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1942 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997
The Physics of the Nonlinear Optics of Plasmas atRelativistic Intensities for Short-Pulse Lasers
W. B. Mori
(Invited Paper)
AbstractThe nonlinear optics of plasmas at relativistic inten-sities are analyzed using only the physically intuitive processes oflongitudinal bunching of laser energy, transverse focusing of laserenergy, and photon acceleration, together with the assumptionof conservation of photons, i.e., the classical action. All that isrequired are the well-known formula for the phase and groupvelocity of light in plasma, and the effects of the ponderomotiveforce on the dielectric function. This formalism is useful when thedielectric function of the plasma is almost constant in the frameof the light wave. This is the case for Raman forward scattering(RFS), envelope self-modulation (SM), relativistic self-focusing(SF), and relativistic self-phase modulation (SPM). In the past,the growth rates for RFS and SPM have been derived in termsof wavewave interactions. Here we rederive all of the aforemen-tioned processes in terms of longitudinal bunching, transversefocusing, and photon acceleration. As a result, the physicalmechanisms behind each are made clear and the relationshipbetween RFS and envelope SM is made explicitly clear. Thisallows a single differential equation to be obtained which couplesRFS and SM, so that the relative importance between eachprocess can now be predicted for given experimental conditions.
I. INTRODUCTION
T
HE nonlinear optics of plasmas was developed exten-
sively in the early 1970s [1][4]. This development
resulted in the identification of numerous so-called para-metric instabilities. Among these were Raman and Brillouin
scattering, so-named because of their close connection to
the processes which occur in unionized gases [5]. In these
original analyses [1][4], the instabilities were formulated in
terms of wavewave interactions and the ponderomotive force.
Using this formalism, general dispersion relations were derived
which can in principle be used to describe the evolution
of arbitrary noise sources. These dispersion relations also
described the well-known filamentation/self-focusing [4], [5],
[7][9] and self-phase modulational instabilities [1][4], [9].
Mechanisms which occur only when the laser oscillates the
electrons at relativistic velocities were also identified in the
early work by Max et al. [4]. However, the relativistic analyses
were confined to the weakly relativistic limit. Almost all of the
early work was undertaken because of its importance to laser
Manuscript received August 22, 1996; revised July 1, 1997. This work wassupported by DOE under Grant DE-FG0392ER40727, LLNL under ContractB291465 and Contract B335241, and the National Science Foundation underGrant DMS-9722121.
The author is with the Department of Electrical Engineering and theDepartment of Physics and Astronomy, University of California of LosAngeles, Los Angeles, CA 90095 USA.
Publisher Item Identifier S 0018-9197(97)07828-7.
fusion which utilized lasers with relatively long pulse lengths
and modest intensities.
With the advent of short-pulse laser technology [10], new
applications [11][13] have evolved and the relevant insta-
bilities of the lasers in plasmas has changed. As a result,
there has been a large amount of new work in the field of
the nonlinear optics of plasmas at relativistic intensities. For
short-pulse lasers, the nonlinear optics of plasmas involves
only electron motion because the ions are immobile during
the transit time of the laser. This significantly limits thenumber of instabilities which can occur. The most important
instabilities for short-pulse lasers are Raman forward scattering
(RFS) [14][30], relativistic self-focusing (SF)[4], [31][33],
and relativistic self-phase modulation (SPM) [4], [27]. Each
of these instabilities can be described conventionally in terms
of wavewave interactions [1][4], [18], [19], [23][27]. In
each case, an incident electromagnetic wave at frequency
decays into two forward moving electromagnetic sidebands at
frequencies (the Stokes wave) and (the anti-
Stokes wave). The frequency corresponds to modulations
to the index of refraction. In RFS, , where is
the plasma frequency, and an electrostatic plasma wave is
generated. This instability is important because the phasevelocity of the plasma wave is nearly the speed of light. This
allows the plasma wave to accelerate electrons to relativistic
energies [11][13]. In filamentation and SPM, , and
the modulation at is caused by relativistic mass corrections
to the electrons motion.
Recently, the importance of RFS type instabilities for short
pulses was clearly demonstrated in fluid simulations [18][22].
In these simulations, finite-width pulses were found to break
apart axially into beamlets separated in time by roughly
. Associated with this breakup was the generation of
a large amplitude plasma wave. This clearly indicates the
occurrence of some form of RFS. In explaining these results,
one group developed a theory which described an envelope
self-modulation for finite-width pulses [28], [29], and the other
used a conventional Raman wavewave analysis [18], [19].
