Coherent Spontaneous- and Stimulated-Superradiance of Bunched Electron Beams

Avraham Gover (Tel-Aviv University), Reuven Ianconescu (Tel Aviv Univ. and Shenkar College), Aharon Friedman (), Claudio Emma, Nick Sudar, Pietro Musumeci Claudio Pellegrini (UCLA, Los Angeles)

Rev. Mod. Phys. 91, 035003 – Published 19 August 2019

Tutorial - FEL International conference Hamburg, August 2019

1 CONCEPTS TO BE CONSIDERED

• Fundamental processes of coherent radiation emission from bunched beam current: -Spontaneous superradiant emission (SP-SR) -Stimulated Superradiant emission (ST-SR).

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

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

• Bunched-beam/radiation Self-interaction.

• Review of applications.

2 DICKE’S SUPERRADIANCE

B (or E) I = (r+m)(r-m+1)I0

V<<3 r = 1/2, 1, 3/2…..N/2 m = -r…..0…..r

Spontaneous emission (quantum)

N=1, r = m = 1/2  I = I0  Superradiant emission (classical) r = N/2 1, m = 0  1 2 I  4 N I0 N Electric or magnetic dipoles prepared by means of a “/2 pulse”

------3 R. H. Dicke, PR, 93, 99 (1954) COHERENT vs RANDOM SUPERPOSITION OF RADIATION WAVEPACKETS

Spontaneous Emission Spontaneous Superradiance (Shot-Noise Radiation) SP-SR

 2/  4 Formulation of Radiation mode Expansion

𝑬 𝒓, 𝜔 = 𝐶 𝑧, 𝜔 𝑬 (𝒓) Spectral (Fourier) ± Frequency domain

𝑯 𝒓, 𝜔 = 𝐶 𝑧, 𝜔 𝑯 (𝒓) ± 𝑑𝐶(𝑧, 𝜔) 1 ∗ =− 𝑱 𝒓, 𝜔 ·𝑬 (𝒓)dA 𝑑𝑧 4𝒫

1 ∗ 𝐶 𝜔−𝐶 𝜔=− 𝑱 𝒓, 𝜔 ·𝑬(𝒓)dV 4𝒫

out in - e C q   C q  

ˆ ~ Er,  Cq z, Eq r q

5 g1

WAVEPACKETS EMISSION BY PARTICULATE CHARGES

For particulate current: 𝐉 𝐫, 𝑡 = −𝑒𝐯(𝑡)𝛿 𝐫−𝐫(𝑡) 1 Radiation wavepackets 𝐶 𝜔=𝐶 𝜔− ∆𝒲 4𝒫

∗ ∆𝒲 =−𝑒 𝒗 ·𝑬 𝒓𝒋 𝑡𝑒 𝑑𝑡

() () Zero order: ∆𝒲 =∆𝒲 𝑒

Spectral radiant energy:

𝑑𝑊 2 = 𝑃 𝐶 𝜔 = 𝑑𝜔 𝜋 = + + /

6 COMPLEX PLANE REPRESENTATION OF SINGLE MODE EXCITATION AMPLITUDES out in  CCqq C  j qj

Spontaneous Emission

Superradiance

Stimulated Emission ( Amplifier)

Stimulated – Super radiance

A. Gover, PRST-AB 8, (030701) 2005 7 g15 COMPLEX PLANE REPRESENTATION OF SINGLE MODE EXCITATION AMPLITUDES out in  CCqq C  j qj

Spontaneous Emission

Superradiance

Stimulated Emission (Laser Amplifier)

Stimulated – Super radiance

A. Gover, PRST-AB 8, (030701) 2005 8 g15 COMPLEX PLANE REPRESENTATION OF SINGLE MODE EXCITATION AMPLITUDES out in  CCqq C  j qj

Spontaneous Emission

Superradiance

Stimulated Emission (Laser Amplifier)

