Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption
The interaction of ultrashort laser pulses with plasma is studied, where the pulse is considered as a photon flux and the plasma is presented by an ensemble of free electrons absorbing photons due to backstopping. The penetration depth is estimated for powerful ultrashort laser pulses at multiphoton...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption / D.V. Zaitsev // Вопросы атомной науки и техники. — 2000. — № 1. — С. 74-76. — Бібліогр.: 11 назв. — англ. |
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| citation_txt | Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption / D.V. Zaitsev // Вопросы атомной науки и техники. — 2000. — № 1. — С. 74-76. — Бібліогр.: 11 назв. — англ. |
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| description | The interaction of ultrashort laser pulses with plasma is studied, where the pulse is considered as a photon flux and the plasma is presented by an ensemble of free electrons absorbing photons due to backstopping. The penetration depth is estimated for powerful ultrashort laser pulses at multiphoton collisional absorption.
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ 2000. №1.
Серия: Плазменная электроника и новые методы ускорения (2), с. 74-76.
74
UDK 533.9
ESTIMATION OF ULTRASHORT LASER PULSE PENETRATION DEPTH
INTO PLASMA AT MULTIPHOTON COLLISIONAL ABSORPTION
D.V. Zaitsev
Central Institute of Physics and Technology Sergiev Posad-7, 141307, Russia
The interaction of ultrashort laser pulses with plasma is studied, where the pulse is considered as a photon flux and
the plasma is presented by an ensemble of free electrons absorbing photons due to backstopping. The penetration
depth is estimated for powerful ultrashort laser pulses at multiphoton collisional absorption.
Inroduction
One of the important aspects in the problems of laser
thermonuclear fission is estimating the powerful laser
penetration depth. In [1], the infinitely penetration depth
of ultrashort laser pulses into conducting media is
substantiated, even when the field spectrum is below the
plasma frequency. In [1], the field dynamic in metals to
be described by the sine-Gordon equation having
undamped soliton solutions. Here due to pulse duration,
ultrashort as compared to the electron relaxation time in
conductors, the beam absorption due to backstopping is
neglected. Meanwhile, it is important to take into
account the collisional absorption by electrons,
otherwise one can overestimate penetration of the pulse
of arbitrary duration, when considering it as a sequence
of ultrashort pulses. In [2], we show, that account the
collisional absorption by electrons in process of ultra-
short pulses interaction with a conductors lead to finite
penetration depth. But in [2] we neglect the processes of
multiphoton absorption. In the present work, we
estimate the penetration depth of ultrashort laser pulses
into plasma at multiphoton collisional absorption.
The theory multiphoton absorption on the basis of
the quantum-mechanical theory is developed (for
example, [1] and reference there). An estimation of n
quantum absorption probability in one act of interaction
is known
Рn=αn In, (1)
here I - intensity of radiation; αn - factor of
proportionality.
However, from the practical point of view, more
important estimation of mathematical expectation and
variation of quantum quantity, absorbed by electron in
one act of interaction, then estimation of probability of
absorption of quantum quantity. Besides the existing
theory multiphoton absorption is developed for
monochromatic field, in this connection the analysis of
interaction of ultrashort pulses with plasma requires
attraction of the new approaches.
In the present work the research of multiphoton
collisional absorption by electron is spent with use of
methods of the Markov processes theory with discrete
states and continuous time. In order to demonstrate it we
shall carry out a substantiation of the offered approach,
that is we shall show compatibility of Markov processes
theory with quantum mechanics. The principles of this
approach was reported in [3].
1. Compatibility of markov processes theory
with quantum mechanics
In work [4-7] is shown, that effects compelled
stimulated Brillouin scattering (SBS), the amplification
and generation of laser radiation, chemical reaction rate
can be correctly described on the basis of the Markov
processes theory with discrete states and continuous
time. In a basis of the given approach the description of
physical effects with the help the graph (Fig.1) is
necessary, each state Si answers a certain condition of
examined process (system). Dynamics of system is
described with the help of the Kolmogorov equations for
probabilities Pi of each state Si.
dPi /dt= -ν2Pi + ν1Pi-1. (1.1)
Fig.1. Graph of a some quantum system
Here ν1, ν1 - random flux intensities transportation's
between states.
