Modeling and Multi-Dimensional Optimization o f a Tapered Free Electron Laser Yi Jiao on behalf of J. Wu, Y. Cai, A.W. Chao, W.M. Fawley, J. Frisch, Z. Huang, H.-D. Nuhn, C. Pellegrini, S. Reiche SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA March 5-9, 2012 @ FLS 2012 Requirements for High-Peak-Power FEL Recent results on single pulse coherent diffraction imaging of proteins and vir uses using an X-ray free electron laser (FEL) show that the resolution can be i mproved by both increasing the number of the coherent photons and simulta neously reducing the pulse duration, i.e.10 fs or less, one TW or larger. Protein nanocrystallography H.N. Chapman, et al., Nature 470, 7377 (2011) Single shot diffraction patterns on virus particles N.M. Seibert, et al., Nature 470, 78-81 (2011) 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 2 Tapered FEL Following Self-Seeding 4m e- chicane 30 m 1st undulator e 200 m 2nd undulator with taper 20 GW - Courtesy of Juhao Wu SASE FEL At the entrance The monochromatic radiation pulse is recombined with the e-beam, and amplified. The seed power is dominant over the equivalent shot noise power. 1~5 MW

Single Single crystal: crystal: C(400) C(400) 1TW Self-seeded tapered FEL e dump Seeded 10 fs, 1 TW Small band width. SASE Great idea from DESY (Geloni, Kocharyan, Saldin, DESY 10-133, Aug. 2010) Limitation of the Energy Extraction Efficiency Studies with three-dimensional (3D), time-dependent codes GENESIS and GINGER sug gest that the available energy extraction efficiency is limited by some combination of d iffraction, refraction, radial dependence of the radiation field, and time-dependent, sli ppage effects. The energy transfer from the electrons to the radiation is below that is predicted by th e 1D KMR theory , suggesting one can not only consider the longitudinal dynamics. We develop a model extending KMR 1D theory to include the physics of diffraction, optical guiding, and radially-resolved particle trapping, with the aim to look insight t he FEL process in a tapered FEL, and clarify how these effects combine together. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 4 KMR Hamiltonian Approach 1D Hamiltonian formulation, take analogy between the FEL process and radio frequency accelerator. Larger Yr, smaller Ft; Yr = 0, Ft1; YYr = p/2, Ft0 Synchronous energy and phase gr 2 ( z ) ks (1 aw2 ( z )). 2k w ( z ) - gr ( z )gr( z ) sin[Y r ( z )] aw ( z )k s | as ( z ) | g . stable unstable

Y 2 p - Y r , cos Y1 Y1 sin Y r gr cos Y 2 Y 2 sin Y r . Assume electrons uniformly distribute along Y dimension, the electron trapping fraction is Ft Y 2 - Y1 2p 02/27/2020 -p Y1 Y. Jiao, FLS 2012, Jefferson Lab @19 slides Y2 p Yr 5 Optical Guiding Approach Scharlemann, Sessler, and Wurtele [Phys. Rev. Lett. 54, 1925 (1985)] highlighted the fa ct that the light being amplified in a FEL can be refracted toward the axis by the elect ron beam. It looks like the light mode is guided by the electron beam (optical guiding). By taking analogy of guiding of light by optical fiber, they define an effective complex i ndex of refraction n, p2 0 rb20 aw e - iY n 1 2 2 [ JJ ] , 2 2 sV2 r=b (n22| -a1)k g and a fiber parameter V by s | s rb . They showed that for a tapered FEL, In the exponential gain regime, gain guiding dominates, Re(n) ~ 0. After this regime, Im(n) ~ 0, while Re(n) describing the refractive guiding, dominates. p2 0 rb20 aw cos Y V ( z ) 2[Re(n) - 1]k r 2 [ JJ ] . c as 0 gr 2

