Journal of Electromagnetic Analysis and Applications, 2012, 4, 419-422
http://dx.doi.org/10.4236/jemaa.2012.410057 Published Online October 2012 (http://www.SciRP.org/journal/jemaa)
419
Proposal for Plasma Wave Oscillator
Ashutosh Sharma1,2,3*
1Department of Education, University of Lucknow, Lucknow, India; 2Centre for Plasma Physics School of Mathematics and Physics,
Queen’s University, Belfast, UK; 3Institute of Photonics Technologies, National Tsing Hua University, Hsinchu, Taiwan.
Email: *a_physics2001@yahoo.com
Received August 6th, 2012; revised September 10th, 2012; accepted September 20th, 2012
ABSTRACT
Generation of radiation by laser pulses in uniform plasma is generally minimal. However, if one considers propagation
in corrugated plasma channels, the condition for radiation generation can be met due to the inhomogeneity of the
plasma channel and the presence of guided waves with subluminal phase velocities. For establishing a large amplitude
plasma wave driven by moderate-power laser, one has to implement a distributed-feedback structure into the plasma
(Plasma Wave Oscillator) with the feedback matching the plasma resonance. In this note the theoretical analysis for
plasma waves driven by moderate-power laser for corrugated waveguide filled with pre-ionized hydrogen plasma has
been developed. The growth of amplitude of plasma waves in corrugated structure, coupled to the laser and sideband
fields has been investigated. The four coupled equations corresponding to laser field, sideband field and forward and
backward plasma waves can be numerically solved for various parameters of the laser field, plasma density, and corru-
gated structure to arrive at experimental design of the Plasma Wave Oscillator, which may be used for the generation of
radiation and particle acceleration.
Keywords: Laser-Plasma Interaction; Generation of Electromagnetic Radiation
1. Introduction
In recent years, the use of plasmas in high power micro-
wave (HPM) devices has been actively researched. Some
HPM devices are found to have a better performance
when filled with plasma; as an example the backward
wave oscillator (BWO) [1,2], has proved to be an effi-
cient HPM source. A corrugated waveguide may be used
to slow down the phase velocity of electromagnetic
waves; in some experiments microwave radiation has
been generated with considerably enhanced efficiency
[3]. When a periodic structure such as a corrugated
waveguide is introduced, the dispersion characteristics
are drastically modified on account of the periodicity of
the structure [4]. The periodic variation of the radius of
waveguide may impose a periodic variation in the plasma
density [5]. It is known that for a straight waveguide or
channel the matched laser spot size and hence the irra-
diance depends strongly on the channel radius. Hence
one expects that a periodic variation of the channel radius
can modulate the plasma density and laser irradiance
along the axis of a channel. The modulation of plasma
density exerts a periodic perturbation on the dielectric
constant. This density variation and hence dielectric
modulation induces distributed feedback (DFB) [6,7],
which in turn influences the wave propagation and the
plasma waves, driven by the laser beam.
In a recent investigation [8] a widely tunable dual
mode laser diode with a single cavity structure is demon-
strated. This novel device consists of a distributed feed-
back (DFB) laser diode and distributed Bragg reflector
(DBR). Continuous wave THz radiation is successfully
generated with low-temperature grown InGaAs pho-
tomixers from 0.48 GHz to 1.5 THz.
The realization of corrugated plasma channel [9] al-
lows for the guiding of laser pulses with subluminal spa-
tial harmonics. These spatial harmonics can be phase
matched to high energy electrons, making the corrugated
plasma channel ideal for the acceleration of electrons.
The simulations [10] performed in a corrugated pre-
formed plasma channel obtain the accelerating gradients
of several hundred MeV/cm for laser powers much lower
than required by standard laser wake field schemes. The
development of corrugated slow-wave plasma guiding
structures with application to quasiphase-matched direct
laser acceleration of charged particles is reported by
York et al. [11]. These structures support guided propa-
gation at intensities up to 2 × 1017 W/cm2, limited at pre-
sent by our current laser energy and side leakage. These
structures remove the limitations of diffraction, phase
matching, and material damage thresholds and promise
*Corresponding author.
Copyright © 2012 SciRes. JEMAA
Proposal for Plasma Wave Oscillator
420
to allow high-field acceleration of electrons over many
centimeters using relatively small femtosecond lasers. It
is investigated in this work by simulations that a laser
pulse power of 1.9 TW should allow an acceleration gra-
dient larger than 80 MV/cm and modest power of only
30 GW would still allow acceleration gradients in excess
of 10 MV/cm.
In the present analysis a moderate-power laser (Ti:
Sapphire) propagating through a corrugated waveguide
(Sapphire) filled with pre-ionized hydrogen plasma has
been considered. The corrugated waveguide is shown in
Figure 1; the wall radius y(z) of waveguide varies sinu-
soidally according to function,

02π
exp
22
yhz
yz j

 


where h and are amplitude and the period of corruga-
tion and y0 is the average radius of corrugated structure.
The basic mechanism, considered in this study is the
parametric instability [12] resulting from the resonant
interaction between three non-linearly coupled waves. A
laser beam, i.e. a transverse electromagnetic wave, pro-
pagating through an unmagnetized, pre-ionized under-
dense hydrogen plasma (frequency l
of incident light >
plasma frequency
p
). This incident laser beam will
reflect off any electron density fluctuation, in particular
perturbations related to electron plasma waves. Under
condition of appropriate phase matching, the scattered
and incident laser field may beat together in such a way
as to reinforce the plasma wave. This reinforced plasma
wave will in turn lead to a higher level of scattering, and
this increased scattered light will lead to a stronger beat-
ing with the incident light, which will further amplify the
plasma wave.
A periodic variation of the channel radius modulates
the plasma density and laser irradiance along the axis of
the channel. The modulation of plasma density exerts a
Figure 1. Plasma Wave Oscillator—where is the period
of sinusoidal perturbation of radius of plasma channel ge-
nerated inside the Sapphire grating structure, h is the
groove depth.
periodic perturbation on the dielectric constant. This
density variation and hence the dielectric modulation
induces distributed feedback (DFB), which in turn influ-
ences the wave propagation and the plasma waves driven
by laser beam.
2. Analysis
The electric fields of the laser are given as,
 
