SASE in plasmas: Analysis and simulation of Raman backscatter from noise

Nuclear Instruments and Methods in Physics Research A 393 (1997) 371-375

NUCLEAR INSTRUMENTS & mmloos IN PHVSICS RESEARCH Sectjon A

SASE in plasmas: Analysis and simulation of Raman backscatter from noise G. ShvetP*,

J.S. Wurteleb

“Princeton Plasma Phvsics Laboratoy. Princeton, NJ 08543. USA “Department qf Physics, Universiy of California. Brrkelq~. CA 94270. USA

Abstract Theoretical techniques from free-electron laser research are modified for analytical and numerical investigations of Raman Backscatter (RBS) in plasmas. The physical system consists of an intense short laser pulse propagating through an underdense plasma. The analogy with the free-electron laser is essentially a correspondence of the plasma electrons with the electron beam, the incident laser pulse with the wiggler and the backscatter pulse with the radiation. This approach has significant advantages over previous theoretical treatments, which were mostly linear fluid theories, and particle-in-cell simulations which are computationally expensive. The problem of backscatter from noise, which is the RBS equivalent of SASE, is analyzed and compared to experimental observations.

1. Introduction The backscatter of laser light as it propagates through an underdense plasma has been the subject of much recent experiments [l] and theoretical and numerical investigations [2]. In this paper we present a new method for studying the backscatter that is similar in many respects to the single-particle approach used in FEL theory [3]. The nonlinear model uses the eikonal approximation to simplify the wave equation, includes spacecharge forces (i.e., the plasma electric field), can be used in both the weakly and strongly coupled regimes, makes no assumption that the electron quiver velocity in the drive field is nonrelativistic and includes electron temperature. The ions are assumed motionless and uniform, and consistent with this, the incident laser pulse length is assumed to be much shorter than the inverse of the ion plasma frequency. A one-dimensional set of equations is derived which can include much of the full three-dimensional dynamics. In particular, both diffraction of the pump field and return current from the backscatter are included in an approximate form [4]. The one-dimensional model is implemented numerically and used to study backscatter in a wide range of parameter space.

*Corresponding author. Tel.: + 1609 243 2609; fax: + 1609 243 2662; e-mail: [email protected]. 016%9002/97/$17.00 Copyright P[I SOl68-9002(97)00515-9

Comparisons are made with analytical theory and with experimental observations of backscatter growth from noise. Recent experiments [l] observed the spectrum of the Raman backscattered radiation from a sub-picosecond 1.05 p intense laser pulse. It was observed that the spectrum of the RBS becomes modulated as the intensity of the laser (and the intensity of the backscattered radiation) goes up. Our calculations show that the nonlinear saturation of the RBS introduces temporal modulations in the output power which translated into modulation of the backscattered spectrum. We also find that the spectral properties of the RBS as predicted by the linear model lead to the spectra1 width AI( % ,/r,is, where r is the temporal growth rate and s is the duration of the laser pulse. The model yields an explicit (closed form) Green’s function for the backscatter in the weakly coupled regime and a simple expression for the Green’s function in the strong pump limit. The plasma density perturbations and radiation spectrum are understood when the plasma noise is white. This paper is organized as follows: Section 1 presents the basic formalism for the evolution of RBS and sets up an initial value problem which is implemented numerically. Linear response is studied in Section 2 and in Section 3 the temporal and spectral properties of linear growth of the RBS from random density noise (corresponding to SASE in an FEL) is investigated. Section 4

8~: 1997 Elsevier Science B.V. All rights reserved V. PRE-BUNCHING!SUPERRADIANCE

G. Shvets, J.S. Wurtele/Nucl.

312 gives a qualitative the RBS.

picture

of the nonlinear

Ins&. and h4eth. in Phvs. Res. A 393 (1997) 371-375

evolution

of

2. Nonlinear equations of evolution We assume the total laser field to be given by the sum of the incident pump a0 and the backscatter ai where a0

=

F

(e, + ie,)eiZo + cc,

ai(rl, ai = 2

z)

(e, - ie,)e’“’ + C.C.

