SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION DOI: /NPHYS2443 Beating the shot-noise limit Supplementary Materials Append. S1. Coulomb expansion rate of bunches with excess density of charged particles Here we show simulations of expansion due to the space charge force of a sphere and of a disk filled up with randomly distributed point charge sample particles. The simulation was carried out using General Particle Tracing code (GPT) of Pulsar Physics. The simulation of the sphere was done with ΔN=5000 particles of charge Q=100 pc and initial sphere diameter d=2 mm. The simulation of the disk was done with the same parameters where d=1 mm is the width of the disk. The simulation shows that the sphere expands by a factor x1.3, and the disk expands by a factor x2 in a time of quarter plasma oscillation calculated for the initial density. Remarkably, the same results are obtained independently of the choice of ΔN, Q, and d (see supplementary video 1, supplementary video 2). Append. S2. Homogenization and noise suppression of an electron beam in free drift Here we show the homogenization effect of the beam electron distribution as viewed in the beam reference frame for the case of uniform beam flow. The lower figure shows the development of the computed current-noise reduction factor along the drift length L. The simulation was made with GPT for the case of e-beam drift with parameters: E = 110 MeV, Q = 0.2 nc, σ = 200 μm, L=18 m. The simulation was carried out with 250,000 sample particles. The top figure shows the charge density development during the drift time t=l/c=π/2ω p =0.6 ns as viewed in the beam reference frame (see supplementary video 3). Append. S3. Interpretation of the experimental data The OTR measurements were carried out at a fixed point (CTR-1) 6.5 m away from the LINAC exit, right after viewers YAG3, YAG4 and the quads of Triplet 2. Keeping the beam spot sizes on viewer YAG4 wide in all experiments ( ) made it possible to measure similar OTR image patterns on the CCD screen in all experiments (see Fig. S1). The beam dimensions on YAG4 and CTR-1 did not change significantly with the variation of bunch charge from 200 pc to 500 pc. However, when the beam energy was changed from 50 MeV to 70 MeV we needed to change the quad settings in order to keep similar spot dimensions, all well within the CCD sensors screen. The imaged OTR spot pattern displayed a smooth nearly Gaussian shape (see Fig 1), similar to the YAG screen patterns. No speckled COTR patterns were observed. The camera traces in Fig. S1 show that the OTR signal levels were well below saturation, but in all cases still much higher than the noise level. Similar OTR measurements were made for reference at CTR-0, right after the LINAC exit. The data of these measurements, displayed in Fig. 3 show sub-linear scaling of the OTR signal at CTR1 and linear scaling at the reference point CTR-0. These are interpreted as evidence for noise suppression due to the theoretically predicted process of collective microdynamics in the beam during its transport in the long free drift section. NATURE PHYSICS 1

2 Fig. S1: OTR image and axis intensity profiles. Photographed from CTR-1 with a macro lens of 1:1 magnification. In normal applications of OTR diagnostics the camera is focused to image the screen. If the beam dimensions are large relative to (which is the case in our experiment) then the imaged OTR pattern on the camera sensors replicates the beam current distribution integrated over the beam bunch duration (which is much shorter than the CCD integration time). Below saturation, the electronic signal from the camera CCD pixels is a linear convolution of the spectral OTR radiant intensity incident on the camera sensors and the spectral response coefficient of the (Basler) camera model sca-1400 used: ( ) ( ) ( ) ( ) ( ) ( ) where ( ) is the electron beam transverse current distribution, normalized to 1, η(ω) is the CCD screen effective electronic charge response per unit radiation power, which is proportional to the quantum efficiency of the sensors, and ( ) is the field of the virtual source of a single electron OTR emitter [S1]. Eq. S1 is valid under the assumption that all