Spatial Coherence Effects in the Transition Radiation Spectrum for Relativistic Charged Beams: Theoretical Results and Beam Diagnostics Implications

Proceedings of the 27th International Free Electron Laser Conference

SPATIAL COHERENCE EFFECTS IN THE TRANSITION RADIATION SPECTRUM FOR RELATIVISTIC CHARGED BEAMS: THEORETICAL RESULTS AND BEAM DIAGNOSTICS IMPLICATIONS

G.L. Orlandi∗, ENEA, C.R. Frascati FIS-ACC, Via E. Fermi 45, 00044 Frascati, Italy

Abstract the instantaneous and highly directional emission of the radiation, the dependence of its angular distribution on the In the electromagnetic radiative phenomena originated beam energy and the possibility to minimize the radiator by relativistic charged beams, angular distortions as well intercepting consequences on the beam thanks to the use of as variations of the photon flux are commonly observed as thin metallic coated dielectric films. Concerning the pos- a function of the ratio between the beam transverse size and sible beam diagnostics applications of the transition radi- the observed wavelength, even at a wavelength shorter than ation originated by relativistic three-dimensional charged the longitudinal bunch length. In the framework of a single beams, it could be interesting to study if and how the an- particle theory of the transition radiation, diffractive alter- gular distribution of the radiation spectral intensity is sen- ations of the spectrum due, for instance, to the finite size sitive to a variation of the transverse size (σ) of the charged of the radiator screen are already known. For relativistic beam with respect to the observed wavelength (λ) belong- three-dimensional charged beams, it could be interesting to ing, for instance, to the optical region. In other words, it check if the transition radiation emission undergoes mod- could be interesting to check if, similarly to what can be ifications depending on the finite value of the beam trans- commonly observed in other electromagnetic radiative phe- verse size with respect to the observed wavelength. Taking nomena - such as the synchrotron radiation or the single into account the beam diagnostics applications of the tran- photon bremsstrahlung - also in the transition radiation a sition radiation in a linear accelerator, such an experimen- diffractive distortion of the angular distribution along with tal check can offer promising perspectives. The theoretical an intensity variation occurs as a function of the ratio σ/λ. background and physical basis of the spatial coherence ef- According to [4], the angular distribution of the transition fects affecting the spectral distribution of the transition ra- radiation should be sensitive to a variation of the ratio σ/λ diation intensity in conditions of temporal incoherence will even though this is observed in a condition of temporal in- be presented. The main beam diagnostics applications will coherence, i.e., at a wavelength much shorter than the lon- be also contoured. gitudinal length of the three-dimensional charged beam. In the following, the transition radiation spectral intensity will INTRODUCTION be derived and possible beam diagnostics applications of the source-size ( ) effects will be contoured. A relativistic charged beam passing through a vacuum- σ/λ metal interface can originate an electromagnetic radiation emission. The steeply local variation of the dielectric properties of the medium imposes a boundary condition to the PSEUDO-PHOTON AND TRANSITION charge electric field at the interface. Thanks to such a con- RADIATION FIELDS straint, the virtual quanta field of the charge can materialize at the vacuum-metal interface and propagate as a radiation In the laboratory reference frame, a relativistic three- field, the so called transition radiation field [1, 2]. Covering dimensional distribution of electrons in a rectilinear and a broad wavelength band extending up to the X-rays region, N uniform motion hits a metallic planar surface located in the spectral distribution of the transition radiation suffers a S the plane of the reference frame and surrounded low frequency diffractive cut-off occuring as the transverse z =0 by a non-dispersive medium [4]. In the visible optical re- extension of the harmonic component of the pseudo-photon gion, ideal conductor properties can be reliably attributed field becomes comparable with the finite size of the radiator to the metallic screen. At the time t=0, the center of mass surface [3]. This manifests in either a modification of the of the electrons, moving with a velocity along the - angular distribution or an intensity variation of the emitted N w z axis, collides with the metallic surface . For a strict and radiation. At a high frequency, a suppression of the sin- S essential description of the radiation emission mechanism, gle electron spectrum of the radiation intensity is instead the spatial distribution of the charged beam can be observable when the metallic screen behaves, under the ac- ρ(r, t) assumed time-invariant and referred to the particle config- tion of the rapidly varying electromagnetic field, as a real uration corresponding to the collision time ( ): conductor with vacuum-like dielectric properties [2]. In a t =0 linear accelerator, transition radiation finds a large and con- solidate application as a tool of beam diagnosis thanks to N N ρ(r, t)= δ(r r )= δ(r r ). (1) − j 0 − 0j ∗ [email protected] j=1 j=1

