Analysis of shot noise suppression for electron beams

Shot noise can affect the performance of free-electron lasers (FELs) by driving instabilities (e.g., the microbunching instability) or by competing with seeded density modulations. Recent papers have proposed suppressing shot noise to enhance FEL performance. In this paper we use a one-dimensional (1D) model to calculate the noise amplification from an energy modulation (e.g., electron interactions from space charge or undulator radiation) followed by a dispersive section. We show that, for a broad class of interactions, selecting the correct dispersive strength suppresses shot noise across a wide range of frequencies. The final noise level depends on the beam's energy spread and the properties of the interaction potential. We confirm and illustrate our analytical results with 1D simulations.


II. INTRODUCTION
In a bunch of random (uncorrelated) electrons, the longitudinal density contains white noise fluctuations, commonly called shot noise.While shot noise drives Self-Amplified Spontaneous Emission (SASE) Free Electron Lasers (FELs), the same density fluctuations may adversely affect FEL operation.For example, the microbunching instability, thought to originate from shot noise, can incapacitate diagnostics and degrade FEL performance [1][2][3][4][5][6][7].Shot noise also competes with external modulations in the operation of seeded FELs [8,9].Recent papers have proposed schemes to decrease the noise level below that of shot noise to aid the FEL process or for other applications [10][11][12].In this paper we use the approach of [13] to study the evolution of noise as the beam travels through a system with interactions between the electrons as well as dispersive regions.To simplify the analysis, we consider a one-dimensional (1D) model system of a generic self-interaction, h, which changes the particle energies, followed by a dispersive region, R 56 , which converts the change in energy to change in position (Fig. 1).We show that for a broad class of interactions, it is possible to suppress density fluctuations below the shot noise level, and we provide 1D simulations to confirm the result.Future theoretical and numerical work will extend these results to 3D models, and explore the feasibility of demonstrating shot noise suppression experimentally.1.Schematic of our model system.Starting with an initial electron distribution function, f0(z, η), the interaction and dispersive regions produce a final distribution function, f f (ẑ, η).The dispersion may be positive or negative.

III. ANALYTICAL MODEL A. Noise Factor
To characterize the level of noise at a wavevector, k, we define the noise factor e ik[zj (s)−z l (s)] (1) where z j (s) is the longitudinal bunch coordinate of particle j at position s in the accelerator, and N is the number of particles in the beam.We note that the noise factor can equivalenty be defined by F (k, s) ≡ N |b(k, s)| 2 , with the bunching factor b(k, s) ≡ j exp[ikz j (s)]/N .The noise factor, F (k, s), is a measure of the correlations between particle coordinates at wavevector k.If the particle positions are uncorrelated, we find the expectation value of shot noise, F (k, s) = 1.On the other hand, if the positions are strongly correlated at wavevector k, we find F (k, s) ∼ N , with N 1 generally; such correlated (or 'bunched') beams are found at the output of an FEL, and as the result of the microbunching instability [6,7].We may also consider the case of an anti-correlated (or 'quiet') beam, with F (k, s) < 1, below the shot noise level.In this paper, we investigate the possibility of producing quiet beams.
Though the noise factor is defined as a function of accelerator position, s, we are particularly interested in the noise level at the output of our system, F (k, s f ).Starting from an initial distribution function at s 0 , we would like to determine the resulting final noise level at s f .
To facilitate an analytical solution, we will study the simplified system of Fig. 1.We assume the particle distribution is a function of position in the bunch, z(s), and relative, normalized energy, η(s) ≡ [E(s) − E 0 ]/E 0 , with average beam energy, E 0 .Though both z and η are functions of s, we are primarily interested in the initial and final coordinates, so for brevity we define z, η ≡ z(s 0 ), η(s 0 ), ẑ, η ≡ z(s f ), η(s f ) and F (k) ≡ F (k, s f ).We can then describe the system as follows.We start with a simple N -particle initial distribution of particles, f 0 (z 1 , ..., z N , η 1 , ..., η N ).After an interaction period, the energies are modified, giving distribution f a (z 1 , ..., z N , η1 , ..., ηN ).A dispersive region (assumed to have zero interaction), then changes the longitudinal positions, giving final distribution f f (ẑ 1 , ..., ẑN , η1 , ..., ηN ).

