FEL Theory for Pedestrians

FEL Theory for Pedestrians Peter Schmüser, Universität Hamburg Preliminary version Introduction Undulator Radiation Low-Gain Free Electron Laser One-dimensional theory of the high-gain FEL Applications of the high-gain FEL equations Typeset by FoilTEX 1

bound-electron laser

Sinusoidal electron trajectory in undulator small longitudinal oscillation leads to odd higher harmonics

Co-moving coordinate system into laboratory system Text Computation by Shintake

Radiation power in laboratory system is the same: P = dw dt = dw dt = P P = e 2 c γ 2 K 2 k 2 u 12πε (1 + K 2 /2) 2 Line shape of undulator radiation Electron passing an undulator with N u periods produces wave train with N u oscillations. 1 I ( ω ) / I.5.8.9 1 1.1 (ω ω ) / ω Spectral intensity : I(ω) 2 sin ξ with ξ = π N u(ω ω ) ξ ω Typeset by FoilTEX 11

Higher harmonics of undulator radiation Complicated issue and not the topic of my FEL lecture For details see J.A. Clarke, The Science and Technology of Undulators and Wigglers Model calculation for a detector at θ = with very small aperture electric field K =.2 spectral intensity K =.2 small undulator parameter only first harmonic electric field K = 2 time spectral intensity K = 2 1 3 5 7 9 ω / ω 1 large undulator parameter harmonics 1, 3, 5, 7... radiation at angles θ > has also even harmonics (see Clarke) U m (ω) [a.u.] detector of finite size centered at θ= aperture matched to bandwidth of n th harmonic 1 2 3 4 photon energy [ev]

Theory of the Low-Gain FEL Energy transfer from electron to light wave Differential equations of the low-gain FEL The pendulum equations FEL gain, Madey theorem Higher harmonics Typeset by FoilTEX 12

Question: can there be a continuous energy transfer from electron beam to light wave? Electron energy W = γm e c 2 changes in time dt by dw = v F dt = ev x (t)e x (t) dt Average electron speed in z direction v z = c 1 1 (1 + K 2 /2) 2γ 2 Electron and light travel times for half period of undulator: <c t el = λ u /(2 v z ), t light = λ u /(2c) Continuous energy transfer happens if ω (t el t light )=π v x E x v x electron trajectory z slippage of light wave 1 optical wavelength per undulator period E x light wave Typeset by FoilTEX 14

From the condition ω (t el t light )=π compute light wavelength λ = λ u 2γ 2 1+ K2 2 Identical with undulator radiation wavelength in forward direction (θ =) Remark: ω (t el t light )=3π, 5π... also possible generation of odd harmonics (λ /3, λ /5...) Note however: ω (t el t light ) = 2π, 4π... yields zero net energy transfer from electron to light wave even harmonics (λ /2, λ /4...) are not present Typeset by FoilTEX 15

Definition of FEL bucket FEL bucket ζ FEL bucket electrons in left half remove energy from light wave (bad) - λ/2 λ/2 ζ

Differential equations of the low-gain FEL Energy transfer from an electron to the light wave dw dt = ev x (t)e x (t) = e ck γ cos(k uz)e cos(k z ω t + ψ ) ecke 2γ [cos ψ + cos χ] Ponderomotive phase ψ rapidly oscillating phase χ ψ =(k + k u )z(t) ω t + ψ χ =(k k u ) βct ω t + ψ Continuous energy transfer from electron to light wave if ψ is constant Optimum value ψ = Neglect longitudinal oscillation, so v z v z The condition ψ = const can only be fulfilled for a certain wavelength ψ(t) =(k + k u ) v z t k ct+ ψ = const dψ dt =(k + k u ) v z k c = Typeset by FoilTEX 16

Insert v z and use k u k to compute light wavelength: λ = λ u 2γ 2 1+ K2 2 Condition for sustained energy transfer yields wavelength of undulator radiation at θ = spontaneous undulator radiation can seed a SASE FEL What about phase χ? The term cos χ averages to zero χ(z) =ψ(z) 2k u z cos χ(z) cos(2k u z) 1 cos ψ cos χ 1 2 3 z/ λ u 1 Typeset by FoilTEX 17