The dependence on rl = (x, J)) allows for diffraction and guiding of the pulses. In this analysis a one-dimensional approximation is made and the fields are assumed to vary only in (z, t). The phases of the wave are a, = (k,z - w,t) and c(r = (klz + w,t). Typically, wc = wr and k. = kl, with any frequency or wave number shift absorbed into the eikonal amplitude ai. The backscatter pulse a1 is driven by the currents induced from the coupling of the ponderomotively bunched plasma (at a wavelength of z2ko) with the quiver motion in the pump (at k,). We assume that the eikonal approximation is valid. This corresponds to an assumption that the growth rate and the plasma frequency are slow compared to the laser frequency. The plasma electric field will be included, so that our model is valid for growth rates larger or smaller than the plasma frequency oPO. In laser-plasma physics this differentiates the strongly coupled and weakly coupled regimes, while for the FEL this differentiates the Compton and Raman regimes. The underlying physics is the same. The plasma ions are taken to be motionless on the time scales of interest. Consistent with the one-dimensional approximation, we assume vanishing canonical momentum in the (x, y) plane and have longitudinal equations of motion for a particle, labeled by the index j: 1 84 I-” l%lZ + Iail y dz iiz 2y2

dPzj -= dz f

eiej

ik,,aea, --5-+&2yz'

Y

a

which is neglected here, will modify these equations [4] somewhat, but will not change any of the basic results discussed below. The one-dimensional approximation is valid for interaction lengths of order a Rayleigh range and the neglect of return current implies pulse widths I c/w,. Note that the one-dimensional approximation in FELs requires a similar limit on interaction length as well a constraint that r,/l, < 1. The latter relation, imposed by transverse gradients in a magnetostatic wiggler field is not needed here, nor is it required for electromagnetic wave wigglers. The field equations, derived as usual through an eikonal expansion in Maxwell’s equations are

(4)

The the d/dr the

numerical implementation of these equations uses variables (5, s = (7 - z) so that particles evolve with = ((a/&) + (a/as)) and the differential operator in field equation becomes ((a/at) - (ii/az)) = ((d/87) +

2(a/h)).

3. Linear model of the RBS In this section we follow the analysis of Ref. [S] and derive a Green’s function response in the linear regime for a cold plasma. In this limit a fluid model is sufficient and

a

(

z+2;

>

a,(z,t)=

2 lq0

)

az,_ - Pzj. a7 We have normalized the momenta by mc and r = ct, where m is the electron mass and c the speed of light, and assumed that longitudinal electron velocity is nonrelativistic. The latter assumption is justified provided the plasma is very underdense, so that the phase velocity of the ponderomotive beat is nonrelativistic. The electrostatic potential 4 has been normalized by mc’. The phase 0, = crO(zj(s), t) + a,(zj(t), t). The plasma return current,

(6)

(7)

The density perturbation

aOal

Sna

4wocy0 no O’

is Sn and the relativistic

factor

YO= Jm. Th e initial conditions are specified at the head of the pump and the amplitude of the backscatter is calculated at the tail of the laser pulse. This motivates the coordinates co-moving with the pump T = ct and s = (ct - z). An analytic Green’s function can be obtained in the limit that the plasma is weakly coupled (the Raman regime in FEL parlance). Combining the above equations yields

(8) where r2 = copooo~ao~2/(4c2y~i4)is the temporal rate of RBS.

growth

G. Shvets, J.S. Wurteleilkf.

fnsb. and Meth. in Phw. Rex A 393 (1997) 371-375

It appears reasonable to assume that RBS in plasma is mainly seeded by the density fluctuations of the plasma encountered by the front of the laser pump. Thus, we specify the initial conditions s = 0 and find a, (T, s) = 01 (T - s/2, s = 0) 2ito~ao

