Avraham Gover (Tel-Aviv Univ.), Reuven Ianconescu (Tel Aviv Univ

Super-radiant Pre-bunched Beam FEL in the Coherent- Spontaneous and Tapered-Wiggler Regimes

Avraham Gover (Tel-Aviv Univ.), Reuven Ianconescu (Tel Aviv Univ. and Shenkar College),

Moore Symposium on compact X-Ray FEL, UCLA Jan 2019

1 eSASE AND HIGH HARMONIC TES OPTION

ALEXANDER A. ZHOLENTS Phys. Rev. ST Accel. Beams 8, 040701 (2005)

e-SASE X-ray Buncher

Uniform Wiggler Tapered Wiggler SP/SR TESSA

Tapering Enhanced Superradiance 2 (a) Spontaneous emission (b) Superradiant emission (coherent spontaneous emission)

3 Formulation of Radiation mode Expansion

푬ෙ 풓, 휔 = ෍ 퐶ሙ푞 푧, 휔 푬෩푞(풓) Spectral (Fourier) ±푞 Frequency domain

푯ෙ 풓, 휔 = ෍ 퐶ሙ푞 푧, 휔 푯෩ 푞(풓) ±푞 ሙ 푑퐶푞(푧, 휔) 1 ∗ = − න 푱ෘ 풓, 휔 ∙ 푬෩푞(풓)dA 푑푧 4풫푞

표푢푡 푖푛 1 ∗ 퐶ሙ푞 휔 − 퐶ሙ푞 휔 = − න 푱ෘ 풓, 휔 ∙ 푬෩푞(풓)dV 4풫푞

out in - e Cq  Cq 

ˆ ~ Er,  Cq z,Eq r q 4 푁

Particulate current: 퐉 퐫, 푡 = ෍ −푒퐯푗(푡)훿 퐫 − 퐫푗(푡) 푗=1 푁 표푢푡 푖푛 1 Radiation wavepackets 퐶푞 휔 = 퐶푞 휔 − ෍ ∆풲ෙ푞푗 4풫푞 푗=1

∞ ෙ ෩∗ 푖휔푡 ෙ (0) ෙ 푠푡 ∆풲푞푗 = −푒 න 풗෥푗 ∙ 푬푞 풓풋 푡 푒 푑푡 = ∆풲푞푗 + ∆풲푞푗 −∞ Spontaneous: ∞ (0) (0) (0) (0) (0) ෙ ෙ 푖휔푡푗0 Where: ෙ ෩∗ 푖휔푡 ∆풲푞푗 = ∆풲푞푒 푒 ∆풲푞푒 = −푒 න 풗푒 (푡) ∙ 푬푞(풓푒 (푡))푒 푑푡 −∞ 푁 푁

표푢푡 푖푛 0 푖휔푡0푗 푠푡 퐶푞 휔 = 퐶푞 휔 + ∆퐶푞푒 휔 ෍ 푒 + ෍ ∆퐶푞푗 푗=1 푗=1 Spectral radiant energy:

푑푊푞 2 2 푑푊푞 푑푊푞 푑푊푞 푑푊푞 = 푃 퐶표푢푡 휔 = + + + 푑휔 휋 푞 푞 푑휔 푑휔 푑휔 푑휔 푖푛 푠푝/푆푅 푠푡−푆푅 푠푡

5 Schematic description of the four radiative emission processes in the complex 표푢푡 푖푛 plane of the mode amplitude 퐶푞 , which is composed of 퐶푞 and the superposition of the particles radiation wave packets ∆퐶푞푗 :

Spontaneous Emission Superradiance

Stimulated Emission Stimulated – Super (Laser Amplifier) radiance

A. Gover, Superradiant and stimulated-superradiant emission in prebunchedelectron-beam radiators PRST-AB 8, (030701) 2005 6 A Pulse Composed of a Periodic Train

of NM Micro-Bunches

f(t-t0)

7 Coherent SP-SR of Undulator Radiation

For a finite train of periodic bunches:

2 2 2 2 푑푊푞 푁 푒 푍푞 푎 퐿 = 푤 푀 휔 2 푀 휔 2푠푖푛푐2 휃퐿Τ2 푑휔 16휋 훽 훾 퐴 푏 푀 푆푅 푧 푚

휔 휔 − 휔 2 0 휔0 ≅ 2훾푧 푐푘_푤 ∆휔 = 휔0Τ푁푤 Detuning parameter: 휃 휔 퐿 = − 푘푧푞 휔 − 푘휔 퐿 ≅ 2휋 푣푧 ∆휔 (Nw=# of undulator periods)