As is known, considered in work [4-7] the effects are
described within the framework of the quantum
mechanics on the basis of the Shrodinger equation
iћ ∂Ψ
∂t
=HΨ, (1.2)
We shall note formal similarity of expressions (1.1)
and (1.2) - the left parts of the equations contain first
derivative on time. Besides Ψ(r,t) 2 - probability of a
finding of system in a vicinity of a point r at the moment
of time t, that is, actually, Pi. There is the hypothesis that
there is the deep interrelation the Kolmogorov equations
system with the Shr dinger equation not only for the
same problems [4-7], but also in a general case. We
shall show, that the system of the Kolmogorov equations
can be received from the Shrodinger equation.
We shall substitute in the Shr dinger equation (1.2)
wave function Ψ=аеiS/ћ (a - slowly varying function; S -
action) [8], we shall receive by making differentiation
a S
t
i a
t
a
m
S i
m
a S i
m
S a
m
a Ua∂
∂
∂
∂
− + ∇ − − ∇ ∇ − + =! ! ! !
2 2 2
02
2
( ) ∆ ∆
In the last equation are available real and imaging terms.
Equating that and other separately to zero, we shall
receive two equations:
∂
∂
S
t m
S U
ma
a+ ∇ + − =1
2 2
02
2
( ) ! ∆ , (1.3)
75
∂
∂
a
t
a
m
S
m
S a+ + ∇ ∇ =
2
1 0∆ . (1.4)
The first equation without the account last term,
containing ћ2, represents the classical Hamilton-Jacob
equation for action S.
The second of the received equations after
multiplication on 2а can be rewritten as (with the
account а2=Рi and ∇ S/m=v - velocity)
dPi/dt= - div v .Pi - v.grad Pi. (1.5)
We shall consider for simplicity one dimension
movement. By choosing dx1=dx2≈∆x, (here ∆x - size of
quantum system) we shall receive, that flux intensities
transportation's between states depend on life-time of
quantum system state ν1=1/dt1 and ν2=1/dt2 accordingly
div v = ∂v/∂x ≈ (v2-v1)/∆x=(dx2/dt2)/∆x-(dx1/dt1)/∆x=
=ν2-ν1,
v.grad Pi = v.∂Pi/∂x ≈ v (Pi-Pi-1)/∆x=(Pi-Pi-1)
(dx1/dt1)/∆x = ν1(Pi-Pi-1).
By substituting the last expressions in (1.5), we shall
receive
dPi/dt= -(ν2-ν1)Pi - ν1(Pi-Pi-1)= -ν2Pi + ν1Pi-1,
That is, Kolmogorov equation (1.1) for probability Pi of
a finding of considered quantum system in a state Si.
Thus, is shown equivalence of the description of
quantum systems on the basis of the Shrodinger equation
and on the basis of the Markov processes theory
together with the Hamilton-Jacob equation.
2. Investigation of multiphoton collisional
absorption on the basis of the Markov
processes theory
Investigation of multiphoton collisional absorption
on the basis of the Markov processes theory we shall
begin from construction the graph (Fig.2).
Fig.2. Graph of multiphoton collisional absorption
process
On Fig.2. are designated:
Si - states of an electron in adequate an absorption i
photons;
σi - cross section of an absorption of i-th a photon;
j - photon flux of radiation interacting with plasma.
For each state Si the Kolmogorov equation for
probability of each state [9] can be written as
dPi/dt= -σi j Pi + σi-1 j Pi-1 (2.1)
with initial conditions
Ро=1, Рi=0, i=1,∞. (2.2)
Us will be interest the probability Pi in an instant t, when
the electron already has absorbed i-1 a photons.
