Abstract
The current shotnoise of an electron beam is proportional to its average current and the frequency bandwidth. This is a consequence of the Poisson distribution statistics of particles emitted at random from any source. Here we demonstrate noise suppression below the shotnoise limit in optical frequencies for relativistic electron beams. This process is made possible by collective Coulomb interaction between the electrons of a cold intense beam during beam drift^{1}. The effect was demonstrated by measuring a reduction in optical transition radiation power per unit of electronbeam pulse charge. This finding indicates that the beam charge homogenizes owing to the collective interaction, and its distribution becomes subPoissonian. The spontaneous radiation emission from such a beam would also be suppressed (Dicke’s subradiance^{2}). Therefore, the incoherent spontaneous radiation power of any electronbeam radiation source (such as freeelectron lasers^{3,4}) can be suppressed, and the classical coherence limits^{5} of seedinjected freeelectron lasers^{6} may be surpassed.
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Main
Current shotnoise suppression in an electron beam in the optical frequency regime is an effect of particle selfordering and charge homogenization on the scale of optical wavelengths^{7}, which statistically corresponds to the exhibition of subPoissonian electron number statistics (similarly to photons in squeezed light^{8}). A dispute over the feasibility of this effect at optical frequencies is resolved in the experiment reported here. We have evidence for electronbeam noise suppression from measurement of the optical transition radiation (OTR) power emitted by a beam on incidence on a metal screen after passing through a drift section. The OTR emission is proportional to the current shotnoise of the incident beam.
In a randomly distributed stream of particles that satisfies Poisson statistics, the variance of the number of particles that pass through any crosssection at any time period T is equal to the number of particles N_{T} that pass this crosssection during the same time T, averaged over different times of measurements. Consequently, the current fluctuation is . When formally calculated, the average beam shotnoise spectral power is:
Here I(t) is the beam current modulation, and the Fourier transform is defined as .
In radiofrequency linear accelerators (RFLINACs) it is usually assumed that collective interparticle interaction is negligible during beam acceleration and transport, and the shotnoise limit (1) applies. However, with recent technological advances in radiofrequency accelerators, and in particular the development of photocathode guns^{9}, highquality cold and intense accelerated electron beams are available, and the neglect of collective microdynamic interaction effects in the transport of such electron beams is no longer justified. Effects of coherent OTR emission and superlinear scaling with I_{b} of the OTR emission intensity were observed in the LCLS (LINAC Coherence Light Source) injector^{10} at SLAC and in other laboratories^{11}. These effects, originally referred to as unexpected physics^{12}, are now clearly recognized as the result of collective Coulomb microdynamic interaction and establishment of phase correlation between the electrons in the beam. In these cases, however, the collective interaction led to shotnoise and OTR power enhancement (gain), and not to suppression. Collective effects were shown to be responsible also for beam instabilities (microbunching instability) that were observed in dispersive electronbeam transport elements^{13,14}. Collective microdynamic evolution of 1 THz coherent singlefrequency current modulation was recently reported^{15}, but no stochastic optical noise suppression effect could be observed.
Noise gain due to collective interaction has been demonstrated in numerous laboratories, but the notion of beam noise suppression at optical frequencies^{1} has been controversial (although analogous effects were known in nonrelativistic microwave tubes^{16}). To explain the physics of the noise suppression, we point out that in the electronbeam frame of reference the effect of noise suppression appears as charge density homogenization. The simple argument that follows shows that the spacecharge force, which is directed to expand higher density charge bunches, has a dominant effect over the randomly directed Coulomb repulsion force between the particles (Fig. 1).