Stimulated – Super radiance

A. Gover, PRST-AB 8, (030701) 2005 9 A TRAIN OF PULSES (MACRO-PULSE)

N N M N bk eeit00jj it jkj111

it 0 NMbM  M   e

Bunch form factor:   NM b '  1 it0 j ''it ' M  eftedtft0 F b    00 0 NM J 1  b  Macro-pulse form factor: N M it sinN  / 1 0 k M b M M  e N M k 1 N Mbsin / 10 SR and ST-SR of UNDULATOR RADIATION

For a finite train of periodic bunches:

𝑑𝑊 𝑁 𝑒 𝑍 𝑎 𝐿 = 𝑀 𝜔 𝑀 𝜔 𝑠𝑖𝑛𝑐 𝜃𝐿⁄ 2 𝑑𝜔 16𝜋 𝛽 𝛾 𝐴

𝑑𝑊 𝑁𝑒 𝑎 2𝑍𝒫 = 𝐶 𝜔 𝐿 𝑀 𝜔 𝑀 𝜔𝑠𝑖𝑛𝑐𝜃𝐿⁄ 2 𝑐𝑜𝑠 𝜑−𝜃𝐿⁄ 2 𝑑𝜔 2𝜋 𝛽 𝛾 𝐴

𝜔 𝜔−𝜔 Detuning parameter: 𝜃 𝜔𝐿= −𝑘 𝜔−𝑘 𝐿≅2𝜋 𝑣 ∆𝜔

∆𝜔 = 𝜔 ⁄𝑁 , 2 /1a 2 , 0z0w2  zw aBkmcwww /

A. Gover, PRST-AB 8, (030701) 2005 11 Bunching coefficient: Finite interaction length Pulse train form-factor:

22 frequency line-shape (NM=# of bunches in t0 2tb forft0  e / 2 tb macropulse)   (Nw=# of wiggler periods)  2 2 sin N / 2 22 M b  tb 2 M  Me sincL ( ) / 2 M 22 b NsinM / b 1 1 1 0,8  0,8  1 0,8  0  N 0,6 0,6 w 0,6 nb nN M 0,4 0,4 0,4

0,2  tb 1/ 0,2 0,2 0 0 0 00,511,52 -10-8-6-4-20 2 4 6 810 01234

 () 2(0 )/   / b  tb

12 g13 Superradiant Emission from a Pulse Composed of a Train of Bunches

ω- ω si nc20 Δω 

 =𝝎𝟎⁄𝑵𝒘  Single harmonic:    n 2 0 0 b |MM()| 0 //Nnwbwb N

nN w

2/tM 

(n-2)b nb (n+2)b

(n-1)b (n+1)b 13 Superradiant Emission from a Pulse Composed of a Train of Bunches

|dWq/d| Single harmonic: 2/tM 0  nb  =𝝎𝟎⁄𝑵𝒘 0 //Nnwbwb N  nN w 0 (n-2)b nb (n+2)b

13 PREBUNCHED – FEM PREBUNCHED-FEM STIMULATED SUPERRADIANCE MEASUREMENT SUPERRADIANCE MEASUREMENT

A. Cohen et al, PRL, 74, 3812 (1995) ; M. Arbel et al, PRL, 86, 256 (2001) M. Arbel et al, NIM A445 (2000)

14 Mesured multi-bunch coherent Smith-Purcell linewidth (MIT - S.E. Korbly et al PRL 2005)

4 fnfGHznb240  ff / n  1/ nN M  1.3 10

14 1.7 GHz 14 550 15 SP-SR HARMONIC POWER EMISSION of A LONG

PERIODICALLY BUNCHED e-BEAM (NMw N )

2 Discrete harmonics: nbz0wn2

Spectral radiant energy:

𝑑𝑊 𝑁 𝑒 𝑍 𝑎 𝐿 = 𝑀 𝜔 𝑀 𝜔 𝑠𝑖𝑛𝑐 𝜃𝐿⁄ 2 𝑑𝜔 16𝜋 𝛽 𝛾 𝐴

Integrate over frequencies Power of harmonic n:

dW ( ) qn n b Ptq,n  M d NM [Bunching parameter : 222  2 Ia00w L w 2 22 Pbsinc()/2q,n n n ntb  /2 8A()0z em,qn  bM(nb n )e ] 16 g16 ELECTRON BEAM BUNCHING SCHEMES

