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

Transcription

1 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

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

3 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

4 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)

5 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, (2005)

6 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 Our answer: on-axis NHG is suppressed [1] H.P. Freund et al. PRL 94, (2005)

7 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

8 NHG as electrodynamical problem

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

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

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

12 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

13 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

14 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 code-given sources

15 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

16 Maxwell equations Space-frequency domain

17 Maxwell equations Space-frequency domain current term

18 Maxwell equations Space-frequency domain gradient term

19 Maxwell equations Space-frequency domain f r

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

21 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

22 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

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

24 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

25 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

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

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

28 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

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

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

31 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

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

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

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

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

36 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

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

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

39 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

40 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

41 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

42 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: k ω + z hk / v = 0 w θ e r x x ( k + hkw) z ωt Azimuthal resonant condition (literature)

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

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

45 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

46 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

47 Conclusions NHG in FELs with helical wigglers: electrodynamical problem

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

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

50 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

51 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

52 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 NIMA & DESY