Assume that in some regions of the beam, there is higher particledensity bunching. Encompassing such a bunch within a sphere of diameter d′, the excess charge in this sphere is eΔN′, and the potential energy of an electron on the surface of the sphere is ɛ_{sc} = e^{2}ΔN′/2πɛ_{0}d′. This potential energy turns in time into kinetic energy of electrons, transferred in the direction of bunch expansion (homogenization). At the same time, the electron also possesses an average potential energy due to the Coulomb interaction with neighbour electrons at an average distance n′_{0}^{−1/3} (n′_{0} is the average density in the beam frame). This potential energy ɛ_{Coul} = e^{2}/4πɛ_{0}n′_{0}^{−1/3} turns into kinetic energy of electrons that are accelerated in random directions. In a randomly distributed electron beam that satisfies Poisson statistics ΔN′ = N′^{1/2}, where N′ = πd′^{3}n′_{0}/6 is the average number of electrons in such a sphere. Therefore, the condition for domination of the directed spacecharge energy over the random energy is
We conclude that a random cold beam of density n′_{0} always tends to homogenize in any spatial scale larger than the average interparticle distance: d′>n′_{0}^{−1/3}.
The expansion trend of a dense bunch would clearly tend to homogenize the charge distribution of an initially cold electronbeam plasma. The timescale for the homogenization process can be connected only to the plasma frequency—the fundamental collective excitation in the beam, namely: ∼π/ω′_{p0}, where ω′_{p0} = (e^{2}n′_{0}/m ɛ_{0})^{1/2}. Interestingly, this is the characteristic expansion time of a bunch of charged particles as can be verified by calculating or simulating the expansion time of a sphere or disc of charged particles (see Supplementary Appendices S1 and S2).
In electronbeam transport under appreciable spacecharge conditions, the microdynamic noise evolution process may be viewed as the stochastic oscillations of Langmuir plasma waves^{1}. In the linear regime, the evolution of longitudinal current and velocity modulations of a beam of average current I_{b}, velocity β c and energy E = (γ−1)m c^{2}, can be described in the laboratory frame by^{17}:
where (ǐ)(ω) = (Ǐ)(ω)e^{iωz/βc}, ()(ω) = ()(ω)e^{iωz/βc},(Ǐ)(ω),()(ω) are the respective Fourier components of the beam current and kineticvoltage modulations. The kineticvoltage modulation is related to energy and longitudinal velocity modulations: () = −(m c^{2}/e)() = −(m c^{2}/e)γ^{3}β(); is the accumulated plasma phase; W(z) = r_{p}^{2}/(ω A_{e}θ_{pr}ɛ_{0}) is the beam waveimpedance; A_{e} is the effective beam crosssection area, θ_{pr} = r_{p}ω_{pl}/β c is the plasma wavenumber of the Langmuir mode; r_{p}<1 is the plasma reduction factor; ω_{pl} = ω_{p0}/γ^{3/2} is the longitudinal plasma frequency in the laboratory frame.
The singlefrequency Langmuir plasma wave model expression (3) can be solved straightforwardly in the case of uniform drift transport. After employing an averaging process, this results in a simple expression for the spectral parameters of stochastic current and velocity fluctuations (noise) in the beam assuming that they are initially uncorrelated^{1,17}:
The beam current noise evolution is affected by the initial axial velocity noise through the parameter , where k_{D} = ω_{pl}/c σ_{β} is the Debye wavenumber and c σ_{β} is the axial velocity spread. Equation (4) suggests that current noise suppression is possible if the beam is initially cold— N^{2}<1, and if plasma phase accumulation of ϕ_{p}(L)∼π/2 is feasible. This condition also assures that Landau damping is negligible^{18}.
The noise suppression demonstration experiment was conducted on the 70 MeV RFLINAC of ATF/BNL (Fig. 2). The beam current noise measurement was made by recording the OTR radiation emitted from a copper screen placed L = 6.5 m away from the accelerator exit. Keeping the camera and screen CTR1 position fixed, the practical way to control the plasma phase ϕ_{p} was to vary the beam pulse charge (200–500 pC) (as in ref. 15) and the beam energy E = (γ−1)m c^{2} (50–70 MeV). The quad settings were readjusted for each beam acceleration energy, and the beam spot sizes σ_{x},σ_{y} were measured at four points (YAG1–YAG4) along the beam transport line (Fig. 2). For reference, the OTR signal was measured independently also on a screen CTR0 preceding the drift section.