The physical mechanism for the envelope self-modulation
(SM) is as follows [13], [22], [28], [29]. First, the laser
pulse creates a plasma wave wake noise source. Next, the
density compressions and rarefactions of the plasma wave
wake transversely focus and defocus laser energy. As a result,
the laser spot size and hence intensity are modulated at nearly
the plasma frequency and at nearly the wavenumber
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MORI: THE PHYSICS OF THE NONLINEAR OPTICS OF PLASMAS FOR SHORT-PULSE LASERS 1943
. The ponderomotive force of the intensity modulation
then excites a larger wake and the process feeds back on
itself. Importantly, no mention is made of the light wave
decaying into other light waves. It was subsequently shown
that the differential equation and growth rate for this envelope
SM are the same as those for RFS at small forward angles,
[23][26]. It was also shown that the above
physical picture is complicated by the fact that the intensity
modulations and the density modulations are not resonantly in
phase [23], [24] and that the envelope analysis cannot describe
RFS in the exact or near forward direction,
[23][26]. RFS in the near forward direction (very small
angles) is distinct from SM by the phase relation between
the density and intensity modulations. We will henceforth use
RFS0 to mean scattering into the near forward direction. It has
also been pointed out [23], [24], [34], [35] that plasma waves
produced by RFS0 can also cause transverse focusing even
when so-called SM has not occurred. In spite of all this new
work to identify the various regimes of RFS, there is still some
confusion and disagreement as to how RFS and envelope self-
modulation are related and which mechanisms are importantfor given experimental conditions.
In this paper, we present a formalism which hopefully
removes some confusion. We analyze RFS0, envelope self-
modulation (SM), self-focusing (SF), and self-phase modula-
tion (SPM) all from the same set of physical phenomena. We
do not use the concepts of Stokes and anti-Stokes waves, rather
we calculate the modulation in laser intensity in terms of the
physically intuitive phenomena of longitudinal bunching of
laser energy, transverse focusing of laser energy, and photon
acceleration [36], [37]. Each of these phenomena arise when
modulations in the index refraction appear stationary in the
light waves frame, i.e., the index of refraction has a relativistic
phase velocity. The local index of refraction determines thelasers group and phase velocity. Longitudinal bunching (LB)
is caused by longitudinal variations in the group velocity,
transverse focusing (TF) is caused by transverse variations
in the phase velocity, and photon acceleration (PA) is the
change in local frequency caused by longitudinal variations
in the phase velocity. Recall that both SF and envelope SM
are typically explained in terms of transverse focusing of
laser energy. We calculate the overall change in the waves
amplitude from all of these effects, by assuming that photon
number, i.e., action is conserved [1][4], [38]. An equation is
then derived which relates the amplitude modulations to the
modulations of the index of refraction. By combining these
equations, we recover the exact growth rates for RFS0, SM, SFand SPM. Therefore, it is possible to precisely identify which
phenomena causes which instability. In addition, because
direct comparison between each phenomenon is now possible,
we may now determine which instability is most important
for given experimental conditions. We will argue that in all
existing experiments [39][42] one-dimensional (1-D) and
higher dimensional (2-D and/or 3-D) effects of RFS0 are more
important than SM. In future experiments, in which shorter
pulse lasers and lower plasma densities are used, then SM may
dominate. Incidentally, these same physical phenemona can be
applied to instabilities in other media so long as the index of
refraction appears stationary in the light waves frame, i.e., the
Stokes and anti-Stokes waves are both important.
II. PHYSICAL PHENOMENA
In this section, we begin by stating the basic assumptionsof this formalism and then describe how the waves amplitude
can be changed by modulations to the index of refraction.
A. Assumptions
In an unmagnetized plasma, the index of refraction of a
linearly polarized light wave is where
, is the plasma density, and
is the normalized vector potential of the incident laser,
. A useful formula is 0.85 10 (W/cm ) ( m)
where is the laser intensity and is the laser wavelength;
is commonly referred to as . It is clear that the
index of refraction can be altered by either modulating the
plasma density, the laser amplitude, or the laser frequency. In
this paper, we consider only small modulations and weakly
relativistic pumps. Therefore, the index of refraction can beexpanded as
(1)
where is the normalized density perturbation, and
represents averaging over the fast laser oscillations. In general,
, so an expression for is
(2)
and an expression for is [43], [44]
(3)
Therefore, it is clear that both and can be modulated
through changes in either density, laser intensity, or frequency.