) Τ  1 푠푖푛 푁푀 휋휔 휔푏  Pulse train form-factor: 푀푀 휔 = 푁푀푠푖푛 휋휔Τ휔푏 nb nN M (NM=# of bunches in macropulse 푁푏 1 2 2 푖휔∆푡푗 −휔 휎 Bunching coeficient: 푀푏 휔 = ෍ 푒 = 푒 푡푏 푁푏 (Nb=# of particles in bunch) 푗=1

A. Gover, PRST-AB 8, (030701) 20058 The Macropulse Form-Factor Function Drawn

for NM = 8

sin NM/ b  MM   NMsin / b 

 1  nb nN M

9 FINITE SIZE BUNCHING FACTOR

A. Gover Erratum: Superradiant and stimulated-superradiant emission in prebunched electron-beam radiators. I. Formulation Phys. Rev. ST Accel. Beams 8, 030701 (2005)] 10 BUNCHING BY LASER MODULATION AND DISPERSIVE SECTION

0.67 n22 /2A Me A/ mod  bn n1/3 0 0.67 nAMe 1/2 max b,nmax 1/3 nmax ALEXANDER A. ZHOLENTS Phys. Rev. ST Accel. Beams 8, 040701 (2005) 11 Phase-merging enhanced harmonic generation

Feng, C., Deng, H., Wang, D., & Zhao, Z. (2014). Phase-merging enhanced harmonic generation free-electron laser. New Journal of 12 Physics, 16(4), 043021. SP-SR HARMONIC POWER

For: NNMw 2 nb0z0wn2   dW()dW() qqn   ddn 2 22 2 2 dWqn ( ) Ne 0aL ww 2  MMb,nM ( ) d16A  (  )  0 zem,q n Power of harmonic n:

dWqn ( ) b Ptq,n p dN M n 

222 2 I0 0 a w L w Mb,n 80  z  A em,q (  n ) 13 ST-SR IN THE NONLINEAR REGIME

2 PNmc1/ebb  T   2 PCq,nq,nq P

Self-consistent nonlinear model formulation for an infinite pulse or finite pulse with zero-slippage:

dp dp q,n e 0 dz dz 14 TILTED PENDULUM EQUATION MODEL FOR TAPERED WIGGLER

N.M. Kroll, P.L. Morton, M.N. Rosenbluth, IEEE J. Quant. Electron., QE-17, 1981 A. Gover, R. Ianconescu, A. Friedman, C. Emma, N. Sudar, P. Musumeci, C. Pellegrini, “Superradiant and stimulated-superradiant emission of bunchedelectron beams” arXiv-2018-1810.07566v1 (accepted Rev. Mod. Physics 2018)

15 ST-SR IN THE NONLINEAR REGIME 2nd ORDER PERTURBATIVE SOLUTION

For short interaction length uz/L  w :

22 P/qREF PE(u)E(0)

ST-SR SR TAPERING

For

16 Uniform wiggler: maximal gain Stimulated Superradiance

17 Tapering Enhanced Stimulated Superradiance

(TESSA): (0) r

18 Tapering Enhanced Stimulated Superradiance (TESSA):(0)/2

19 Superradiance in the nonlinear regime (self interaction) - uniform wiggler

20 Tapering Enhanced Superradiance (TES)

E. C. SNIVELY, J. XIONG, P. MUSUMECI, A. GOVER “Broadband THz Amplification and Superradiant Spontaneous Emission in a Guided FEL” (SUBMITTED) 21 Phase – space trajectories of a realistic bunch in a tapered wiggler FEL

N. M. Kroll, P. L. Morton, M. N. Rosenbluth, IEEE J. Quant. Electron. 17, 1436 (1981) Y. Jiao et al, “Modeling and multidimensional optimization of a tapered free electron laser” PRST-AB 15, (050704) (2012) E. A. Schneidmiller, M. V. Yurkov, “Optimization of a high efficiency fel amplifier” PRST-AB 18, 03070 (2015)

22 TESSA: EXTENTION TO DISTRIBUTED BUNCH

Bunches get trapped in the tapered undulator in ponderomotive buckets of the SP-SR generated harmonic radiation.

f:t fraction trapped depends on phase between beam and radiation

TAPERING SR

Need to optimize ftrmodr (,)sin

N. Sudar, P. Musumeci et al “High Efficiency Energy Extraction …Tapered Undulator” PRL 117, 174801 (2016) OPTIMAL TAPERING PHASE Assume:

mod   trap For n ( A), 2    max z nmax

1 r 2

2mod

2trap

*RIVER ROBLES - STUDENT UCLA PHYSICS DEPT. A Model Problem with a SAMURAI beam

3.2 μm

1 GeV, 800 A microbunched beam Focusing optics 320 nm radiation Short wiggler Tapered wiggler (10 periods, 24 cm) (125 periods, 3 m) K10 K(0)10 Non-tapered Undulator Radiation