JACoW / eConf C0508213 576 21-26 August 2005, Stanford, California, USA Proceedings of the 27th International Free Electron Laser Conference

tr The corresponding Fourier transform [4] ρ(k, ω) screen, Ex,y can be expressed by means of the Helmholtz- Kirchhoff integral theorem, which, in the far ﬁeld approxi- 1 i(k r ωt) mation ρ(r, t)= dkdωe · − ρ(k, ω) (2) (i.e., wavelength λ R), can be represented as the (2π)4 ps Fourier transform of Ex,y with respect to the coordinates ρ =(x, y) of the planar surface S [1]: reads

N tr k ps iκ ρ E (κ, ω)= dρE (ρ, ω)e− · (9) ik r x,y x,y ρ(k, ω)=2π e− · 0j δ(ω w k). (3) 2πR ⎛ ⎞ − · S j=1 ⎝ ⎠ In the Fourier space, according to the gauge of Lorentz, the with κ =(kx,ky). From the previous equation and from Eqs.(7,8) the radiation ﬁeld can be ﬁnally derived: propagation equations of the 4-potential (A, Φ) are:

N ω2 2 4πe ( 2 + k )A(k, ω)= wρ (k, ω) tr i(ω/w)z0j v c Ex,y(κ, ω)= e− Hx,y(κ, ω,ρ 0j ) (10) − ω2 2 4πe ( 2 + k )Φ(k, ω)= ρ(k, ω) j=1 − v Therefore, the charge electric ﬁeld reads with

iω iτ ρ0j E (k, ω)= ikΦ(k, ω)+ A(k, ω)= (4) iek τx,y e− · i(τ κ) ρ H (κ, ω,ρ )= dρdτ e − · . (11) − c x,y 0j 2π2Rw τ 2 + α2 N 2 2 ik r0j S i8π e (k ωw/v )( e− · ) = − j=1 δ(ω w k) − k2 (ω/v)2 − · − It is worth to be noted [4] that the longitudinal i(ω/w)z0j iτ ρ0j and the harmonic component of the charge electric field in (e− ) and transverse (e− · ) components of the the transverse plane can be finally obtained: particle phase factor in Eq.(3) play a different role in the definition of the transition radiation field. The transverse ie component of the particle phase factor contributes - accord- E (r, ω)=− (5) x,y π × ing to the Huygens-Fresnel principle - to define the ampli-

N ik r0j tude term Hx,y(κ, ω,ρ 0j ) of the radiation field. The longi- ik r kx,y( j=1 e− · ) dke · δ(ω k w). tudinal one instead, reflecting the temporal sequence of the × k2 (ω/v)2 − · − particle collisions on the metallic screen, determines the At the vacuum-metal interface (z =0), the transverse com- relative phase of the field amplitude terms and takes part in ponent of the electric field resulting from the relativistic the definition of the causality correlation between them. charge and the induced charge satisfies the following con- tinuity condition TRANSITION RADIATION SPECTRUM ps Ex,y(x, y, z =0,ω)+Ex,y(x, y, z =0,ω)=0, (6) The angular distribution of the transition radiation spectral intensity - i.e., the radiated energy per solid angle and which allows the harmonic component of the pseudo- frequency units - is proportional to the square module of photon (virtual quanta) field of the relativistic charge in the Poynting vector of the radiation field: the vacuum ( =1) to be finally derived: 2 2 d I cR 2 2 N = Etr(k ,k ,ω) + Etr(k ,k ,ω) . ps i(ω/w)z 2 x x y y x y E (x, y, z =0,ω)= e− 0j K (x, y,ρ ,ω) (7) dΩdω 4π x,y x,y 0j (12) j=1 With reference to Eqs.(10,11), the following expression for with the spectral density of the transition radiation intensity for a bunch of N relativistic electrons can be obtained: iτ ρ0j ie iτ ρ τx,ye− · Kx,y(x, y,ρ 0j ,ω)= dτe · , (8) wπ τ 2 + α2 d2I cR2 N = ( H 2 + dΩdω 4π2 | μ,j| where ρ =(x ,y ) with j =1, .., N are the electron μ=x,y j=1 0j 0j 0j coordinates in the transverse plane and τ =(τx,τy). N i(ω/w)z0j i(ω/w)z0l Owing to the passage of the relativistic charge, the + e− e Hμ,jHμ,l∗ ) (13) vacuum-metal interface behaves like the scattering sur- j,l(j =l)=1 ps face of the pseudo-photon field (Ex,y). According to the Huygens-Fresnel principle, this propagates as a radiation with Hμ,j = Hμ(κ, ω,ρ 0j ) and μ = x, y. By averaging tr field (Ex,y) from the metallic screen in the backward and the previous expression with respect to the continuous dis- forward half-spaces. At a distance R from the metallic tribution functions [ρx(x0j), ρy(y0j) and ρz(z0j)]ofthe

