Models of longitudinal space-charge impedance for microbunching instability

A 1D model of space-charge impedance, assuming a transversely uniform beam with circular cross section, has been proposed and is being extensively used in the modeling of the microbunching instability of relevance for the beam delivery systems of x-ray free-electron lasers. In this paper we investigate the limitation of the model when applied to studying the effect of shot noise—one of the sources of the microbunching instability. We make comparison with a fully 3D calculation and identify the upper end of the frequency spectrum for applicability of the 1D model. Relaxation of the assumptions regarding axis symmetry and uniformity of the transverse density is also reviewed.


I. INTRODUCTION
A successful design of the beam delivery systems for xray free-electron lasers (FELs) requires control and hence reliable and efficient modeling of the microbunching instability [1].At present, two distinct approaches are being pursued for modeling: macroparticle simulations [2 -4] and direct solution of the Vlasov equation (in its full [5,6] or linearized form [7,8]).
Macroparticle simulations in combination with particle in cell techniques allow for accurate determination of the fields for a given charge density but are sensitive to the unphysical fluctuations arising from using a number of macroparticles smaller than the bunch population.Direct methods to solve the full Vlasov equation are attractive in that they are immune from this source of noise but have limitations on their own.Because of the strong dependence of the computational load on the dimensionality of the system, they are most efficient in a reduced phase space.Indeed, solvers implemented so far have been limited to a 2D (longitudinal) phase space [5,6] and are necessarily based on a simplified model of beam dynamics and, in particular, of space-charge effects.
In this paper we discuss the simplified model of space charge proposed in [9] and adopted in [2,5,6].This model assumes an infinitely long beam in free space with uniform transverse density and circular cross section and yields an on-axis longitudinal electric field in Fourier space Ẽz k ÿZk Ĩk, which is exclusively determined by the beam current I and an impedance Zk [10].
The model has the pleasant feature that Zk can be cast into a handy analytical expression but has two obvious limitations: (i) the reduced dimensionality (1D); (ii) the assumption regarding uniformity and symmetry of the transverse density.Our main goal here is to address (i) with regard to the electric field generated by shot noise, the most fundamental source of density fluctuations seeding the microbunching instability.At high frequency one may expect that an evaluation of Ẽz k based on a 1D beam model would fail when the wavelength of the charge perturbation (in the beam frame) becomes comparable to the beam transverse size.In Sec.IV we identify this critical wavelength by making a comparison between the expectation values hj Ẽz kj 2 i as calculated from a 3D and the 1D model.We find that the two quantities start to diverge significantly for kr b = * 0:5, where r b is the radius of the beam transverse cross section.This is the regime where the transverse correlation length for Ẽz k becomes comparable to, or smaller than, the beam transverse size.(In the above equation the average hi is taken over the random realizations of the charge density due to shot noise.) As for the remainder of the paper, in Sec.III we review the 1D model of space-charge (SC) impedance for uniform axis-symmetric beam, while in the last two sections we briefly discuss beams with transverse Gaussian densities and non-axis-symmetric profiles, and for completeness we review the well-known effect of conducting walls (which limits the validity of simplified 1D models of space charge in free space from the low-frequency side of the frequency spectrum).

II. BASIC EQUATIONS FOR THE ELECTRIC FIELD
Consider an infinitely long electron beam with density x; y; z, where is the (uniform) linear particle density.More specifically, we will consider densities of the form x; y; z ?x; y z z with normalization R ?x 0 ; y 0 dx 0 dy 0 1 and R L=2 ÿL=2 dz z z L for L ! 1.The beam moves in free space with constant velocity c in the z direction (we shall assume ' 1).In the lab frame the longitudinal electric field generated by such a beam is given by E z x e=4" 0 R Gx; x 0 x 0 d 3 x 0 with Green function Gx; x 0 z ÿ z 0 =x ÿ x 0 2 y ÿ y 0 2 z ÿ z 0 2 2 3=2 .When working in cylindrical coordinates it is convenient to make use of the expansion [11] Gx; where following Jackson's notation r < (r > ) denotes the smaller (larger) between r and r 0 , and I m and K m are the modified Bessel functions.The Fourier component Ẽz k 2 ÿ1 R 1 ÿ1 E z xe ÿikz dz of wave number k then reads with obvious meaning of the shorthand notation I < m and K > m .

