Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse
On the basis of analytical solution of the time-dependent Schrödinger equation the excitation of residual current density (RCD) in a gas ionized by two-color laser pulse is studied. We find general analytical expression for the RCD for arbitrary values of the Keldysh parameter, which coincides with...
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irk-123456789-1122142017-01-19T03:02:31Z Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse Romanov, A.A. Silaev, A.A. Vvedenskii, N.V. Нелинейные процессы в плазменных средах On the basis of analytical solution of the time-dependent Schrödinger equation the excitation of residual current density (RCD) in a gas ionized by two-color laser pulse is studied. We find general analytical expression for the RCD for arbitrary values of the Keldysh parameter, which coincides with the semiclassical calculations in the case of tunneling regime of ionization. На основі аналітичного розв’язку нестаціонарного рівняння Шредингера досліджується збудження залишкової густини струму в газі, що іонізований інтенсивним біхроматичним лазерним імпульсом. Знайдено загальний аналітичний вираз залишкової густини струму для довільних значень параметру Келдиша, який співпадає з результатами напівкласичних розрахунків при тунельному режимі іонізації. На основе аналитического решения нестационарного уравнения Шрёдингера исследуется возбуждение остаточной плотности тока в газе, ионизируемом интенсивным бихроматическим лазерным импульсом. Найдено общее аналитическое выражение остаточной плотности тока для произвольных значений параметра Келдыша, которое совпадает с результатами полуклассических расчетов при туннельном режиме ионизации. 2015 Article Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse / A.A. Romanov, A.A. Silaev, N.V. Vvedenskii // Вопросы атомной науки и техники. — 2015. — № 4. — С. 286-289. — Бібліогр.: 18 назв. — англ. 1562-6016 PACS: 52.50.-b, 32.80.Rm http://dspace.nbuv.gov.ua/handle/123456789/112214 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах |
| spellingShingle |
Нелинейные процессы в плазменных средах Нелинейные процессы в плазменных средах Romanov, A.A. Silaev, A.A. Vvedenskii, N.V. Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse Вопросы атомной науки и техники |
| description |
On the basis of analytical solution of the time-dependent Schrödinger equation the excitation of residual current density (RCD) in a gas ionized by two-color laser pulse is studied. We find general analytical expression for the RCD for arbitrary values of the Keldysh parameter, which coincides with the semiclassical calculations in the case of tunneling regime of ionization. |
| format |
Article |
| author |
Romanov, A.A. Silaev, A.A. Vvedenskii, N.V. |
| author_facet |
Romanov, A.A. Silaev, A.A. Vvedenskii, N.V. |
| author_sort |
Romanov, A.A. |
| title |
Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse |
| title_short |
Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse |
| title_full |
Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse |
| title_fullStr |
Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse |
| title_full_unstemmed |
Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse |
| title_sort |
strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2015 |
| topic_facet |
Нелинейные процессы в плазменных средах |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112214 |
| citation_txt |
Strong-field approximation for analytical calculation of the residual current density excited by gas ionization with an intense two-color laser pulse / A.A. Romanov, A.A. Silaev, N.V. Vvedenskii // Вопросы атомной науки и техники. — 2015. — № 4. — С. 286-289. — Бібліогр.: 18 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-08T03:33:00Z |
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2025-07-08T03:33:00Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №4(98) 286
STRONG-FIELD APPROXIMATION FOR ANALYTICAL
CALCULATION OF THE RESIDUAL CURRENT DENSITY EXCITED BY
GAS IONIZATION WITH AN INTENSE TWO-COLOR LASER PULSE
A.A. Romanov1,2, A.A. Silaev1,2, N.V. Vvedenskii1,2
1Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia;
2Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
E-mail: vved@appl.sci-nnov.ru
On the basis of analytical solution of the time-dependent Schrödinger equation the excitation of residual current
density (RCD) in a gas ionized by two-color laser pulse is studied. We find general analytical expression for the
RCD for arbitrary values of the Keldysh parameter, which coincides with the semiclassical calculations in the case
of tunneling regime of ionization.
