Laser News Vol.8 No.1 January 1997
ARTICLES
Physics of Laser- matter interaction at ultra-high intensities
Prasad A. Naik
Laser Programme, Centre for Advanced Technology,
Indore, 452 013
In this article, we shall briefly outline the principle of
some of these new ideas. These include: Above Threshold
Ionization by multi-photon ionization, Non relativistic odd
harmonic generation, Above Threshold Ionization by optical
field ionization, Beyond critical density penetration of laser
light, Relativistic harmonic generation, Non linear Thomson
scattering, and Harmonic generation in solid targets. Some
applications based on processes occurring at ultrahigh
intensities will also be discussed. These include: Laser Wake
Field Accelerator, Optical guiding of laser beams, X-ray
lasing to ground state, Fast ignitor concept, and Attosecond
lasers.
Introduction:
The advent of Table Top Terawatt (TTT) lasers has
opened up an hitherto unavailable range of laser intensities
for laser-matter interaction studies. Before the arrival of
these lasers, the high power laser arena was dominated by
the huge lasers operating in nanosecond to several hundreds
of picosecond pulse duration. These multi kilojoule lasers
were not only gigantic in size, they were extremely
expensive to build, cumbersome to operate, and had a very
low repetition rate. The availability of these lasers was
confined to few big laboratories in the world. Even with
these lasers, it was not possible to carry out experiments in
A.
Journey towards Table Top Terawatt
( TTT ) Lasers:
Studies on interaction of laser light with matter started
soon after the first laser was made by Maiman in 1960.
With the advent of high peak power pulsed lasers, focussed
power exceeding the damage threshold of matter became
achievable. This opened up a new arena in physics: the laser
plasma interaction. The possibility of achieving in
laboratories extreme conditions like that in the core of the
Sun and other stars made this field very attractive. With the
aim of getting out more energy by fusion of D-T ions than
that put in, several high power laser chains were developed
in big laboratories around the world. Initially this quest for
stellar energy by Inertial Confinement Fusion scheme7 was
tried out with nanosecond lasers as well as multi picosecond
lasers. However, it was soon clear that nanosecond laser had
a better chance of achieving fusion than the multi
picosecond lasers from compression point of view.
Fig.1: Chirped Pulse Amplification Scheme
After the invention of the first pulsed laser, the
breakthrough in increasing the laser power came with the
introduction of the technique of Q-switching. The second
jump came with the advent of the technique of Mode
Locking. After that, for about one and a half decade, there
was not much progress in pushing up of the laser power. In
this period, several high power laser systems came up. The
focussed intensities of these lasers were in the range of 10 12
to 1014 W/cm2. Several phenomena8 were discovered in this
intensity regime. These included Stimulated Brillouin
Scattering (SBS), Stimulated Raman Scattering (SRS), Two
Photon decay, Parametric decay, creation of Mega Gauss
magnetic fields, Inverse Bremsstrahlung heating, emission
of radiation from far infra red to x-ray region and so on. It
was not possible to push the intensity beyond 10 15 W/cm2 for
long pulse lasers because this required not only high laser
energy, but the effective intensity was also limited due to the
increase in laser focal spot and increased back reflection of
the laser light from the target. The stagnation in laser power
was overcome with the advent of the technique of Chirped
Pulse Amplification (CPA). This technique is depicted in
Fig.1.
the >1015 W/cm2 intensity regime due to problem of increase
of focal spot size and strong stimulated back reflection. The
versatile and affordable short pulse Table Top Terawatt
lasers broke these financial and scientific barriers to open up
the high intensity regime. Achieving very high intensities
>1017 W/cm2 is now well within the reach of any moderate
resource laboratory.
To get an idea of the electric fields involved, consider the
electric field inside a Hydrogen atom. It is about 5 x10 9
V/cm. To get this electric field using lasers, one needs a
focussed intensity of about 3.4 x1016 W/cm2 which is easily
achievable with TTT lasers. When plasma is subjected to
the laser electric field, the electrons oscillate due to the laser
field. The energy of this oscillatory motion can be of the
order of the rest mass energy of the electron at intensities of
the order 1019 W/cm2 . Even these fields are now achievable.
As a result of the ultrashort pulse duration and ultrahigh
intensities, there has been an explosion of new physical
processes1-6 hitherto unthought. Some of these are
theoretically predicted, some experimentally verified, and
many more are expected. This has opened up an altogether
new field of physics referred to as “High Field Physics”.
