Introduction of Physics related to ultra-shortpulse laser-plasma
Hitoki Yoneda
Institute for Laser Science, University of Electro-communications
Chofugaoka, Chofushi, Tokyo 182-8585, Japan
Abstract. This lecture deals with the basic physics of the laser-plasma interaction. Specially, the
subjects related to the experiments of solid-state target with ultra-short-pulse lasers are chosen.
Since these interaction processes are very complex and it needs a lot of time to overview whole
of this area, only simple and basic physical models are used for discussion in this article. The
common sense with normal solid-states and/or optics is also introduced for understanding optical
response for high density plasma.
^INTRODUCTION
Recently, chirped pulse amplification has made it possible to achieve high intensity
electromagnetic field in the laboratory. Even with commercial base lasers, their output
power can exceed multi TW level. When these laser lights are focused, very high
concentration of optical field can be obtained easily. The materials having the extreme
condition are achieved with these lasers so that these research tools has opened new
research areas in nonlinear optics, plasma physics, and new material processing.
However, at the same time, the physics inside these interaction areas becomes more
complex and user of these lasers, sometimes, escape from consideration of details
physics and/or creation of models by themselves.
In this lecture, the basic idea for understanding the laser-plasma interaction is
introduced for beginner scientist of this field. In this article, I'll show some common
sense between these interactions of high optical field and solid-states physics and
normal optical physics.
2.BASIC PROCESS IN LASER PLASMA INTERACTION
When the laser light illuminates the solid-state materials, thin surface layer of the
target is heated, melted and evaporated. Finally, plasma is created on the surface. In
this stage, electrical field of laser heats up material with Joule heating mode. After
creation of the plasma, energy of the laser is absorbed with electrons in the plasma.
The absorbed energy drives thermal conduction wave into cold target material, and the
plasma expansion toward vacuum area with normal direction to the surface. This
energy is also used for excitation and ionization of ions. These ions will loss the
energy with emitting radiation in late stage. Changing density and temperature and
CP634, Science of Superstrong Field Inter actions, edited by K. Nakajima and M. Deguchi
© 2002 American Institute of Physics 0-7354-0089-X/02/$ 19.00
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changing ionization state and/or the populations of excitation levels cause change of
the optical properties (n(co), k(co)) of the plasmas. Then, the laser propagation is
modified due to this change of the optical constant. Figure 1 shows relation of abovementioned processes. This feedback rout, sometimes, produces strong non-linearity in
the laser plasma interaction. These complexly also put difficulty in prediction of the
whole phenomena.
To remove from this difficulty, some models use the assumption of 'quasi-steady
state' for this interaction. For example, with nanosecond laser pulse, several selfsimilar solutions [1] has been proposed for analysis of thermal expansion and or heat
waves. In these models, laser light interacts with the plasma having given profile of
density and temperature, which has slowly changed in time. In the case of interaction
with ultra-short-pulse lasers, since pulse duration of laser is so small that some plasma
parameters are frozen and are ignored during the interaction time. Among the
processes shown in figure 1, for example, hydrodynamics and some of ionization
process are slower than pulse duration of femtosecond lasers. Even relaxation time
from electron system and ionic one, sometimes, become slower than pulse duration.
In this case, initial parameters of plasma are kept during laser heating, and it is enough
to consider only some dominant processes. Therefore, laser light wave can interact
with plasmas having quasi-constant optical parameters.
Optical
properties of
plasma n, k
FIGURE 1. The process inside laser-plasma interaction. These 'feedback' loops, sometimes, result in
strong nonlinear feature of laser plasma interaction.
In usual treatment in optics, light propagation is considered with following Maxwell
equations.
= ^oE-^E
c
c
(1)
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In these equations, a harmonic light wave (E = E(x)exp[-icot]) was considered as
incident laser light. Only a is a material parameter. To estimate this value, the free
response of the free electrons under the harmonic wave is considered. They are
quivering with electric field of laser —^ = ——E(;t)exp[-/a#] and induced current
dt
me
density becomes J = -ne(x)eue. According to simple ohm's law, conductivity of
e2n
0)2p
plasma is represented with a = i
because — = ——LE - ——E
dt
dt
co^2
andJ = oE = i
Including this discussion, eq.(l) rewrite a wave equation such as,
c
co
(2)
If variation of the dielectric constant 8 is slow in space, the second term of left-hand
side can be neglected. This assumption maybe reconsider carefully in the case of ultrashort-pulse laser interaction with matter because its density scale length of plasma is,
sometimes, shorter than wavelength of laser. It is also treat this thin layered plasma
carefully since it changes absorbance and mechanism of absorption, easily. Dispersion
relation of light in plasma is deduced from eq.(2), that isco2 = co2pe + k2c2. Figure 3
shows this relation. Any elastic and non-elastic scattering will be discussed on this
plot, (for example, Stimulated Brillouin scattering process is denoted with vectors )
Fotward.
loo acoustic
FIGURE 2. Dispersion relations in laser plasma interaction. The light has CO = COpe + k C and the
other waves have the other formula. (For help of understanding, that of ion acoustic wave is also plotted
in this figure.) Phase matching condition between two waves can be treated on this figure.
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To proceed the discussion of propagation problem in plasma, linear density gradient
(ne-nc—) are considered in following part. Dielectric constant of the plasma
/-/
becomes
(3).
CO
The wave equation then, rewrites V2E+ ^r\l-—\E = 0. The solution of this
c \ L)
equation [2] is already known as mathematics subject and that is combination with two
Airy functions of Ai(x) and Bi(x) (shown in Figure 3).