• Photo-cathode emission (sub-pSec) • Klystron (RF cavity) (Superradiance) • Optical klystron oscillator (Stimulated-Superradiance)(Vinokurov and Skrinsky in 1977) • Ultrafast laser bunching: HGHG (Li Hua Yu) EEHG (Stupakov) E-SASE (Zholent)

Extensive reference list: Gover et al Rev. of Mod. Phys. Vol/EID: 91/035003 (August 2019) 17 BUNCHING BY LASER MODULATION AND DISPERSIVE SECTION

Energy distribution: Energy modulation: Dispersive section:

2 2 1  j 002  R56 fej 0 ttjj'  j  0   ibj0modsin t c 2 0 

f0 (,) jjz

  0 2   0    2 0 2 tb

 t  t bt b b 18 BUNCHING OPTIMIZATION

 j  0  modR 56   0 Normalized parameters *: pABjb,,  00c 0  2 1 pAsin tBp  2 fpt,  ejbjj 0 jj 2

itt ',Tb 2  b enj jj dt dp f t, p njjjjT 2 0 b

𝑅 𝜎 222 bJnABe ntb 𝜎 = nn 𝑐 𝛾

0.67 222 A B  1 ntb Maximal bunching for :: ben  1 n 3

*Hemsing E., Stupakov G., Xiang D., & Zholents A., Rev. Mod. Phys., 86(3), 897 (2014)

19 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 , 16(4), 043021. 20 ULTRA-COMPACT X-RAY FEL BASED ON e-SASE*

Towards Ultra-Compact X-ray FEL, J. Rosenzweig, UCLA Moore Foundation Workshop 1/22-25/2019

EGeV=1 ()  2000bM 3.2 Q  200 pC I M  80 0 A

with w  1.2 mm  1.57Å

A. A. ZHOLENTS Phys. Rev. ST Accel. Beams 8, 040701 (2005) 21 EXERSIZE # 1: HIGH HARMONIC UV SUPERRADIANT SOURCE

Buncher E-SASE

e-SASE X-ray Buncher

UV

Uniform Wiggler SP-SR

22 A Model Problem with a SAMURAI beam: 10th harmonic SR

75m w  24cm

nb10 /0.3nm  zb 3 μm

1 GeV, 800 A microbunched 10th harmonic radiation beam wiggler

 2 a 10, N 10, L  2.4m ww w wn2242 10 cm  1 aw zL  Lz 2( ) A Rn w10 wRn10 em, q n 24

IL22 PGW00w 2.4 qTE0,0 ,10 n 8 0 Aem

*RIVER ROBLES - STUDENT UCLA PHYSICS DEPT. 23 Steady-state (single frequency) phasor formulation for periodic pre- bunching

24 SINGLE-MODE PHASOR MODEL for SP-SR /ST-SR ()0  nb

1 z ik z Cz C0,  Jr   rezq drdz2 qq0 0 q 4Pq  2 PqqqzCz P  Pq()zP q 0 PzP SR   ST SR  z

2 2 1 22z 0 z PZIFcSP SR q m sin  32Aem  2

1 rb 0 z PIEFzzcST SR m 0cos000 2sin 42 Relative weight of ST-SR and SP-SR Power

1 2Zq IPFzmin PAAST SR4 Aem em27q em 2 880PEin  in  2 PZIFzAZIFzSPSR 1 2 z qm em qm ZIqm F 32 Aem ------25 E. A. Schneidmiller, M. V. Yurkov, “Optimization of a high efficiency FEL amplifier” PRST-AB 18, 03070 (2015) Ratio of 0-order ST-SR to SP-SR (calculated for the parameters of the tapered section of LCLS X-FEL)