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Laser News Vol.8 No.1 January 1997
In this technique, a short mode locked pulse (few ps to
sub picosecond) is stretched in time to few hundreds of
picoseconds using a long (~a kilometer) fibre. Essentially,
the pulse gets chirped due to group velocity dispersion
(GVD) in the fibre. Equally important is the role of self
phase modulation (SPM), which increases the frequency
bandwidth of the pulse. Further chirping is usually done
using a pair of antiparallel gratings. In the case of a
Ti:sapphire laser, which has a large bandwidth, only
chirping is required. In this case, fibre is not used and the
chirping is done with grating pair alone. This temporally
stretched pulse of large bandwidth is then amplified in a
broadband amplifier chain (like Nd:glass, KrF etc.) to the
maximum possible intensity and energy. This long amplified
pulse is then compressed using a pair of parallel gratings to
a short pulse of extremely high power. The duration of this
pulse could vary from tens of femto seconds to ~1 ps
depending on the gain band width of the amplifiers. Such
lasers due to their small size (compared to the huge
nanosecond terawatt chains) are referred to as Table Top
Terawatt (TTT or T3 ) lasers.
At present, there are several TTT lasers operating in tens
of Terawatt range3. These include: P102 Nd:Glass laser at
Limiel, France, operating at 55 TW, Vulcan (Nd:glass) at
RAL, UK, operating at 35 TW, Glass based laser at ILE,
Japan, operating at 30 TW, Nd:glass laser at LLNL, USA,
operating at 10TW, full Ti:sapphire laser (LIF) at LOA,
France, operating at 10TW. Titania (Ti:sapphire-KrF) laser
at RAL, UK, is being upgraded to 15 TW. At Saclay,
France, a full Ti:sapphire 10 TW laser is under construction.
LULI in France (Nd:glass), and UCSD, USA (full
Ti:sapphire), are building 100 TW laser systems. LLNL has
a Petawatt (1000TW) laser under construction. It may be
noted that the full Ti:sapphire based lasers operate in reprate mode (10Hz), something that was unthinkable for the
earlier nanosecond terawatt systems which fired at a
maximum rate of a shot per day.
Apart from these, there are more than a dozen systems
operating at 1 < P <10 TW. Since the pulse duration is now
very short, there is no spread of the focussed energy and
hence very short focal spots are possible. Moreover, due to
short pulse duration, the back reflection processes do not get
any time to build up. Due to these two reasons, intensities ~
1018 W/cm2 are routinely available with these Table Top
Terawatt lasers. Intensities ~ 1021 W/cm2 will be achievable
after completion of bigger systems now under construction.
Ponderomotive energy: Let us consider an electron oscillating in the field of a
laser. E = E0 ( cos t X +  sin t Y) where  = 0 and 1
correspond to linearly polarized light and circularly
polarized light respectively. It can be easily shown that the
oscillation energy, referred to as Ponderomotive energy is
given by
Up = ( e2 E02 / 4 m 2) ( 1+ 2 )
In terms of laser wavelength (in microns) and intensity (in
1014 W/cm2), Up = 9.33 I14 m2.
For example,for a Nd:glass laser of 1 m wavelength,
focussed to an intensity of I ~ 10 16 W/cm2, Up is of the
order of keV.
Keldysh Parameter:Keldysh Parameter (compares the ionization energy
with the ponderomotive energy5. It is defined as  = IP /
2 Up where IP is the ionization potential and Up is the
ponderomotive energy. For H-like ions, Keldysh parameter
is given by =( 0.73 Z2 ) / [ (1+2) I14 m2] where I is in
1014 W/cm2 and  is in microns.
For  >1, i.e. IP > Up >>h 0 , ionization occurs by
absorption of more than one photon. This situation is
referred to as Multi photon ionization. On the other hand, 
<1 corresponds to the case where IP < Up wherein the fields
are very high and the ionization is by Optical Field
ionization.
Laser strength Parameter:At high laser intensities, the electron motion becomes
relativistic. It is convenient to define a parameter called the
laser strength parameter4 : a0. This is defined as a0 = p / m0c
, where p is the electron momentum and m0 is the electron
rest mass. The laser strength parameter can be re-written as
a0 = eE/m0cor in terms of the wavelength a0 = eE/
2m0c2. In terms of the laser intensity and wavelength we
have a0 = 0.857 x 10-9 (m)  I (W/cm2). This expression
tell us that a0 will be significant (  1) for intensity I > 1018
W/cm2 (for  ~ 1m). a0 is related to the electron velocity
(v) as a0 = mv/m0c =  (2 -1) which gives the relation
between the of the electron and the laser strength
parameter as  =  ( 1 + a02).
2.
B.
Laser Matter Interaction at High
Intensities:
In the multiphoton ionization of an atom with ionization
potential IP by radiation of photon energy h0, the energy of
the free electron on ionization is given by, s h - IP. At
normal intensities of light, the value of s (the number of
photons absorbed) is such that (s-1) h0  IP  s h0 and
 h0. Further, the probability of absorption of s+1 photons
is much less than that of s photons. However, in a number of
experiments in low density gas targets at laser intensity of I
~ 1012-1014 W/cm2, it was observed that the free electron
energy is much in excess of the above limit. In fact, the
energy spectrum of the electrons (Fig.2) showed many
distinct peaks at energies
To understand the laser-matter interaction at high fields,
it is useful to define some parameters.
