The Physics of the Nonlinear Optics of Plasmas at Relativistic Intensities for Short-Pulse Lasers.pdf

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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

00189197/97$10.00 1997 IEEE


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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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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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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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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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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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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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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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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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