PACS: 52.50.-b, 32.80.Rm
INTRODUCTION
This work is focused on analytical investigation of
the excitation of quasi-dc residual current density
(RCD) due to gas ionization by ultrashort laser pulses.
At present, this phenomenon is of great interest due to
the possibility of using it to convert efficiently laser
pulses into low-frequency radiation, in particular, into
the radiation of terahertz frequency band [1 - 10].
Various ionization-driven mechanisms for generating
residual currents in the plasma are being considered at
present. When multicycle laser pulses are used, the RCD
can be generated due to gas ionization by two-color laser
pulses [3, 4, 6 - 10] or due to the asymmetry of the
ionized medium [11]. In the case of using few-cycle laser
pulses, free electrons can be accelerated by the electric
field of the ionizing laser pulse itself [1, 2, 4, 12 - 15].
Previous analytical studies of this phenomenon were
based on the so-called semiclassical approach, which
includes the hydrodynamic equation for the plasma
current density, and the adopted model expression for
the tunneling ionization probability per time unit [4, 10,
15]. However, the range of applicability of the
semiclassical approach is limited by the parameters of
laser pulses corresponding to the tunneling regime of
ionization, when the Keldysh parameter [16] (defined
by the ratio of atomic ionization energy and the average
kinetic energy of an electron in a laser field) is much
less than unity [12].
In this work we calculate RCD analytically by
solution of the time-dependent Schrödinger equation
using the strong-field approximation used in the pioneer
work of Keldysh [16]. We assume that the quasi-dc
RCD is generated due to gas ionization by two-color
laser pulse, which contains a strong field at the
fundamental frequency and a low-intensity field at the
doubled frequency. We find the general analytical
expression for the time derivative of the low-frequency
current density and simplify it in the case of the low
Keldysh parameter. In this case we obtain closed-form
formula for the RCD and show that it coincides with the
corresponding formula obtained on the basis of
semiclassical approach.
1. STATEMENT OF THE PROBLEM
We assume that the electric field )(tE of the laser
pulse is polarized linearly along the z axis. In order to
ensure equality to zero of the integral of )(tE , the
electric field is given via the vector potential )(tA :
0
0
0 0
1 A ˆE( ) , A( ) z ( )
( ) = ( ) sin( ) sin(2 ) .
2
cEdt t a t
c dt
a t f t t t
ω
αω ω φ
= − = −
+ +
(1)
Here, ẑ is the unit vector along the z axis, 0E is the
peak amplitude of the main field, 1<<α is the ratio of
the amplitudes of the additional and main fields, 0ω is
the fundamental (carrier) frequency, ϕ is the phase
shift between the carriers of additional and main fields,
)(tf is the pulse envelope, and c is the speed of light
in vacuum. For the sake of certainty, we will assume
that the envelope has the Gaussian form
( )22/2ln2exp=)( pttf τ− . (2)
Here, pτ is the intensity full-width at half maximum
(FWHM). We neglect the interaction of atoms with each
other assuming that the gas density is sufficiently low.
In addition, we do not take into account the polarization
response of plasma assuming that the maximum density
of plasma is much less than the critical density and
plasma frequency is 1−<< pp τω .