320 nm PSP-SR

3.2 μm

● In undulator, beam slips back 10 radiation wavelengths ● Choosing tenth harmonic for resonance avoids “gaps” in the resulting laser distribution MODEL PROBLEM RESULTS

● Beam generates 2.4 GW of SP-SR radiation in the uniform wiggler which acts as seed radiation for the tapered wiggler ● An additional 13.4 GW is produced in the tapered wiggler ● Energy extraction efficiency: 2% Conclusions 1. High power coherent spontaneous superradiant emission (SP-SR) may be feasible at high UV harmonic (x30) of modulating laser. 2. High power extraction may be attainable in a Tapering Enhanced Superradiant (TES) scheme where the bunched beam is trapped by its own generated SP-SR. 3. The advantage of TES is self phase matching to the bunch train. 4. Can be companion to e-SASE facility. 5. Consider option of oscillator (TESO). No need for seed.

28 RESERVE

29 Analytic Output Predictions

● Spontaneous power in normal wiggler: 2.4 GW ● Power gained in tapered wiggler: 13.4 GW ● Radiation wavelength: 320 nm ● Radiation pulses: 250 ● Pulse spacing: 320 nm Experimental Setup Numbers

● Microbunching period: 3.2 microns ● Normal wiggler period: 24 mm ● Input beam energy: 1 GeV ● Wiggler periods: 10 ● Average bunch current: 800 A ● Resonant wavelength: 320 nm ● Bunch charge: 200 pC ● Normalized vector potential: 10

● Tapered wiggler period: 24 mm ● Wiggler periods: 125 ● Norm. vector potential (entrance): 10 ● Resonant wavelength: 320 nm EXTENTION TO DISTRIBUTED BUNCH

1/2 1/2 Pq  8P q (0)  P SR  P SR

2 2 1 02 L w a w (0) PSR I 0 f r ,A / n) 40 A em (  n )   0

A/  mod 0

z/  2A0

2 z/  nnA 0 maxmax ALEXANDER A. ZHOLENTS Phys. Rev. ST Accel. Beams 8, N.040701 Sudar, (P.2005 Musumeci) et al “High Efficiency Energy Extraction …Tapered Undulator PRL 117, 174801 (2016) 32 eSASE AND HIGH HARMONIC TES OPTION

ALEXANDER A. ZHOLENTS Phys. Rev. ST Accel. Beams 8, 040701 (2005)

X-ray Buncher

Uniform Wiggler Tapered Wiggler SP/SR TES

33 Phase-merging enhanced harmonic generation

Feng, C., Deng, H., Wang, D., & Zhao, Z. (2014). Phase-merging enhanced harmonic generation free-electron laser. New Journal of 34 Physics, 16(4), 043021. Mesured multi-bunch coherent Smith-Purcell linewidth (MIT - S.E. Korbly et al PRL 2005)

4 for  mfb  240GHz f  fb / NM  28MHz f / for 1/ mNM 1.310

14 1.7 GHz 1.7 GHz 550 14 550 35 SR and ST-SR of Undulator Radiation

For a finite train of periodic bunches:

2 2 2 2 푑푊푞 푁 푒 푍푞 푎 퐿 = 푤 푀 휔 2 푀 휔 2푠푖푛푐2 휃퐿Τ2 푑휔 16휋 훽 훾 퐴 푏 푀 푆푅 푧 푚

푑푊푞 푁푒 푎 2푍푞풫푞 = 퐶ሙ푖푛 휔 푤 퐿 푀 휔 푀 휔 푠푖푛푐 휃퐿Τ2 푐표푠 휑 − 휃퐿Τ2 푑휔 푞 2휋 훽 훾 퐴 푏 푀 푆푇−푆푅 푧 푒푚푞

휔 휔 − 휔 2 0 휔0 ≅ 2훾푧 푐푘_푤 ∆휔 = 휔0Τ푁푤 Detuning parameter: 휃 휔 퐿 = − 푘푧푞 휔 − 푘휔 퐿 ≅ 2휋 푣푧 ∆휔 (Nw=# of undulator periods)