B. Expectation Value of Noise Factor
To calculate the expectation value, we break F (k) into incoherent (j = l) and coherent (j = l) portions.First, we treat the incoherent portion.With j = l, the phases cancel and we find N terms, all equal to 1, giving which is simply the noise level due to shot noise.Next, we calculate the coherent portion.To find the expectation value at the final accelerator position, we integrate F (k) over the final particle distributions, f f (ẑ 1 , ..., ẑN , η1 , ..., ηN ).In general, f f may be a complicated function of all 2N variables.However, if we assume the electrons are initially uncorrelated, then we can write the initial distribution function as with the single particle distribution functions for a beam with Gaussian energy spread of σ η and uniform longitudinal density of length L given by for −L/2 < z < L/2 0 elsewhere We then express the final coordinates in terms of the initial coordinates (ẑ, η → z, η), and integrate over the product of N simple initial distributions, f (1) .In the interaction region, we assume the bunch is longitudinally frozen (z, η → z, η), and likewise in the dispersive region we assume there is zero interaction (z, η → ẑ, η).To further simplify the calculation, we ignore any transverse effects.(The validity of the 1D approximation will depend on the interaction of interest.)Our resulting map from initial to final coordinates then is with dispersive strength R 56 , and h(z j , z i ) the change in energy of particle j due to the interaction with particle i.We can now write the coherent portion of F (k) in terms of the initial coordinates, and integrate over each single particle distribution, f i ≡ f (1) (z i , η i ), to find the expectation value where we have assumed the N 2 − N ≈ N 2 1 coherent terms of the sum in Eq. 1 are identical, and we have chosen j = 1, l = 2 without loss of generality.
Our approach (following [13]) will be to explicitly separate the z 1 , z 2 terms.We assume the interaction depends only on the distance between the particles, h(z 1 , z 2 ) = h(z 1 − z 2 ), so we change variables, z 1 , z 2 , z l , z m → ζ, Z, τ l , τ m with ζ ≡ z 1 − z 2 , Z ≡ (z 1 + z 2 )/2, and τ l,m ≡ z l,m − z 2 .Finally, we assume that the interaction is nonzero only within a charactersitic distance, L h , which is much shorter than the bunch length, L. We can then integrate over Z and η 1 ...η N to find where we have defined the 1D particle density n 0 ≡ N/L and we have used L h L to both ignore edge effects and set the ζ integral limits to infinity.First, we note that the N − 2 integrals over τ i are separable and identical.Second, we assume kR 56 h 1 so we can linearize the exponentials, yielding with definitions where we have expanded Γ 1 and Γ 2 in powers of the small parameter, kR 56 h.For Γ 1 , the term linear in kR 56 h is nonzero, so we drop all higher order terms.However, from our assumption of a long bunch, the linear order terms in Γ 2 cancel after the integration, so we must also keep the quadratic term for Γ 2 .Combining the two square terms, dτ h(−τ We may be tempted to drop Γ 2 , because it is second order in kR 56 h.However, Γ 2 is also raised to the power of N , and with N 1 generally, Γ 2 may even be the dominant term (as for the microbunching instability, see e.g.[6,7]).In this paper, we keep both terms, and will see that noise suppression occurs when Γ 1 and Γ 2 are comparable.