OTR photons that are emitted from the OTR screen within the spectral response range of the CCD sensors arrive into the camera. Namely, that the image on the CCD sensors is not distorted or limited by the camera aperture and by the optics transmission (unity MTF). The lens transmission in the camera sensitivity range (0.4-1 µm) is fairly flat, and the opening angle of the 100 mm diameter macro lens, used at 1:1 magnification, was about 1 radian, much larger than the 4/ opening angle of the OTR radiation lobe [S2] (and also larger than a transverse coherence diffraction angle /2 x ). The imaged spot dimensions on the sensors screen (Fig. S1) were kept sufficiently smaller than the CCD chip (11mm diagonal) to capture the entire imaged spot. The current noise measurement was based on integration of the CCD signal distribution over all pixels (Eq. S1). This parameter is therefore proportional to the average current noise ( ) in the beam within the 5 ps bunch duration and within the spectral response range of the CCD sensors. In the absence of collective interaction, this integrated signal should scale linearly with the beam charge Q b. The reference OTR measurement data at CTR-0 follow this scaling law (Fig. 3), but the measurement data at CTR-1 displays sub-linear scaling, and indicates current shot noise suppression. The error of the camera integrated signal measurement 3%, was determined from the variance in the value of the signal due to pulse to pulse variation,

3 measured repeatably while keeping all beam control parameters fixed. The charge measurement error is 1%. The error bars in Fig. 3 represent these errors. Several precautions were taken to assure that the measured 20-30% relative suppression of the current noise is immune from systematic errors. The measured OTR signal in the beam charge range pc was well below saturation of the CCD sensors. The dark current, including the small contribution of stray X-Ray photons that penetrated the camera lead shielding, was subtracted from the measured signals of the frame grabber before integration over all pixels. This dark current subtraction was necessary, since at low beam-charge levels (down to 200 pc) the dark current was not negligible any more. The bunch charge was measured at entrance using a Faraday cup with 1% accuracy. The e-beam size was less than 2 x =3 mm along the entire transport line. No significant charge loss is expected in the straight-line transport through the 34 mm diameter aperture vacuum pipe. The temporal distribution of the e-beam charge was measured by a standard technique of off-phase acceleration and energy spectrometer [S3], and is shown in Fig. S2 for different levels of bunch charge. The pulse shape measurements show that the pulse duration did not change significantly, and was about 5 ps long for bunch charges in the pc range. The pulse shape has a fairly flat top part, but certainly its variation would introduce inaccuracy into a modeling attempt of noise suppression that is based on a coasting beam uniform current. For this reason, and because the noise suppression and OTR emission are both wide frequency band processes, and weakly dependent on current variation, and since the short pulse radiation is integrated within the camera s response time, the suppression data in Fig 3 is presented in terms of the beam charge and not its current. Therefore, it only indicates the average current noise suppression during the entire beam pulse and the corresponding suppression of the total OTR radiation energy emission due to the charge homogenization effect. Other difficulties that limit exact modeling of the noise suppression rate are possible development of transverse coherence in the beam and excitation of higher order Langmuir plasma wave modes. Due to these limitations and the substantial variation of the beam dimensions along the drift transport line, the simple uniform beam model of Eq. 4 does not suffice to describe the noise suppression rate. Rather, we explain in the next Appendix S4 the qualitative features of the observed relative suppression rate and its scaling only through a model computation case, based on solution of the varying beam parameters equations 3.