Internal bunch coordinate zeta and ponderomotive phase psi bucket center at ζ =, ψ = - π/2 Reference particle: ψ = - π/2 zero energy transfer between electron and light wave electron trajectory light wave FEL case: ψ = energy transfer from electron to light wave Laser-acceleration: ψ = - π energy transfer from light wave to electron

The pendulum equations Lasing process in undulator is started by monochromatic light of wavelength λ Resonance electron energy W r = γ r m e c 2 defined by λ = λ u 2γ 2 r 1+ K2 2 γ r = λ u 2λ 1+ K2 2 (Electrons with energy W = W r emit undulator radiation with wavelength λ = λ ) Consider off-resonance electron γ = γ r, define relative energy deviation η = γ γ r γ r ( < η 1) Ponderomotive phase no longer constant for η =. Also η changes due to interaction with radiation field dψ dt =2k uc η dη dt = ee K 2m e cγ 2 r cos ψ Typeset by FoilTEX 19

Define shifted phase ϕ = ψ + π/2 to see analogy with mathematical pendulum FEL pendulum dϕ dt dϕ dt = 2k u c η = 1 m 2 L dη dt = ee K 2m e cγ 2 r dl dt = mg sin ϕ sin ϕ R otation angular momentum L fixpoint separatrix φ φ π π Os cillation angle φ Small angles: sin ϕ ϕ pendulum carries out a harmonic oscillation: ϕ(t) =ϕ cos(ωt), L(t) = m 2 ωϕ sin(ωt) elliptic phase space curves Large angular momentum motion unharmonic. Very large angular momentum: rotation (unbounded motion) Typeset by FoilTEX 2

FEL phase space curves γ = γ r γ > γ r η = W / W 1.5.5 1 phase φ / π 1.5.5 1 phase φ / π On resonance (γ = γ r ): net energy transfer zero Above resonance (γ > γ r ): positive net energy transfer from electron beam to light wave Typeset by FoilTEX 21

Gain Function of Low-Gain FEL Madey Theorem: the FEL gain curve is proportional to the negative derivative of the line-shape curve of undulator radiation spectral line of undulator.5 FEL gain gain of FEL function intensity I(ω) gain G(ω) -.5 ω ω ω ω The normalized lineshape curve of undulator radiation and the gain curve of a typical low-gain FEL Note: gain of FEL amplifier is G(ω)+1 Typeset by FoilTEX 22

One-dimensional Theory of the High-Gain FEL Microbunching Basic Elements of the 1D FEL Theory Radiation Field and Space Charge Field The Coupled First-Order Differential Equations of the High-Gain FEL The Third-Order Equation of the High-Gain FEL General analytic solution of the third-order equation Typeset by FoilTEX 25

Microbunching Essential feature of high-gain FEL: very many electrons radiate coherently Radiation grows quadratically with the number of particles P N = N 2 P 1 big problem: concentration of 1 9 electrons into a tiny volume is impossible, L bunch λ Simulation of microbunching by Sven Reiche, UCLA (code GENESIS).2 (a).2 (b).2 (c) x [mm].1.1 x [mm].1.1 x [mm].1.1.2 ζ.2 ζ.2 λ! ζ Electrons losing energy to light wave travel on a sinus orbit of larger amplitude than electrons gaining energy from light wave Result: modulation of longitudinal velocity Typeset by FoilTEX 26

Microbunching and exponential gain Simulation of microbunching (Sven Reiche) Measured FEL pulse energy at 98 nm wavelength length of undulator

Basic elements of the one-dimensional FEL theory 1D FEL theory: dependency of bunch charge density and electromagnetic fields on transverse coordinates x, y is neglected. Also betatron oscillations and diffraction of the light wave are disregarded. Complex notation Note: this is a constant E in the low-gain theory Ẽ x (z,t) =Ẽx(z) exp[ik z iω t] E x (z,t) = Re Ẽx (z) exp[ik z icω t] Complex amplitude function Ẽx(z), grows slowly with z Analytic description of high-gain FEL (1) coupled pendulum equations, describing phase-space motion of particles under the influence of electric field of light wave (2) inhomogeneous wave equation for electric field of light wave (3) evolution of a microbunch structure coupled with longitudinal space charge forces Typeset by FoilTEX 27