Then Pi ≈ 0 and the equation (2.1) can be rewritten as
dPi/dt= σi-1 j Pi-1. (2.3)
By dividing the last expression on intensity of
photon flux j, we shall receive (by definition [10] P.439)
cross section of an absorption i photons
σi= σi-1 Pi-1. (2.4)
The system of the equations (2.1) with the initial
conditions (2.2) allows the analytical solutions
Рk=
σ
σ σ
σ
l
l
k
i m
m
m i
k i
i
k
jt=
−
=
≠
=
∏
−∏
−∑ 0
1
0
0 ( )
exp( )
,
. (2.5)
Expression (2.5) allows to carry out estimations of
mathematical expectation of quantity absorbed photons
by electron at its collision with an ion in a field of a
powerful pulse of laser radiation. However for this
purpose an estimation of section σi is necessary. In work
[11] with use of methods of quantum electrodynamics
we estimate cross section of collision absorption photon
by electron for unrelativistic case
σ(ac)= 1
2π
Z2α re
2 λ3ni
c
v
F( v
v' ,θ), (2.6)
Where Z - the charge number of an ion;
α=1/137 - constant of a thin structure;
re - classical radius of an electron;
λ - wavelength of radiation;
ni - concentration of ions in plasma;
c - light velocity;
v, v’ - velocities of a colliding and scattered electron
accordingly;
F(θ) - factor of the order 1;
θ - angel between photon and colliding electron.
The estimation σi on the basis (2.6) turns out by
substitution of velocities of a colliding and scattered
electron v v’ before and after absorption i-th a photon.
The further analysis the most simple to carry out for
a case of that answers case of interaction of photons
with wavelength up to a soft X-radiation with low- and
high-temperature plasma with Т < 108 К. The
consideration of relativistic case does not cause basic
difficulties. For unrelativistic case it is possible to
consider that σi≈const. Then from (2.1) instead of (2.5)
easily to receive the following analytical solutions for
each probability Рk
Pi=
( )
!
σ τj
i
i
exp(-σjτ), (2.7)
here τ=min{τc,τu} - minimum time from time of impact
of an electron with an ion and ultrashort laser pulse
duration. As the intensity of photon flux j is proportional
to an radiation intensity I (j=I/ћω in a laboratory system,
where the ion is stopped), probability of an absorption
of n photons, as follows from (2.7), is proportional In
pursuant to (1), that testifies to adequacy of the offered
approach to the analysis effect of a multiphoton
absorption. The evaluation of probabilities (2.7) allow
us to receive an analytical evaluation of expectation of
number of absorbed photons in one act of interaction in
case of a powerful ultrashort laser pulse
k=ΣiPi=σjτ. (2.8)
The obvious result - average of absorbed photons
number equally to number photons witch interacted with
electron in this case received, that is a corollary of
approximation σi = const. In more general case σi≠const
76
the given approach also allows to receive an analytical
evaluation of an average of absorbed photons number.
3. Estimation of ultrashort laser pulse
penetration depth into plasma at
multiphoton collisional absorption
To evaluate the penetration depth for ultrashort laser
pulses, let us consider the light-matter interaction in the
framework of a next concept. We consider the
interaction of laser photons flux with plasma electrons
which periodically colliding with the ions ones within
the average time
<t>=1/ νei, (3.1)
where vei is the frequency of those collisions. Note that
such an approach naturally relieves the problem of time
representation for the field of laser pulse with a duration
of few field oscillations or even shorter than one period.
To simplify further analysis, we assume the medium
parameters (ρ, T, P, ne etc.) to be unchanged during
irradiation, which is not correct in the case of long
(nanosecond and longer) laser pulses. However, this
assumption is valid for the pulse duration comparable or
shorter than the average time of collision between
quasifree electrons with ions. On this basis, the
"ultrashort" laser pulse is defined as a pulse during
which the medium has no time to change its parameters.
These parameters vary after passing the ultrashort laser
pulse through the medium. Since <t>=1/vei≈l0-13s, this
definition embraces femtosecond and shorter pulses.
The flux intensity j after passing the thin layer
plasma dx is additionally reduced by νeinek dx
dj=-νeinek dx, (3.2)
where k is average number of absorbed photons in one
act of interaction in case of a powerful ultrashort laser
pulse. After substituting k from (2.8) easily to receive
the following equation
dj=-νei neσ jτ dx, (3.3)
which, in turn, is integrated to yield
j=j0 exp(-σ neνei τ x) (3.4)
from here the penetration depth for powerful ultrashort
laser pulses is estimated as
l= 1 1
σ ν τne ei
(3.5)
For case τc <τu the penetration depth is not depended
from pulse duration l= 1 1
σ ν τne ei c
. The pulse of
arbitrary duration, when considering it as a sequence of
ultrashort pulses, would have the same penetration
depth. However pulses longer than the ultrashort ones
heat the medium, thus changing the beam penetration.