C. Analytical Expression: Weak Interaction
If we consider a weak interaction under the stronger assumption, Γ 2 1, we can solve for the noise level analytically.Adding in the shot noise term again and expanding Eq. 8, we find We are interested in k = 0, and so will drop the δ function.(The δ function arises from our assumption of L → ∞. For finite L, we will have a term that is nonzero for k < 1/L, but even so our focus is on much shorter wavelengths.) We can now identify the three regimes for F (k) .For zero interaction, we are left with only the leading shot noise term, F (k) = 1, which is simply the white noise of an uncorrelated bunch.The Γ 2 contribution is positive-definite, so for Γ 2 Γ 1 , we find a correlated beam with F (k) > 1.Finally, for Γ 1 ∼ Γ 2 , the term linear in R 56 cannot be neglected.If R 56 is chosen so that Γ 1 < 0, it is possible to create an anti-correlated beam, with the noise factor suppressed below the shot noise level, F (k) < 1.In this paper we consider the third regime.
Identifying the ζ integral as a Fourier transform (FT), we rewrite the noise factor as where h(k) denotes FT{h(τ )}.We drop the second term in the remaining integral because it has no ζ dependence, and so its Fourier transform is nonzero only for wavelengths longer than the bunch (k < 1/L).The first term is the autocorrelation of h(τ ), which has Fourier transform | h(k)| 2 , yielding If the energy spread is small (σ η → 0), and the interaction has purely imaginary Fourier transform, h(k), we can write the noise factor as a perfect square We suppress the noise factor below the shot noise level when the suppression parameter is in the range 0 < Υ < 2 and the noise disappears completely for Υ = 1.(We note that partial noise suppression is possible even if the interaction contains a real component.)We are particularly interested in interactions that can be approximated as step functions near ζ = 0: h(ζ) → AH(ζ)+ const, with Heaviside function H, and interaction strength, A. For such interactions, we find h(k) ∝ 1/k for high frequencies, so that Υ is independent of k.We are then able to simultaneously suppress bunching at a wide range of frequencies.
We can draw a broad lesson from Eq. 13; a quiet beam is attainable from any interaction with primarily imaginary Fourier transform, e.g. from step function interactions (for k = 0).We will treat the special cases of space charge and undulator interactions later, but here emphasize that any interaction with imaginary Fourier transform will suffice.For example, the wake from a linac with periodic structures also satisfies these conditions [14].We have assumed negligible energy spread here; see the appendix for a discussion of the effect of energy spread on noise suppression.
For a physical interpretation of the requirement for imaginary Fourier transform, we consider a test particle in front of localized density spike of width 1/k.If h(τ ) > 0 for τ > 0, the test particle will receive positive energy change.A positive dispersive region then causes the test particle to move forward and away from the dense region.Likewise, a test particle at the back of a dense region (τ < 0) loses energy relative to the front particle for h(k) imaginary, and moves backward and away in a positive dispersive region.The end result is a reduction in the density spike and thus a reduction in the noise.If h(τ ) < 0 for τ > 0, as is the case for an undulator, we have the identical argument, but require negative dispersion.The process is illustrated in Fig. 2.

Space Charge Undulator
FIG. 2. Schematic of an interaction near a density spike (solid green line).At left, for the space charge case, particles in the front half of the spike gain energy, while particles in the back half lose energy, and in positive dispersion, the density spike shrinks (dotted green line).We have a similar result for an interaction due to undulator radiation (right).At high frequency (spike much shorter than undulator resonant wavelength), all particles lose energy, but following a dispersive region with negative R56 we still find a reduction in the density spike (dotted green line).

D. Numerical Approximation: Strong Interaction
For stronger interactions, we may not be able to approximate Γ 2 1.If it is not possible to evaluate Eq. 8 analytically for an arbitrary h, we can carry out the integrals numerically.Using the less stringent approximation Γ 2 N (satisfied even for simulation parameters with relatively small N ) we take (1 + Γ 2 /N ) N ≈ exp(Γ 2 ) to obtain For physical interactions, Γ 1 → 0 as ζ → ∞, so the second term, e Γ2(ζ) Γ 1 (ζ), converges and can be integrated numerically.We cannot directly integrate the first term, exp[Γ 2 (ζ)], because the h 2 (−τ ) in Eq. 10 has no ζ dependence; in the limit ζ → ∞, we find Γ 2 (ζ) → Γ2 = 0, and the integral diverges.However, the divergence occurs only for k = 0; otherwise, Γ2 exp[ikζ] integrates to zero (which is why we dropped the h 2 (−τ ) term from Eq. 12).Following the same reasoning here, with we explicitly remove the constant term, exp[ Γ2 ] to find where we've used Eq. 10 to see that Γ 1 (ζ) and Γ 2 (ζ) are respectively odd and even functions of ζ.We can then integrate Eq. 17 numerically.

