Theory of Nonlinear Harmonic Generation In Free-Electron Lasers with Helical Wigglers

Theory of Nonlinear Harmonic Generation In Free-Electron Lasers with Helical Wigglers Gianluca Geloni, Evgeni Saldin, Evgeni Schneidmiller and Mikhail Yurkov Deutsches Elektronen-Synchrotron DESY, Hamburg NIMA & DESY 07-58 http://arxiv.org/abs/0705.0295

Motivation Is on-axis NHG in FELs with helical wiggler suppressed? Answer important for both main FEL development paths

Motivation Is on-axis NHG in FELs with helical wiggler suppressed? Answer important for both main FEL development paths NHG is beneficial for X-ray SASE sources

Motivation Is on-axis NHG in FELs with helical wiggler suppressed? Answer important for both main FEL development paths NHG is beneficial for X-ray SASE sources NHG is a detrimental effect for high average power oscillators (UV harmonics damage to mirrors) Answer in literature 1 : on-axis NHG is strong [1] H.P. Freund et al. PRL 94, 074802 (2005)

Contents In this talk I will 1. Describe our theory of NHG in FELs with helical wigglers 2. Treat a particular example-case 3. Explain why, in our view, literature is incorrect

NHG as electrodynamical problem

NHG as electrodynamical problem FEL self-consistent code: Interaction I harmonic & beam

NHG as electrodynamical problem Bunching at the h th harmonic FEL self-consistent code: Interaction I harmonic & beam

NHG as electrodynamical problem Bunching at the h th harmonic FEL self-consistent code: Interaction I harmonic & beam Maxwell wave equation

NHG as electrodynamical problem Bunching at the h th harmonic FEL self-consistent code: Interaction I harmonic & beam Maxwell wave equation Solution for the h th Harmonic problem

NHG as electrodynamical problem Bunching at the h th harmonic FEL self-consistent code: Interaction I harmonic & beam Maxwell wave equation Solution for the h th Harmonic problem Solution of the NHG problem means solution of Maxwell equations with God-given sources

Maxwell equations Space-frequency domain We are interested in solving Maxwell Equations in paraxial approximation with respect to the F.T. of the field ) E (ω

Maxwell equations Space-frequency domain

Maxwell equations Space-frequency domain current term

Maxwell equations Space-frequency domain gradient term

Maxwell equations Space-frequency domain f r

Maxwell equations Space-frequency domain r f = r r r f ( z; (c) ) from code; r r (c) f r accounts for helical motion

Maxwell equations Space-frequency domain r f = r r r f ( z; (c) ) from code; r r (c) f r accounts for helical motion λ w η r (c) z kw = 1/ D w

Solution of paraxial equation on-axis Resonance approximation N w >>1

Solution of paraxial equation on-axis Resonance approximation N w >>1 Define detuning parameter as C h ω = 2 2γ z hk w = ω k ω r w ( ω = hω ω) r

Solution of paraxial equation on-axis Resonance approximation N w >>1 Define detuning parameter as C h ω = 2 2γ z hk w = ω k ω r w ( ω = hω ω) r ω Ch << kw i. e. << 1 ω r

Solution of paraxial equation on-axis Far zone solution of paraxial Maxwell equation on-axis:

Solution of paraxial equation on-axis Far zone solution of paraxial Maxwell equation on-axis: C h << k w

Solution of paraxial equation on-axis Far zone solution of paraxial Maxwell equation on-axis: Slowly varying function of z over λ w C h << k w

Solution of paraxial equation on-axis Far zone solution of paraxial Maxwell equation on-axis: Only h=1 survives, on axis.

Solution of paraxial equation: second harmonic Why second harmonic? Only as a particular case. Similar reasoning holds for all harmonics.

Solution of paraxial equation: second harmonic Why second harmonic? Only as a particular case. Similar reasoning holds for all harmonics. Circularly polarized field, vanishing on-axis

Simple model To get further results, we need to treat a particular case C 2 =0;

Simple model To get further results, we need to treat a particular case C 2 =0; Gaussian transverse profile

Simple model To get further results, we need to treat a particular case C 2 =0; Modulation wave front orthogonal to z

Simple model To get further results, we need to treat a particular case C 2 =0;

Simple model: II harmonic directivity diagram ˆ θ = D θ 2 L w ˆ η = D η 2 L w N 2 = σ D 2L w

Simple model: II harmonic power 5 F 2 4 3 2 1 0 0,0 0,5 1,0 1,5 2,0 N

Criticism to literature y x e r θ y r P e r y e r r z θ e r x x

Criticism to literature y x e r θ y r P e r y e r r r E z θ e r x r r r, t) = E ( e + ie )exp[ iφ ] φ = kz + hθ ωt wave phase ( o r θ h h x

Criticism to literature y x e r θ y r P e r y e r r r E z r r r, t) = E ( e + ie )exp[ iφ ] φ = kz + hθ ωt wave phase ( o r θ h θ = k w z h Azimuthal electron motion in helical wiggler θ e r x x

Criticism to literature y x e r θ y r P e r y e r r r E z r r r, t) = E ( e + ie )exp[ iφ ] φ = kz + hθ ωt wave phase ( o r θ h θ = k w z h Azimuthal electron motion in helical wiggler Phase along ptc trajectory: θ e r x x ( k + hkw) z ωt

Criticism to literature [1] [1] H.P. Freund et al. PRL 94, 074802 (2005)

Criticism to literature k ω + z hk / v = 0 w consequence of y θ = k w z Incorrect kinematical picture O r θ x

Criticism to literature 2 w r DL w << 1 electron rotation radius ~ r w y electron beam area ~ σ 2 Particles have nearly constant azimuthal position θ. There is no special resonance condition in helical undulators. r θ x Simply, NHG emission on-axis vanishes at h>1. single electron diffraction area ~ D θ k w z θ = constant L w

Criticism to literature 2 w r DL w << 1 electron rotation radius ~ r w y electron beam area ~ σ 2 Particles have nearly constant azimuthal position θ. There is no special resonance condition in helical undulators. r θ x Simply, NHG emission on-axis vanishes at h>1. k k + z hk w ω / v = 0 k w ω / v = 0 + z single electron diffraction area ~ D θ k w z θ = constant L w

Conclusions NHG in FELs with helical wigglers: electrodynamical problem

Conclusions NHG in FELs with helical wigglers: electrodynamical problem Coherent superposition of filament beams in space-frequency

Conclusions NHG in FELs with helical wigglers: electrodynamical problem Coherent superposition of filament beams in space-frequency Developed a general theory

Conclusions NHG in FELs with helical wigglers: electrodynamical problem Coherent superposition of filament beams in space-frequency Developed a general theory Treated a particular example

Conclusions NHG in FELs with helical wigglers: electrodynamical problem Coherent superposition of filament beams in space-frequency Developed a general theory Treated a particular example Criticized literature: on-axis NHG is suppressed