;,;‘4(‘Qc

~~,~

dr’ G,(T - r’, s)ar(r’, s = 0)

r-s!2

2

lQpOUo

+p

s s r-.,:2

_____

dr’ G,(r - T’, s)

4too’ioc

&(r’, s = 0)

~--s

and the Green functions

n0

(9)

for noise and amplitude

I,[F(2r ~~T~T~~~~2

G,(T. s) =

S)"2(S - T)"'],

GJt, s) = I,[T(Zr

- .s)“~(s - r)“‘],

(10)

One of the convenient ways to describe a random time-dependent noise is to introduce ensemble (or time) averaged correlations. The correlation function of a random quantity ,f(t) is defined as + r) = W(T).

(12)

0

Here the first two expressions are related by the ergodic theorem. The correlation function W(t) is assumed to decay over a correlation time z, < T. If g(w), the Fourier transform of w(z), is frequencyindependent, then the random quantity f(t) is white noise, and has a very short correlation time. A straightforward way to model noise is to represent it by a sequence of equally spaced spikes of random sign. The distance between the spikes is roughly the correlation distance. To understand the correlation properties of the output signal, it is instructive to consider the following simple example. Suppose that the output signal is convolution of an input signal with a simplified Green’s function G,(r) = J rg G, = 0

for 0 < t <

otherwise.

Tpr

output signal ccl/J% and the output signal changes sign of the order N steps, corresponding to an increase in correlation length from T, to rg. This example illustrates a well-known property of any similar linear system: the correlation time of a convolution of an input signal with a Green function (with a much longer correlation time than the input noise) is equal to the correlation time of the Green function. From Eq. (9) we see that for particle noise

where ^ denotes

Fourier

transform

and

(11)

4. RBS growth from noise

{‘di f(t-)f(t’

The output signal is the arithmetic mean of N = r8,ir, input signals in the limit 7c < 5%. If the input signal is a string of delta functions with random sign. the rms

are

Eq. (9) describes the linear evolution of the backscattering starting from an initial density perturbation or lieid seed. The closed form for the Green function was possible to obtain because of the weakly coupled assumption. A closely related closed-form Green’s function was recently obtained by us [S] for an FEL amplifier in a strongly Raman regime.

((.f‘(t,j(t + 5))) =;

373

(13) (14)

(16)

For high-gain

systems

(r, 8 1) the spectral

width

is

Ao = J8rols,. The corresponding”cooperation length” I, = l/Au. which is just equal to the width of the Green function in a. For high-power RBS experiments, I, is typically much shorter than the distance the laser propagates through plasma (of order a Rayleigh length). The seeding noise for the RBS is much broader in frequency than the spectral width of the Green function. This does not imply that the noise spectrum of plasma density is truly “white noise”, but rather that, at spatial frequencies of order 2ko there is no appreciable variation in noise level over a frequency range of order Ao. The Green function approach allows us to analyze the backscattered radiation that is seeded by white plasma density noise. The resulting electric field consists of spikes whose width is of order 1, and amplitudes are random. This spiky signal is observed in numerical simulations. We average the numerically predicted power of the backscattered signal over many random realizations of the noise to get a smooth spectrum. The details of the simulation technique and the need to average will be discussed elsewhere [4]. The importance of averaging simulation output in SASE has also been recently recognized [6].