1.
Above Threshold ionization by multi photon
ionization and Odd harmonic Generation:
Interaction Parameters:
Laser- matter interaction in high fields greatly depends
on the kinetic energy of the electrons oscillating in the field,
and the ionization energy of the atoms / ions. Three
parameters4: Ponderomotive energy, Keldysh parameter, and
Laser strength parameter, are commonly used to characterize
the interaction.
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Laser News Vol.8 No.1 January 1997
However, once again, before it can escape, field changes its
direction. As a result, the electron keeps oscillating in the
vicinity of the atom and keeps gaining energy from the field
in the process as depicted in Fig.5.
Now two processes are possible. As the energy of the
electron increases, its probability of tunneling through the
raised potential becomes
Fig.2 : ATI Electron energy spectrum
given by= (s+n) h0 - IP , where n = 1,2,3..... In some
cases, the value of n is observed to be as large as 10. It is
now established that the maximum value of n is related to
the ponderomotive energy as nmaxh0 = 3 - 3.5 times Up.
This clearly means that electrons prefer to absorb more
photons than necessary to become free on ionization. This
process of ionization is called Above Threshold Ionization
(ATI)2,5.
It is found that along with ATI electrons, the plasma
(created by ionization of a gas) also emits odd harmonics of
0 such that h = N h0 where N = 1, 3, 5,.... (Fig. 3). This
process is called Odd Harmonic Generation. Here too,
similar to ATI, the maximum value of N is related to the
ponderomotive energy as N max h0 = IP+ (3-3.5)
Fig.5: Graphical explanation of ATI and OHG.
significant. It is clear from Fig.5 that such an electron will
have energy much above the zero energy (threshold) and
hence this process is called above threshold ionization.
Alternatively, the energetic electron can lose its energy
by emission of a photon. An energetic electron may be
visualized as being in a virtual state reached by absorption of
a certain number (N) of photons (from the radiation field) by
the ground state atom. Therefore, the electron can come
down to the ground state by emission of a single photon of
energy Nh. However, since, L = 1 for dipole transition,
only odd values of N (for which the parity of virtual and
ground state would be different) are allowed. Consequently,
the spectrum of emitted radiation would consist of only odd
harmonics (Fig.3).
Fig.3 : Odd Harmonic spectrum from plasma
times Up. It is therefore not surprising that the two processes
of above threshold ionization and harmonic generation are
related. Occurrence of these processes can be primarily
understood by considering the atomic potential well and its
cyclic distortion due to the strong field of the radiation.
(comparable to the atomic field). Fig.4 shows a simplified
picture of the potential well and the linear potential due to
radiation field.
3. Above threshold ionization by optical field ionization :
Consider the case when the Keldysh parameter is small (
i.e.  < 1). This happens, for example, when we have a
Nd:glass laser beam (= 1 m) focussed to an intensity I >
1015 W/cm2. Here the electric field is quite strong compared
to the ionization potential and the ionization of the atom is
by the optical field.
We have seen earlier that the oscillatory energy of an
electron in a radiation field of the form
E = E0 ( cos t X +  sin t Y) is given by
 (t) = (e2E02/2m2) (sin2 t + 2 cos2 t )
For a linearly polarized light,  = 0. Here,
 (t) = (e2E02/2m2) sin2 t = 2 Up sin2 t
Fig.4: Potential well distortion by laser field.
The free energy is due to phase mismatch between the
maximum of the electric field and the instant of ionization. It
may be noted that the electron energy is maximum when the
electric field is minimum and vice versa. Since the
ionization rate is maximum at the peak of the field (due to
optical field tunneling), maximum number of electrons are
produced with small energy. The maximum energy is 2U p.
When the potential is lowered on one side due to the peak
of the field, the bound electron tries to escape from the atom
as it has more energy than that necessary to escape in that
direction. However, before it can escape, the field direction
gets reversed and the potential is raised on that side and
lowered on the opposite side. So the electron is pushed back
into the atom and it tries to escape from the opposite side.
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Laser News Vol.8 No.1 January 1997
However, the average energy is found to be about 0.1 U p.
The electron energy spectrum in this case is shown in Fig.6.
Fig.6: ATI Spectrum for linearly polarized light.
Fig.9: Electrons produced at zero electric field.
The behaviour of the electron energy spectrum for
circularly polarized light is in sharp contrast to that for
linearly polarized light. In this case, an electron liberated at
any time will be at the peak of one component of the electric
field and will gain energy = 2Up at the cost of the other
component of the electric field. Hence all electrons,
irrespective of their time of liberation, will have the same
energy (=2Up).