1

26 SR and ST-SR IN THE NONLINEAR REGIME

27 PERIODIC TIGHT BUNCHING MODEL

dC () z 1 2 q J(()r) E rd 2 r   q  PCq,n P q q,n dz 4Pq dγ Q Nmc2 =b  Er ,t()z 2 b e PNmc1/Teb  b dz z Self-consistent nonlinear model formulation for an infinite pulse (or finite pulse with zero-slippage): dP dP q,n e 0 dz dz 28 NONLINEAR SR AND ST-SR INTERACTION OF A PREBUNCHED BEAM IN AA WIGGLER

TAPERING-ENHANCED STIMULATED SUPERRADIANT AMPLIFICATION - TESSA

N.M. Kroll, P.L. Morton, M.N. Rosenbluth, IEEE J. Quant. Electron., QE-17, 1981

N. Sudar, P. Musumeci et al “High Efficiency Energy Extraction …Tapered Undulator” PRL 117, 174801 (2016) A. Gover, R. Ianconescu, A. Friedman, C. Emma, N. Sudar, P. Musumeci, C. Pellegrini, “Superradiant and stimulated-superradiant emission of bunchedelectron beams” Rev. Mod. Phys. 91, 035003 – Published 19 August 2019 29 SYNCHRONIZM CONDITION OF A TRAPPED BUNCH  pm wave    zkzkz0   ()  n vzwzq 0 b  z Controlled trap dynamics  zkz2 w    r z r z 

Resonant energy of a trapped bunch:

2 2 1azw   k0 eBw () z  ()z  0  r z  ()azw  2 kzw  kzmcw 

Phase of ponderomotive (pm) wave relative to bunches:

z izq  zzdzz''   0 qq  0  Czqq Cze  0  0

30 RADIATION EMISSION AND BUNCH DYNAMICS – UNIFORM WIGGLER

Single mode:   Ez Cqq z   0   r  const

dE bsin dz  2 d 2 d zr r 2 Kzs sin Kzsin dz s dz k0 db db k  cos  0  cos dz E 2  dz zr r E

ke 2 0  Iabw z Z q Kzsw 22 2 azEz  b  2zr r mc 2Aem, q r

31 The  - phase-space trajectories of the pendulum equation

32 RADIATION EMISSION AND BUNCH DYNAMICS – TAPERED WIGGLER

ddk0 d r r  zz  2  dz dz dz zr r

dE bsin dz  d 2 2 d zr r 22 Kzs sin sin r KK z sin sin dz s sr dz k0 db db  cos   cos  dz E dz E

k0 d r sin r 22 zr rKdz s

33 PHASE-SPACE TRAJECTORIES IN A TILTED PENDULUM

N.M. Kroll, P.L. Morton, M.N. Rosenbluth, "Free-Electron with Variable Parameter Wigglers", IEEE J. Quant. Electron., VOL. QE-17, NO. 8, AUGUST 1981 34 The potential energy for a tapered wiggler

Untrapped

Trapped

 r

35 SEPARATRIX OF TRAP IN A TAPERED WIGGLER

2 trap

 2

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

For short interaction length in normalized parameters (u  z / L w ) :

22 P/PqREF E(u)E(0) 2uE 0 sin 0 sin u22 sin 0 2 uE  0 sin   rr  ST-SR SR TAPERING

For

22 PPem REF20sinsin uE   r  u  r

37 UNIFORM WIGGLER: HIGH GAIN ST-SR

Ψ(0)= 0 π/2

39 Uniform wiggler: maximal extraction

Ψ(0)= 0 π/2

41 Tapered wiggler: maximal gain ST-SR

Ψ(0)= 0 Ψr π/2 Ψ(0)=π/2 -12 ΔPem(TOT)

ΔPem(ST-SR)

ΔPem(SR)

ΔPe(SYN )

ΔPe(γr )

ΔPe(TOT)

43 Tapered wiggler: best trapped bunch

Ψ(0)= 0 Ψr π/2

45 Uniform wiggler: self-interaction (NO RADIATION REACTION FORCE IS INVOKED) Ψ(0)= 0 π/2

Abraham-Lorentz Radiation Reaction, Dirac, P. A., 1938, Proc. R. Soc. A 167, 148. 47 SELF INJECTION: SR TO TESSA Ψ(0)= 0 π/2

Snively, E. C., Xiong, J., Musumeci, P., & Gover, A. Optics Express, 27(15), 20221(2019). Broadband THz amplification and superradiant spontaneous emission in a waveguide FEL.

