LIENARD-WIECHERT POTENTIALS AND METHOD OF IMAGES IN RF FREE ELECTRON LASER PHOTOINJECTOR W. Salah; The Hashemite University, Zarqa 13115, Jordan, R.M. Jones; Cockcroft Institute, Daresbury, WA4 4AD, UK; University of Manchester, Manchester, M13 9PL, UK.

Abstract Based on Lienard-Weichert retarded potentials and the τ<<1/ν (where ν is the RF frequency) and the beam potential due to the image of charges on the cathode, a radius a is small compared to the cavity radius ℜ . For rigorous relativistic description of the beam transport the “ELSA” photoinjector ν = 144 MHz, ℜ = 60 cm, inside the RF-photoinjector is presented. The velocity πa2=1 cm2; the first condition provides the pulse duration dependent effects are taken into account. Simulations are τ << 7 ns. Under these conditions, the beam velocity presented for parameters of the "ELSA" photo-cathode. G G β(z,t) and acceleration η (z,t) can be shown [3] to be G INTRODUCTION parallel to E and independent of time: 0 G β=βG RF-photoinjectors are used as a source of low- (z,t) (z)uz emittance and ultra-high brightness electron beams. (1) There are a limited number of codes which take wakefield η=ηG G effects into account in computing electron transport in (z) (z) uz photoemission. Although there are notable exceptions (1+− Hz(t))2 1 which do include these effects numerically [1]. The β=(z) (2) + electromagnetic wake in a photoinjector is different from 1Hz(t) the standard case of a coasting ultrarelativistic beam due η(z)=+ 1 Hz (3) to the rapidly changing velocity. In this situation the 1 z(t)=+−−( 1 (Hc(t t ))2 1) (4) influence of the acceleration-radiation field, or z retardation, must be taken in to account in addition to the H 2 − mc image charges on the cathode. H 1 = (5) The aim of the present paper is to treat the wakefield of eE0 an intense electron beam strongly accelerated inside a where: m and e are the rest mass and charge of the cylindrical cavity similar to that of a photoinjector. We electron, respectively, z(t) is the longitudinal coordinate employ both Lienard-Weichert potentials and the method of an electron at time t, and tz is the time at which an of images in order to derive an analytical expression for element z of the beam leaves the photocathode. the field driven by the beam. Retarded position expressions are computed for the “ELSA” photoinjector electron trajectory Present position G ′ G W (t ) R zˆ

P electron G r yˆ xˆ Figure 2: Field driven by an electron. GG The electromagnetic fields (E,B) generated at time t

and point P, by an electron, that is moving on a specified Figure 1: ''ELSA'' photoinjector (144 MHz cavity). trajectory depend on the position W(t’) of the electron at

time t’ (Fig. 2). These fields are driven from the rebuilt facility [2] schematized in Fig. 1. Furthermore, by G Φ applying the principle of causality we are able to simplify scalar and vector potentials and A , respectively. the effects associated with the actual cavity, illustrated in Taking into account the boundary condition imposed on Fig. 1, to an analysis of the electromagnetic fields in a the cathode by the equipotential and causality, these fields are given by Lienard-Weichert expression as pill-box cavity. G G −β ′ The beam pulse is assumed to be axisymmetric, of G A ()R(t)R radius a, emitted by the cathode from t = 0 to t = τ E(P,t| W)=−G G (c + c(R−β R. (t′ ))3 γ (t′ ) (where τ is the time at which the photoemission ends), G with a constant and uniform J. The GGG ∂β(t′ ) G R×−β× {(R (t′ )R) }) (6) ∂ ′ acceleration RF- E0 may be considered as t constant and uniform provided, the beam pulse duration G GG1R Using equations (10) and (6), the field components E , B(P,t|W)=× E(P,t|W) (7) z cR Er and Eθ on the axes of Fig. 3 are given as −1/2 γ=−β(t′′ )() 1 (t )2 (8) ζ−′′ +β22 + ζ− ′ = A( z s ( z) ) E(P,t|W)z,β (11) -1 ′′′′22 2 23 where A=(4πε0) , ε0 is the of free space, γ+ζ−+βζ−(s ( z) ( z)) ==−′′ RW(t)Pc(tt)is the magnitude of the vector ′ . from the retarded position W to the field point P, and t is β′ 2 = As the retarded time. E(P,t|W)z,β (12) 22′′′ 23 The first term in the parentheses in equation (6) is the c( s+ζ− ( z) +βζ− ( z) ) velocity field while the second one is the acceleration or radiation field. The former falls off as 1/R2 while the later θ = As cos falls off as 1/R. E(P,t|W)r,β (13) γ+ζ−+βζ−′′′′22(s ( z) 2 ( z)) 23 DEVELOPMENT OF LIENARD - . βζ−′′ θ WIECHERT POTENTIALS =− A( z)scos E(P,t|W). (14) r,β c( s22+ζ− (′′′ z ) +βζ− ( z) 23 ) The components of the electromagnetic field driven by an electron within the beam and an image of the charge on the cathode can be obtained by the projection of θ Lienard-Wiechert fields given by equations (6) and (7) on = Assin E(P,t|W)θβ, (15) the axes shown in Fig. 3. This projection is applied in the γ′′′′22(s + ( ζ− z) 2 +β ( ζ− z)) 23