The quantum-mechanical approach for calculation of
the RCD is based on the solution of time-dependent
Schrödinger equation for the electron wave function ψ :
ψψ
−+∇−
∂
∂ rEr eU
mt
i )(
2
= 2
2
. (3)
Here, is the Planck constant, )(rU is the potential of
the parent ion. For the sake of simplicity, we assume
that the gas consists of hydrogen atoms and )(rU is the
Coulomb potential, reU /=)( 2−r . The RCD of free
electrons is written as
RCD ˆj = | p |g
f f t
eN
m
ψ ψ
→∞
〈 〉 , (4)
where gN is the undisturbed gas density, p̂ = i− ∇ is
the momentum operator, and fψ is the projection of
the wave function on the continuum states.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 287
2. ANALYTICAL RESULTS
Since the duration of the laser pulse is sufficiently
large, the density of free electrons increases during
many periods of the electric field. It allows one to
approximate the RCD by the following integral
∫
∞
∞− ∂
∂ dt
t
jj =RCD (5)
where
pdtW
m
eN
t
g 3),(= ppj
∫∂
∂ (6)
is the average (low-frequency) growth rate of RCD,
which is equal to the time derivative of the low-
frequency current density j . Here, ),( tW p is the
averaged over the field period momentum distribution
of the ionization probability per unit time.
In order to calculate ),( tW p we assume that the
envelope of the pulse is constant, i. e., the vector
potential is
0 0
0
ˆA ( ) = z [sin( ) ( / 2)sin(2 )]f
f
cE
t t tω α ω φ
ω
− + + , (7)
where the field amplitude )(= 0 tfEE f is the function
of the "slow" time. We use the strong-field
approximation used in the Keldysh work [16]. In this
approximation the interaction of free electrons with the
parent ion is neglected. At the same time, it is assumed
that the laser pulse intensity is small enough to neglect
the depletion of the atomic ground state. In this case the
momentum distribution of the ionization probability per
unit time is expressed as the sum of the probabilities of
n-photon processes:
−
∆∞
∑ nELtW
nn 0*=
2
0
2
)(2=),(
ω
δ
ω
π
pp . (8)
Here, 〉+〈 1)/(~=* 0ωpIn is the minimum possible
number of absorbed photons (the expression 〉〈x
denotes the integer part of the number x ),
ppp UII +=~ , pI is the atom ionization energy,
/4)(1= 2
0 α+pp UU is ponderomotive energy in the
two-color field, 2
0
22
0 /4= ωmEeU fp is ponderomotive
energy in the field of fundamental field,
pImpE ~/2= 2 +∆ is the energy for detachment and
acceleration of the electron.
The function )(pL in the formula (7) describes the
envelope of the momentum distribution of the ionization
probability. Taking into account that the photon energy
is much smaller than the ionization energy, i. e.,
1/= 00 >>ωpIn , the function )(pL is written as
)(
)/
/21
2=)(
,(1/2
2
3/23
0
s
stiS
stp
p
tE
e
mIp
I
e
iL
p
p ∑
+ ⊥
πω . (9)
Here,
( )
∫ ′
+
+t
p
f tdI
m
ce
tS
0
2
2
/
=),(
Ap
p . (10)
is the part of the action of a free electron that is
independent of the coordinates, ⊥p is the module of the
transverse momentum, st is the stationary points of
),( tS p , and s is the index numbering these points. The
values of st satisfy the equation
0=),(
st
t
tS
∂
∂ p , (11)
and have a positive imaginary part and a real part lying
in the interval 0[0, 2 /π ω .
The action of the second harmonic field is taken into
account in the phases in Eq. (9) by considering terms
linear with α . Under the condition 1, −<< γγα the
stationary points of action are found in the absence of
additional field:
−
++ ⊥
−
.)(/=
,1arcsin=
*
102
21
01
tt
qiqt z
ωπ
γγω
(12)
Here, where zq is the projection of dimensionless
momentum pmI2/= pq on the z axis, ⊥q is the
module of the transverse dimensionless momentum,
0/2= pp UIγ is the adiabacity parameter of Keldysh.
Following the work [16] we assume that the main
contribution in ),( tW p is given by the small values of
final electron momentum, 12 <<q . It makes possible to
neglect the momentum dependence of the pulse in the
pre-exponential factors and use a Taylor series
expansion of q up to the quadratic terms in the
exponential factor:
( )
( )[ ] .),()(2cosh
)(2exp
1
42
)(
0=
0
02
2
0
2
3
*
−
∆
+×
+
≈
∑
∞
nEan
bn
m
I
W
nn
p
ω
δγε
γ
γ
ω
π
qq
qp
(13)
Here,
( )
.