A. Space Charge Interaction
So far we have not specified the interaction term, constraining only that the energy change, h, is a function of ζ, the distance between the particles.We now consider the Coulomb interaction between two particles.We assume the interaction occurs over a distance L a in the accelerator, during which the particles are frozen longitidunally.We consider a 1D system, treating the particles as uniform, rigid sheets of charge with radius a, valid in the limit a γ/k [15].To calculate the relative change in energy due to the longitudinal E-field (E z ), we integrate over the sheets of source and test particles, 3/2 (18) with average particle energy, γmc 2 , electron charge, e, area of sheet, S = πa 2 , and One of the θ integrals trivially gives a factor of 2π, and the remaining integrals can be solved numerically to produce the interaction h sc (ζ) shown in Fig. 3.We note that the interaction will go to zero for ζ a/γ, as required in our derivation of Eq. 7. In the limit of infinite sheets (a → ∞), the E z field is simply so that the interaction causes an energy change per charge, e, of with classical electron radius r e ≡ e 2 /4π 0 m e c 2 .

B. Space Charge Fourier Transform
In the simplified case of infinite sheets, the step function at ζ = 0 dominates h(k), and we find a purely imaginary Fourier transform, We calculate the energy modulation to the test sheet by averaging over the entire sheet (solid blue curve).Though Ez (and thus hsc) is not constant everywhere in the test sheet, we note that there is relatively little variation near the center of the sheet, as can be seen from hsc evaluated at radius r = 0 (dashed green line) and r = a/2 (dotted red line).
with definition Following Eq. 14, we then define the suppression parameter for space charge, Υ sc ≡ n 0 R 56 A sc , and we expect broadband suppression for Υ sc = 1.
In the finite sheet model, when ζ a/γ the interaction falls off as 1/ζ2 .The cutoff for h(ζ) as ζ → ∞ determines the noise suppression at low frequencies; the approximation of h(k) ∝ 1/k breaks and we expect suppression to be frequency dependent for small k.Averaging the energy modulation across the disc gives (see e.g.[16,17] with modified Bessel functions I 1 (x), K 1 (x).As k → ∞, we find h(k) → iA sc /k, reproducing the result for the infinite sheet (Eq.22).However, as k → 0, we find h(k) → 0, and we expect weaker noise suppression.

C. Space Charge Simulation
To check our analytical result, we simulate the interaction between particles in a 1D code.We load N particles randomly within a bunch length L, with initial energy spread, σ η .A particle at location z 0 interacts with all particles within the range z 0 − L h < z < z 0 + L h , and we choose the interaction distance L h so that L L h a/γ.To avoid edge effects from a finite bunch, we enforce periodic boundary conditions on the interaction.Following the interaction, the longitudinal positions shift according to ẑ = z + R 56 η, where the relative energy η is solely determined by the interactions of the first stage.We can then calculate the noise factor (or equivalently the FFT) of the resulting distribution, though even by eye it is apparent we have suppressed high frequency noise (Fig. 4).In the limit of a cold beam, the 1D space charge interaction results in regularly spaced particles, each separated by the local inverse density, 1/n 0 (Fig. 5).
We check the analytical solution (Eqs.13, 24) against the simulations in Fig. 6.For all space charge simulations, units of length are normalized to the sheet radius, a, and for now we assume zero initial energy spread, σ η = 0. 5. Longitudinal distribution of particles in simulation before (×) and after ( * ) the noise suppression process.For ση 1/R56n0 and a 1D beam, it is possible to show that the initially uncorrelated distribution gives way to a regularly spaced beam with inter-particle spacing 1/n0 (see appendix, Section B).The regular structure amplifies bunching at very high frequencies, k = 2πn0 and its harmonics, while suppressing F (k) at frequencies below 2πn0.

