Longitudinal phase space properties of a photoinjector beam are important in many areas of high-brightness beam applications such as bunch compression, transverse-to-longitudinal emittance exchange, and high-gain free-electron lasers. In this paper, we discuss both the rf and the space charge contributions to the uncorrelated energy spread of the beam generated from a laser-driven rf gun. We compare analytical expressions for the uncorrelated energy spread and the longitudinal emittance with numerical simulations and recent experimental results

Dowell, D.

Emma, P.

Huang, Z

Limborg-Deprey, C.

Stupakov, G.

Wu, J.

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UNCORRELATED ENERGY SPREAD AND LONGITUDINALEMITTANCE OF A PHOTOINJECTOR BEAM∗Z. Huang† , D. Dowell, P. Emma, C. Limborg-Deprey, G. Stupakov, J. WuSLAC, Stanford, CA 94309, USAAbstractLongitudinal phase space properties of a photoinjec-tor beam are important in many areas of high-brightnessbeam applications such as bunch compression, transverse-to-longitudinal emittance exchange, and high-gain free-electron lasers. In this paper, we discuss both the rf andthe space charge contributions to the uncorrelated energyspread of the beam generated from a laser-driven rf gun.We compare analytical expressions for the uncorrelated en-ergy spread and the longitudinal emittance with numericalsimulations and recent experimental results.INTRODUCTIONHigh-brightness electron beams generated by laser-driven rf guns have important applications in x-ray freeelectron lasers (FEL) and advanced accelerator and lightsource R&D. While the transverse phase space propertiesof these beams have been studied extensively, the uncor-related energy spread and the longitudinal emittance haveonly been characterized recently at a few rf gun facili-ties [1, 2, 3]. Unlike the common desire to minimizethe transverse emittance, the measured uncorrelated en-ergy spread of a photoinjector beam is in fact too smallto suppress a microbunching instability associated with ac-celeration and magnetic compression designed for short-wavelength FELs [4, 5]. To this end, a laser-based beamheater that increases the uncorrelated energy spread is de-signed in the Linac Coherent Light Source (LCLS) tocontrol the beam quality degradation due to the insta-bility [6]. Nevertheless, some other beam manipulationschemes such as transverse-to-longitudinal emittance ex-change [7] and variable-wavelength high-gain harmonicgeneration FEL [8] require very small uncorrelated energyspread and longitudinal emittance. Therefore, a general un-derstanding of these longitudinal beam parameters will beuseful for a variety of high-brightness beam applications.In this paper, we study the origins of the uncorrelatedenergy spread in an rf gun, taking into account both therf and the space charge effects. We show that the maincontribution to the uncorrelated energy spread comes fromtransverse variations of the longitudinal space charge fieldexperienced by an intense electron bunch. We also discussthe longitudinal emittance and compare analytical estima-tions with simulations and experiments.∗Work supported by US Department of Energy contract DE-AC02-76SF00515.† Email: zrh@slac.stanford.eduRF CONTRIBUTION TO THEUNCORRELATED ENERGY SPREADIt is known that the energy gain for an ultra-relativisticparticle through an axially symmetric accelerating cavity isindependent of the transverse offset of the particle. Nev-ertheless, in an rf gun electrons are accelerated from restto relativistic velocities starting in the middle of a full-cellcavity. Thus, the energy gain due to the rf acceleration candepend on the transverse offset of the electron and may giverise to the uncorrelated energy spread of the beam.To quantify the rf contribution, we use the paraxial rep-resentation of the rf accelerating field near the cavity axis(see, e.g., Ref. [9]). In MKS units, the longitudinal field isEz(r, z, t) ≈[1− r24(∂2∂z2− 1c2∂2∂t2)]Ez(0, z, t) ,(1)where r =√x2 + y2 is the radial offset of the electron,and z is the longitudinal distance from the cathode. Theradial