10
-14
4x10
-14
6x10
-14
8x10
-14
1x10
-13
10
-3
10
-2
10
-1
10
0
10
1
10
2
I,
W/cm
2
t-z/c,
s
z =
50 cm
z =
10 cm
z
= 1 cm
b
Figure 7: Spatial dependence of generated visible radiation
at different instants of retarded time (a), temporal
dependence of generated visible radiation at different
propagation distances (b).
Typical dependences obtained from (10) for
propagation length of
25=L
cm for a number of
transitions
snp 2
corresponding to visible
radiation are presented at fig.8. One can see
considerable enhancement of a seed pulse up to 25
times at a given distance.
0
1x10
-14
2x10
-14
3x10
-14
4x10
-14
0
5
10
15
20
25
1
4
3
seed pulse
6p --> 2s
7p --> 2s
8p --> 2s
I
, a.u.
t-z/c,
s
2
Figure 8: Amplification of visible radiation in a plasma
channel. Curve (1) is the input pulse. Curves 2-4
correspond to the transitions
snp 2
, n=8 (2), 7 (3), 6
(4). Propagation length L=25 cm.
The obtained data demonstrate that the
amplification and generation of rather short pulses
with duration less than the ionization time of
Rydberg states looks fairly effective. Such a method
can compete with the conventional method of HHG
having been widely explored since pioneer papers
(L’Huillier et al, 1993; Lewenstein et al, 1994;
Becker et al, 1994).
4 LASING AND
AMPLIFICATION OF
ELECTROMAGNETIC
RADIATION DURING THE
LASER PULSE
In this section we would like to draw attention to
another way of obtaining lasing from population
trapped in Rydberg states with higher frequency that
differs dramatically from the above described one. It
is known that during the interaction of high-intensity
Ti-Sa laser pulse with trapped atoms their Rydberg
levels are shifted significantly that results in channel
closing and essential increment of the ionization
potential (see fig.9). In this case the position of
Rydberg levels can be estimated as
22
0
4
ωε
+
nn
EE
(11)
For example, if four channels of ionization are
closed (in argon such situation appears to exist for
the intensity
14
108~ ×
W/cm
2
– see fig.3), the
energy of the quanta
Ω
=
for the transition
snp 1
will be of order of 30 eV. It means that lasing in
XUV frequency band is also possible. The width of
the stabilized ac-Stark-shifted state np is determined
by the ionization process and in the intensity range
near the threshold of the IS can be written as
Γ
ΓνΔ
2
)(
2
1
==
nn
IS
EE
.
(12)
As the threshold for the IS stabilization for the 2
nd
harmonic of Ti-Sa laser radiation is close to
14
10
W/cm
2
, for the intensity
14
810× W/cm
2
the cross
section
ph
σ
for
~6 8n
can be estimated as
17
10
ph
σ
cm
2
. If one assumes that the inverse
population is
16
10ΔN
cm
-3
the gain factor is
estimated to be 0.1 cm
-1
that is very large value for
XUV radiation. In comparison with the emission in
the after-pulse regime pulses of XUV radiation can
be emitted only during rather short time interval near
the peak intensity of the laser pumping. Probably, it
is one more way to generate the pulses of sub-
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
296
femtosecond duration. As the Rydberg wave packet
is continually located in the vicinity of the atomic
nucleus and doesn’t spread in time, the proposed
effect of the up conversion should be much more
efficient than the emission of the XUV radiation
during the re-scattering of an electron on the parent
ion (Corkum, 1993).
Figure 9: Schematic illustration of the ponderomotive shift
of Rydberg states and the continuum boundary.
5 CONCLUSIONS
In this paper we propose to use the population
inversion between set of Rydberg states and low-
lying excited or even ground states appearing as a
result of the interference stabilization phenomenon
in strong laser fields for the amplification and
generation of visible, VUV and XUV frequency
pulses. Estimations for the possible gain factor are
performed at it is demonstrated that for typical
conditions of IS gain factor for the visible and VUV
radiation can reach ~ 0.1 - 1 cm
-1
which is very large
value implementing the possibility to obtain
effective emission in plasma channels produced by
high intensity laser pulse. We explore the process of
generation and amplification of different pulses
using the approach of rate equations. The problem of
generation of XUV pulses during the interaction
with high intensity pumping laser pulses is also
observed. We would also like to note that a given
lasing effect based on the IS stabilization
phenomenon can probably be observed in filaments
at rather far away distances from the source of
pumping laser radiation.
ACKNOWLEDGEMENTS
This work was supported by the Russian Foundation
for Basic Research (projects no. 15-02-00373, 16-
32-00123). Numerical modeling was performed on
the Lomonosov supercomputer.
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