,,
,Re exp
lsls ls
ztE zjt
,

E (1a)
and the complex amplitude of the laser and sideband
field
,,
exp
ls lsls
Ez Azjkz
,
(1b)
where A(z) is the slowly varying field envelope, sub-
scripts l and s stand for laser and sideband field.
The electromagnetic wave propagation in plasma as
governed by the wave equation,
2
2
22 2
14π
t
ctc

E
J
E (2)
where e
e
N
v is the current density.
Taking into account the laser and sideband field the
plasma wave equation may be obtained as,
2
22 2
.
2thp epond
vNF
t

 


(3)
where

0
.
e
pond
NB
Fmc

v
12
2
0
4πe
p
N
m


is the plasma frequency,
N0 is the background unperturbed plasma density, m is
the electronic mass, e is the electronic charge, c is the
speed of light in vacuum.
Following the phase matching condition [
p
ls

and
p
ls
kkk
], the coupled wave equations for laser
field, sideband field and the plasma wave can be written
as,
22 2
22 2
4πe
ll l
ll es
s
e
EE
zc mc


NE
(4)
22 2
*
22 2
4πe
ss s
s
s
l
e
EE
zc mc


el
N
E
(5)
and

 

2
22 2
2
2
2
0
2
eexp
thp e
sl lsp p
els
vN
t
Nkk
A
zA zjkzjt
m






(6)
Copyright © 2012 SciRes. JEMAA
Proposal for Plasma Wave Oscillator 421
The solution for the plasma wave Equation (6) can be
written as,


 
,,, exp
,,
ee ee
e
NNtyzNyz jt
Nyz UyzFz




(7a)
where,
 
,SinπUyz yyz (7b)
 


00
exp exp
F
zAzjzBz jz
 (7c)
The radius of the plasma channel can be expressed as


0exp
22 g
yh
yz jkz
where 2π
g
k,
is the period of sinusoidal perturbation of radius of
plasma channel generated inside the Sapphire grating
structure, h is the groove depth.
One can expand y(z) by assuming that the perturbation
is small, as follows:

22
0
0
112
1expg
hjk z
y
y
yz

 

 

(8)
Using the expansion of y(z) from Equation (8), one can
express U(y,z) as


0
2
0
ππ
,Sin Cos
22
4π
expg
yy
Uyz yy
yhjkz
y
 

 
 




0
t
(9)
On substituting Equation (7a) in Equation (6) one ob-
tains,


22
0
,
exp
e
ppe
Nyz
Cjkzj





(10)
where

 
2
2
0
02
eslls
els
Nkk
CA
m

zAz
,
and
22
2
2
ep
th
v
.
Substituting for Ne(y,z), U(y,z) and F(z) from Equa-
tions (7a), (7c) and (9) in Equation (10), multiplying both
sides of the resulting equation by

0
Sin2π
y
y and
integrating the equation. from –y0/2 to y0/2 and following
the Bragg condition 0
2
g
k
 

0
00
22
2
0
3
00
00
2exp 3
π
3π
24
Az Bz
jC h
jkz
zz
hh
jB zy
y










y





(11)
and
 


0
00
22
2
0
3
00
00
2exp
π
9
3π
24
Bz Az
jC h
jkz
zz
hh
jA zy
y



 






y





(12)
where ls
kkk
.
On substituting the laser and sideband fields from
Equation (1b) in Equations (4) and (5), one obtains the
laser and sideband field coupled with the plasma wave
 
0
exp
ssl
Az jC AzAzjkz
z
 
(13)
and
 
exp
lls
Az jCBzAzjkz
z

(14)
where

22
2
0
πe1
1exp
2
s
sg
ls
h
Cj
ky
mc




kz
and

22
2
0
πe1
1exp
2
l
lg
sl
h
Cj
ky
mc




kz
.
We have developed an analytic model for laser pulse
propagation in a corrugated plasma waveguide, using the
slowly varying envelope approximation. The background
plasma was assumed to be a cold non-relativistic fluid
that responded linearly to the laser field. The small
enough amplitude of corrugation (as shown by Cs and Cl
above) may impose the periodic variation on plasma
wave derived by coupled laser fields. Equations (11) and
(14) may be used to study the growth of plasma wave in
a corrugated plasma channel and hence to design a
plasma wave oscillator.
3. Conclusion
An analysis of the growth of amplitude of plasma waves
in a corrugated structure, coupled to the laser and side-
band fields has been made; the resulting equations viz.
Equations (11)-(14) can be numerically solved for vary-
ing parameters of the laser field, plasma density, and
corrugated structure to arrive at a design of Plasma Wave
, one finds two coupled
equations for forward and backward plasma waves,
driven be the laser and sideband field,
Copyright © 2012 SciRes. JEMAA
Proposal for Plasma Wave Oscillator
Copyright © 2012 SciRes. JEMAA
422
Oscillator which may be useful for generation of radia-
tion and particle acceleration.
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