electric field and the azimuthal magnetic field aregiven byEr ≈ −r2∂Ez(0, z, t)∂z, Bθ =r2c2∂Ez(0, z, t)∂t. (2)The electron motion is governed by the Lorentz forced(γmv)dt= e(E+ v ×B) , (3)where v = (dr/dt, dz/dt) is the velocity vector. The en-ergy gain can be calculated fromd(γmc2)dt= ev ·E . (4)We take the on-axis longitudinal field of the formEz(0, z, t) = E0E¯(z) sin(2pif0t+ φ0) , (5)where f0 is the rf frequency, E0 is the maximum rf fieldamplitude, φ0 is the launch phase of the laser pulse rel-ative to the rf zero-crossing, and E¯(z) is the dimension-less longitudinal field profile of a specific rf gun design.Equations (3) and (4) can be integrated numerically for agiven E¯(z). For instance, the LCLS injector uses an S-band 1 1/2-cell gun with f0 =2856 MHz and E¯(z) given byOmega3P simulations [10] as shown in Fig. 1. The on-axiselectron energy γmc2 and the energy deviation ∆γmc2 ofan off-axis electron that leaves the cathode at r = 1 mmare shown in Fig. 2, for E0 = 120 MV/m and φ0 = 30◦.SLAC-PUB-11240May 2005Contributed to Particle Accelerator Conference (PAC05), Knoxville, TN, May 16-20, 20050 5 10 15z HcmL-1-0.500.51E HzLFigure 1: Dimensionless longitudinal field E¯(z) as a func-tion of distance z from the cathode in the LCLS gun.0 2 4 6 8 10 12 14z HcmL123456Γmc2HMeVL0.0.51.1.52.DΓmc2HkeVLγmc2∆γmc2Figure 2: On-axis electron energy γmc2 (dashed bluecurve) and energy deviation ∆γmc2 for an off-axis elec-tron that leaves the cathode at r = 1 mm (solid red curve).The rf induced uncorrelated energy spread for an electronbeam with the transverse size about 1 mm remains small(< 1 keV) for a wide range of the initial phase φ0 that pro-duces a rapid acceleration in the gun. We will therefore ne-glect its contribution in comparison with the space chargeeffects discussed below.SPACE CHARGE CONTRIBUTION TOTHE UNCORRELATED ENERGY SPREADSpace charge is the main collective effect in the rf gun.The transverse variation of the longitudinal space chargefield can give rise to the uncorrelated energy spread. To es-timate this contribution, we follow Kim [11] who observesthat due to the typical 1/γ2-dependence of the space chargeforces, the dominant space charge contribution in the guncomes from the region where γ ≈ 1. In this approximation,the total space charge induced energy change is [11]∆γ(r, s) ≈ pi2mc2eE0 sinφ0I0IAEz(r, s) , (6)where I0 is the peak current, IA ≈ 17 kA is the Alfve´n cur-rent, s is the longitudinal coordinate relative to the bunchcenter (with the bunch head at s < 0), and the normalized-2 -1 0 1sc HpsL0510rmsenergyspreadHkeVLFigure 3: Uncorrelated rms energy spread as a function ofbunch position s/c (head at s < 0) from Parmela (blacksymbol) and from Eq. (8) (red curve).longitudinal space charge field (at γ = 1) isEz(r, s) = sσ2s∫ ∞0dξexp[− r2A22σ2x(A2+ξ) +s22σ2s(1+ξ)](1 + ξ)3/2(A2 + ξ)(7)for a three-dimensional Gaussian charge distribution witha transverse rms size σx = σy and an rms bunch lengthσs = σx/A. As expected, the longitudinal space chargefield at any bunch position s depends on its transverse offsetr. Taking rms over the transverse Gaussian distribution,the space charge induced uncorrelated energy spread as afunction of the longitudinal bunch coordinate s isσγ(s) =√〈∆γ2〉r − 〈∆γ〉2r≈pi2mc2eE0(sinφ0)σsI0IAF(sσs), (8)whereF 2(u) =u2∫ ∞0dξ1e−u2/[2(1+ξ1)](1 + ξ1)3/2∫ ∞0dξ2e−u2/[2(1+ξ2)](1 + ξ2)3/2× 13A4 + 2A2(ξ1 + ξ2) + ξ1ξ2−[∫ ∞0dξe−u2/[2(1+ξ)](1 + ξ)3/2(2A2 + ξ)]2. (9)Let us compare Eq. (8) with the uncorrelated energyspread obtained from the Parmela [12] space charge sim-ulation of the Gun Test Facility (GTF) at SLAC with thefollowing parameters: E0 = 110 MV/m, φ0 = 30◦,I0 = 130 A, and a Gaussian laser pulse with σs = 0.27mm (0.9 ps). We approximate the uniform transverselaser pulse with 1-mm radius by a Gaussian shape withσx = 1/√3 ≈ 0.58 mm to obtain the bunch aspect ratioA = σx/σs ≈ 2. As shown in Fig. 3, the uncorrelated en-ergy spread predicted from Eq. (8) agrees reasonably wellwith the simulation results obtained at the exit of the gun,indicating the transverse variation of the longitudinal spacecharge field