2
11
2
1
sin
1
=)(
,
3
2cos=)(
arcsh2
2
2
2
2
γ
γ
γ
γ
ϕαγ
γ
γ
ϕαγ
++−
+
+−
+
+ ⊥
q
q
q
b
qa
z
zq
q
(14)
The term ),( γε q in Eq. (13) is associated with the
intercycle interference of two electron trajectories
originating from electron ionization from neighboring
half-cycles. It is a rapidly oscillating function of the
momentum for arbitrary values of γ . Therefore, when
calculating the integral characteristics such as average
ionization probability and current density the term ε
can be neglected. The rest of the function )(pW is a
product of the smooth envelope and the sum of delta
functions corresponding the spheres defined by the
energy conservation law. The maximum of the smooth
envelope is located at the some optimal momentum:
ISSN 1562-6016. ВАНТ. 2015. №4(98) 288
+− − γγ
ϕαγ
arcsh2112
sin2
= p
opt
mI
zp . (15)
It can be seen that for 0=α the optimal momentum
is 0=optp and the function )(pW is symmetric in the
longitudinal momentum. Therefore, in the absence of the
second harmonic the average photoelectron momentum is
zero. The addition of a small field at the doubled
frequency breaks the symmetry of the momentum
distribution of the ionization probability. It leads to the
excitation of nonzero residual current density.
Substituting the expression (13) in Eq. (6) we obtain
an expression for the derivative of the low-frequency
current density
( ) .2exp
12
1
~2
exp
1
8
=
arcsh
arcsh
2
0
*=
2
2
0
2
2
2
3
nn
nn
p
p
g
Pqn
I
m
I
eN
t
γ
γ
γ
γ
γ
ω
γ
γ
−
∞+
×
+
+
−−×
+∂
∂
∑
zj
. (16)
Here, 1/22
0 ))1/2(1/(= γ+−nnqn is the dimensionless
momentum of the electron that absorbed n photons and
( )
( ) .sin
1
2
exp
)(2/3cos2cosh=
z
2
z2
0
22
0
zz
q
q
znn
dqqqqn
qqnP
n
n
−
+
×
−+∫
−
ϕαγ
γ
γ
ϕαγ
(17)
The expression obtained is rather complicated.
However, it can be significantly simplified in the limits
of high and small Keldysh parameter γ . In this work
we simplify the obtained expression for 1<<γ , which
corresponds to the tunneling regime of ionization. In
this case, the derivative of the low-frequency current
density is represented as a product of the average
ionization probability per unit time pp 3),(= dtWw ∫ ,
and the most most probable electron velocity
moptopt /= pv :
weN
t optg vj =
∂
∂ , (18)
where
ϕ
ω
α
sin
2
3=
0m
Ee f
opt zv . (19)
Average ionization probability per unit time is
calculated by the method similar to that is used to
calculate the t∂∂ /j ,
,cos
3
2
cosh
3
2
exp
22
3=
1/2
0
×
−
ϕ
α
π
f
a
f
a
a
fp
E
E
E
E
E
EI
Qw
(20)
where 1=0Q . Note, that the strong-field approximation
neglects the interaction of the free electron with its
parent ion. The account of this interaction leads to the
addition of the correction factor
)/)(2/16(= 1/4
0 fa EEQ π [16, 17].