III. 1D MODEL OF SHOT NOISE
Assume a transversely uniform density with circular cross section of radius r b and an observation point located on-axis (r 0).Only the m 0 term in (2) contributes, while the radial integration can be carried out explicitly An impedance (per unit length) Zk is defined as where Ĩk is the Fourier transform of the current Iz ec z z, with z k 2 ÿ1 R 1 ÿ1 z ze ÿikz dz.We have where b kr b =.It is interesting to report the limiting form of (4) at high and low frequencies.For x ! 1, K 1 x decreases exponentially, and therefore Zk ! 1 iZ 0 =kr 2 b .For small x we have The granularity of the elementary charge gives rise to random fluctuations of the beam current (shot noise).We are interested in investigating how charge density fluctuations translate into electric field fluctuations and determining their statistics.
For convenience of calculation consider a beam with long (i.e.longer than any length scale involved in the problem) but finite length L. Consider a subdivision of L into N intervals of length z L=N .Denote with N b L the total number of electrons in the beam and N j the population of electrons in the interval z 2 zj ÿ 1; j.The occupation number N j is a random process obeying the Poisson statistics, which we model as [12] N j hN j i hN j i 1=2 j ; (6) where the expectation value over noise realizations is hN j i z N b z=L and j is a univariate normal random process with vanishing average and variance equal to unity h j i 0, h i j i ij .The last equation expresses the assumed lack of correlation between the number of electrons populating different intervals.From z z j z N j it follows and where denotes complex conjugation.Because L is finite it is understood that only a spectrum of discrete wave numbers is allowed: k k n 2n=L.From ( 3) and ( 8), h Ẽk i 0 follows, and after taking the limit where for brevity we have introduced the notation Ẽk Ẽz k.At low frequency b !0, we have the limiting form

IV. A 3D MODEL OF SHOT NOISE
Next, we want to contrast (9) to the result obtained from a 3D model of the shot noise.Again, we consider a beam with circular cross section but with transverse density that is uniform only on average: h ?ri ??0 1=r b for r < r b (and vanishing for r > r b ).Consider an elementary cell of volume r i rz centered at r i , ' , z j .Denote with N i'j the number of electrons populating this volume cell.Similarly to (6) we have N i'j hN i'j i hN i'j i 1=2 i ' j (10) with the average number of electrons given by hN i'j i ?0 r i rz.
To calculate h Ẽk Ẽ k 0 i we first discretize the volume integral in (2), R dV 0 !P i'j r i rz, make use of hr i ; ' ; z j r i 0 ; ' 0 ; z j 0 i 1 ii 0 jj 0 '' 0 =hN i'j i, and take the limit L ! 1 in the end.We find (for k Þ 0 and k 0 Þ 0)

MARCO VENTURINI
Phys.Rev. ST Accel.Beams 11, 034401 (2008) For b !0, we have The relative difference between ( 9) and ( 11) vanishes in the zero-frequency limit and becomes significant only for b * 0:5 (see Fig. 1).In the high-frequency limit (11) tends to a constant, (9), which decreases as 1= 2 b .We conclude that the 1D model gives a good approximation of the field fluctuations for long wavelengths down to ' 4r b =.
An interesting quantity is the radial correlation h Ẽk r Ẽ k 0i.Again, from (2) we have where kr=, indicating that the radial correlation of the Fourier spectrum components of the longitudinal fields decreases exponentially at high frequencies ( b 1), see Fig. 2 where the quantity h Ẽk r Ẽ k 0i=hj Ẽk 0j 2 i is plotted as a function of r=r b (solid lines).
Equation (12) should be contrasted to the radial correlation from the 1D model (see dashed lines in Fig. 2): At low frequencies ( b 1) the correlation (13) tends to (12) and they both tend to unity-a confirmation of the validity of the 1D model in this regime.
For ( b 1) the limiting form of ( 12) is well approximated by h Ẽk r Ẽ k 0i=hj Ẽk 0j 2 i ' K 1 , where the modified Bessel function K 1 ' =2 p e ÿ for of the order of, or larger than, unity.If we define the correlation length ' c as the radial distance over which the correlation decreases by 1=e, we find ' c ' 1:66=k ' 0:26.The 3D effects start to become important when the correlation ' c is comparable to or smaller than the beam transverse size r b .
The model considered in this section presupposes a frozen beam where interparticle distances do not change significantly in comparison with the correlation length ' c .In contrast, the 1D model of the previous section only assumes that the longitudinal projection of the interparticle distance does not vary.The time scales over which these assumptions hold are different.Outside dispersive regions of the lattice, the time scale for longitudinal density variations is set by the longitudinal plasma oscillations.In a relativistic regime ( * 100), say past the injection section in a 4th generation light source, the time scale as measured in terms of the distance traveled by the beam is of the order of 100 m [2].On the other hand, the 3D model is also sensitive to thermal motion in the transverse plane.One can expect that this may affect transverse correlations of length ' c over a time of the order ' c =v ?[or equivalently over a distance s c ' c' c =v ?], where v ? is the magnitude of the particles transverse velocity.For example, assuming that ' c be a fraction of the transverse beam size, say ' c r b =5, with r b ' " x = p and v ?' c "= x p , we find s c ' x =5.For typical values of the betatron function x , s c would be at most of the order of a few meters.This relatively short distance, however, could be sufficient in certain circumstances to induce detectable 3D effects.
The two curves are proportional to the expectation value hj Ẽk j 2 i as determined from the 1D model [black curve, Eq. ( 9)] and 3D model [red curve, Eq. ( 11