On the other hand, for circularly polarized light ( = 1),
(t) = 2 Up . In this case, all electrons will have the same
energy irrespective of the instant of their ionization. The
electron energy spectrum would therefore have a peak at =
2 Up (Fig.7).
4. Beyond critical density penetration :
An electromagnetic wave propagating through an
inhomogeneous plasma can penetrate only upto a certain
maximum density (nc ) where the plasma frequency p [ p2
= (4nee2)/m0 ] becomes equal to the laser frequency. At
small laser intensity, (=1)the mass of the electron is
essentially the rest mass m0. However, at large intensities,
the plasma frequency at a given density will decrease due to
relativistic increase in mass.
The modified dispersion relation for the laser e.m. wave
in plasma is given by 2 = k2c2 + p2/ This gives the
relativistic refractive index of the plasma for the laser wave
as  =  (1- p2 / 2) =  (1 - ne/nc). The
electromagnetic wave (laser) can propagate up to 2  0
i.e. up to ne   nc or ne > nc.
This means that the laser light can penetrate even beyond
the critical density. At low intensities (laser light can
penetrate only up to the critical density.
Physically, this is because p2rel = (4nee2)/m <
(4nee2)/m0. As the electron motion becomes relativistic, m
increases and hence plasma frequency decreases. The e.m.
wave sees a lower plasma frequency and it propagates till
laser frequency equals the local plasma frequency which
now occurs at a higher density than in non relativistic case.
Fig.7. : ATI spectrum for Circularly polarized light
The width of the distribution (which determines the
temperature), is very small. Thus, whereas the gas is highly
ionized due to strong radiation, the electron temperature is
rather very low. This plasma is therefore in a highly nonequilibrium ionization state. As we shall see later, such a
plasma is an ideal medium for x-ray lasing to ground state.
The above said features of the electron energy spectrum can
be realized from Fig.8 and 9, which depict the time variation
of the field and electron velocity during the pulse and as the
pulse is over. It is seen from Fig.8 that an
5.
Relativistic harmonic generation :
The wave equation for an e.m. wave in plasma is 2 A (1/c2) 22t = - (4ne2/m0 c2) A /
where n = n0 + n. n is the perturbation in density due to
the vxB force on the electrons. A is the field vector potential
(defined as B= xA).
The laser field can be written as
E = E0 (cos X + sin Y) ;  = (t - kz).
v = eE0/m ( sin X -  cos Y)
B = kE0/ (- sin X + cos Y)
vxB = (eE02k/m)
sin2 (1-2) Z

2
= (ek/m0c )(A /) sin2(1-2) Z
Case I : Plane polarized light (  = 0)
In this case, the wave equation becomes
 2A()= - (4n0e2/m0 c2) A2/)sin2(t - kz)} A() / 
Fig.8: Electron produced at the peak of electric field.
electron produced at the peak of the laser electric field will
have no energy left after the pulse is over. On the other
hand, as seen from Fig.9, electrons produced at the zero of
the laser electric field have a residual energy equal to twice
the ponderomotive energy.
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Laser News Vol.8 No.1 January 1997
where d is some constant. Here the driving term on the
R.H.S. is a function of odd multiple of  {f ( [2M+1]); M
= 0,1,2,..}. As a result, one gets odd harmonic generation.
This is depicted in Fig.10
For high laser intensities, a0 >>1, the velocity of
electrons becomes relativistic. Classically, a charged particle
moving in a curved path will emit bremsstrahlung radiation.
In addition, if the motion is periodic, the emitted radiation
will contain harmonics. The scattered radiation contains of
harmonics of  up to a maximum frequency given by: 4
max = Nmax 2 c/
where  = √(1+a02) and  = instantaneous electron radius.
Case I : Circularly pol. laser light : scattering from plasma
In this case, the motion of the electron is circular and
hence is constant. Here the force due to electric field on
the electron is balanced by the centripetal force due to the
circular motion of the electron i.e.: eE0 = m2.
Fig.10.: Linearly Pol. Light: Harmonic generation
The origin of the odd harmonic emission in this case
can be understood as follows: As shown in Fig.10, in the
non-relativistic case, the motion of the electron is one
dimensional simple harmonic motion at a frequency and
as a result, the emitted radiation has the same frequency .
However, in the case of an electron in a relativistic field, the
trajectory is a figure of 8 due to the vxB force. Due to this
motion which has a 2 frequency component in the Z
direction, higher order frequencies (sidebands of )
separated by 2 ( , 3, 5,...) are produced giving rise to
odd harmonic generation.
Fig.12.:
light.
Case II : Circularly polarized light ( = 1)
Thomson scattering of circularly polarized
Also, we have a0 = eE0/m0c0 . This gives the value of the
radius of curvature as  = a0c / 0 which corresponds to a
value of Nmax = 34 / a0. For large    a0 which gives
Nmax  33  3a03 i.e. : we have a Table Top Synchrotron.