D. Validity of 1D model
Throughout the paper we use a 1D model of sheet particles (sheets distributed with random longitudinal positions), so we would like to check that the resulting interaction, Eq. 24, is a reasonable approximation of a 3D distribution of particles.We may look to Ref. [15], which studies the difference between 1D and 3D models of longitudinal space charge in the high frequency limit.Though the 1D and 3D distributions of longitudinal fields diverge at high frequency (see Eqs. 9,11-13 from Ref. [15]), we find that when averaged transversely, the two models give approximately equal results.
The assumption of rigid 1D sheets may also overestimate the noise suppression.Past work on noise suppression resulting from plasma oscillations has found that 3D models lead to weaker noise suppression [10,11].In our 1D model we assume a rigid sheet of charge that moves uniformly due to the average longitudinal field, whereas in reality each particle moves independently.To check the validity of our 1D model, we have written a 3D version of the space charge simulation.We confirm the existence of noise suppression for Υ = 1, but with somewhat weaker level of suppression.The 3D theory and simulations will be published elsewhere.Simulation, n 0 a=10 3 ,Y=1 Analytical Estimate (Weak Limit) Shot Noise Level FIG. 6.A comparison of simulation and analytical results shows noise suppression as a function of frequency.With Υ ≈ 1 at high frequency, we find strong suppression.At low frequencies (k 2π/a), we no longer have hu(k) ∝ 1/k, so suppression is weaker for the given parameters.

A. Undulator Radiation Interaction
As a second example, we consider the case of a beam traveling through an undulator.In the 1D limit, we can write down a simple, closed form solution for the interaction due to a helical undulator [18], providing a convenient system for studying noise suppression.For this reason, we neglect the space charge component in the following analysis, though we will see that in the absence of an amplifier [12] the space charge effect is generally dominant.We then find the undulator interaction (see Section A 1 of the appendix for a derivation) with definition undulator strength parameter, K, length, L u , period, λ u , and resonant wavelength, λ 0 .This 1D expression is valid in the limit with a the transverse beam size (see appendix, Eq.A20).

B. Undulator Fourier Transform
From Eq. 13, noise suppression originates from the imaginary component of the Fourier transform.For the undulator case, with N u assumed to be an integer.At high frequencies (m 1), we neglect the second term, and find a purely imaginary FT hu As in the space charge case, we use Eq. 14 to define the suppression parameter Υ u = −A u n 0 R 56 .In general, hu (k) is not purely imaginary, as stipulated in Eq. 13.However, at high frequencies, the undulator interaction looks like a step function (with the purely imaginary Fourier transform in Eq. 30), and the physical picture in Fig. 2 applies here as well.Again, Υ u has no k dependence, so we expect broadband suppression.At low frequencies, the approximation in Eq. 30 fails and the Fourier transform will be complex.If we take the limit of m → 1, then from Eq. 29 we find which is approximately real.We then find F (k) ∼ 1 + | hu (k 0 )| 2 and consequently expect bunching to increase at low frequencies.Note that | hu (k 0 )| = (N u π/2)Υ u , so for N u 1, we can expect an enhancement of ∼ N 2 u π 2 /4 at the fundamental when Υ = 1.
It is interesting to note that at high frequencies, the undulator interaction is strictly weaker than space charge (Eq.26 vs. Eq.23).Because the interactions have opposite sign, the undulator would only act to dampen the noise suppression from space charge.

C. Undulator Simulation
To check our analytical result, we again run the simulation code but with the undulator interaction (Eq.25) instead of space charge.We load N particles randomly within a length L N u λ 0 , and for the undulator case a particle at location z 0 interacts with all particles within the range z 0 − N u λ 0 < z < z 0 .
The simulations confirm both the analytical solution (Eq.13, valid for Γ 2 1) and the numerical integral (Eq.17).In all undulator simulations, we normalize units of length to the resonant wavelength, λ 0 , and we assume zero initial energy spread, σ η = 0. (In the appendix, we consider the effects of initial energy spread and energy modulation to the beam.)