Note that when the relativistic term is included [43],
[44].
In addition, we assume that within a local volume the
photon number, i.e., the classical action, is conserved. The
conservation of the classical action in laser-plasma instabiliteshas been discussed extensively in the literature [1][4], [38],
[45]; however, for completeness, we provide a derivation inAppendix A. Conservation of action can be stated as
constant (4)
where represents averaging over the fast oscillations, is
the spot size, and is some initial longitudinal extent. The
lasers vector potential can therefore be modulated by only
three ways:
1) modulate longitudinal bunching;
2) modulate transverse focusing;
3) modulate photon acceleration.
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MORI: THE PHYSICS OF THE NONLINEAR OPTICS OF PLASMAS FOR SHORT-PULSE LASERS 1945
Fig. 2. A physical picture for photon acceleration.
III. GROWTH RATES
In this section, we consider specific modulations to andand derive the growth rates for each instability.
A. Raman Forward Scattering
In order to derive growth rates, we need to identify how the
index of refraction is being modulated. In the 1-D limit, the
laser intensity can only be modulated from either longitudinal
bunching or photon acceleration. This is illustrated in Fig. 3.
Therefore, can be written as
(17)
and evolves in time as
(18)
Substituting (9) and (16) into (18) gives
(19)
In RFS0, the modulations to , i.e., and , are solelythe result of modulations to . Therefore, from (2) and (3)
we can rewrite (18) as
(20)
It is of interest to note that longitudinal bunching and photon
acceleration contribute equally to the modulation of . We
define and
where depends slowly with both and , to obtain
(21)
Therefore, in RFS0 the modulations to are out of
phase with the density response, . In order to derive
a growth rate, we need an equation which describes how
Fig. 3. A physical picture for RFS0. RSF0 is due to equal amounts oflongitudinal bunching and photon acceleration.
modulations to cause density perturbations. This is well
known to be a harmonic oscillator equation of the form [1][4]
(22)
where the right-hand side is the divergence of the ponderomo-
tive force. In the speed of light variables, this becomes
(23)
which can be rewritten as
(24)
where
and depends slowly on . In reducing (23) to (24), we are
not considering the strongly coupled (others call it the short-
pulse) regime of RFS0 [13], [18], [19], [23][26], [28][30],
because we are interested in the portions of the pulse with themost -foldings. Combining (24) with (21) gives
(25)
where . This is identical to [25, eq. (10)]
which was obtained from the conventional wavewave anal-
ysis using Stokes and anti-Stokes sidebands. It describes
the so-called four-wave resonant regime. Therefore, we have
shown that in RFS0 the density modulations of the plasma
wave are out of phase with , and the perturbations
to are caused equally by longitudinal bunching and photon
acceleration. The longitudinal bunching is caused by the
changes to . Note that the changes to from photon
acceleration also change but these modulations do not have
the correct phase to be reinforced.
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1946 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997
Fig. 4. The physical picture for relativistic SPM, which is photon accelera-tion followed by longitudinal energy bunching.
Equation (24) can be solved in closed form for specified
boundary and initial conditions [23][26]. Here, we simply
give its asymptotic solution which is
(26)
As noted earlier, this growth corresponds to four-wave res-
onant RFS0 and more research is required to obtain the
four-wave nonresonant growth of RFS0 using these simple
physical pictures.
B. Relativistic Self-Phase Modulation
In relativistic self-phase modulation, the index of refraction
is modified by the relativistic term in (2). This leads
to a modulation in from photon acceleration. However,unlike RFS0, the modulation to which results from this
does not have the correct phase to reinforce the original
perturbations of . Instead the modulation of causes
to vary, causing energy to bunch longitudinally. This is
illustrated in Fig. 4. This is a two-step process and this is why
SPM has a lower growth rate than RFS.
To calculate the growth rate, we first write the change in
from only longitudinal bunching,
Therefore, evolves in time as
(27a)
Substituting (9) into (27a) gives
(27b)
Assuming that the changes in are due to the term in
(3) results in
(28)
where we have substituted .