. βζ−′′ ϑ =− A( z)ssin E(P,t|W). (16) θβ, c( s22+ζ− (′′′ z) +βζ− ( z) 23 ) . ∂β where the indices β and β= denote the field ∂ t components due to the velocity and acceleration; respectively. According to the cylindrical symmetry, the

integration of the component Eθ over the whole beam Figure 3: Cylindrical coordinates s,θ and z. gives zero.

laboratory frame. The point where we observe the field will be taken as the origin of this frame. The cylindrical GENERATION OF GLOBAL coordinates (s, θ, z) of a W′ are defined in Fig. 3. FIELDS FROM INDIVIDUAL The vector from the retarded position of the electron W COMPONENTS ( t′ ) to the field point p is −θscos G We generate the global fields driven by the beam using Rssin=− θ (9) the field components driven by an individual electron and z − ζ′ corresponding image charge. For seek of simplicity, we show how we can generate the longitudinal component Ez where the superscript (′) denotes that the values are taken of the global field, since the other components are at time t’. Since a paraxial approximation is used for the identical to the longitudinal one. Consider a cylindrical beam dynamics, the beam velocity β(t) and the beam beam pulse, with radius a, carrying a current I, emitted by ∂β(t) the cathode with a constant and radially uniform current acceleration are in the same direction. Therefore, ∂ t density J, moving along the z-axis with velocity β(t) that the double cross product in equation (6) reads varies with time. For seek of simplicity, we assume that G the shape of the beam does not change during the ∂β −−s(zζ′ )cosθ acceleration. If n(W, t) is the density of electrons or image ∂ t charge at time t then the longitudinal component of the GG GGG ∂β1 ∂β global field at the point P is R((RR)×−β ×==− ) s(z)sin −ζ′ θ (10) 3 2 E (P,t)=+ n(W,t)E (P,t|W)d W ∂∂ttR zz∫ G D ∂β (17) 2 n(W,t)E (P,t|W)d3 W s ∫ z ∂ t D where E(P,t|W)z and E(P,t|W)z are the field APPLICATION OF METHOD components due to an electron and image charge; We apply the method to several emission regimes [4] respectively, D and D represent an ensemble of electrons from the photocathode. A particularly interesting case is and image charges; respectively having an antecedent at that which occurs at the end of photoemission, the retarded time t’and t’’. corresponding to complete extraction of the beam (i. e at The components E(P,t|W)and E(P,t|W)can be z z the instant t = τ = 30 ps). This is illustrated in Fig 4. in written in term of W(t) and W(t) using the following which the axial electric field is displayed as a function of ⎧ ss′ = Z = Hz for the following parameters I =100 A, E0= 30 ⎪ MV/m. This field is compared to that due to the space W(t′′ )=ℑ (M) = θ =θ (18) z,t ⎨ charge (or self-field) and the image of charge on the ⎪ ′ ⎩ζ =θf(s, ,ζ ) cathode. At the centre of the cathode (r = 0, z = 0) the beam self field and the field driven by the image of ⎧ ss′ = charges on the cathode are similar. However, the field of ′′=ℑ =⎪ θ ′ =θ W(t )z,t (M) ⎨ (19) 4 ⎪ζ′ =θf(s, ,ζ ) Global Field ⎩ Self Field Hence 3 Charges Image Field 3 E(P,t)=ℑ+ n(W,t)E(P,t|W) (W)dW τ = 30 ps zzz,t∫ D E = 30 MV/m (20) 2 0 n(W,t)E (P,t|W)ℑ (W)d3 W ∫ zzz,,tt D Since the integral will be carried out with respect to 1 WW(t)′′= and WW(t)′′′= , we can write dW313=Ω ( ℑ− )dW (21) 0 dW313=Ω ( ℑ− )dW′ (22) -1 with -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

− ββ−ζ′′(z ) Z = Hz (m) Ωℑ()1 = (1 − ) (23) β′ 22 s(z)+−ζ′ Figure 4: Axial electric field Ez within beam at the end of photoemission. − ββ−ζ′′(z ) Ωℑ()1 = (1 − ) (24) β′ s(z)22+−ζ′ the beam is dominated by that due to the image charges on the cathode as one moves from the tail to the head of Ωℑ−1 Ωℑ−1 ℑ−1 where () and () are the Jacobeans of the beam. Far from the cathode the self-field dominates. and ℑ−1 ; respectively. By means of equations (17-24), equation (17) becomes: ACKNOWLEDGMENT ζ−′′ +β22 +ζ− ′ = J zs(z) E(P,t)z A ( One of the authors, Wa’el Salah has benefited from a ∫ ecβ γ′′′′22 + ζ− 2 +β ζ− 3 D(P,ζ ,t) (s ( z) ( z)) Cockcroft visiting fellowship. The majority of the ββ′ s2 research presented was completed during the tenure of +×) this fellowship. c( s22+ζ− (′′′ z) +βζ− ( z)) 3β′ β−ζ′′(z ) REFERENCES (1−θζ+ ) sds d d ' 22′ s(z)+−ζ [1] A. Candel et al., MOP104, This conference. 22 [2] S. Joly and S. Striby, 1998, Revue Chocs n18, CEA. J ζ−′′zs(z) +β + ζ− ′ A( + [3] W. Salah and JM. Dolique, 1999, Nucl. Inst. and ∫ β ′′′′22 2 3 D(P,ζ ,t) ecγ+ζ−+βζ−(s ( z) ( z)) Meth. A431. [4] To be published. β′ s2 β ) × c( s2 + (ζ−′ z)23 +β′′ ( ζ− z)) β′ β−ζ′′(z ) (1−θζ ) sds d d ′ s(z)22+−ζ′ (25)