Initial conditions: uniform charge distribution in bunch at z =, lasing process started by seed laser Interaction with periodic light wave gradually produces density modulation periodic in ponderomotive phase ψ (resp. internal bunch coordinate ζ with period λ ) ρ(ψ,z)=ρ + ρ 1 (z)e iψ j(ψ,z)=j + j 1 (z)e iψ Oscillatory part in longitudinal velocity is neglected: z(t) = βct Higher harmonics are ignored Radiation field Wave equation for E x field 2 z 2 1 2 c 2 t 2 E x (z,t) =µ j x t + 1 ε ρ x 1D FEL theory: charge density independent of x neglect ρ/ x High-gain FEL: complex amplitude Ẽx(z) depends on path length z in undulator E x (z,t) =Ẽx(z) exp[ik (z ct)] Ẽ x () = E Typeset by FoilTEX 28

First goal: find differential equation for field amplitude Ẽx(z) Slowly varying amplitude (SVA) approximation: change of amplitude within one light wavelength (growth rate) is small change of growth rate is negligible Ẽ x(z) λ Ẽx(z) Ẽ x(z) k Ẽ x (z) Ẽ x(z) k Ẽx(z) Ẽ x(z) is negligible Result: Differential equation for slowly varying amplitude dẽx dz = iµ 2k j x t exp[ ik (z ct)] Question: What is the transverse current j x? j = ρv j x = j z v x /v z j z K γ cos(k uz) dẽx dz = iµ K 2k γ j z t exp[ ik (z ct)] cos(k u z) Typeset by FoilTEX 29

j z t = j z ψ ψ t = iω j 1 e iψ = iω j 1 exp[ik (z ct)+ik u z]. The derivative of the transverse field becomes dẽx dz = µ ck 2γ = µ ck 4γ j 1 exp[ik (z ct)+ik u z] exp[ i(k z ct)] eikuz + e ikuz j 1 {1 + exp(i2k u z)} 2 The phase factor exp[i2k u z] carries out two oscillations per undulator period λ u and averages to zero dẽx dz = µ ck 4γ j 1 Typeset by FoilTEX 3

Space charge field (longitudinal field) Electric field created by modulated charge density is computed using Maxwell equation E = ρ/ε Rapidly oscillating field: E z z = ρ 1(z) ε Amplitude of longitudinal electric field is exp[i((k + k u )z ω t)] i Ẽ z = ε (k + k u ) ρ 1 iµ c 2 j 1 ω Ẽ z = iµ c 2 ω j 1 Typeset by FoilTEX 31

The Coupled First-Order Differential Equations Low-gain FEL: Evolution of ponderomotive phase ψ and of relative energy deviation η described by pendulum equations (note that we use z = β ct as our quasi-time) dψ dz =2k uη, dη dz = ee ˆK 2m e c 2 γr 2 cos ψ High-gain FEL: field amplitude is z dependent dη dz light wave = e ˆK 2m e c 2 γ 2 r Re(Ẽxe iψ ) Add energy change due to interaction with space charge field: dη dz space charge = e m e c 2 γ r Re(Ẽze iψ ) Typeset by FoilTEX 32

Combining the two effects yields dη dz = e m e c 2 Re γ r ˆKẼ x 2γ r + Ẽz e iψ Goal: study phase space motion of electrons as in low-gain case, but take growth of field amplitude Ẽx(z) into account and also evolution of space charge field Ẽz(z). Both are related to modulation amplitude j 1 (z) of electron beam current density: dẽx dz = µ ck 4γ j 1 (z) Ẽ z (z) = iµ c 2 ω j 1 (z) Obvious task: compute j 1 for a given arrangement of electrons in phase space Subdivide electron bunch into longitudinal slices of length λ corresponding to slices of length 2π in phase variable ψ Distribution function for N particles per slice S(ψ) = N n=1 δ(ψ ψ n ) ψ, ψ n [, 2π] Typeset by FoilTEX 33