21-26 August 2005, Stanford, California, USA 577 JACoW / eConf C0508213 Proceedings of the 27th International Free Electron Laser Conference particle coordinates (x0j,y0j ,z0j) [4], the transition radia- and analogously for ρy(τy). Concerning the temporal co- tion spectrum herent part of the spectrum, the average operation reads:

2 2 i(ω/w)z0j i(ω/w)z0l d I cR 2 e− e Hμ,jHμ,l∗ av = av = (N Hμ,j av + (14) dΩdω 4π2 | | i(ω/w)z0j i(ω/w)z0l μ=x,y e− av e av Hμ,j av Hμ,l∗ av = i(ω/w)z i(ω/w)z 2 +N(N 1) e− 0j e 0l H H∗ ) = F (ω) Hμ(κ, ω,ρ 0j ) av (23) − μ,j μ,lav | | where H (κ, ω,ρ ) 2 is given by Eqs.(20, 21) and assumes the following form [4]: | μ 0j | av i(ω/w)z i(ω/w)z 2 2 S F (ω)= e− 0j e 0l = (24) d I d I av av av = [N + N(N 1)F (ω)] , (15) dΩdω dΩdω − i(ω/w)z0j i(ω/w)z0l = dz0jρz(z0j)e− dz0lρz(z0l)e where is the longitudinal form factor in Eq.(16). 2 i(ω/w)z F (ω)= dz ρz(z)e− (16) is the charge longitudinal form factor, deﬁned as the square module of the Fourier transform of the particle distribution function along the longitudinal axis, and

d2IS cR2 c(ek)2 = H 2 = dΩdω 4π2 | μ,j| av 16π6w2 × μ=x,y 2 τμ ρx(τx)ρy(τy) i(τ κ) ρ dρdτ e − · (17) × τ 2 + α2 μ=x,y S is the spectral distribution function of the radiation intensity in the case of a metallic surface S with a generic shape and size. With respect to the case of radiation emitted by a single electron, Eq.(17) is a function of the charge transverse form factor, which, depending on the Fourier trans- forms ρx(kx) and ρy(ky) of the charge in the transverse plane, in the case of gaussian beams reads:

2 2 k sin θ (σ2 cos2φ+σ2 sin2φ) ρx(kx)ρy(ky)=e− 2 x y . (18)

Details of the average operation are below described [4]:

2 H = H H∗ = H H∗ (19) | μ,j| av μ,j μ,jav μ,jav μ,jav where H with μ =(x, y) is given by μ,jav Figure 1: For an observed wavelength λ =0.5 μm and a beam energy of 300 (a) and 500 (b) MeV, the polar angle iek τ e iτ ρ0j μ − · av i(τ κ) ρ (20) distribution of the transition radiation intensity dI(θ)/dθ is Hμ,j av = 2 dρdτ 2 2 e − · . 2π Rw τ + α here reported as a function of the transverse size σ of the S charged beam: 125 ( ), 100 ( ), 75 ( ) and 50 μm ( ). with the average of the particle phase factor given by

i(τxx0j +τy y0j ) iτxx0j iτy y0j e− av = e− av e− av = SPATIAL COHERENCE AND BEAM = ρx(τx)ρy(τy) (21) DIAGNOSTICS IMPLICATIONS In a linear accelerator, transition radiation is observed in where the relation between ρx(τx) and ρy(τy) and the av- iτ x iτ y a condition of temporal incoherence (F (ω)=0) for most erage of the particle phase factors e− x 0j and e− y 0j is: part of the beam diagnostics applications. For a measurement of the beam energy and transverse size, the radiation iτ x iτ x ρ (τ )= e x 0j = dx ρ (x )e− x 0j (22) x x av 0j x 0j is mainly detected in the visible optical region by means

JACoW / eConf C0508213 578 21-26 August 2005, Stanford, California, USA Proceedings of the 27th International Free Electron Laser Conference of a quite standard experimental set-up [5]. By focusing, the beam transverse size (λ<σ), the spectral intensity and for example, the backward emitted transition radiation on the angular distribution of the transition radiation depend the pixel matrix of a charge-couple-device (CCD) camera, not only on the energy but also on the transverse size of the the transverse size of the charged beam can be estimated charged beam. Such a result, illustrated here in the case from the measured image of the light spot. An estimation of an infinite metallic screen (S = ), can be interpreted of the beam energy can be obtained from the angular distri- as an intrinsic diffractive effect due∞ to the finite transverse bution of the radiation measured by a CCD camera placed extension of the charged beam with respect to the observed at the focus of a convergent lens or at a given distance from wavelength. In the framework of a single particle theory of a lens with the focus on the metallic screen. For such ob- the transition radiation [3], similar diffractive effects of the servation conditions, even in the case of ultra-relativistic radiation angular distribution and intensity are expected as charged beam, an infinite surface (S = ) can be in prac- the finite transverse size of the metallic screen S becomes tice assumed for the metallic screen. In∞ such a case, for a comparable with the extension of the pseudo-photon field. gaussian beam with a cylindrical symmetry, the transition From the point of view of the possible beam diagnostics ap- radiation spectrum reads: plications, a dependence on σ/λ of the angular distribution