The radiation will be emitted predominantly in the XY plane
and will be plane polarized the same plane. Odd as well as
even harmonics will be present.
Consider a circularly polarized laser beam with
wavelength =1m, pulse duration  =1ps, focussed to an
intensity of 1020 W/cm2 corresponding to a0 = 8.5, in a gas
of density ne=1019 cm-3. For these parameters, Nmax = 1842
corresponding to a min = 6 Å which is in the soft x-ray
region.
Here
(1- 2) = 0 which gives vxB = 0. This means that
there is no modulation of the electron density and hence no
harmonic generation. Here the driving term is also at  and
hence there no emission at harmonics of .
Physically, this is because, in this case, electron velocity
is parallel to the magnetic field (v  B ). Here, the electron
undergoes circular motion at frequency  in the XY plane
(see Fig.11). Since modulation at higher harmonic
frequencies (like at 2 in the plane polarization case) is
absent, there is no harmonic generation.
Case II : Linearly pol. laser : scattering by plasma
As discussed earlier, due to the strong vxB force at
high fields, the motion of the electron in a linearly polarized
optical field is a figure of 8. Moreover, the velocity of the
electron is also relativistic. As a result, (see Fig.13), the reemitted (scattered) radiation will be polarized in the XZ
plane. In general, in any direction in the XZ plane, there will
be odd as well as even harmonic emission. However, in the
Z direction, there will be only even harmonics as the
radiation is emitted in this direction twice during one period
of the laser field.
Fig.11: Harmonic generation in circularly polarized
light.
6. Non Linear Thomson Scattering :
Scattering of light by electrons is referred to as Thomson
scattering. For low intensity lasers ( a0 <<1), the frequency
and power of light scattered by an electron is given by  
0 and dP/d = ½ re2 (1+cos2) where re = e2/m0c2 is the
classical electron radius. For single electron scattering,
Thomson cross section is given by T = (8/3) re2. For a
laser pulse of duration scattered by a plasma of density ne ,
the ratio of the total power scattered by the plasma to that by
a single electron is given by P/P0 = (8/3) re2(nec
Fig: 13: Thomson scattering of linearly polarized. light.
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Laser News Vol.8 No.1 January 1997
odd) up to 8th order with short pulse laser incident on a glass
/ aluminium targets at 450
CaseIII: Scattering of light by a counter propagating
electron beam:
Consider an intense laser beam propagating in a
direction opposite to an e.beam. The e.beam will scatter the
laser beam. The frequency of the scattered radiation will be
upshifted and will be given by max = 3M4 0/a0 ( 2 =
1+a02) where
M=(1+b)2/(1+a02/2)(1-b2) and b= vbeam/c
For  >>1 i.e. a0 >>1, Nmax  24b2a0.
The scattered radiation will be collimated within a narrow
cone of angle  1/b in the direction of the e.beam.
For example, consider an electron beam of energy Ebeam
= 5 MeV (b11) interacting with the laser in the previous
example ( a0 = 8, =1m). In this case, we get Nmax  23230
corresponding to a min  0.5 Å which is in the hard x-ray
region.
7. Harmonic generation in solid targets:
C.
Some applications of ultrahigh intensity
phenomena:
Before discussing the applications, it is necessary to
understand a force called Ponderomotive Force which
arises whenever there is an intensity gradient. In short pulse
lasers, these gradients are very high and this force becomes
very significant.
The Lorentz force on an electron subjected to a space
varying electric field E(x,y,z) can be written as F = e {
E(x,y,z; r1=0) + r1.  Er1=0 + vxB} where r1 is the first order
displacement of the electron ignoring the vxB term. The first
term gives the force due to the electric field to the first order.
The second and third are non-linear second order force
terms. One can calculate the time average of this non-linear
force as
<FNL>= -{e2(1+2)/4m2} E2(x,y,z)
= - { e2 E2(1+2)/4m2} = - Up.
This force arising due to gradient of ponderomotive energy
is called the ponderomotive force [ - I (x,y,z)].
In the case of focussed high intensity laser beam in
plasma, ponderomotive forces arise due to two
Fig.14: Density profile for long pulse plasma.
Fig.14 depicts the density profile of plasma produced by a
long pulse (~ ns) laser. The density gradient here is of the
order of laser focal spot size which is few hundred times the
laser wavelength. For a short pulse (1ps) laser, this picture
gets modified to the one shown in Fig.15.
Fig.16: Radial Ponderomotive force.
gradients. As seen from Fig.16, due to the variation of
intensity in the radial direction, there is a ponderomotive
force on the electrons directed radially outwards. Also, in
time, the intensity is varying. As a result, there is a
ponderomotive force on the electron in the direction of beam
propagation. This is depicted in Fig.17.