Differentiating (28) with respect to time leads to
(29)
The evolution of with time is given by (16) provided
that the modulations in phase velocity are the result of the
relativistic term in (2). This results in
(30)
which upon substitution into (29) gives
(31)
If then
grows in time as
(32)
Thus, it would appear that grows exponentially intime with a growth rate which increases indefinitely
with . Note that if this is times smaller
than the RFS0 growth rate. In reality, there is a term which
balances this growth for large . To understand this, we must,
for the moment, realize that the intensity modulation can be
represented as the superposition of a pump at frequency and
two sidebands at . The two sidebands cannot both be
exactly resonant because the linear dispersion relation of light
is quadratic, i.e., . Therefore the beat pattern of
is not stationary in the light waves frame, but appears
to have a real frequency. A real frequency term would appear
on the right-hand side of (32) but with the opposite sign.To calculate the effective frequency, we assume that is
of the form
(33)
where and. We assume the s are the
same, and that to obtain
(34)
Using a trigonometric identity, this can be rewritten as
(35)
Using the definitions for the s gives
(36a)
and
(36b)
Therefore, consists of a beat pattern moving at the
speed of light times a temporal oscillation which means that in
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MORI: THE PHYSICS OF THE NONLINEAR OPTICS OF PLASMAS FOR SHORT-PULSE LASERS 1947
the speed of light frame, oscillates at . Since
is the same for both the Stokes and anti-Stokes waves, we
can Taylor expand about to find
(37a)
and
(37b)
Therefore, dispersion leads to
(38)
Since in the absence of any growth must oscillate at this
frequency, we can add
(39)
to the right-hand side of (32) to obtain the final equation for
relativistic SPM
(40)
Therefore, the threshold for instability is and
the maximum growth rate is
(41)
and it occurs for
These results are in agreement with [4]. Therefore, the
physical picture of SPM is as follows. It requires first for the
frequency, i.e., the phase, of the wave to be modulated and
second for the laser energy to bunch longitudinally because
of the group velocity modulations associated with the changes
in . It is due to the relativistic corrections to , not to .
This physical picture was given qualitatively in [4]. Note that
the other mechanisms which modulate the laser amplitude do
not have the correct phase to be reinforced. In particular, the
relativistic terms in do not lead to an instability, but rather
to group velocity steepening.
C. Relativistic Self-Focusing
In the next two sections, we consider the effects of trans-
verse focusing. We begin with relativistic self-focusing and
show that our physical arguments for deriving an equation for
the evolution of the spot size yield the exact SF equation.
We start from (12) and assume that varies only from the
relativistic term in (2) to get
(42)
This is not the whole story since is the absence of variations
the spot size increases because of diffraction. Therefore, we
need to add the diffractive term on the right-hand side of (42).
The diffractive term in SF is analogous to the dispersion term
in SPM. The evolution of the spot size from diffraction for a
Gaussian beam is well known to be [5]
(43)
where is the Rayleigh or diffraction time,
is the laser wavelength, and is the spot size
at the focus. In Section II-C, we considered nearly planar
wavefronts, i.e., regions near the focus. Near the focus, (43)
can be differentiated twice to get
(44)
Adding this term to the right-hand side of (42) yields
(45)
Therefore, self-focusing occurs if the term in brackets is
negative. The threshold condition for SF is therefore
(46)
This is identical to the result obtained from more formal and
more complicated derivations using source-dependent expan-
sions [48], [49] or variational techniques [50]. We emphasize
that in the derivation presented in Section II-C the position of
the outer position of the wavefront was somewhat arbitrarily
assumed to be one spot size away from the axis. The more
rigorous derivations must therefore be done at least once.The threshold condition is in terms of the product ,
which is proportional to the laser power. Therefore, (46) can
be viewed as a power threshold condition where
the critical power is [31]
GW (47)
Note that if only transverse focusing occurs that is a
constant, i.e., the laser power is conserved.
D. Envelope SM (Laser Sausaging)
Transverse focusing can also be caused if the index ofrefraction is varied by the density pertubation term in (12).