The measured signal S_{OTR} was the integrated charge accumulated in all of the pixels of a CCD (chargecoupled device) camera exposed to the OTR emission on the incidence of single microbunch electronbeam pulses on the screen. The measured data of S_{OTR}/Q_{b} in CTR0 and CTR1 are shown in Fig. 3 as a function of the varied bunch charge Q_{b} in the range 200–500 pC. The pulse duration in all experiments remained approximately the same (5 pS) corresponding to an average current of 40–100 A (see Supplementary Information). In a shotnoisedominated beam, in the absence of collective microdynamics, the current noise and consequently S_{OTR} are proportional to I_{b} (1). Our measured data of S_{OTR}/Q_{b} in CTR0 lie approximately on a horizontal line, confirming the absence of collective microbunching and currentvelocity noise correlation or noise suppression before this point. On the other hand, the data measured on CTR1 exhibit a systematic drop as a function of charge (200–500 pC) for all beam energies (50–70 MeV). As the measurement conditions at the two measurement positions and at different beam energies could not be made identical, the absolute suppression level between the two points could not be determined. However, the normalized data of all measurements of noise per unit charge depicted in Fig. 3 show the scaling with charge for all beam energies. This demonstrates attainment of 20–30% relative noise suppression, and confirms the predicted effect of collective microdynamic noise suppression process in the drift section.
Beyond observation of noise suppression, interpreting the measured data and the noise suppression rate in terms of a simple theoretical model as presented by equation (4) would be very crude. For N^{2}≪1 equation (4) predicts maximum suppression by a factor N^{2} at ϕ_{p} = π/2, but this is true only for uniform beam transport. In the present transport configuration the beam was focused to a tight waist in the section between triplet 1 and triplet 2 (the location of a turnedoff chicane). According to a theorem of ref. 1 a plasma phase increment ϕ_{p} = π/2 is accumulated along a beam waist, but this applies only if the beam transport is fully spacecharge dominated. This is not the case for the present experiment conditions in which the beam angular spread due to emittance and beam focusing is significant, and the collective microdynamic noise suppression process is compromised. Solving the more general equation (3) under conditions of varying beam crosssection and increased angular spread (the focusing at triplet 1 at different beam energies led to rather high values of N^{2}: 0.3–0.8) shows that the noise suppression expected in a model configuration similar to the present experimental configuration is substantially smaller than anticipated in the uniform beam model (4) (see Supplementary Appendix S4). This model calculation also shows that the noticeable weak dependence of the relative suppression rate on the beam energy observed in Fig. 3 is consistent with calculations at the experimental conditions: the downscaling of ω_{pl} with γ^{3/2} is offset by the increased current density at the focused beam waist at higher beam energies. This weak energy dependence is consistent with a point of view that in the beam rest frame the beam envelope expansion and the charge homogenization effects are related^{1}. Other factors that can reduce the collective microdynamic suppression rate may be threedimensional effects and excitation of higherorder Langmuir plasma waves of different wavenumber values θ_{pr} (refs 7, 19). These and other deviations from ideal conditions can explain why the relative noise suppression effect measured in this experiment in the range of charge variation (200–500 pC) is quite modest (20–30%).
The observation of current shotnoise suppression at optical frequencies, although small, is of interest from the point of view of both fundamental physics and of applications points of view. The electronbeam shotnoise expression (1) is a direct consequence of the Poisson statistics of random particle number distribution, and therefore is widely considered to be an absolute limit. The experiment demonstrates that at least at optical frequencies this limit may be surpassed in charged particles owing to Coulomb collective interaction. The suppression of OTR emission is also a noteworthy demonstration of a fundamental physical process, as it is a vivid demonstration of spontaneous emission subradiance in Dicke’s sense^{2} in the classical limit. This subradiance is attributable to the more uniform rate of incidence of electrons on the OTR screen after the beam charge is homogenized. This same process would be expected also to suppress spontaneous and selfamplified spontaneous emission in any freeelectron radiation device, including undulators and freeelectron lasers (ref. 5), as was first suggested in ref. 20. Furthermore, microdynamic control over electronbeam noise may be also of practical use in particle beam physics, and help in controlling beam instability in the transport of intense highquality beams.