49 EXTENTION TO DISTRIBUTED BUNCH

50 Phase – space trajectories of a realistic bunch in a tapered wiggler FEL

Phase [rad]

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)

51 TESSA: EXTENTION TO DISTRIBUTED BUNCH Bunches get trapped in the tapered undulator in ponderomotive buckets.

f:t fraction trapped.

2 aaww00Z0  2 PzPErad00 fIz tsin r   fIz t sin r 004Aemq 

TAPERING SR

Need to optimize f(tr , mod )sin  r

N. Sudar, P. Musumeci et al “High Efficiency Energy Extraction …Tapered Undulator” 52 PRL 117, 174801 (2016)  TRAPPING EFFICIENCY

   20trap   0mod fdt  dpfpp0 , A 10trap     00



C. Emma et al Phys. Rev. A-B 110701 (2017) 53 EXERCISE # 2 - HIGH HARMONIC UV RADIATION TESSA SOURCE

Buncher

e-SASE X-ray Buncher

UV

Uniform Tapered Wiggler SP/SR Wiggler TESSA

54 A Model Problem with a SAMURAI beam: 10th harmonic SR and TES*

3.2 μm

Focusing optics 1 GeV, 800 A Short wiggler microbunched Tapered wiggler (10 periods, 24 cm) 320 nm radiation beam (125 periods, 3 m) a10w  a(0) 10

2 aa00Z  2 PETESSA ww fILsin 0  fIL sin rad 0 trww tr 004Aemq  TESSA Summary: PWout,10 n []

TESSA SR PPin out 2.4 GW TESSA PGWout 13.4 TESSA TESSA PoutPP in out 16 GW

*RIVER ROBLES - STUDENT UCLA PHYSICS DEPT. 55 APPLICATION OF SUPERRADIANCE:

Extensive reference list: Gover et al Rev. of Mod. Phys. Vol/EID: 91/035003 (August 2019)

56 Energy Retrieval LINAC (ERL) JEFERSON LAB FEL COHERENT SYNCHROTRON RADIATION

Terahertz beamline transports Continuous average THz power: Pav  20W

Carr, G. L. et al, 2002, Nature 420, 153. 57 TERA-HERZ ELBE (TELBE) HELMHOLTZ ZENTRUM DREZDEN ROSENDORF

CONTINUOUS TRAIN OF SINGLE PULSES FROM SRF ACCELERATOR THZ SOURCES: SR UNDULATOR RADIATION, SR TRANSITION RADIATION

Beam energy: 24 MeV Charge/pulse: 100 pC Number of wiggles: 8 THz frequency: Up to 1.4THz THz energy/pulse: 1μJ

Green, B., et al., 2016, Sci. Rep. 6, 22256. 58 g19 FLASH – DESY THz Beamline

tunable: 10 - 300 μm; up to 100 μJ/pulse; ~10% bandwidth,

broadband at 200 μm; up to 10μJ/pulse; ~100% bandwidth

B. Faatz et al NIM A 475 (2001) 363 M. Gensch et al Infrared Physics and Technology 51, 423-425 (2008) 59 UVSOR OKAZAKI JAPAN

THz SR EMISSION IN A STORAGE RING

• E=750MeV • The energy of storage beam pulses is modulated in an undulator by interaction with a THz modulated laser beam. They emit Synchrotron SR at the bend.