02/27/2020 2 2 s b Y. Jiao, FLS 2012, Jefferson Lab @19 slides 6 Optical Guiding and Focusing Sprangle, Ting, and Tang [Phys. Rev. Lett. 59, 202 (1987)] used the so-called source-de pendent expansion (SDE) approach to obtain a general, self-consistent formalism to d escribe the phenomena of radiation guiding and focusing in FEL. With assumptions that the radiation beam profile remains approximately Gaussian, th ey obtain an envelope equation for the radiation beam, rs K 2 rs 0, with the optical focusing parameter K2 in terms of V2, G, Y 4 1 4 2 sin Y 2 1 2 d (V 2G / cos Y ) 2 -4 K 2 (- 1 V G V G k r sin Y ) r , s s s 2 ks 4 cos Y 4 dz 2 G (1 - f ) / (1 f ) 2 , f ( z ) (rb / rs ) 2 , --- filling factor. p2 0 rb20 aw cos Y V ( z ) 2[Re( n) - 1]k r 2 [ JJ ] . --- after the exponential gain regime. c as 0 gr 2 2 2 s b

Y ---ponderomotive phase of the electrons. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 7 Radially-Resolved Physical Model Slicing treatment in radial dimension y Ignore Same energy 0.8 0.7 x Uniform ditrib. in Y each slice Electron No. |as | 0.9 Arb. Unit betatron motion 1 0.6 0.5 0.4 0.3 0.2 = e-iYr(r) each slice sin[ Y r ( r , z )] e 0.1 0 0 0.5 1 r2 rs2 ( z ) 1.5 r/r

At a specific rmax, Yr(rmax) = p/ 2, Ft = 0. 2 2.5 b At different radial position r, electrons experience different radiation field amp., the ponderomotive poential is different, Yr(r) is different, bucket size is different, and therefore local trapping fraction Ft(r) is different. Electrons at different r have different detrapping rate, and detrap more rapidly at larger radius. The overall Ft and is calculated by taking average over r. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 8 3 Formulation of our Physical Model 1, Energy conservation 2 p0 2 2 b0 s as20 ( z z )rs2 ( z z ) - as20 ( z )rs2 ( z ) r Ft ( z )( g( z ) - g( z z ) ), 2, Envelope equation for radiation field 2 rs K rs 0, Ft(z), g(z), rb(z), (z) (z) 4 1 4 2 sin Y 2 1 2 d (V 2G / cos Y ) 2 K 2 (- 1 V G V G k s rs sin Y )rs- 4 . 2 ks 4 cos Y 4 dz 2 3, E-beam (trapped) energy variation (ignore energy spread ~ r), g(z) = gr(z) gr 2 ( z )

ks (1 aw2 ( z )). 2k w ( z ) 4, E-beam (trapped) synchronous phase, depends on r - gr ( z )gr( z ) sin[Y r (r , z )] . aw ( z ) k s | as ( r , z ) | aw(z), as0(z), rs(z) 5, Radially-resolved local electron trapping fraction Ft ( r , z ) [ Y 2 (r , z ) - Y1 (r , z )] / 2p , 6, Overall electron trapping fraction and microbunching term 1 rmax 1 rmax Ft ( z ) Ft (r , z ) f 0 (r )2p rdr , eiY ( z ) eiY r ( r , z ) Ft ( r , z ) f 0 (r )2p rdr. Ne 0 Ne 0 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 9 Property of our Physical Model It combines the KMR analysis and optical guiding approach with explicit formulation. It adopts radially-resolved synchronous phase and local electron trapping fraction to couple the longitudinal and transverse dynamics. Since the model is not derived strictly from Maxwell equations, it well describes the evolution of the radiation field within rmax, more or less 3rb0, while not considering th e radiation propagating outside this region. To simplify the analytical study, it is assumed that the undulator is tapered from the initial satura tion location. At initial saturation (the end of the exponential gain regime), Psat rPbeam, we have approximately rs,sat = rb0, Yr,sat = 0, Ft,sat =1, gsat = g0, aw,sat = aw0, and w 9 Psat 2(1 aw2 0 ) r 2 Lsat ln( ), as 0, sat , Pin 4p 3r aw0 [ JJ ] Given a taper profile aw(z) ) [and rb(z)] and values of (as0,sat, rs,sat, Yr,sat, Ft,sat), one can it erate the equations to evolve those parameters in z along the undulator. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 10 GENESIS Scan with LCLS-II Like Parameters