D. Undulator Numerical Integration
While we already know the noise factor in the weak-interaction limit from Eq. 14, we would like to calculate Γ 2 explicitly to evaluate the numerical integral.For Υ ≈ 1, we find the weak interaction limit is equivalent to n 0 k 2 L u .While the weak approximation is valid for many realistic examples, to facilitate simulations we use low particle numbers, where the approximation fails.For that reason, we use the numerical integration, Eq. 17, to check our simulations without the assumption of weak interaction.
Plugging the undulator interaction into Eq. 10 yields (see appendix, Section A 2) Plugging into Eq.16 gives constant term for the undulator interaction and then from Eq. 17 we find which can be integrated directly.Simulations for the case of N u = 1 show good agreement with both the analytical result, Eqs. 13 and 29, and the numerical integration of Eq. 34, though as expected the analytical result fails for n 0 ∼ k 2 L u (Figs. 7, 8).For a case with a longer undulator (N u = 10), the numerical integration is essential for comparison with simulations (Fig. 9).At this point we can also explicitly confirm the result from Section III C by plugging h(k) and Γ Simulation, N u =1,n 0 λ 0 =10 3 ,Y=1 Analytical Estimate (Weak limit) Numerical Integral Shot Noise Level FIG. 7. A comparison of simulation, analytical result and numerical integral shows noise suppression at high frequency for Υ = 1.At low frequencies (m ∼ 1), we find hu(k) is approximately real (Eq.31), and bunching increases to  1 is valid, the noise scales as (1 − Υ) 2 .We have chosen m so that hu(k) is approximately imaginary.

VI. EXAMPLE PARAMETERS
Though the focus of this paper is strictly theoretical, we calculate the interaction strength for SLAC's Next Linear Collider Test Facility (NLCTA) to illustrate the scale of parameters involved.For the case of space charge over a length of L a ∼ 10m with beam cross section S ∼ 10 −6 m 2 and energy γ ∼ 100MeV, we find A sc ≡ 4πreLa Sγ ≈ 2 × 10 −9 .A beam of 20A (n 0 = 4 × 10 11 m −1 ), then needs R 56 ∼ 2mm to produce Υ = 1.We note that we are within the 1D limit even for optical wavelengths (k 0 σ/γ > ∼ 25).
For the undulator radiation to dominate over the space charge interaction, we may use an amplifier, as proposed by Litvinenko [12].The increase in the interaction strength also has the benefit of decreasing the required dispersion, R 56 , allowing for larger energy spreads and higher frequency suppression.However, the larger modulation may increase the beam energy spread (see appendix, Section C).

VII. CONCLUSION
We present a longitudinal 1D model of shot noise suppression for a simplified system of an interaction region followed by a dispersive region.In the limit of small energy spread (|kR 56 σ η | 1), interactions with primarily imaginary Fourier transforms can suppress the noise factor below the shot noise level.We work out the specific cases of undulator and space charge interactions, and confirm both results with a 1D simulation.We note that a wide range of imaginary impedances (e.g.linac wakefields) may also reduce shot noise.In the 1D limit with small energy spread, the suppression process may amplify bunching at very high frequencies near the inter particle spacing, 1/n 0 .