We consider first the special case of a perfectly uniform longitudinal distribution of the electrons in the bunch and continue the function S(ψ) periodically. The more realistic case of a random longitudinal particle distribution is investigated later. Fourier series S(ψ) = c 2 + Re c k exp(ikψ) k=1, c k = 1 π 2π S(ψ) exp(ikψ)dψ The modulated current density at the first harmonic is j 1 = ecn e 2 N N n=1 exp( iψ n ) Typeset by FoilTEX 34

Coupled first-order equations dẽx dz j 1 = n e ec 2 N = µ c K 4γ j 1 N n=1 exp( iψ n ) dψ n dz dη n dz = 2k u η n, n =1...N = e m e c 2 γ r Re K Ẽ x 2γ r iµ c 2 ω j 1 exp(iψ n ) Coupled first-order equations describe time evolution of 1) modulated current density 2) light wave amplitude Ẽx 3) ponderomotive phase ψ n of electron number n (n =1...N) 4) relative energy deviation η n =(γ n γ r )/γ r Many-body problem without analytical solution Typeset by FoilTEX 35

The Third-Order Equation of the High-Gain FEL Main physics of high-gain FEL is contained in the coupled first-order equations Drawback: they can only be solved numerically. Goal: find differential equation containing only the electric field amplitude Ẽx(z) of light wave. For a small periodic density modulation the quantities ψ n and η n characterizing the particle dynamics in the bunch can be eliminated by defining a normalized particle distribution function F (ψ, η,z) = Re F (ψ, η,z) = F (η) + Re F1 (η,z) e iψ η ζ η ψ ψ Typeset by FoilTEX 36

F (ψ, η, z) obeys the Vlasov equation, a generalized continuity equation df dz = F z + F ψ ψ z + F η η z = After many mathematical steps one finds the third-order equation Ẽ x Γ 3 +2i η ρ FEL Ẽx Γ 2 + k 2 p Γ 2 η ρ FEL 2 Ẽ x Γ i Ẽx =. Three important parameters: simplest form gain parameter Γ = space charge parameter k p = ω p c FEL parameter ρ FEL = Γ 2k u µ K 2 e 2 k u n e 4γrm 3 e 1/3 2λ γ r λ u, ω p = FEL bandwidth n e e 2 γ r ε m e Typeset by FoilTEX 37

Third-order differential equation is solved analytically by trial function Ẽ x (z) =A exp(αz) Special case η =and k p =, i.e. energy on resonance and negligible space charge: α 3 = i Γ 3 α 1 = iγ, α 2 =(i + 3)Γ/2, α 3 =(i 3)Γ/2 Second solution leads to exponential growth of Ẽx(z). Power of light wave grows as exp( 3Γz) exp(z/l g ) Power gain length L g = 1 = 1 3Γ 3 4γ 3 rm e µ K2 e 2 k u n e 1/3 Typeset by FoilTEX 38

General analytic solution of the third-order equation Third-order differential equation is solved by assuming a z dependence of the form exp(αz). Cubic equation for exponent α has three solutions α 1, α 2, α 3. Field amplitude is linear combination of the three eigenfunctions Ẽ x (z) =c 1 V 1 (z)+c 2 V 2 (z)+c 3 V 3 (z) V j (z) = exp(α j z) First and second derivative Ẽ x(z) = c 1 α 1 V 1 (z)+c 2 α 2 V 2 (z)+c 3 α 3 V 3 (z) Ẽ x(z) = c 1 α 2 1V 1 (z)+c 2 α 2 2V 2 (z)+c 3 α 2 3V 3 (z) Since V j () = 1 the coefficients c j can be computed by specifying the initial conditions for Ẽx(z), Ẽx(z) and Ẽ x(z) at the beginning of the undulator at z =. The initial values can be expressed in matrix form by Ẽ x () Ẽ x() Ẽ x() = A c 1 c 2 c 3 with A = α 1 α 2 α 3 1 1 1 α1 2 α2 2 α3 2 Typeset by FoilTEX 39