provides the dominant contribution. Parmelasimulations also show that the uncorrelated energy spreaddoes not increase much after the rf gun (see, e.g., Fig. 8 ofRef. [13]).The uncorrelated energy spread varies along the bunchlength. For convenience, we may define an average uncor-related energy spread as(σ¯γ)unc ≡[∫ ∞−∞ds√2piσsσ2γ(s) exp(− s22σ2s)]1/2=pi2mc2eE0(sinφ0)σsI0IAG(A) , (10)whereG2(A) =A4∫ ∞0dξ1∫ ∞0dξ2(3 + 2ξ1 + 2ξ2 + ξ1ξ2)3/2× 14A4 + 2A2(ξ1 + ξ2) + ξ1ξ2× 13A4 + 2A2(ξ1 + ξ2) + ξ1ξ2. (11)Equation (10) yields (σ¯γ)unc ≈ 9 keV for the above GTFparameters. For the LCLS gun, we use E0 = 120 MV/m,φ0 = 30◦, I0 = 100 A, σx = 0.57 mm, and a Gaussianlaser pulse with σs = 0.87 mm (2.9 ps). Then the averageuncorrelated energy spread is (σ¯γ)unc ≈ 5 keV. Parmelasimulations of the LCLS injector using a flattop laser pulsewith the same rms bunch length (about 10 ps fwhm) show3 keV rms uncorrelated energy spread, which is consistentwith this estimation since the longitudinal space charge ef-fects are somewhat reduced for a flattop current profile.Finally, the uncorrelated energy spread is measured at theTTF photoinjector [1] with the following parameters: E0 =40 MeV, φ0 = 40◦, σx = 3/√3 ≈ 1.7 mm, σs ≈ 2 mm(about 7 ps), and I0 = 4000/(2√pi×7) ≈ 228A for a 4-nCbunch. From Eq. (10), we obtain (σ¯γ)unc ≈ 10 keV, whichis about twice the measured average (close to 5 keV) [1].LONGITUDINAL EMITTANCEThe rms longitudinal emittance characterizes the longi-tudinal phase space area of the beam enclosed by a simpleellipse. For a typical short electron bunch, the nonlinears-dependent rf effect is negligible, and the rms longitudi-nal emittance is determined by both the r-dependent andthe nonlinear s-dependent space charge effects in the gun,given approximately by [11]²s ≈ pi2mc2eE0 sinφ0I0IA1.1(1 + 4.5A+ 2.9A2). (12)Since the main effects occur in the region where γ ≈ 1, wetake the initial rms bunch length σs (given by the laser pulselength) in calculating I0 and A even though the measuredfinal rms bunch length (σs)f varies with the bunch charge.Figure 4 shows the longitudinal emittance measured in theGTF injector [2] for various bunch charges, in comparisonwith Eq. (12) using parameters discussed in the above sec-tion. We can also define an effective uncorrelated energyspread by (σγ)eff = ²s/(σs)f and compare the theoreticalexpectations with the measured data [2] in Fig. (5). In both100 200 300bunch charge HpCL51015longitudinalemittanceHkeV*psLFigure 4: Longitudinal emittance measured in the GTF(black symbol) and obtained from Eq. (12) (red curve).100 200 300bunch charge HpCL51015rmsEnergySpreadHkeVLFigure 5: Effective uncorrelated energy spread measured inthe GTF (black diamond) and the corresponding theoreticalvalue (red square).cases, the experimental values are somewhat larger than theanalytical estimations. This can be understood because theelectron bunch continues to evolve nonlinearly in the gunas well as in the subsequent accelerator sections.REFERENCES[1] M. Hu¨ning and H. Schlarb, Proceedings of PAC2003, 2074(2003).[2] D. Dowell et al., Nucl. Instrum. Methods A 507, 331 (2003).[3] H. Loos et al., Nucl. Instrum. Methods A 528, 189 (2004).[4] M. Borland et al., Nucl. Instrum. Methods A 483, 268 (2002).[5] E. Saldin, E. Schneidmiller, and M. Yurkov, Nucl. Instrum.Methods A 528, 355 (2004).[6] Z. Huang et al., Phys. Rev. ST-AB 7 074401 (2004).[7] M. Cornacchia, and P. Emma, Phys. Rev. ST-AB 5, 084001(2002).[8] T. Shaftan, and L.-H. Yu, Phys. Rev. E 71, 046501 (2005).[9] R. Helm, and R. Miller, in: Linear Accelerators, eds. P.M.Lapostolle and A.L. Septier (North-Holland, 1969), p. 115.[10] Z. Li, private communication.[11] K.-J. Kim, Nucl. Instrum. Methods A 275, 201 (1989).[12] J. Billen, LANL Report LA-UR-96-1835 (1996).[13] C. Limborg et al., Proceedings of PAC2003, 3216 (2003).

Uncorrelated Energy Spread and Longitudinal Emittance of a Photo injector Beam

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Uncorrelated Energy Spread and Longitudinal Emittance of a Photo injector Beam

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