VIII. ACKNOWLEDGMENTS
We would like to thank K.J. Kim and R. Lindberg for helpful discussions.This work was supported by U.S. DOE Contract No. DE-AC02-76SF00515.In this appendix we will derive Eq. 25 for the interaction of two slices of a bunch separated by distance ζ during passage through a helical undulator and obtain an applicability condition for our 1D approximation.For now, we only consider the transverse field, E ⊥ ; the contribution from the longitudinal space charge field, E z , is given in Section IV, with γ replaced by γ z = γ/ 1 + K 2 /2 due to the presence of the undulator [19].
Our derivation is based on the paraxial approximation for the field of a relativistic particle from Ref. [20].The Fourier component (indicated by hat) of the field of a point charge q (which we call a source charge) at a point with coordinates x, y, z is given by the following formula (see Eq. 28 in [20]) where and H is the step-function.In these equations v ⊥ (z) is the transverse component of particle's velocity as a function of coordinate z, x 0 (z) and y 0 (z) define the particle trajectory, s(z) is the length of the trajectory as a function of z, and v is the absolute value of particle velocity that is assumed constant.The vector a(z ) is with x and y the unit vectors in corresponding directions.The step function under the integral (Eq.A1) is missing in Eq. 28 of [20]-a mistake that was corrected by the authors in a later publication [19].
Let us consider a helical undulator of length L u .Inside the undulator, 0 < z < L u , the transverse velocity of the particle and its orbit are where K is the undulator parameter, γ is the Lorentz factor and k u = 2π/λ u with λ u the undulator period.Note that since the transverse velocity of the source charge is zero outside of the undulator, the integration over z in Eq.A1 is actually limited to the interval 0 < z < L u .
Let us now consider a test particle of charge e travelling in front of the source particle on a parallel trajectory shifted in the transverse direction by vector X x + Y y in such a way that it passes through each point z earlier than the source particle by time T > 0. The current density associated with the test particle is where we've defined the initial distance between particles, ∆z k = z k+1 − z k , and assumed that the average spacing, ∆z , is small to make the Taylor expansion in the second step.(Specifically, we assume 1, everywhere except at the step function.)Hence the energy difference between the two neighboring particles is To compute the last sum, we replace the summation by an integration (assuming, as before, a uniform longitudinal distribution of particles in the beam).Skipping over the region where we've explicitly assumed there are no particles, for a longitudinally uniform beam we find where we've approximated h(∆z ± k ) ≈ h(0 ± ).For an interaction with a step function at The energy difference, ∆E k+1 − ∆E k , depends linearly on the initial distance between the particles (Fig. 11).The relative modulation between neighboring particles, E k+1 − E k , is proportional to the initial distance between the particles (sheets), ∆z.Simulation is for the undulator interaction, with Υ = 1, n0λ0 = 10 3 .Particles that are closer (farther) than the inter-particle spacing, ∆z < 1/n0, lose (gain) energy relative to the previous particle, and move away (closer) in negative dispersion.13.Energy modulation, ∆η , induced by the interaction (Eq.C3) is compared to the result of repeated simulations for the undulator case.The final energy spread is given as a function of the particle per wavelength, n0λ0, for Nu = 1 and 10, and initial energy spread ση = 0.

Energy Spread for FEL
Our goal is to create a quiet beam, so we would like to consider the extent to which reducing shot noise will amplify energy noise.For example, FELs require energy spreads smaller than the Pierce parameter, ρ, giving ∆η 10 −3 for current XFEL designs [22].
Quiet beams may be useful for controlling FEL start-up, which is driven by noise, F (k), for SASE FELs, and from an external radiation field for seeded FELs.However, there is also a contribution to the FEL start up from the energy noise [23], We note that F η (k) scales as η2 , which is always small.However, if ηj (z) is longitudinally periodic (as can be seen in Fig. 12 for k = k 0 ), F η (k) will also scale as the number of particles, N , which is generally very large.To claim a quiet start up for an FEL, we must ensure that F η (k) F (k).
FIG.1.Schematic of our model system.Starting with an initial electron distribution function, f0(z, η), the interaction and dispersive regions produce a final distribution function, f f (ẑ, η).The dispersion may be positive or negative.

2 aFIG. 3 .
FIG.3.Space charge from a source sheet produces a change in energy (hsc) in a test sheet located at a distance ζ.We calculate the energy modulation to the test sheet by averaging over the entire sheet (solid blue curve).Though Ez (and thus hsc) is not constant everywhere in the test sheet, we note that there is relatively little variation near the center of the sheet, as can be seen from hsc evaluated at radius r = 0 (dashed green line) and r = a/2 (dotted red line).
FIG.8.A closeup of Fig.7shows agreement with the analytical expression starts to fail for m > ∼ 2, but the numerical integral matches well everywhere.
Appendix A: Derivations for Noise Suppression for the Undulator Interaction 1. Helical Undulator Interaction in the 1D Limit FIG. 11.The relative modulation between neighboring particles, E k+1 − E k , is proportional to the initial distance between the particles (sheets), ∆z.Simulation is for the undulator interaction, with Υ = 1, n0λ0 = 10 3 .Particles that are closer (farther) than the inter-particle spacing, ∆z < 1/n0, lose (gain) energy relative to the previous particle, and move away (closer) in negative dispersion.

Following 4 FinalFIG. 12 .
FIG.13.Energy modulation, ∆η , induced by the interaction (Eq.C3) is compared to the result of repeated simulations for the undulator case.The final energy spread is given as a function of the particle per wavelength, n0λ0, for Nu = 1 and 10, and initial energy spread ση = 0.