Coefficient vector is given by c 1 c 2 c 3 = A 1 Ẽ x () Ẽ x() Ẽ x() Consider now the simple case η =and k p =, i.e. beam energy on resonance and negligible space charge. Then the eigenvalues are α 1 = iγ, α 2 =(i + 3)Γ/2, α 3 =(i 3)Γ/2 A 1 = 1 3 1 i/γ 1 /Γ 2 1 ( 3 i)/(2γ) ( i 3 + 1)/(2Γ 2 ) 1 ( 3 i)/(2γ) (i 3 + 1)/(2Γ 2 ) Start FEL process by an incident plane light wave of wavelength λ and amplitude E E x (z,t) =E cos(k z ω t) with k = ω /c =2π/λ Typeset by FoilTEX 4

Initial condition is Ẽ x () Ẽ x() Ẽ x() = E All three coefficients have the same value, c j = E /3 Ẽx(z) = E 3 exp( iγz) + exp((i + 3)Γz/2) + exp((i 3)Γz/2) First term oscillates along undulator axis, third term carries out a damped oscillation. Second term exhibits exponential growth and dominates at large z. FEL power grows asymptotically as P (z) = P 9 exp( 3Γz) = P 9 exp(z/l g) for z 32L g P power of incident seed light wave Typeset by FoilTEX 41

FEL startup by seed laser radiation, incident power P 1 1 5 1 1 4 P(z) / P() 1 1 3 1 1 red curve: analytic solution of third-order equation blue line: approximation P(z)= (P/9) exp(z/lg) P 1.1 2 4 6 8 1 12 number of gain lengths lethargy regime Solid red curve: analytic solution about 3 gain lengths Dashed blue line: P (z) = P 9 exp(z/l g) Typeset by FoilTEX 42

Applications of the High-Gain FEL Equations FEL gain curve Consider electron beam which is not on resonance but has still energy spread zero γ = γ r η = σ η = Lasing process seeded by incident plane wave Gain G(η,z) as a function of the relative energy deviation η and the position z in the undulator is 2 Ẽx (η,z) G(η,z)= 1 (Field Ẽx inside undulator depends implicitely on η through η dependence of the eigenvalues α j ) E remember: gain function G is defined as G=gain-1 Typeset by FoilTEX 43

Short undulator: low-gain limit Take undulator magnet that is shorter than one gain length L und L g.7 gain function G(η) Red curve: high-gain theory Blue circles: low-gain theory -.7 -.4.4 relative energy deviation η Note: maximum gain is only 1.5 (gain (G = gain 1=.5) -1= low-gain regime. The quantitative agreement proves that the low-gain FEL theory is the limiting case of the more general high-gain theory. Typeset by FoilTEX 44

Long undulator: high-gain regime Red: high-gain theory, blue: low-gain theory.6 2 L g 8 4 L g gain G(η) 4 -.6 -.2.2 4 8 L g - 4 -.2.2 16 16 L g gain G(η) -1 -.2.2 relative energy deviation η -.2.2 relative energy deviation η For z L g : maximum amplification near η =(on resonance) Typeset by FoilTEX 45

Bandwidth of FEL Analysis of third-order equation shows that FEL gain drops significantly when relative energy deviation exceeds the FEL ρ parameter η > ρ FEL = 1 4π 3 λu L g 1 2 L g normalized gain G(η).5 z dependent energy bandwidth -2-1 1 2 η/ρ η(z) =3 πρ FEL Lg z Normalized gain at z = 2 L g as a function of η/ρ FEL Gain curve has a FWHM 1. ρ FEL Typeset by FoilTEX 46 high-gain FEL acts as a narrow-band amplifier typical bandwidth about.1

analytic solution of third-order equation power gain P(z) / P() 1 1 6 4 1 numerical solution of coupled first-order equations 1 1 2 3 position in undulator z / L g

Comparison of different input powers of seed radiation. FEL power [arb. units] 1 6 1 4 FEL power in arbitrary units 1 1 1 1 1 1 1.1 Red: seed power.1 Blue: seed power 1.1.1 5 1 15 number 1of gain lengths 2 3 z / L g FEL power depends linearly on input power in exponential regime However: saturation level is independent of input power FEL power oscillates in the saturation regime between electron beam and light wave. energy is pumped back and forth Typeset by FoilTEX 5