4 Fig. S2: Temporal pulse profile of the e-beam bunch at different bunch charge levels. Pulse duration was approximately 5 ps for all experiments. Alternative models have been considered for excitation of microbunching and COTR effects relating them to processes in the photocathode gun and early acceleration stages, Note that our modeling of the collective microdynamic process starts at the drift section right after the beam acceleration. The reference OTR measurement at CTR-0 (Fig 3) justifies this assumption. However, there is interest to present here some discussion on possible beam microdynamic processes in the gun and the accelerator, and explain why they can be ignored. Several models for micro-dynamic processes at the early stages of the beam emission from the photocathode gun have been considered. In SCSS, microbunching and COTR effects have not been observed [S4]. Since the beam in this facility is generated with a thermionic cathode, it has been suggested that the optical microbunching effects in photocathode gun injectors originate from non-uniformities in the laser illumination or in the photoelectric quantum efficiency across the cathode surface. Another near-cathode known effect, which has been considered, at least in the microwave tube theory, to influence the noise development, is the space charge potential barrier present there in the space charge dominated regime of electron gun operation. We assert that whatever optical frequency microbunching processes may take place at the early stages of acceleration, they would be washed away in the higher energy acceleration stage. The process of energy spread growth in the early acceleration stages of the beam in the e-gun and the Linac [S5], and the partly independent process of collective microdynamics in these sections are not yet thoroughly understood and controlled. The electron beam is emitted from the cathode with very small ( ev order) local energy spread and kept focused with an axial magnetic field emittance-compensating coil [S6]. The beam energy spread grows up in the e-gun and the first Linac by two or three orders of magnitude due to several processes: kilovolt range transverse space charge potential depression across the beam combined with angular spread due to finite emittance and wake-field interaction, Longitudinal-Transverse equipartitioning in a space-charge dominated beam [S7] and the Boersch effect [S8] act to distribute the energy distribution across the beam. Even though the Coulomb collision time [S7, S8] is much longer than the beam transit time, these effects cause the electron beam to get partly thermalized, and when accelerated on crest, its distribution is commonly approximated at the Linac exit by a Gaussian distribution function with E ~3 KeV [S9]. Whatever are the thermalization processes and the beam energy distribution, we assert that the Coulomb microdynamic processes in the acceleration section may be neglected, and at least at optical frequencies, the beam is current shot-noise dominated when emerged at the accelerator exit. The collective microdynamic processes of microbunching (either instability or noise suppression) take place then only if the beam is then focused to a waist, and in particular when it is subsequently transported through a dispersive section. [S10, S11] A necessary condition for a current shot-noise dominated beam is, where ( ) ( ) ( ) ( ) This earlier defined parameter can be written explicitly in general in terms of the axial velocity spread:

5 ( ) ( ) In the case that the axial velocity spread is attributed to energy spread: ( )( ) ( ) where I A =17 ka is the Alfven current. Exemplary parameters I b =100 A, x =0.5 mm, =1 µm, E=3 KeV substituted in Eqs. S2, S4 result in that the parameter N 2 exceeds unity at optical wavelengths (namely ) up to acceleration energy of few MeV. Therefore any microdynamic plasma oscillation processes, that may take place at optical frequencies at the photocathode gun and the first few MeV acceleration section, would be washed away by Landau damping [S12, S13]. Coherent plasma oscillation starting from the cathode has been observed at low energies (4 MeV) only at THz frequencies in UCLA s Pegasus [S14], but optical frequency noise suppression could not be observed at these low energies. The noise dynamics in an injector Linac can be summarized then as follows: In the early acceleration stages, mostly in the photocathode gun, the beam slice energy spread grows up to a few KeV, and since at optical frequencies, any possible microdynamic (microbunching) processes in this region are damped. At the higher acceleration energies (to MeV) the energy spread gets frozen, but because of the strong inverse dependence of N 2 on (Eq. S4), one arrives to the limit, or, and at the accelerator exit velocity-current plasma oscillation dynamics can start taking place at optical frequencies without damping. At this same condition (see Eq. S2) the beam current shot-noise dominates over velocity noise (as generally accepted [S15]), and therefore current noise suppression effect can take place in a subsequent drift section. If axial velocity spread due to emittance and beam focusing (Eq. S3) does not break the condition, then a sufficient condition for microbunching dynamics and noise suppression to take place is space-charge dominated transport of the beam through a waist in a subsequent drift section [S10]. When this is satisfied, the interaction takes place in a nearly uniform beam, where Eq. 4 is valid. With the velocity noise (second term in Eq. 4) being negligible (a current shot noise dominated beam), significant noise suppression would be attainable after a quarter plasma wavelength drift. We stress that if the beam emittance is not negligible, and the beam is subjected to tight focusing after the acceleration (as in the present experimental setup), large angular spread and consequently large longitudinal velocity spread are produced in the drift section, and the inequality may not be strong. In this case, a reduced effect of noise suppression should be expected. This is exemplified in the model computation of the next section, and presented as a plausible explanation for the moderate value of experimentally measured suppression effect. Append S4: Model computation of collective microdynamics in varying cross-section beam transport As explained above, at the exit of the Linac, the beam is current-noise dominated, and. However, due to finite emittance, the axial velocity spread becomes significant as the beam is focused and consequently N 2 (Eq. S3) is no longer negligible. Furthermore, at tight focusing the uniform beam expression (Eq. 4) cannot be used and in order to calculate the noise suppression, one should solve differential equations (Eq. 3) for variable beam parameters. There may be various reasons why in the current experiment the measured current noise suppression was modest relative to the simple model prediction.