In this case, (12) reduces to
(48)
Note that if is negative, focusing occurs while, if
is positive, defocusing occurs. In order to couple the evolution
of the spot size to that of the density pertubation, we need to
relate to . From (5)
(49)
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1948 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997
from which it follows
(50)
Equation (50) combined with (23) completely describes enve-
lope SM, and as before we are not considering the strongly
coupled regime. Using the same definitions as before for the
field envelopes gives
(51a)
(51b)
Combining (51a) and (51b) gives a single equation for :
(52)
We rewrite the right-hand side in terms of and to get
(53)
This is identical to the equation obtained by differentiating [28,
eq. (4)] with respect to , which was derived using the source-
dependent expansion method. This differential equation has no
closed form solution. However, using saddle point integration
techniques, the asymptotic form for the solution can be shownto be [23][26], [28], [29]
(54)
Before proceding, there are several points to make. First,appears in the differential equation because of the
choice for normalizing the parameters; the relativistic SF terms
have been neglected. We will comment more on this shortly.
Second, an identical differential equation and hence asymptotic
solution was derived using a conventional wavewave RFS
analysis [23], [24] for scattering angles of .
Therefore, the scattered light is not in the direct forwarddirection but is at well-defined angles. This results in the
generation of higher order Gaussian modes. Third, in SM,
only transverse focusing and defocusing occurs. The intensity
modulations arise from reductions and increases in the spot
size, so sometimes SM is referred to as laser sausaging
due its analogy with an electron beam instability. Fourth,the asympotic solution also has an imaginary part to the
phase, which means that the density pertubation and intensity
modulations are not out of phase [23], [24], and the phase
velocity of the plasma wave gradually decreases in time [25],
[26]. The decrease in phase velocity means that SM is less
likely to generate ultrarelativistic electrons. Last, transverse
focusing is not equivalent to SM, it can also occur during
RFS0. Since in RFS0, and are still out of
phase, transverse focusing leads to contour shapes in which
one-half is narrow and the other half is wide. These shapes
have been refered to as inverse D shapes [24], [34], [35]. This
Fig. 5. The effects of TF from RFS0 without SM. If n and p
are = 2 outof phase and the wave is moving to the right, then intensity contours becomeinverse D-shaped, so a signature of transverse focusing in RFS0 is inverseD-shaped contours.
is shown in Fig. 5, and such shapes cannot be generated inthe SM process.
Envelope SM is directly related to RFS in the so-called
four-wave nonresonant regime and is distinct from RFS0.
The term nonresonant refers to the fact that because of the
extra phase shift, the plasma wave does not oscillate
at exactly . The difference between four-wave nonresonant
RFS0 and SM lies in which term dominates in the mismatch
quantity of [23], [24], i.e., if then RFS
dominates. Similar conclusions were independently reached by
Andreev et al. [25], [26]. In the nonresonant regime of RFS,
multiple cascading is less likely to occur [25], [26], [34], [35],
[51]. Therefore, an experimental diagnostic of RFS0 is the
observation of strong multiple cascading.In the original SM paper by Esarey et al. [28], [29], it
was indicated that relativistic SF was critical to envelope SM.
They stated that needed to be larger than roughly 1/2
(depending on the sharpness of the rise time) in order for
strong SM to occur. We believe this to depend on the parameter
space under consideration. In the preceding analysis and in the
conventional wavewave analyses [23][26], the relativistic
SF terms have been neglected. This assumption is valid so long
as the spot size changes little from SF or diffraction before the
instability saturates. In particular, it is obvious from (53) that
for any given value of a pulse length can be chosen such
that SM (or RFS0) can still occur within a Rayleigh time. In
PIC simulations, beam breakup is observed even for0.5 [51][53].
The SF and diffraction terms can easily be included by
combining (45) and (48) to get
(55)
Combining (54) with (49) now gives
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MORI: THE PHYSICS OF THE NONLINEAR OPTICS OF PLASMAS FOR SHORT-PULSE LASERS 1949
If a matched beam is assumed, i.e., , or equivalently
a preformed channel is used [20], [21], then the equations
given earlier in this section are still valid. The physical
mechanism of the instability is not due to relativistic SF,
but rather is completely due to transverse focusing in plasma
waves. The inequality, , can be important if
relativistic guiding is required to increase the interaction length
beyond several Rayleigh lengths.
IV. COMPARISON BETWEEN RFSAND ENVELOPE SELF-MODULATION
In the previous section, we carefully described the differ-
ences between longitudinal bunching (LB), photon accelera-
tion (PA), and transverse focusing (TF). We also showed that
when these processes are caused by plasma waves, they lead
to RFS0 and SM. In this section, we make detailed compar-
ison between the relative importance of the 1-D phenomena
(LB and PA) and the 2-D phenomenon (TF), with particular
emphasis given to discerning which processes dominate for
given experimental conditions.