2 of the transition radiation means that a beam energy estima- d2I (eβ)2 sin2θe (kσsinθ) ∞ = − , (25) tion cannot be disjoined from a measurement of the beam dΩdω π2c (1 β2cos2θ)2 − transverse size. Moreover, in the case of a non-symmetric charged beam (σ = σ ), the angular distribution of the ra- with β = w/c [see Eqs.(17,18)]. For a constant value of the x y observed wavelength λ =0.5 μm and in correspondence diation spectral intensity is expected to show an azimuthal of a beam energy of 300 and 500 MeV, the angular dis- angular dependence deriving from the transverse form fac- tribution of the radiation intensity has been calculated on tor [see Eqs.(17,18)]. The observation of the transition ra- the basis of Eq.(25) for different values of the beam trans- diation with the aim to retrieve information about the en- verse size σ = 125, 100, 75 and 50 μm, as it is reported in ergy and the size of the charged beam implies a careful Fig(1). As the transverse size σ of the charged beam is re- spectral analysis of the angular and the intensity distribu- duced with respect to the observed wavelength λ, the radiation of the radiation, which requires the use of either CCD tion intensity increases as well as the angular distribution of cameras to detect the angular distribution or photomultipli- the radiation undergoes a broadening, tending as a limit to ers to measure the intensity of the transition radiation. the single electron shape. Such a behavior is quite reason- able if the radiation emission, being originated in the end CONCLUSIONS by the interaction of a relativistic charge with the conduc- Diffractive source-size effects affecting the spectral dis- tion electrons of the metallic screen, is compared with other tribution of the transition radiation intensity are here illus- similar electromagnetic radiative phenomena involving rel- trated. Depending on the ratio between the transverse size ativistic charged beams, such as the synchrotron radiation of the charged beam and the observed wavelength, they can or the single photon bremsstrahlung. Moreover, if the scat- modify the radiation angular distribution and intensity with tering nature of the transition radiation, resulting from the respect to the ideal single particle values. Such effects can diffusion of the virtual quanta field on the metallic surface, be useful diagnosis tools of the energy and the transverse is considered, then such a behavior finds a natural explana- size of a relativistic charged beam. tion. Since the radiation field wave front results from the interference at the observation point of the spherical wave fronts, which, according to the Huygens-Fresnel principle, REFERENCES are originated by all the points of the metallic screen that [1] M.L. Ter-Mikaelian, High-Energy Electromagnetic Pro- are enveloped by the virtual quanta field, the transition ra- cesses in Condensed Media, Wiley, New York (1972). diation is expected to be sensitive to the transverse distribu- [2] V.L. Ginzburg, V.N. Tsytovich, Transition Radiation and tion of the electrons. As the observed wavelength is small Transition Scattering, Adam Hilger, Bristol (1990). compared with the beam transverse size, the radiation field [3] W. Barry, 7th Beam Instrumentation Workshop, AIP Confer- bears a phase information about the charge distribution in ence Proccedings, Argonne, 390 (1996) 173; N.F. Shul’ga, the transverse plane and, consequently, a diffractive distor- S.N. Dobrovol’sky, JETP Lett. 65 (1997) 611; A.P. Potylit- tion of the angular distribution and intensity of the emit- syn, Nucl. Instr. Meth. B 145 (1998) 169; M. Castellano et ted photons should be observed. As the wavelength be- al., Nucl. Instr. Meth. A 435 (1999) 297. comes comparable with the beam transverse size, the parti- [4] G.L. Orlandi, Opt. Commun. 211, (2002) 109; G.L. Orlandi, cle phase information is lost and the radiation field behaves submitted to Optics Communications. like a field generated by a single macro-particle. Conse- quently, the radiation intensity reaches a coherent thresh- [5] L. Wartski et al., J. Appl. Phys. 46, 8 (1975); D.W. Rule, Nucl. Instr. Meth. B, 24/25 (1987) 901; A.H. Lumpkin et al., old with the typical angular distribution of a single elec- Nucl. Instr. Meth. A 285 (1989) 343; M. Castellano et al., tron hitting the metallic screen. According to [4], even for Part. Acc. Conf. Proc., 3 (1999) 2196. a condition of temporal incoherence F (ω)=0and pro- vided that the observed wavelength is suitable to resolve

21-26 August 2005, Stanford, California, USA 579 JACoW / eConf C0508213