In the case of af EE /<<α the expression (18)
exactly coincides with the analogous formula obtained
by semiclassical approach [18]. Using Eqs. (18) and
(20) it is easy to find an analytical expression for the
RCD. To do this, we use the method used previously in
[10, 15, 18]. In this method, the time dependence of the
average ionization probability is approximated by a
Gaussian function with a characteristic scale
1/2
000 )]()/('(0)2[= −EwEwEfi
τ . As a result, the RCD
is
ϕασ sin
2
3
oRCD scj≈j , (21)
where iEw τπσ )(/2)(= 0 is the final degree of
ionization and 00
2
o /= ωmENej gsc is the maximum
oscillatory current density in the field of fundamental
harmonic.
CONCLUSIONS
The excitation of residual current density due to gas
ionization by ultrashort laser pulses was studied on the
basis of the strong-field approximation for the solution
of the time-dependent Schrödinger equation. It is
assumed that the laser pulse contains the main field at
the fundamental frequency and the additional field at the
doubled frequency. We have found the general
analytical expression for the time derivative of the low-
frequency current density, which is significantly
simplified in the case of Keldysh parameter 1<<γ
corresponding to tunneling regime of ionization. In this
case the photocurrent is determined by the product of
the average ionization probability per unit time and the
most probable velocity of the electron (corresponding to
the maximum of the velocity distribution function), in
good agreement with the results given by the
semiclassical approach. When the condition 1<<γ is
not satisfied, such factorization is impossible and the
dependence of the residual current density on the laser
pulse parameters may differ significantly from the
results obtained by the semiclassical approach.
ACKNOWLEDGEMENTS
The derivation of general analytical expressions for
ionization probability and time derivative of low-
frequency current density was supported by the
Government of the Russian Federation (Agreement No.
14.B25.31.0008) and the Russian Foundation for Basic
Research (Grants Nos. 14-02-00847 and 14-02-31722).
The derivation of closed-form analytical formula for
RCD in the case of small Keldysh parameter was
supported by the Russian Science Foundation (Grant
No. 15-12-10033).
ISSN 1562-6016. ВАНТ. 2015. №4(98) 289
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Article received 01.06.2015
ПРИБЛИЖЕНИЕ СИЛЬНОГО ПОЛЯ ДЛЯ АНАЛИТИЧЕСКОГО РАСЧЁТА ОСТАТОЧНОЙ
ПЛОТНОСТИ ТОКА, ВОЗБУЖДАЕМОГО ПРИ ИОНИЗАЦИИ ГАЗА ИНТЕНСИВНЫМ
БИХРОМАТИЧЕСКИМ ЛАЗЕРНЫМ ИМПУЛЬСОМ
А.А. Романов, А.А. Силаев, Н.В. Введенский
На основе аналитического решения нестационарного уравнения Шрёдингера исследуется возбуждение
остаточной плотности тока в газе, ионизируемом интенсивным бихроматическим лазерным импульсом.
Найдено общее аналитическое выражение остаточной плотности тока для произвольных значений парамет-
ра Келдыша, которое совпадает с результатами полуклассических расчетов при туннельном режиме иониза-
ции.
НАБЛИЖЕННЯ СИЛЬНОГО ПОЛЯ ДЛЯ АНАЛІТИЧНОГО РОЗРАХУНКУ ЗАЛИШКОВОЇ
ГУСТИНИ СТРУМУ, ЗБУДЖУВАНОГО ПРИ ІОНІЗАЦІЇ ГАЗУ ІНТЕНСИВНИМ
БІХРОМАТИЧНИМ ЛАЗЕРНИМ ІМПУЛЬСОМ
А.А. Романов, А.А. Силаєв, Н.В. Введенський
На основі аналітичного розв’язку нестаціонарного рівняння Шредингера досліджується збудження зали-
шкової густини струму в газі, що іонізований інтенсивним біхроматичним лазерним імпульсом. Знайдено
загальний аналітичний вираз залишкової густини струму для довільних значень параметру Келдиша, який
співпадає з результатами напівкласичних розрахунків при тунельному режимі іонізації.
Introduction
1. Statement of the problem
2. Analytical results
Conclusions
ACKNOWLEDGEMENTS
references
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