Theory predicts that much bigger factors of shotnoise suppression are possible in more favourable configurations, and schemes have been suggested for producing the noise suppression process in shorter lengths using dispersive transport^{17,21} (dispersive transport noise suppression has just been demonstrated experimentally in the LCLS (ref. 22)). We have demonstrated noise suppression at optical frequencies—four orders of magnitude higher than the previous microwave noise suppression works. Further research and beam quality improvements are required to determine the shortwavelength limits of applicability of this process^{23}, a limit that is ultimately bound at Xray wavelengths by the beam charge granularity interparticle spacing limit (2).
Methods
The experimental setup is presented in Fig. 2. The electron beam, which was generated in a photocathode by ultrafast pulses (t_{pulse} = 5 pS) of a Nd–YAG laser, was accelerated by an RFLINAC to beam energies of 70, 64, 57 and 50 MeV. It was accelerated at the oncrest phase to avoid chirped energy variation along the beam.
The noise measurements were carried out by recording the integrated OTR radiation from the beam in screen CTR1, positioned 6.5 m after the LINAC exit. The beam was allowed to drift freely between two quad triplets (QUAD1, QUAD2) that focused the beam and were readjusted for different beam energies. A reference OTR measurement was done in a separate setting (CTR0), just after the LINAC exit, showing a lack of noise suppression effect at this point.
To control the collective microdynamic process we varied the plasma phase by varying the beam charge Q_{b} in 50 equal increments (in the reference measurement 8 increments) between 0.2 and 0.5 nC in each of the four beam energy experiments. The photocathode current was varied by attenuating the incident laser beam using variableangle crossed polarizers. This method made it possible to fix the OTR screen and camera at the same position in all experiments, and enabled stable comparative data collection in all experiments. There was no change in the beam spot dimensions as the beam charge was varied in the range 200–500 pC. As the acceleration energy was varied, the beam spots on the screens were readjusted and kept small by controlling the two quad triplets.
We used a (Basler) scA1400 CCD camera equipped with a (Nikkor) macro lens (100 mm F/# = 2.8) to obtain 1:1 image magnification of the OTR Copper screen (CTR1). The screen was placed at 45° to the beam line. The current noise measurement is based on its proportionality to the integrated OTR photon number, measured by integrating all of the pixels using a frame grabber. The camera aperture opening angle, operating in screen imaging mode, was wide enough to collect the entire OTR radiation lobe (of opening angle ∼ 4/γ) in all experiments. The photographed 1:1 image of the OTR spot on the 11 mm (diagonal size) CCD chip was small enough to assure full collection of photons in the measured wavelength range. As the optical spectrum of the electronbeam noise (and the OTR photons) is quite uniform in the spectral range of the camera sensitivity (λ = 0.4–1 μm), and the integrated OTR photon number is only weakly (logarithmically) dependent on the beam energy^{24}, the proportionality factor between the measured integrated pixels charge, S_{OTR}, and the current noise power within the measured spectral range was nearly the same for all experiments.
The quad currents were varied as the beam acceleration energy was changed, to focus the beam within the drift section, and keep the beam spot well within the camera frame in all experiments. The beam spots on screens YAG1 to YAG4 were recorded. The measured spot dimensions at these points were used to evaluate the beam profile dimensions along the drift section at different acceleration energies, employing a threedimensional numerical code simulation with spacecharge effects (General Particles Tracer^{25}).
References
Gover, A. & Dyunin, E. Collective interaction control of optical frequency shotnoise in charged particle beams. Phys. Rev. Lett. 102, 154801 (2009).
Dicke, R. H. Coherence in spontaneous radiation processes. Phys. Rev. 93, 99–110 (1954).
Emma, P. et al. First lasing and operation of an ångstromwavelength freeelectron laser. Nature Photon. 4, 641–647 (2010).
Saldin, E. L., Schneidmiller, E. A. & Yurkov, M. V. The Physics of Free Electron Lasers (Springer, 1999).
Gover, A. & Dyunin, E. Coherence limits of free electron lasers. J. Quant. Electron. 46, 1511–1517 (2010).