FIGURE
3. Analytical
(
AiryW ^
\c
solutions
of eq.(3)
are (a)
AiryAi[
jc L
(z-L)]
and (b)
(z-L)]|l-AiryAi[-~ | L]C2
Vc L
As shown in Fig. 3, one of them (b) is
not correct physical meaning so that
only Ai(x) function can be solution of
this subject.
Whenx«-l,
' - —1 and
(^
X)
cosine function can be rewrite with a
pair of exponential functions
&a>L/c)(l-z/L)m -i3
(e
and
4
e 3
). This denoted
that the solution is made with both of
the forward and reflected backward
light waves.
FIGURE 4. Oblique incidence of laser light
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If the incident laser light enters the target with oblique condition, analysis of ray
trajectory becomes important. Again, we start from the wave equation (eq.2) with two
orthogonal vector components for E and B. When the laser light has s-polarization,
and axis notation is chosen in the manner of Fig.4, this equation can be solved with
variable separation method. If we can rewrite Ex(y9z) = F(y)G(z), the equation
becomes
1
1
rG(z) + -re(z)G(z) = const.
(4).
The first equation produces a solution of harmonic wave for y-direction so that the
light can propagate with constant ky (=k0sin0) vector during the interaction for this
direction. The second one rewrites with above ky.
d2
G)2
With similarity with eq. (2), it is easy to image that the laser light can penetrate into
the position of ne = nc\\ - sin2 6). The ray of this laser trajectory is calculated with
integral of dz/dy with its path, therefore,
dy
k
sin0
(6).
Typical results of this analysis are shown in Figure 5. It is important to consider
the adaptability of this assumption of variable separation. It is much more clear when
this solution is compared with ray tracing in a graded index fiber. In the latter case,
variable separation can't be used so that its solution is different from Fig.5.
FIGURE 5 Ray tracing of oblique incident laser in plasma
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3.ABSORPTION PROCESS IN PLASMA AND COMPARISON
WITH OPTICAL RESPONSE OF METALS
There are several absorption mechanisms in the laser-plasma interaction. Among
them, collisional absorption and resonance absorption are dominant processes at
moderate intensity condition. From dispersion relation of Fig.2 and ray tracing of
Fig.3, the laser wave can penetrate into the point having density of nc(l-sin20).
However, beyond this point, the electrical field can penetrate into higher density
region with evanescent-like mode(see Fig.2(a)). This component can be coupled with
electron-plasma wave at the critical density point (ne=nc). With this resonance, the
amplitude of electrical field at this point becomes huge value and any small dissipation
can absorb the laser energy, effectively. This is explanation of resonant absorption.
About the collisional absorption, we can start eq. (1) and (2) with ohm's law. When
the laser light enter the plasma, the electrons are driven by its electric field. If some
resistive force can be assumed in the plasma, equation of motion of the electrons is
When the incident laser is harmonic wave, the solution of this equation is
e
«[*] = -*
(8).
Therefore, conductivity of the plasma is written with a = i————— and its
4n(co + ivei)
2
co
dielectric constant is with e -1 - ——~——. With the same procedure in Sec.2,
dispersion relation is
a)^co:f\l-i—
Q)
(9)
Therefore, this electrical field damped in the resistive plasma and the coefficient is
FIGURE 6. Damping properties of optical electric field in the plasma (a) (see manuscript for its
assumption) and normal metals (b).
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V 0)
7= — sL —if. In this discussion, we used an assumption of smaller damping coefficient.
2 co
Therefore, whole solution of this wave is damping oscillation with 0) frequency, (see
Fig.6(a))
It is very interest to compare this absorption and the interaction with normal metals.
As well-known, for interaction of light with metals the similar Drude models can be
used. The incident light also penetrates into metal with evanescent mode (skin depth).
The dispersion relation of metals [3] is Arc2 = 0)2/LL\ e + i
Since any metals at
co )
normal condition (at room temperature) has larger conductivity i
47TC7
cos
' 1), k has to
be square root of pure imaginary value. It means
co
.TTTTj
(10).
Therefore, it can be concluded that the 'propagation wave' inside metals has strongly
damping feature and its oscillating frequency and its damping coefficient are equal
each other, (see Fig.6 (b))
laser
(a)
v~106m/s
balistic electron motion
e-e collision
104m/s
e-ph collision
hot electron diffusion
thermal equilibrium
thermal diffusion
FIGURE 7. Post process of ultra-short-pulse laser heated materials
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If this normal metal is illuminated with ultra-short-pulse laser, it will be heated up
to so called plasma temperature in the meanwhile. During this heating process, both
curves in Fig.6 will be identical. How does it change? These comparisons and
questions will produce one of the important evidence when people create common
sense between solid-state physics and plasma physics.
4. ENERGY RELAXATION PROCESS AFTER ABSORPTION
As mentioned before, the absorption process and laser plasma interaction itself,
sometimes, are able to separate other physical process. So that, after energy absorption
in electron system, relaxation, conduction, and hydrodynamics expansion will
occurred as a post process in some case of ultra-short-pulse laser experiments (Fig.7).
Among them, they may separate each other in some condition. Typical time scales of
each process are Tee, Te_ph, xei respectively. If we can assume Tee« Te_ph,«Tei , the
discussion of this post process will be simplified. More explanation will be appeared
in future.
REFERENCES
1. For example, references in text book " Physics of laser driven plasmas" edited by
H.Hora, A Wiley-Interscience Pub., NewYork 1981 Chapter 5 p.58.
2. W. Kruer, "The Physics of Laser Plasma Interactions", Addison-Wesley, Redwood
city 1988 p.32-35.
3. See textbook of M.Born and E.Wolf, "6th edition Principle of Optics", Pergamon
Press Oxford 1980 Chapter XIII p.611-664.
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