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

and A. M. Popov

1,2

D. V. Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, 119991, Russia

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 ﬁlament 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

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=

= 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

probability

intensity, W/cm

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

102 ×≈I

W/cm

, which

exactly corresponds to the condition that the

ponderomotive shift of the ionization threshold

ωε

pond

reaches the value of the photon

energy

= ,

pond

. 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

102 × W/cm

, next one at

104× W/cm

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

W/cm

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

harmonic of Ti-Sa laser pulse.

2x10

4x10

6x10

8x10

1x10

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

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

210⋅

W/cm

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

- type transitions

repopulating the Rydberg states with even angular

momentum. As laser intensity grows (

104 ×

W/cm

), 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

0481216

1E-4

1E-3

0,01

population

principal quantum number

Figure 4: Population distribution of Rydberg states with

different angular momentum in dependence on principal

quantum number for argon atoms. Laser intensities are

102× W/cm

(a) and

104 × W/cm

(b).

types of radiation. For example, the transition

1,,

±→ lnln ( 1,1

211

=−=Δ>> nnnn ) with the

quanta

1.01

761

≈≈

−=

eV (see fig. 4)

represents the down conversion of the 2

harmonic

of Ti-Sa laser radiation into the far IR frequency

band. Similar, for

−=n , 10

≈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

=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

may

be emitted effectively after the irradiation of atom

by the intense laser radiation of intensities above

107 × W/cm

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

1nl n l→± reads

nl n l

πν

→±

(1)

where

is the transition wavelength,

1nl nl

→±

the Einstein coefficient for the spontaneous emission

and

is the width of the emitted line

1,,

2121

±→±→

lnlnlnln

(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

np s

dnn

→

≈

(

1~l ). Then, for example, for visible radiation one

obtains

103

2121

lnlnlnln

−

±→±→

⋅

≈≈

(3)

The width

of the Rydberg state arising from the

collisional broadening and for

10=N cm

-3

can be

estimated as

103

−

×≈Δ

(in atomic units). Then

the cross section can be estimated as

1517

1010

−−

÷≈

. If one assumes, that the

population inversion is

10≈ΔN cm

-3

the gain

factor is calculated as

101.0 −≈

×= Ng

-1

It means that if

5.0=g

-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

: 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:

zIg

⎟

⎠

⎞

⎜

⎝

⎛

∂

),()(

),(

ττ

(4)

ion

τω

−Δ−=

),(

(5)

Here

)(

g is the gain factor, czt −=

is the

retarded time,

and

ion

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

στ

(6)

This estimation was performed in the assumption

that the electronic density is ~10

-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

−Δ=Δ

exp

(7)

Formulas (4) and (7) give us the following

dependence of the generated signal intensity:

()

[]

.1expexp

−−Δ−

⎟

⎠

⎞

⎜

⎝

⎛

ionph

phs

ττσ

στ

(8)

For the Rydberg states with principal quantum

number 106

−=

ion

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

104 ×

W/cm

while for VUV radiation generation

the laser intensity was chosen significantly higher (

105.9 ×

W/cm

) 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

W/cm

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(

tzI

τπτ

(9)

012345

-1

, W/cm

t-z/c

=30 fs

=100 fs

2x10

-14

4x10

-14

6x10

-14

8x10

-14

1x10

-13

z =

1 cm

z =

2 cm

z =

5 cm

, W/cm

t-z/c,

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=

fs. Then general solution of (4) can be

written in a form

()

[]

ionph

ττσ

τπττ

−Δ

×=

expexp

)2(sin),(

(10)

I(z,t-z/c), W/cm

z, cm t-z/c, s

Interference Stabilization and Possibility of Ampliﬁcation and Lasing in Plasma Channel Formed in Gas by Intense Femtosecond Laser Field

295

0 1020304050

100

150

200

250

W/cm

, cm

τ =

−

z/c=

= 30 fs

= 20 fs

2x10

-14

4x10

-14

6x10

-14

8x10

-14

1x10

-13

-3

-2

-1

W/cm

t-z/c,

z =

50 cm

z =

10 cm

= 1 cm

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.

1x10

-14

2x10

-14

3x10

-14

4x10

-14

seed pulse

6p --> 2s

7p --> 2s

8p --> 2s

, a.u.

t-z/c,

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

ωε

+→

(11)

For example, if four channels of ionization are

closed (in argon such situation appears to exist for

the intensity

108~ ×

W/cm

– see fig.3), the

energy of the quanta

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

ΓνΔ

)(

1−

−

(12)

As the threshold for the IS stabilization for the 2

harmonic of Ti-Sa laser radiation is close to

W/cm

, for the intensity

810× W/cm

the cross

section

for

~6 8n −

can be estimated as

−

≈ cm

. If one assumes that the inverse

population is

10≈ΔN

-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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