V. BEAMS WITH NONUNIFORM TRANSVERSE DENSITY
For an axis-symmetric beam with Gaussian transverse density given by ?x; y e ÿx 2 y 2 =2 2 =2 2 , the onaxis Fourier component of the longitudinal electric field of wave number k reads where k= and Eix ÿ R 1 ÿx dte ÿt =t is the exponential-integral function, yielding the impedance In deriving (14) we made use of the result At large frequencies Zk ! 1 iZ 0 =2 2 k [having used the asymptotic expansion Eiÿx e ÿx x ÿ1 . .., valid for large x].In the low-frequency limit, since Eiÿx ' E logx for small x, we find which gives a good approximation for Zk up to ' 0:5.By comparing ( 5) and ( 16), one can define an ''equivalent'' uniform beam with radius r b 2 p e 1ÿ E =2 ' 1:747 to represent the longitudinal impedance for a Gaussian beam with rms size , see Fig. 3.
If the Gaussian beam has unequal horizontal and vertical rms sizes, similar considerations show that the on-axis 1D space-charge impedance can be approximated with that of a round uniform beam [14] with r b ' 1:747 x y =2, which is close to the prescription r b 1:7 x y =2 proposed in [2] on the basis of numerical fitting.

VI. EFFECT OF BOUNDARIES
To estimate the effect of boundaries consider again a model of beam with transverse uniform density and round cross section.In the presence of a perfectly conducting pipe of radius r p concentric with the charge distribution, one can easily show that the impedance reads The high-frequency limit k ! 1 is the same as for the free-space case since both K 0 b r p =r b and the ratio I 1 b =I 0 b r p =r b decrease exponentially with b (as r p =r b > 1).The low-frequency limit yields the more familiar expression [15] which can be recovered from ( 17) using the limiting expressions K 0 x ' ÿ E logx=2, I 1 x ' x=2, and I 0 x ' 1, in addition to that for K 1 x reported in Sec.III, for x !0. The effect of the boundary becomes significant at low frequencies below b ' r b =r p or wavelengths * 2r p =.For typical pipe apertures this latter quantity is generally larger than the scale of the wavelength of interest for microbunching.

VII. CONCLUSIONS
We have shown that charge density fluctuations due to shot noise in the transverse plane of the beam can translate into significant fluctuations of the longitudinal electric field with wavelengths smaller than 2r b = (measured in the laboratory frame).This effect is missed by a purely 1D model of impedance, which yields a longitudinal electric field that responds exclusively to variations in the longitudinal line density.In this paper we have quantified these fluctuations, using a beam model with uniform averaged transverse density and circular cross section, for which a calculation can be carried out analytically.We found that the on-axis expectation value for j Ẽk j 2 generated by shot noise tends to a constant at large wave number k.In contrast, in the same limit the 1D model predicts a power-law decay as 1=k 2 .At low frequencies j Ẽk j 2 is largely correlated in the transverse plane while at small wavelengths the correlation length ' c ' 0:26 can become a small fraction of the transverse beam size.The divergence in the behavior between the 3D and 1D models sets a frequency range delimited by kr b = & 0:5 for the validity of the 1D model.
On the low end of the frequency spectrum where the 1D model is accurate, we showed that the impedance derived for a beam with transversely uniform density and circular cross section of radius r b can be used to reproduce with good accuracy the on-axis longitudinal field of a Gaussian beam provided that the parameter r b be adjusted appropriately.
We end with a few words about the practical implications of these findings.
In applications to 4th generation light sources, beam density fluctuations of very small length scale may be ignored if the smoothing effect of finite uncorrelated energy spread and horizontal emittance are significant.If this is the case, the frequency upper limit we have identified may not pose a particularly restrictive limitation to the applicability of the 1D model of space charge for microbunching.
For example, for FERMI [16] right after injection where ' 200 and r b ' 200 m, the 1D model is valid for wavelengths longer than 4r b = ' 12 m.This is below the wavelength range of the microbunching gain function through the bunch compressors, which is peaked at about 60 m and negligible below 20m [6], provided that the relative uncorrelated energy spread, as resulting from use of a laser heater, is not smaller than about 10 ÿ4 .However, turning off the laser heater [17] would move the range of significant microbunching gain to a region well below 20 m, where the applicability of the 1D model could become questionable.The validity of the 1D model is also in doubt at the end of the FERMI linac where the relative uncorrelated energy spread is small (because of the large beam energy) and microbunching on the 1 m scale could develop in the spreader [18] region.(For a discussion see [19].)Finally, we should mention that evidence of microbunching in mediumenergy, longitudinally cold beams that appears to be inconsistent with a simple 1D model of longitudinal space charge has been gathered in recent measurements of coherent transition radiation signals at Linac Coherent Light Source (LCLS) [20].These measurements point to the presence of correlations in the beam transverse plane with correlation lengths smaller than the beam transverse size [21].

ξ b r b k 10 ξ b r b k 0. 1 FIG. 2 .
FIG. 1. (Color)The two curves are proportional to the expectation value hj Ẽk j 2 i as determined from the 1D model [black curve, Eq. (9)] and 3D model [red curve, Eq. (11)].The relative difference is less than 10% up to b r b k= ' 0:5.

FIG. 3 .
FIG. 3. (Color)The longitudinal (on-axis) impedance for an axis-symmetric Gaussian beam with rms size (solid black curve) at low frequencies is very close to that of a uniform beam of radius r b 1:747 (red curve).The dashed curve is the lowfrequency limiting form(16).