Labat, M. et al. Highgain harmonicgeneration freeelectron laser seeded by harmonics generated in gas. Phys. Rev. Lett. 107, 224801 (2011).
Nause, A., Dyunin, E. & Gover, A. Optical frequency shot noise suppression in electron beams: 3D analysis. J. Appl. Phys 107, 103101 (2010).
Walls, D. F. Squeezed states of light. Nature 306, 141–146 (1983).
O’Shea, P. & Freund, H. Free electron lasers: Status and applications. Science 292, 1853–1858 (2001).
Loos H. et al. Observation of coherent optical transition radiation in the LCLS LINAC Proc. FEL08 THBAU01 (2008).
Lumpkin, A. et al. First observations of COTR due to a microbunched beam in the VUV at 157mm. Nucl. Instrum. Methods Phys. Res. A 528, 194–198 (2004).
Akre, R. et al. Commissioning the linac coherent light source injector. Phys. Rev. STAB 11, 030703 (2008).
Shaftan, T. & Huang, Z. Experimental characterization of a space charge induced modulation in highbrightness electron beam. Phys. Rev. STAB 7, 080702 (2004).
Huang, Z. et al. Suppression of microbunching instability in the linac coherent light source. Phys. Rev. STAB 7, 074401 (2004).
Musumeci, P., Li, R. K. & Marinelli, A. Nonlinear longitudinal space charge oscillations in relativistic electron beams. Phys. Rev. Lett. 106, 184801 (2011).
Haus, H. A. & Robinson, N. H. The minimum noise figure of microwave amplifiers. Proc. IRE 43, 981–991 (1955).
Gover, A., Dyunin, E., Duchovni, T. & Nause, A. Collective microdynamics and noise suppression in dispersive electron beam transport. Phys. Plasmas 18, 123102 (2011).
Marinelli, A., Hemsing, E. & Rosenzweig, J. B. Three dimensional analysis of longitudinal plasma oscillations in a thermal relativistic electron beam. Phys. Plasmas 18, 103105 (2011).
Venturini, M. Models of longitudinal spacecharge impedance for microbunching instability. Phys. Rev. STAB 11, 034401 (2008).
Dyunin, E. & Gover, A. The general velocity and current modulation linear transfer matrix of FEL and control over SASE power in the collective regime. Nucl. Instrum. Methods Phys. A593, 49–52 (2008).
Ratner, D., Huang, Z. & Stupakov, G. Analysis of shot noise suppression for electron beams. Phys. Rev. STAB 14, 060710 (2011).
Ratner, D. & Stupakov, G. Observation of Shot Noise suppression at optical wavelengths in a relativistic electron beam. Phys. Rev. Lett. 109, 034801 (2012).
Kim, KJ. & Lindberg, R. R. Collective and individual aspects of fluctuations in relativistic electron beams for free electron lasers, Proc. FEL2011 TUOA2 (2011).
Ginzburg, V. L. Transition radiation and transition scattering. Phys. Scr. 2, 182–191 (1982).
GPT User Manual, Pulsar Physics, Flamingostraat 24, 3582 SX Utrecht, The Netherlands. http://www.pulsar.nl/gpt/index.html.
Acknowledgements
We acknowledge advice and assistance in conducting the experiment from V. Yakimenko and the ATF team, and helpful discussions with T. Shaftan, P. Muggli, G. Andonian, Z. Schuss, A. Arie and A. Eyal. This work was supported in part by the Israel Science Foundation Grant No. 353/09 and by a Rahamimoff grant of the USIsrael BSF.
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A.G. contributed the theoretical and experimental concept. A.N. contributed to the experimental conception, designed and performed the experiment, data analysis and numerical simulations. E.D. contributed to the theoretical analysis of the concept and the experiment. M.F. contributed to performing the experiment on the ATF accelerator.
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Gover, A., Nause, A., Dyunin, E. et al. Beating the shotnoise limit. Nature Phys 8, 877–880 (2012). https://doi.org/10.1038/nphys2443
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DOI: https://doi.org/10.1038/nphys2443
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