ELECTROMAGNETIC MODE CONVERSION BY ULTRAFAST IONIZATION OF GASES AND CONDENSED MATTER
V.B.Gildenburg, N.V.Vvedenskii
Institute of Applied Physics, Russian Academy of Sciences
E-mail: vved@appl.sci-nnov.ru
Abstract. We study in this paper the effect of shock excitation of leaky
and surface modes by the p-polarized electromagnetic wave incident on
the rapidly ionized gaseous or solid layer. Estimations fulfilled show
the potentialities for applying this effect for generation of X-ray and
THz radiation.
1. Introduction
The phenomena of ionization-induced spectrum conversion of
electromagnetic wave attracts the researchers attention in connection
with the problems of transformation and generation of radiation in
some badly mastered or hardly accessible frequency bands (THz, UV,
X-ray). The most theoretical works concerned these problems dealt
with the wave propagation in quasi-homogeneous time-varying plasma
(created by an external ionization source or by the wave itself) or with
the reflection of the normal-incident wave by moving ionization front
or suddenly created plasma [1]. It was considered also effects of “bulkto-surface” mode conversion (with frequency down-shifting) at sudden
[2] or slow (adiabatic) [3] creation of plasma layers. Frequency upshifting of re-radiated waves caused by excitation (and following adiabatic conversion) of Langmuir oscillations in plasma layers was analyzed only for the case of slow ionization [4], when this effect occurs to
be depressed strongly due to collision or radiation damping of this oscillations.
We study in this paper the effect of shock excitation of natural
(leaky and surface) electromagnetic modes by the p-polarized electromagnetic wave incident on the rapidly ionized gaseous or solid layer.
The model considered is based on the following assumptions: (i) the
ionization leads to the formation of homogeneous plasma layer with
Langmuir frequency which is much greater than the incident wave frequency, (ii) the time of ionization (unlike the cases considered in [4, 5])
is much smaller than the period of plasma oscillations, (iii) the layer
thickness is much smaller than the space scale of the incident wave in
plasma. It has been found that besides the known phenomenon of the
symmetric surface waves excitation [2], the rapid ionization leads to
effective generation of low-frequency antisymmetric surface waves and
symmetric leaky waves with strongly up-shifted (Langmuir) frequency.
2. Formulation of the problem and basic equations
Let the electromagnetic field before plasma creation ( t  0 ) is
given in Cartesian coordinates as a plane p -polarized wave of frequency 0
B 0  z 0 B0 exp( i0t  ik 0 cos  x  ik 0 sin  y ) ,
E0  B0 ( x 0 sin   y 0 cos ) exp( i0t  ik 0 cos  x  ik 0 sin  y ) ,
incident at an angle  on the infinite layer of transparent (unionized at
t  0 ) medium, occupying the space between the planes x   d .
At the time instant t  0 the layer is ionized suddenly by some
external source (for example, high intensity laser pulse), so that the
plasma density within the layer grows instantly from 0 to N =const.
The approximation of instant plasma creation, supported by a number
of theoretical and experimental studies [1] is valid if the electron density rise time is much shorter then the period of plasma oscillation. The
spatiotemporal evolution of the electromagnetic field after the plasma
creation (at t  0 ) is governed by Maxwell’s equations
E y
1 Bz
 ik0 sin  E x  
,
(1)
x
c t
4
1 E x
,
(2)
ik0 sin  Bz 
jx 
c
c t
B
4
1 E y
 z 
jy 
,
(3)
x
c
c t
with current density equations for the cold collisionless plasma
2
j y 2p
j x  p

Ey .

Ex ,
(4)
t
4
t
4
Here  p  4e 2 N / m is the electron plasma (Langmuir) frequency,
j x and j y are the x and y components of the electron current density.
Initial conditions (at t  0 ) for the field and electron current are
the temporal continuities of E x , E y , B z and j x (0)  j y (0)  0 (newly created electrons have zero velocity at t  0 ). Boundary conditions
(at x   d ) are spatial continuities of E y and B z .
Method of the solution is Laplace transform of Maxwell’s and
current density equations:

L[ f ( x, y, t )]   f ( x, y, t ) exp(  st ) dt ,
0
where s is Laplace variable, f is a component of the fields or current
density.
3. Laplace transforms
Applying Laplace transform to equations (1)-(4) gives the following equations for the electromagnetic field and current density
transforms b  LB z  , e y  L[ E y ] , e x  LE x  , J x  L j x  ,
J y  L[ j y ] :
 2b
x 2

2p  02 sin 2   s 2
c2
ex 
ey 
s
2p
 s2
b
i0 s  s 2  2p
sc2
B0 expik0 cos  x  ik0 sin  y 
sin i0b  B0 exp(ik0 cos  x  ik0 sin  y ) 
 cos B exp(ik cos  x  ik cos  y )  c b 