5. Nonlinear regime As the intensity of the pump laser increases so does the growth rate of the RBS. The exponential evolution of the instability eventually becomes arrested by various nonlinear phenomena such as wave-breaking, particle trapping and other processes that introduce energy spread on plasma electrons. The growth of the RBS does not stop

V. PRE-BUNCHINGiSUPERRADIANCE

374

G. Shvets. J.S. Wurtele /Nucl. Ins@. and Meth. in Phys. Rex A 393 (1997) 371-375

Fig. 1. Nonlinear competition between superradiant spikes produced by two identical short density clumps, spatially separated by a variable distance AZ. Time t scaled to l/w,, distance AZ scaled to c/q,.

completely once the steady-state “saturation” amplitude is reached. We observe spiking similar to the superradiant spikes studied in the context of FELs [7]. In the plasma context the growth beyond the steady-state saturation amplitude can be easily understood. A single radiation spike grows from a very short density clump. As the RBS radiation propagates through the plasma, it continuously slips into “fresh” plasma electrons and leaves behind plasma electrons with a substantial energy spread. Roughly, in the strongly coupled regime, the width of a spike decreases with its amplitude until plasma electrons make half a bounce in the ponderomotive potential of the spike as it passes over them. Even though a single spike continues growing nonlinearly, a spike trailing close behind it will not be able to reach as large an amplitude unless it is sufficiently far behind the leading spike. It appears to us that a second closely spaced spike will grow to roughly the steady-state saturated value and then stop growing as it encounters plasma that has energy spread due to the nonlinear interaction with the first spike. If the second spike is far enough behind to have already slipped from under the laser pulse by the time it encounters the “damaged” plasma, it will reach the same value as the leading spike. As the power of the pump laser increases, the saturation length of the RBS approaches the pulse width. Before that happens separate spikes do not nonlinearly affect each other and the results of the linear calculation remain valid. This is illustrated in Fig. 1. So long as the RBS is linear a smooth spectrum that mimics the Fourier transform of the Green function is

I

l.O$

i,,,,,~,,,,,,“,,,,,,,‘,‘,““l,,‘,l””I

-4

-3

-2

-1

0

1

2

3

4

Fig. 2. RBS power spectrum in linear regime a, = 0.15, average over C’ = 290 random noise realizations.

observed numerically. This is demonstrated in Fig. 2, where an average over 290 random realizations of the noise is taken. This averaging is justified by the transverse incoherence of the RBS signal, as explained in detail in Ref. [4]. In essence, the RBS signal consists of a large number of transversely uncorrelated filaments, which are averaged over by a spectrometer. As the saturation length, 1, becomes shorter than the pulse width, sO, a small power modulation appears because the spikes of width I, become separated by a distance of the order s0 - Ls. As the saturation length becomes a small fraction of the pulse length, the power

G. Sheets. J.S. Wurtele/Nucl.

Instr. and Meth. in Phvs. Res. A 393 (1997) 371-375

375

Acknowledgements This work was supported by United States Department of Energy, Division of High Energy Physics. One of us (G.S.) acknowledges the support of a US DOE Postdoctoral Fellowship.

References [l]

Fig. 3. Average over

regime; flat-top so = 2Oq%J,.

I = 9 single-shot spectra in nonlinear

circularly

polarized

laser with a,, = 0.45,

[2] [3] [4]

[S]

is strongly modulated and a number of peaks appears in the spectrum. This general behavior is shown in

[6]

Fig. 3

[7]

C. Rousseaux et al.. Phys. Rev. Lett. 74 (1995) 4655; C.A. Coverdale et al., Phys. Rev. Lett., submitted; V. Malka et al., Phys. Plasmas 3 (1996) 1682. G. Shvets, J.S. Wurtele. Bull. Amer. Phys. Sot. (1995). W.B. Colson, SK. Ride, Phys. Lett. A 76 (1980) 379. G. Shvets. J.S. Wurtele, B.A. Shadwick, Analysis and simulation of Raman backscatter in underdense plasmas, Phys. Plasmas, accepted. G. Shvets. J.S. Wurtele, Nucl. Instr. and Meth. A 358 (1995) 147. P. Pierini. W.M. Fawley, Nucl. Instr. and Meth. A 375 (1996) 332; Saladin et al., Nucl. Instr. and Meth., submitted. R. Bonifacio et al.. Phys. Rev. Lett. 73 (1994) 70.

V. PRE-BUNCHING/SUPERRADlANCE