37 RUBICON/NOCIBUR EXPERIMENT

N. Sudar, P. Musumeci et al “High Efficiency Energy Extraction …Tapered Undulator” PRL 117, 174801 (2016) 38 Single Electron Emission Frequency Domain

v  E r  out w q 0 iL 2 it0 j Cqj   e Lsinc L 2 e e 8vz

       0     t  2 0     kz 0  kw    0  sl  vz  

Where: 0 0    k z 0   k w  0 - Synchronism v z 2 L L tsl    - Slippage Time  vz vg (vg=c) 39 Spectral energy flow of super radiant emission for rectangular 55 mm waveguide

Eb = 2.8 MeV 2.5 Qb = 10 pCl

2.0 [nJ/GHz]

f 1.5

/ d d / SR

W 1.0 d d

0.5

0.0 980 1000 1020 1040 1060 1080 Frequency [GHz] 40 PB – FEM SUPERRADIANCE MEASUREMENT*

*A. Cohen et al, PRL, 74, 3812 (1995) ; M. Arbel et al, NIM A445 (2000) 41 PB-FEL STIMULATED SUPERRADIANCE MEASUREMENT

42 SR and ST-SR of Undulator Radiation from a Single-bunch/bunch train Beam

 1  n nN b M 43 Formulation of Radiation mode Expansion

푬ෙ 풓, 휔 = ෍ 퐶ሙ푞 푧, 휔 푬෩푞(풓) Spectral (Fourier) ±푞 Frequency domain

푯ෙ 풓, 휔 = ෍ 퐶ሙ푞 푧, 휔 푯෩ 푞(풓) ±푞 ሙ 푑퐶푞(푧, 휔) 1 ∗ = − න 푱ෘ 풓, 휔 ∙ 푬෩푞(풓)dA 푑푧 4풫푞

표푢푡 푖푛 1 ∗ 퐶ሙ푞 휔 − 퐶ሙ푞 휔 = − න 푱ෘ 풓, 휔 ∙ 푬෩푞(풓)dV 4풫푞

p j out in - e Cq  Cq 

j ˆ ~ Er,  Cq z,Eq r q 44 Photocathode RF-LINAC

RF-Linac cavity can work for: Electron Energy at the exit: 6 MeV Macrobunch charge: 1 nC Micropulse Rep.Rate: 1 THz Macropulse Duration: 12 ps Macropulse Rep.Rate: 100 Hz

3500

TE01, TM01

3000 TE03, TM03

TE21, TM21 2500

2000 Frequency [GHz] Frequency [GHz] 1500

1000

See poster: A. Friedman, “Program and status 500 2 2.5 3 3.5 4 4.5 5 5.5 6 of the THz Facility in Ariel University” Beam energy [MeV] 45

Tapered wiggler: fully trapped bunch

- 8

46 Tapered wiggler: maximal gain ST-SR

- 12

47 Concepts to be considered

- Fundamental processes of coherent spontaneous radiation emission from charged particles and bunched beam current.

- Radiation mode expansion formulation.

- Superradiant (SR), Stimulated Superradiant emission (ST-SR).

- Tapering Enhanced Superradiant (TES) and Stimulated Superradiance (TESSA)

- Nonlinear SR, ST-SR dynamics of a trapped bunched beam.

- Bunched-beam/radiation Self-interaction. 48 APPLICATION OF SUPERRADIANCE:

49 Taper enhanced superradiance (TES) and Taper enhanced stimulated superradiance Amplification (TESSA) [2]

NOCIBUR

푧 ′ ′ 1 휺෤푞(0) 푎푤(푧 ) 푧 퐸 ′′ ′′ሚ ሚ 푖휑푏 ′ 푖 ׬0 휃 (푧 )푑푧 퐶푞 푧 = 퐶푞 0 − 퐼0퐹 푒 න ′ ′ 푀푏(푧 ) 푒 푑푧′ 8 풫푞 0 훾 푧 훽푧(푧 )

퐸 휔0 휃 푣푧, 푧 = − 푘푤 푧 − 푘푧푞 푣푧(푧, 퐸(푧))

Expand around the phase velocity of the ponderomotive wave: 풗풛풓= 흎ퟎΤ 풌풛풒 + 풌풘(풛) :

퐸 훿훽푧 훿훾(푧, 퐸) 푣푧 = 푣푧푟 + 훿푣푧 푧, 퐸0 ⇒ 휃 푧 = 휃 푣푧푟, 푧 − 푘푑 2 ≈ −푘0 2 훽푧푟 푧 훾푧푟 푧 훾푟(푧) ------J. Duris, A. Murokh, P. Musumeci “Tapering enhanced stimulated superradiant amplification” New J. Phys. 17 (2015) 063036 50