0
0
0
x 
s 
s
2p
2
Jx 
2p
4 s
ex , J y 
2p
4s
ey ,
Boundary conditions for the Laplace transforms are spatial continuities
of b and e y at x   d .
The solution for the transform of the magnetic field inside the
plasma ( x  d ) has the form
b {
2p  s 2  i0 s
B0 exp(ik 0 cos  x ) 
s (2p  02  s 2 )
 D1 coshx   D2 sinh x }exp(ik 0 sin  y ) ,
D1  iB0
2p ( 2p  s 2 )
s( s  i0 )(2p  02  s 2 )

  2 sin 2   s 2 cos(k cos  d )  s 2 cos  sin( k cos  d ) 
 0 0

0
0



,
 ( 2  s 2 ) 2 sin 2   s 2 cosh(d )  s 2 c sinh( d ) 
 p

0


D2   B0
2p ( 2p  s 2 )
s( s  i0 )(2p  02  s 2 )

 s 2 cos  cos(k cos  d )   2 sin 2   s 2 sin( k cos  d ) 


0
0
0
0



,
 ( 2  s 2 ) 2 sin 2   s 2 sinh( d )  s 2 c cosh(d ) 
 p

0



2p  02 sin 2   s 2
c2
.
4. Thin layer approximation
If the following inequalities are fulfilled
p
d  1 ,  p  0 sin  ,
k0 d  1 ,
c
the magnetic field inside the layer can be presented in the form
Bz  Bsym  Basym exp(ik0 sin  y ) ,
where symmetric term Bsym does not depend on x , antisymmetric term
Basym ~ x and
Bsym  A1 exp( i0t )  A2 exp(is1t ) 
 A3 exp( is1t )  A4 exp(il1t )  A5 exp(il 2t ),
Basym  k0 cos  x B1 exp( i0t )  B2 exp(is 2 t )  B3 exp( is 2 t )  .
The waves with the frequencies  s1,2 are the surface waves propa-
gating in the forward (  y ) and backward (  y ) directions, respectively. The waves with the frequencies l1,2 are the leaky waves emitted
into vacuum ( Im( l1,2 )  0 ). The amplitudes of the these waves are
determined by the reduces of D1 and D 2 at the corresponding poles
( s  i ).
The frequencies and amplitudes of antisymmetric and simmetic
waves are
2s 2  2p k0 d sin   02 sin 2  .
B1  iB0
2p  02
02
B2  B3  iB0
,
2p
202
,

2p  02 cos  1  ik0d cos 
A1  B0
,
2p  02 cos   ik0d 2p  02 cos2 
2s1  02 sin 2 1  k02d 2 sin 2  ,
A2  B0
A3  B0


2p k02d 2 sin 4 

2 2p  02 cos2  1  sin 
,
2p k02d 2 sin 4 

2 2p  02 cos2  1  sin   sin 3  k02d 2 / 2

,
 l1   p  i0 sin 2  k0d / 2 ,  l 2   p  i0 sin 2  k0d / 2 ,
A4  iB0
 p k0d sin 2 
 p k0d sin 2 
, A5  iB0
.
2 p  0 
2  p  0  i0 sin 2  k0d / 2

The leaky waves are emitted into vacuum under the angles

l1,2   arcsin( 0 sin ) .
p

5. Discussion
The phenomenon of leaky wave excitation may have important
applications for development of X-ray and THz lasers.
For example, if N ~ 10 23 sm-3 (solid density, multiple ionized
plasma), the basic frequency 0 ~ 2  1015 s-1 , d ~ 10 nm , sin  ~ 1
we find, that the wavelength of leaky wave  ~ 100 nm is in soft x
rays band. The intensity of x rays radiation in this case is I ~ 10 3 I 0 ,
where I 0 is the intensity of incident (basic) radiation.
If N ~ 1016 sm 3 (low density gaseous plasma), the basic frequency 0 ~ 2  1011 s -1 , d ~ 30 μm , sin  ~ 1 we find, that the wavelength of leaky wave  ~ 300 μm is in THz band. The intensity of THz
radiation in this case is I ~ 10 4 I 0 .
This work was supported by Russian Foundation for Basic Research (Grant Nos. 02-02-17271 and 04-02-16684) and Russian Science
Support Foundation.
References
1. IEEE Trans. on Plasma Science, 1993, v.21, No.1. Special Issue on
Generation of Coherent Radiation Using Plasmas.
2. M.I. Bakunov, A.V.Maslov.// Phys.Rev.Lett., 1997, v.79, p.4585.
3. V.B. Gildenburg, N.A.Zharova, M.I.Bakunov.//Phys.Rev. E, 2001,
v.63, p.066402.
4. M.I. Bakunov, V.B.Gildenburg, Y.Nishida, N.Yugami.// Phys. Plas
mas, 2001, v.8, p.2987.
5. N.V.Vvedenskii, V.B.Gildenburg.// JETP Lett., 2002, v.76, p.380.