For simplicity, we begin by assuming a constant amplitudeplasma wave and compare the relative contributions to
from 1-D and 2-D effects. This simple exercise is very illus-
trative and it has direct implications to beat wave excitation of
plasma waves where the plasma wave amplitude changes little
in . Focusing in constant amplitude plasma waves is related
to cascade focusing [54]. If is constant in at a value
, then (21) and (50) can be integrated in to give
(56a)
and
(56b)
It is immediately clear that grows as while
grows as . The reason is that TF is a two-step
process which requires the wavefronts to first curve before
energy can be focused. Therefore, the 1-D effects always
dominate early in time while transverse focusing dominates
late in time. In terms of the magnitudes of (note the
phases are different), the transition occurs when
(57)
For tenuous plasmas, i.e., , this is typically a
small number. In beatwave excitation, is generally so
low, and the interaction time so large that TF almost alwaysdominates. From (56b), 100% modulations to occur
within a Rayleigh length from TF when is near unity.
This could have deleterious consequences to the multiple pulse
excitation process [55][58] since it requires accurate phase
relationships to be maintained between each subsequent pulse.
On the other hand, in single-frequency experiments, the
RFS0 instability may have saturated before the transition to
the TF dominated regime occurs. In this case SM, which is
entirely the result of TF, may not occur. We reiterate that TF
can occur in RFS0. The signature for the SM regime is not
that TF occurs, but that the phase shift between and
be , and that TF dominates. To be more precise,
we next derive a differential equation which couples both the
1-D and 2-D effects and then calculate its asymptotic response.
This equation does not include the nonresonant regime of
RFS0, which is reasonable if , i.e., small
[23], [26].If this inequality is not satisfied, then RFS0 always
dominates.
We start by defining , where
the subscript stands for total. Therefore,
(58)
and we substitute (21) and (50) into the right-hand side to get
(59)
Substituting the earlier definitions for and and (24)
into (59) gives
(60)
where we have defined and . Note that
if (no TF) we recover (25) while if (no LB
or PA) we recover (52).
Equation (60) cannot be solved exactly. However, we can
calculate its asymptotic behavior from stationary phase argu-
ments. We assume solutions of the form for
and obtain the dispersion relation
(61)
The stationary phase conditions are
(62a)
and
(62b)
The solutions to (62b) can be seperated into two regimes
depending on whether the parameter is large
or small compared to unity. In the small limit
(63)
Substituting (63) and (62a) into the phase factor
leads to the asymptotic behavior
(64)
In the opposite limit where is small,
(65)
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MORI: THE PHYSICS OF THE NONLINEAR OPTICS OF PLASMAS FOR SHORT-PULSE LASERS 1951
electrons because the phase velocity of the plasma wave it
generates has a spatio-temporal-dependent phase velocity.
We close this section with a few comments. First, this
analysis is only stricly valid for and for times before
saturation occurs. In some of these experiments, ,
and this can lower the growth rate [23][26]. Second, the
asymptotic solutions have restrictions. The front part of the
pulse, i.e., smaller values of , makes a transition to SM
earlier than the back of the pulse. This will obviously effect
how the back of the pulse evolves. The asymptotic solutions
given here only include the stationary phase part. Saddle point
integration provides other factors to the asymptotic solutions.
Third, the theory assumes diffraction-limited beams, while
the beam quality varies from experiment. Last, the time of
saturation and the transverse profile depends on the noise
source. There have already been several types of noise sources
which have been identified [22], [25], [26], [34], [35], [38],
[51][53]. Therefore, to understand current experiments, PIC
simulations [34], [35], [51][53] are required. These indicate
that RFS0 plays an important role and that the dominant
saturation mechanism is wavebreaking. It is worthwhile tonote that some of these PIC simulations were carried out
before many of the above experiments and that many of theirpredictions have been born out in the subsequent experiments.
VI. SUMMARY AND FUTURE WORK
In this paper, we have given simple physical pictures
for the nonlinear optics of plasmas at relativistic intensities.