6 However, in this section we show through a model case computational solution of Eq. 3, that the axially varying beam cross section and the excess axial velocity spread due to angular spread can account for a substantial moderation of the noise suppression effect that was realized in the experiment. This example also explains some parameter scaling features of the measurement data. Fig. S3: Beam envelope dimensions σ x (z), σ y (z) along the collective interaction region computed using GPT for the experimental beam parameters at E=50,70 MeV We turn the generic coupled current and kinetic voltage equations (Eq. 3) into explicit equations in z by substituting ( ) ( ) ( ): ( ) and ( ) where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) This set of equations can be solved explicitly numerically if ( ) ( ) ( ) is given, and the initial conditions ( ) ( ) are specified, and the solution at the end of the drift section L is a linear combination of the initial conditions: ( ) ( ) ( ) ( ) ( ) Assuming that the current modulation noise and the focusing enhanced velocity noise are uncorrelated at z=0, we set ( ) ( ) ( ) ( ) ( ) ( )

7 ( ) ( ) ( ) ( ) ( ) To compute the noise evolution for the parameters of the reported experiment it is necessary to estimate the cross section dimensions σ x (z), σ y (z) along the interaction length. This was done based on the measurements on screens YAG-1 to YAG-4, the recorded quad excitation parameters, and performing full 3-D simulation (with space-charge) using GPT (see Fig. S3). The beam axial velocity spread due to energy spread ( ) is small, and its effect on the initial velocity noise is negligible (Eq. S4). We assume that the initial velocity noise is determined by the standard deviation of the axial velocity spread (Eq. S9), which was calculated for each quad setting from the angular spread standard deviations due to the focusing and emittance (ε x, ε y ~ 2-5 µ [S3]): ( ). The coefficients A(z=L), B(z=L) were computed by iterative integration of the coupled linear differential equation (Eq. S5) with the initial conditions (Eq. S8),(Eq. S9). In Figures S4 and S5 we display the results of the model computation of the variable parameters differential equations. The curves show the noise suppression as function of length and of the beam current for the two extreme energy cases E=50 MeV, 70 Mev. The relative suppression factor in the range Q b = nc at the drift section exit L=6.5 m is displayed in Fig. S6 for the two beam-energy examples. Fig. S4: Noise suppression along the collective interaction region as calculated from solution of the coupled differential equations with σ x (z), σ y (z) of 70 MeV example of Fig. S3.