Simulation of microbunching The coupled first-order differential equations permit to study microbunching Use typical parameters of ultraviolet FEL FLASH.1 12 L g. 6 1 power gain 1 4 1 I I -.1 1 2 3 4 5 6 2. 2 1. 1 1 2 3 number of gain lengths. 1 2 3 4 5 6 2 Typeset by FoilTEX 51

Numerical study of microbunching in a long undulator magnet Martin Dohlus, DESY consider 3 slices start with uniform distribution Important Note: the FEL result: buckets move, Microbunches mainly in the lethargy are located regime in the right half of the slices microbunches are formed in the right halves of the buckets

What happens if the undulator is too long? z = 2 Lg z = 23 Lg -.1 2 norm. charge density ρn -.1 1 norm. charge density ρn.1 relative energy deviation η relative energy deviation η.1 5 ponderomotive phase ψ 1-2π 2π ponderomotive phase ψ electrons move into left half of FEL buckets and take energy out of light wave

Evolution of particle phases along undulator axis 2π ponderomotive phase ψ -2π -4π 5 1 15 2 25 position in undulator z/l g phases of field Ex and current j1 Vorschlag zu Fig. 5.9 central bucket microbunch bucket center central bucket phases of field and current 2π π field current 5 1 15 2 25 z / L g 6 12 18 position in undulator z/l g

Self Amplified Spontaneous Emission Modulated current density resulting from shot noise in electron beam j 1 = 2 e I ω π Sb S b beam cross section FEL power [W] 1 1 1 9 1 8 1 7 1 6 1 5 1 4 1 3 FEL power [ W ] 1 1 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 1 1 5 1 15 2 25 Continuous 1 red curve: computed SASE FEL power Dashed blue curve: FEL startup by seeding with a laser field of E =.7 MV/m 1 1 5 1 15 2 25 Typeset by FoilTEX 54 z / L g z / L g SASE computed analytically with third-oder equation Laser seeding computed numerically with coupled first-oder equations

Measured power rise in LCLS at a wavelength of 1.5 Angström Figure courtesy Zhirong Huang.2 a) b) c).2.2.1.1.1 x / mm x / mm x / mm.1.1.1.2 2π 2π 4π ψ.2 2π 2π 4π ψ.2 2π 2π 4π ψ

Acknowledgements and references I want to thank Martin Dohlus and Jörg Rossbach for numerous fruitful discussions The FEL lectures are mainly based on the book Ultraviolet and Soft X-Ray Free-Electron Lasers by P. Schmüser, M. Dohlus and J. Rossbach, Springer 28 Furthermore, the following references have been extensively used: Duke, P.J.: Synchrotron Radiation: Production and Properties. Oxford University Press, Oxford (2) Wiedemann, H.: Particle Accelerator Physics II, 2nd ed., Springer, Berlin (1999) Clarke, J.A.: The Science and Technology of Undulators and Wigglers. Oxford University Press, Oxford (24) Wille, K.: The Physics of Particle Accelerators. An Introduction. Oxford University Press, Oxford (21) Murphy, J.B., Pellegrini, C.: Introduction to the physics of the free electron laser. In: Colson, W.B., Pellegrini, C., Renieri, A. (eds.) Laser Handbook vol. 6, Free Electron Lasers, p. 11. North Holland, Amsterdam (199) Colson, W.B.: Classical free electron laser theory. Laser Handbook vol. 6, p. 115. (199) 292, 1853 (21) Pellegrini, C., Reiche, S.: The development of X-ray free-electron lasers, IEEE J. Quantum Electron. 1(6), 1393 (24) Pellegrini, C., Reiche, S.: Free-Electron Lasers. The Optics Encyclopedia, p. 1111 Wiley-VCH (23) Saldin, E.L., Schneidmiller, E.A., Yurkov, M.V.: The Physics of Free Electron Lasers. Springer, Berlin, Heidelberg (2) Huang, Z., Kim, K.-J. Review of x-ray free-electron laser theory. Phys. Rev. ST Accel. Beams 1, 3481 (27)