Thus, in the present work on the basis the Markov
processes theory with discrete states and continuous
time receives expectation of number of absorbed
photons and receives estimation of ultrashort laser pulse
penetration depth into plasma at multiphoton collisional
absorption. The given results can be of interest for
research of heating dynamics of plasma by powerful
laser radiation.
References
1. E.M. Belenov and A.P. Kanavin. Quantum Electron.
1993, vol.23, p.335.
2. D.V. Zaitsev and N.S. Zaharov. Estimation of
ultrashort laser pulse penetration into conductors.
Bulletin of the Russian Academy of Sciences. Physics.
1997, vol. 61, №. 8, p. 1203-1206.
3. D.V. Zaitsev. About compatibility of Markov
processes theory with quantum mechanics. Scientific
session of MSEPI. Moscow -2000. Proceedings of
MSEPI. 2000, vol. 5, p. 193-195.
4. D.V. Zaitsev, V.M. Nikitin. Statistic characteristics of
the Stokes signal in stimulated Brillouin scattering.
Bulletin of the Russian Academy of Sciences.
Physics. 1994, vol. 58, № 2, p. 304-306.
5. D.V.Zaitsev. Mathematical model of laser generation
process on the basis of Markov processes theory.
.Proceedings of SPIE, 1995, vol. 2713, p. 2-7.
6. D.V. Zaitsev, N.S. Zakharov. Mathematical model of
initiation starting stage of inflammable mixture on
the basis of Markov processes theory. XI
Symposium on combustion and burst.
Chernogolovka, 1996,. vol. 2, p. 105-106.
7. D.V. Zaitsev. Variance estimation of radiation
intensity on laser amplifier output. Proceedings of
SPIE. 1998, vol. 3574, p. 783-790.
8. Landay L.D., Lifshitz E.M. Quantum mechanics.
Moscow: Nauka, 1989. (in Russian).
9. Ventsel E.S. and Ovcharov L.A. Theory of Random
Processes and Its Engineering Applications.
Moscow: Nauka, 1991. (in Russian).
10. Berestetskii V.B., Lifshitz E.M., Pitaevskii L.P.
Quantum electrodynamics. Moscow: Nauka, 1989.
(in Russian).
11. D.V. Zaitsev, N.S. Zakharov. Quantum
electrodynamics approach to collision absorption of
powerful ultrashort pulses. Proceedings of SPIE.
1998, vol. 3574, p.742-749.
|
| id | nasplib_isofts_kiev_ua-123456789-81609 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:30:47Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Zaitsev, D.V. 2015-05-18T12:42:12Z 2015-05-18T12:42:12Z 2000 Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption / D.V. Zaitsev // Вопросы атомной науки и техники. — 2000. — № 1. — С. 74-76. — Бібліогр.: 11 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/81609 533.9 The interaction of ultrashort laser pulses with plasma is studied, where the pulse is considered as a photon flux and the plasma is presented by an ensemble of free electrons absorbing photons due to backstopping. The penetration depth is estimated for powerful ultrashort laser pulses at multiphoton collisional absorption. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Газовый рaзряд, ППР и их применения Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption Article published earlier |
| spellingShingle | Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption Zaitsev, D.V. Газовый рaзряд, ППР и их применения |
| title | Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption |
| title_full | Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption |
| title_fullStr | Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption |
| title_full_unstemmed | Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption |
| title_short | Estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption |
| title_sort | estimation of ultrashort laser pulse penetration depth into plasma at multiphoton collisional absorption |
| topic | Газовый рaзряд, ППР и их применения |
| topic_facet | Газовый рaзряд, ППР и их применения |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81609 |
| work_keys_str_mv | AT zaitsevdv estimationofultrashortlaserpulsepenetrationdepthintoplasmaatmultiphotoncollisionalabsorption |