Fig.15: Density profile for short plasma.
In this case, the density gradient is a very small fraction
of the laser wavelength. As a result, there is a sharp vacuum
- plasma boundary. When a laser beam is incident at an
angle on this interface, due to the component of the e.m.
field normal to the interface, in one half of the cycle of the
e.m. wave, the electrons are pulled out of the plasmavacuum boundary into the vacuum. In the next half of the
cycle, the electrons are pushed back into the plasma. At the
interface, the motion is strongly anharmonic. This results in
emission of harmonics. The harmonic generation in the
direction of laser light. Unlike in the case of gas targets, here
odd as well as even harmonics are emitted. This is so
because the emission is taking place in an environment with
broken inversion symmetry (as expected in any interface
process). Linde et al9 observed harmonics (even as well as
Fig.17: Longitudinal ponderomotive force.
This ponderomotive force is used in the concept of Laser
Wake Field Accelerator.
i)
Laser Wake Field Accelerator :
Electromagnetic field being transverse in nature, cannot
be used to accelerate electrons as such. However, there are
various means of using the transverse laser field to generate
longitudinal field gradient to accelerate electrons. One way
is to use a laser to excite Langmuir waves in plasma. These
waves are longitudinal waves and hence can be used to
accelerate electrons4.
Consider an intense laser pulse of duration  focussed
in a gas. The front edge of the pulse produces plasma and
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Laser News Vol.8 No.1 January 1997
the rest of the pulse propagates through it. Due to strong
variation of intensity in the direction of propagation of the
pulse, the electrons in front of the pulse are pushed ahead
and those in the trailing part of the pulse are pushed behind.
If the length of this disturbance (cmatches with theplasma
wavelength then high amplitude wake fields are produced
(see Fig.18, 19) in the plasma. This wake field, driven
Fig.20: Radial focussing of electron bunches
The advantages of a wake field accelerator are: 1) It is a non
resonant process unlike the plasma beat wave accelerator5
(PBWA). Hence the restrictions on laser pulse width and
plasma density uniformity not very stringent; 2) No need to
have a pre-formed plasma like in a PBWA. The rising
portion of the pulse itself produces plasma. 3) There is a
radial focussing of the electron bunches.
In absence of optical guiding, the interaction length is
limited to
Lint ~ ZR , where ZR = r02/ is the vacuum Raleigh range.
The maximum energy given to the e.beam =  ~Emax x Lint
which can be written in terms of the laser power as
(MeV) ~ 580 (0/p) P(TW)
or 1.8 {0(m) / (ps)} P(TW)
For example, consider a laser with following parameters:
0=1m,  = 1ps, P=10 TW focussed to a spot of r0=30m,
[ a0 = 0.72] in a gas of density n0=1016cm3. The maximum
field gradient in this case will be Emax = 2 GV/m . Since in
this case, Lint = 0.9 cm, it is possible (in principle) to
accelerate electrons by  = 18 MeV.
Fig.18: Wake Field due to a short pulse.
by the ponderomotive force is in the direction of propagation
of the laser pulse and travels with the pulse with phase
velocity equal to the group velocity of the laser pulse.
ii) Optical Guiding of Laser Beams:
It is possible to increase the interaction length Lint by
optical guiding. In the following section, we shall consider
two methods of optical guiding.
A) Optical guiding by density channels:
Optical guides can be formed in plasma by propagating
a low current e.beam or a low intensity laser beam in the
plasma. In either case, plasma is expelled by the beam
leaving a density depression on the axis. In the case of
e.beam, Coulomb force does the job, and in the case of laser,
it is the nkT pressure which does the job (Fig.21). The
radial variation of the refractive index of plasma for laser
light is given by4:
(r) =1 - ½ (0/p)2 [ n(r)/n0]
(r) =1 - ½ (0/p)2 [n(r)/n0]
Fig.19: Density and field variation inside plasma.
Since high amplitude wake fields are produced when c
p/2 i.e. when the spatial length of the pulse = plasma
wave half wavelength), one gets an optimum laser pulse
duration related to the plasma density given by
opt ~ 0.6
(ps) [1016/no(cm-3)]
The maximum wake field amplitude generated by
linearly polarized laser pulses is given by
Emax (GV/m) ~ 3.8x10-8 n0(cm-3) a02 / (1 + a02/2)
There is an interesting proposal that if another laser
pulse of same duration is injected instead of the e.beam at a
lag of 1.5 p, the wake field of this pulse will be opposite
and will try to cancel out the field of the previous pulse. As a
result, the energy of the photons in the second pulse will get
increased. This is the concept of a Photon Accelerator10.