8 Fig. S5: Noise suppression along the collective interaction region as calculated from solution of the coupled differential equations with σ x (z), σ y (z) of 50 MeV example of Fig. S3. Fig. S6: Relative noise suppression rate computed from Eq. S5 for E=50 MeV, 70 MeV compared to the experimental suppression results. Fig. S6 can only be regarded as a model explanation for the reduced relative noise suppression effect observed in the experiment. As mentioned, there can be additional factors affecting the suppression rate. It is noted that the computed curves display significantly larger relative suppression in the range Q b = nc than the experimental curves, but in either case there is only little dependence on the acceleration energy. In the simple uniform beam model of Eq. 4 one would expect dependence of the noise suppression factor on the beam energy due to the 3/2 power dependence of the plasma frequency on γ, which corresponds to smaller plasma phase accumulation at higher beam energy. However, the beam envelope had to be varied in the experiment in different beam energies, and one must keep in mind, that at the higher energy the beam focuses into a tighter waist due to the reduced space charge effect

9 (see Fig. S3). This tends then to increase the plasma frequency at the waist, where most of the microdynamic process takes place. This observation is consistent with the point of view that the beam envelope expansion and the beam charge homogenization process are in essence the same process of excess charge beam expansion (see Append. S1 above) when viewed in the beam rest frame (independent of the acceleration energy). This is consistent with the plasma phase accumulation theorem of constant plasma phase accumulation in a space charge dominated beam waist [S10]. This provides qualitative physical explanation for the weak dependence of the relative noise suppression on the beam energy in the MeV range, as depicted in both the experimental (Fig. 3) and model calculation (Fig. 6) curves. REFERENCES S1. Geloni, G. Kocharyan, V. Saldin, E. Schneidmiller, E. Yurkov, M. Theory of edge radiation. PartI: Foundations and basic applications, Nuclear Instruments and Methods in Physics Research A605 (2009) S2. Ginzburg, V.L. Transition radiation and transition scattering. Physica Scripta T, 2:182 (1982). S3. S4. Ferrario, M. Shintake, T. High performance electron injectors, Rev. of Accelerator Science and Technology, Vol. 3 (2010) 221 S5. Moody, J. T. Musumeci, P. Gutierrez, M. S. Rosenzweig, J. B. and Scoby, C. M. Longitudinal phase space characterization of the blow-out regime of rf photoinjector operation, Phys. Rev. ST-AB 12, (2009) S6. Serafini, L. Rosenzweig, J. Envelope analysis of intense relativistic quasilaminar beams in rf photoinjectors: A theory of emittance compensation, Phys.Rev. E 55, 7565 (1997) S7. Zou, Y. Cui, Y. Reiser, M. and O Shea, P. G. Observation of the Anomalous Increase of the Longitudinal Energy Spread in a Space-Charge-Dominated Electron Beam, Phys. Rev. Lett. 94, , (2005). S8. Reiser, M. Theory and design of charged particle beams, Weinheim: Wiley-VCH, (2008). S9. Ratner, D. Chao, A. Huang, Z. Three-dimensional analysis of longitudinal space charge microbunching starting from shot noise, Proceedings of FEL08, TUPPH041, Gyeongju, Korea S10. Gover, A. Dyunin, E. Collective interaction control of optical frequency shotnoise in charged particle beams. Phys. Rev. Lett., 102, (2009). S11. Gover, A. Dyunin, E. Duchovni, T. Nause, A. Collective Microdynamics and Noise Suppression in Dispersive Electron Beam Transport. Phys. of Plasmas, 18, (2011). S12. Marinelli, A. Hemsing, E. Rosenzweig, J.B. Three dimensional analysis of longitudinal plasma oscillations in a thermal relativistic electron beam. Phys. Of Plasmas 18, (2011). S13. Jackson, J.D. J. of Nuclear Energy, Part C Plasma Physics, 1 (1960) 171. S14. Musumeci, P. Li, R. K. Marinelli, A. Nonlinear Longitudinal Space Charge Oscillations in Relativistic Electron Beams. Phys. Rev. Lett. 106, (2011). S15. Saldin, E.L. Schneidmiller, E.A. Yurkov, M.V. The physics of free electron lasers, Berlin : Springer, 2000.