Bielawski, S., et al., 2008, Nat. Phys. 4, 390. 60 TeraHertz Superradiant FEL – ARIEL/TAU ()

RF-Linac cavity can work for: Electron Energy at the exit: 3-6.5 MeV Macrobunch charge: 1 nC Microbunch duration sub-pSec Hybrid photocathode gun PBPL UCLA Alesini et al, EPAC, Edinburgh 2006 Micrbunch Rep.Rate: single/3.5 THz Macrobunch Duration: 12 ps Macropulse Rep.Rate: 100 Hz

3500

TE01, TM01

3000 TE03, TM03

TE21, TM21 2500

2000

FrequencyFrequency [GHz][GHz] 1500

1000 Status of the FEL Facility in Israel A. Friedman et al, ID: 2151 - THP077 500 2 2.5 3 3.5 4 4.5 5 5.5 6 Delhi Source (DLS): A Compact FEL-THZ facility - Subhendu Ghosh Beam energy [MeV] 61 NOCIBUR TESSA EXPERIMET UCLA-ATF

Sudar, et al Physical review letters 117.17 (2016): 174801. 62 30% RADIATIVE ENERGY EXTRACTION EFFICIENCY IN THE NUCIBUR EXPERIMENT @  10.5

63 TAPERING ENHANCED STIMULATED SUPERRADIANCE OSCILLATOR (TESSO)

J. Duris, P. Musumeci, N. Sudar, A. Murokh, and A. Gover. Phys. Rev. Accel. Beams 21, 080705 (2018)

P. Musumeci, “Efficiency and High Gain Amplification at 266 nm” THP073 Thursday Poster Session, 29.8.19. 64 Tapered wiggler optimization in seed injected FEL

Krol, Morton, Tosenbluth(1981) D. Prosnitz et al PRA (1981); Scharlemann T. et al., PRL (1981)

𝐼(0) 𝑃 0 Bunched-Beam Radiation zR

z Constant parameters FEL Tapered wiggler section

Effects degrading the fundamental processes: Evolving solutions: Duris, Murokh, and Musumeci (2015) -1d theory [1] -Fresh bunch scheme [11,12] -Diffraction[2-7] -Phase shifter [13,14] -Sideband instability [1,3,4,7,…] -Gain modulation [8] -Spectral effects; Shot-Noise [9,10] -Shot-noise suppression [9,10] -Phase space spread of injected beam [3,7,8] -Experimental tests [15]

[1] KMR (1981); Bonifacio, Casagrande (1988) [2] Fawley (1996); [3] Jiao et al (2012); [4] Emma, Pellegrini (2014); [5] Schneidmiller, Yurkov (2015); [6] Tsai, Wu et al (2018); [7] N.Sudar (2019) [8] Emma, Sudar (2017) [9] Gover and Dyunin (2009); [10] Ratner, Huang, Stupakov (2011) [11] Ben-Zvi et al (1992) [12] Emma C. et al (2017) [13] Ratner, D., et al., 2010 [14] Duris, Murokh, and Musumeci (2015) [15] Wu (2017); N. Sudar et al., (2018)

Extensive reference list: Gover et al Rev. of Mod. Phys. Vol/EID: 91/035003 (August 2019) 65 Conclusions

1. Fundamental radiation emission processes of bunched beam at zero order: -Spontaneous Superradiance (SP-SR)  Nz22, -Stimulated Superradiance (ST-SR)  NzE,, 0 2. Model of periodical tightly bunched e-beam interaction with a single radiation mode in the nonlinear regime: -SR and ST-SR in a uniform wiggler -Tapering Enhanced Superradiance (TES), Tapering Enhance Stimulated Superradiance Amplification (TESSA) and Oscillator (TESSO). 3. Self interaction of a bunched beam in a uniform wiggler and seedless TESSA. 4. Application in THz superradiant sources based on SR emission of sub-picoSec bunches. 5. Applications of TESSA, TESSO in the THz to UV frequencies range. 6. Optimization of tapering strategy in the tapered wiggler section of X-Ray FELs.

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