Coming back to electron acceleration, as seen from
the adjacent figure, the ponderomotive force due to the
transverse electric field gradient of the laser beam will push
the plasma electrons radially outwards, thereby creating a
radial field which will focus the electron beam axially. This
is shown in Fig.20.
Fig.21: Optical guiding by density channel
The condition for guiding is that the divergence due to
diffraction should balance the
convergence due to the
refractive index gradient. i.e.:
 = 1 / (ZR 0 re)
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Laser News Vol.8 No.1 January 1997
one focuses a 1 m TTT laser to an intensity of I~10 16
W/cm2 in nitrogen gas, the heating is mostly by optical field
ionization and one can get He-like ions of Nitrogen at a low
temperature of ~ 20 - 30 eV. In such a case, as the plasma
temperature is very low, the recombined ions cannot remain
in that ionization state for long and recombine to next lower
species till a state is reached which is supported by the
plasma temperature. As a result, the ground states of high
ionization species remain practically empty and hence
inversion to ground state is possible. Thus, high degree of
ionization at very low temperature makes it possible to get
lasing to ground state. Using this scheme, Nagatasa et al12
have shown lasing of Ly  at 135Å in H-like Lithium.
Durfee et al11 made a plasma column of 6mm length using
a Nd:glass (=1m) laser of 100ps duration in 30 torr
Xenon. When another beam of intensity 10 14 W / cm2 (ZR =
300m) was made incident on this channel, the beam was
guided for entire 6mm length of plasma channel.
B) Relativistic Optical Guiding
The refractive index of the plasma for an intense laser
beam is given by the expression :
(r) =1 - ½ (0/p)2 [ 1/ (1+a02)]
On the axis, a0 is highest as the beam intensity is
highest. Hence is higher on axis of the beam than in its
wings (Fig.22). As a result, the beam can be self guided.
The critical power of the laser beam necessary for this
guiding to take place is given by4 Pcrit (GW) = 17.4 (0/p)2
For example, for a plasma density of ne=1018cm-3
(corresponding p= 35m), using a laser of wavelength 0 =
1m , one gets critical
The main advantage of lasing to ground state is that
one gets a shorter wavelength as wavelength of a Ly  is
about ten times smaller than that of corresponding Ba
transition
11. Fast Ignitor Concept:
In the standard Inertial Confinement Fusion scheme7, a
number of high power laser beams are made incident on a
D-T filled microballoon (pellet). The plasma formed
expands outwards, which as a reaction compresses the rest
of the pellet inwards (implosion). The shock wave converges
to the centre compressing the D-T fuel to a high density. At
the centre, ignition takes place, thereby starting the fusion
reaction. The alpha particles liberated heat the rest of the
pellet thereby burning the whole pellet and giving out fusion
energy. However, for practical realization of this scheme,
one needs to have megajoule class lasers. Moreover, the
requirement on implosion symmetry is quite stringent (better
than 1%).
Fig.22: Relativistic optical guiding
power of the laser to be Pcrit ~ 20TW, which is easily
possible with TTT lasers.
iii)
X-ray Laser: Lasing to Ground State:
Now we shall see how lasing to ground state is made
possible using short pulse intense lasers. In this scheme,
using lasers, a cold, fully ionized plasma of a low Z element
is produced. This plasma undergoes rapid recombination by
the process of three body recombination. In this process, an
ion recombines with a free electron and the excess energy is
given to another free electron. Since this electron is given
back to the plasma, its energy is within kTe. As a result, the
recombining electron also goes within kT e from the
ionization level i.e. the H-like ions formed have electrons in
high principle quantum number levels. As they cascade
down, they create population inversion in levels below a
certain level called the collision limit. In energy levels above
the collision limit, no inversion is possible as the levels are
collisionally mixed.
In order to achieve population inversion in this manner,
it is desirable that the three body recombination occurs at a
fast rate. Since the rate of this process varies as ne3 Te-9/2, it
follows that a plasma of high density and low temperature is
required.
With solid targets one gets high electron density (ne ~1021
-3
cm ). However, the temperature of the plasma is also
moderately high as the heating is by inverse bremsstrahlung.
This is the normal approach to x-ray lasers based on
recombination scheme (e.g.: Al10+ : 254Å, 205Å). Here it is
not possible to have population inversion to ground state.
If one uses gas targets (lower ne ~1015-1019 cm3) and
uses intense, short pulse lasers to produce plasma, then the
ionization can be mainly due to optical field ionization
giving rise to a highly ionized plasma at very low
temperature (as seen in earlier sections). If, for example,
As a shortcut to this scheme, Tabak and his coworkers13
have proposed a fast ignitor scheme. This is a three stage
scheme, as explained below:
I) In the first stage, compression of the fuel to R ~
0.4 ( ~  particle range) is obtained using conventional
kilojoule type long pulse (few nanoseconds) lasers.