LCLS-II like parameters Fixed taper start-point z0 = 12 m, GENESIS scan taper ratio = 1-aw(Lw)/aw(z0), and electron beam radius rb 0.25 2.2 2 1.8 0.2 1.6 1.4 0.15 1.2 1 0.8 0.1 Parameters E-beam energy E-beam current Normalized emittances Yex, n/ Yey, n Energy spread E-beam pulse length (FWHM) Normalized undulator parameter aw0 Undulator period w Undulator length Lw Radiation wavelength s Peak radiation input power Pin Value 13.64 4000 0.3/0.3 Unit GeV Ampere m-rad 1.3 10

MeV fs 2.3832 3.2 120 1.5 cm m Angstrom 5 MW 0.6 0.4 0.05 0.2 14 16 18 20 r (m) b 22 24 26 Max. power of 2.2 TW with = 0.12 and rb = 15 mm, Case A Radiation power depends on both the taper profile and transverse focusing, Achieving high radiation power requires strong transverse focusing. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 11 Verification of the Physical Model (Case A) Evolution of radiation power (within 3rb), on-axis radiation field as0, radiation 60 1 40 2.5 0 x 10 40

0 60 80 100 120 z (m) Solid lines : GENESIS single-frequency simulation 20 0 20 40 60 80 100 120 z (m) 0 20 40 60 80 100 120 z (m) -5 1 2 0.8 1.5 0.6 1 0.4 0.5 0 0 02/27/2020 20 t s s 0.5

0 On-ax is |a | r (m) 1.5 F P ow er (T W) beam radius rs, trapping fraction Ft with z. 0.2 20 40 60 80 100 120 z (m) 0 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 12 Dashed lines : Physical model Verification of the Physical Model (Case A) Transverse distribution of radiation field, trapped electron No., and Yr (or 2 x 10 s |a | 1.5 1 0.5 Initial distrib. 0.1 0 0 100 0.8 80 0.6

10 20 30 40 T rans vers e radius r ( m) 60 40 r 0.4 Solid lines : GENESIS single-frequency simulation 0.05 1 0.2 0 02/27/2020 0 10 20 30 40 T rans vers e radius r (m) (Degree) s Normlized |a | 0 T rapped e- No. (Norm.) from GENESIS) at z = 50 m. -5 0 10 20 30 40 T rans vers e radius r (m) 20 0 0 10 20

30 40 T rans vers e radius r (m) Y. Jiao, FLS 2012, Jefferson Lab @19 slides 13 Dashed lines : Physical model Applications of the Physical Model The physical model enable us to understand the mechanism of saturation of the radiation intensity and po wer, the decreasing of refractive guiding and trapping fraction is the major ca use. to understand that a controlled variation in electron beam radius (transverse focusing) can be helpful in improving the overall energy extraction efficiency. to propose multi-dimensional optimization scheme Eight parameters, z0, d, , z1, z2, Kq(z1), Kq(z2) and Kq(Lw) aw ( z ) aw ( z0 ) [1 - c ( z - z0 )d ], K q ( z1 ), with 0 z z1 K q ( z ) K q ( z1 ) [1 - f ( z - z1 )], with z1 z z 2 , K ( z ) [1 - g ( z - z )], with z z L , 2 2 w q 2 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 14 200 m Hard X-ray FEL w/o Break Sections Optimization with rb-variation (red) vs. with constant rb (green). 100 1 4 80 0.8 60 t F 40 0.4 s 2