Here, relativistic refers to either the electrons oscillating at
relativistic energies or the plasma dielectric function having
a relativistic phase velocity. In particular, Raman forward
scattering (RFS0), envelope self-modulation (SM), relativis-
tic self-focusing/filamentation (SF), and relativistic self-phasemodulation (SPM) were analyzed. The analyses used only the
concepts of longitudinal bunching (LB), transverse focusing
(TF), and photon acceleration (PA), together with the as-
sumption of conservation of photons, i.e., the classical action.
Using just these concepts and the well-known phase and group
velocity of light, the growth rates for RFS0, SM, SF, and SPM
were rederived.
Direct comparison between each instability is now possible
because they were derived from the same set of physical
phenomenon (although the derivations are not rigorous). It wasshown that RFS0 is caused by equal amounts of LB and PA
(both are 1-D effects). On the other hand, SPM is caused by
PA followed by LB. It is therefore a two-step process so ithas a lower growth rate than RFS0. It was also shown that
SF is caused by TF from relativistic mass corrections to the
light waves phase velocity. Finally, SM is the result of TF
in a nonresonant plasma wave. Importantly, it was shown
that TF in RFS0 generated plasma waves is not the same as
SM. A comparison between RFS0 and SM showed that in
self-trapped electron experiments RFS0 plays a key role. This
could change in future experiments.
We close with a few remarks regarding future work. This
analysis is only strictly valid in the weakly relativistic limit,
. Based on the success of this weakly relativistic
analysis, it seems worthwhile to try and extend it to the fully
relativistic limit, . This will require precise knowledge
of the fully nonlinear phase and group velocities of light in
plasmas. To date we have not made a successful treatment
for . Another area to investigate is the role of photon
acceleration during SPM. In particular, a modulation to
could lead to a from PA, which in turn could lead to
further modulations to from PA. This two-step process
was neglected in Section II-B, but it may be important. Still
another area for future research is implementing hosing and the
nonresonant regime of RFS0 into these physical pictures. Last,
we point out that this analysis not only allows one to rederive
known growth rates, it also makes it possible to identify new
processes and to properly analyze old problems. For example,
it is possible for SF to occur because of transverse variations
to caused by PA [61], and the phenomenon of resonant
relativistic self-focusing [33] can be coupled to SM [62].
Note Added in Proof: The field of nonlinear optics of plas-
mas is still developing rapidly, as illustrated by the numerous
experimental results on electron acceleration which have been
obtained since the original submission data, e.g., [63][65].
APPENDIX A
LOCAL ACTION CONSERVATION
In this paper, we have assumed that the action, i.e., photon
number, is conserved locally. In this appendix, we derive alocal conservation law for the action using similar reasoning
as others when they derived a global conservation law [38],
[45]. We begin with the basic quasi-static equation for the
normalized vector potential
(A1)
where and are the speed of light frame
variables defined in the paper. Next, we assume to be of the
form to obtain the envelope equation
(A2)
We next define the operator
and take the linear combination of (A2)
to get
(A3)
To obtain a global conservation law, (A3) can be integrated
over all space, i.e., ; but to obtain the more powerful
local conservation law, we need to rewrite several terms in
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1952 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 11, NOVEMBER 1997
terms of complete derivatives. The first two terms of (A3) can
be combined as
(A4)
which can be rewritten as
(A5)
The third term of (A3) can also be rewritten as a complete
derivative
(A6)
Recombining (A5) and (A6) gives a local conservation law in
the coordinates
(A7)
Integrating (A7) over and gives the global conservation
law
(A8)
The term in brackets can be identified as the action by
rewriting ; substituting this form for
into the term in brackets in (A8) gives
(A9)
which is times the action, , since is the
instantaneous frequency.
A local conservation of law of a quantity, , has the generic
form
(A10)
where is the flux of and is the flow velocity of .
Therefore, we can interpret
(A11)
and
(A12)
Note, to lowest order the velocity of the action is equal to
the group velocity of light, so in the speed of light frame
and . Using these
definitions, it is straightforward to verify that interpretations
(A11) and (A12) are correct.
In conclusion, in this Appendix, we have derived the local
conservation law
(A13)
and have given arguments that it is a conservation law for theaction.
ACKNOWLEDGMENT
The author acknowledges useful discussions with K.-C.
Tzeng and Dr. T. Katsouleas, Dr. C. D. Decker, Dr. C. Joshi,
Dr. J. M. Dawson, Dr. E. Esarey, and Dr. G. Shvets.
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