II) In the second stage, a prepulse of few hundreds of
picoseconds is given. This pulse is focussed to an intensity
of I~1016 to 1018 W/cm2 in the plasma. Due to the strong
intensity gradient, a strong ponderomotive force
(corresponding to a transverse pressure of ~ 600 I 18 Mb) is
generated. For I ~ 1018 W/cm2, this pressure will be as high
as 600 megabars. This intense pressure pushes the plasma
and bores a hole into the plasma up to the solid density.
III) In the final stage, an intense laser of pulse duration of
few tens of femtoseconds is focussed to an intensity of 10 20
W/cm2 in this hole. At these intensities, MeV electrons are
produced14, which heat the core (compressed fuel) to fusion
temperature and start the thermonuclear burn, which then
propagates in the rest of the fuel, giving out fusion energy.
iv)
Attosecond Lasers
This idea as proposed by Paul Corkum of NRC, Canada,
based on polarization dependence of harmonic generation is
as follows:
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Laser News Vol.8 No.1 January 1997
As seen earlier, a circularly polarized laser cannot
produce harmonics from gas plasma. The variation of
efficiency of harmonic generation with ellipticity parameter
of the laser beam is shown in Fig.23.
successive linearly polarized pulses, will be about 1ps. The
interval t is adjusted to get linear polarization only once
during the femtosecond pulse. At this instant, a burst of
harmonics will be emitted. At all other times, harmonic
generation will be insignificant. The duration of the coherent
odd harmonic burst will be of the order of few attoseconds
i.e we have an Attosecond Laser.
Summary:
To summarize, we have seen that:
a) Electrons prefer to absorb more energy than necessary for
ionization under intense laser fields;
b) Odd harmonics are emitted by multi photon absorption of
laser light by quasi-ionized electrons;
c) At high intensities, the laser light can penetrate into a
plasma even beyond the critical density;
d) A short pulse, p-polarized laser incident at an angle on a
solid target can produce odd as well as even harmonics;
e) At high intensities, vxB force on the electrons can make
them emit odd harmonics of the incident laser frequency;
f) The relativistic electrons from laser produced plasma can
emit synchrotron type radiation;
g) Scattering of intense laser light by a relativistic electron
beam can give rise to very short wavelength radiation;
h) Wake fields of short intense laser pulses give rise to
strong longitudinal electric fields which can accelerate
electrons;
i) Acceleration length can be increased by optical guiding
of laser beams;
j) In the cold plasmas produced by intense lasers, X-ray
lasing to ground state is possible;
k) Short pulses can bore hole in plasma and produce MeV
electrons in solids; and
l) Attosecond lasers, based on polarization dependence of
harmonic generation, are possible using femtosecond lasers
Fig.23: Harmonic generation efficiency as a function of
ellipticity.
Fig.24 shows the scheme of generating attosecond laser
pulses. In this scheme, one starts with two femto second
perpendicular to each other.
They are focussed to
overlaplasers of equal intensity. Both are linearly polarized
References:
1) C.Joshi and P.Corkum, Phys. Today, 36, 1995
2) M.R.Hutchinson, Contemp. Phys., 5, 355,1989
3) P.Gibbon and E.Forster, Plasma Phys. and
Controlled Fusion, 38, 769, 1996
4) P.Sprangle and E.Esaray, Phys. Fluids B, 4
(7), 1992
5) B.Luther -Davies et al., Sov.J.Quant. Electronics, 22, 4, 1992
6) A.L’Huillier et al, in “Atoms in Intense Laser Fields”, M.Garvila, ed.,
Academic , NY, 1992
7) K.A.Bruckner and S.Jorna, Rev. Mod. Phys. 46, 325 (1974);
J.A.Nuckolls et al, Nature, 239, 139 (1972)
8) W.Kruer, “The physics of Laser Plasma
Fig.24: A scheme to generate attosecond laser
pulses
Interactions” , Addison-
Wesley, 1987.
9) Von der Linde et al, Phys.Rev.A, 52, 257,1995
well as temporally. The frequencies of the two lasers differ
by a small amount. Due to the frequency difference, there is
a time dependent phase lag between the two. As a result, the
net polarization will keep cycling between linear 
elliptical  circular  elliptical  linear. The initial phase
is adjusted to get linear polarization at the peak of the
femtosecond pulses. For a frequency difference  =
0.08%, = 1m, t, the time interval between two
10) S.Wilks et al, Phys. Rev.Lett., 62, 2699, 1989
11) C.Durfee and H.Milchberg, Phys. Rev. Lett., 71, 2409, 1993
12) Y. Nagatasa et al, Phys.Rev.Lett., 71, 3774, 1993
13) M.Tabak et al, Phys. Plasmas, 1, 1626, 1994
14) J.Kmetec et al, Phys. Rev. Lett., 68, 1527, 1992
16