0.6 rs b 3 1 x 10 50 100 z (m) 150 0 200 0 50 100 z (m) 150 0 200 0 50 100 z (m) 150 200 0 50 100 z (m) 150 200 -5 w

s 4 Normalized a 3 2 1 0 0.2 rb 0 02/27/2020 50 100 z (m) 150 200 3.5 0.8 3 Bunc hing fac tor 5 0 20 q 0 On-axis |a | r & r (m) 5 &K P ow er (T W) Higher power, 4.9 TW vs. 4.4 TW, increase by 11%, Higher on-axis radiation field, small radiation radius; higher F t. 2.5 2

1.5 Kq 1 aw 0.5 0 50 100 z (m) 150 200 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 0.6 0.4 0.2 0 15 200 m Hard X-ray FEL w/Break Sections, 1m/4.4m 100 1 4 80 0.8 rs 0.6 t 60 40 0.4 s 2 F b 3 1

x 10 50 100 z (m) 150 w s 3 2 1 0 0 50 100 z (m) 150 0 200 0 Increase in rs and reduction in as0, cascade into increase in Yr and decrease in Ft. 50 100 z (m) 150 200 50 100 z (m) 150 200 -5 4 0

0 200 0.2 rb 50 02/27/2020 100 z (m) 150 200 3.5 0.8 3 2.5 Bunc hing fac tor 5 0 20 q 0 On-ax is |a | r & r (m) 5 Normalized a & K P ow er (T W) Optimization for FEL w/o break-section (red) vs. with/break section (black). Power reduction, 46% (from 4.9 TW to 2.7 TW), partially due to less magnetic length (23% reduction), partially due to vacuum diffraction of radiation in break sections. Kq 2 1.5 1 aw 0.5

0 50 100 z (m) 150 200 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 0.6 0.4 0.2 0 0 16 Impact of Sideband Growth in Hard X-ray FEL 200 m Hard X-ray FEL w/o break sections with time-dependent effects (blue) vs. w/o time-dependent effects (red). ---time-dependent effects lead to significant detrapping by around 100 m, and the average radiation power and on-axis |as| reach saturation much earlier in z and, most importantly, at much smaller values than is true for the single-frequency run. 5 4 4.5 3.5 4 1 0.9 0.8 0.7 2.5 2 1.5 0.6 2 t 3 2.5 F s On-axis |a |

P ow er (T W) -5 3 3.5 0.3 0.2 0.5 0.5 0 02/27/2020 50 100 z (m) 150 200 0 0.5 0.4 1.5 1 1 0 x 10 0.1 0 50 100 z (m) 150 200 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 0 0 50 17

100 z (m) 150 200 No Seed vs. Seeded Power Spectrum SASE: blue Seeded: red 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 18 Seeded, Sideband Spectrum 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 19 Conclusions A new physical model of tapered FEL, has good success in predicting the beh avior with z of the electron and radiation beams; Multi-dimensional optimization based on GENESIS single-frequency simulati ons; A reasonable variation of the transverse focusing can to some extent further enhance energy extraction efficiency. Preliminary study for the impact of the sideband growth. A general description of FEL process and saturation mechanism; Dependence of the available max. radiation power on various parameters(not shown her e). 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 20 Thanks for Your Attention! 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 21 Backup slices 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 22 KMR vs. SDE vs. Our Model

Radiation beam KMR SDE Our Model No transverse variation, Radiation field and e-beam overlap perfectly rs=rb=const. Only consider variation of as with z Solve the evolution of the field from wave equation, whose source function is in term of n. Consider only the fundamental mode Energy conservation 2 s0 2 s0 a ( z z ) - a ( z ) 2p 0 Ft ( z )( g( z) - g( z z ) ), s2 Synchronous energy (aver. energy of the trapped electrons) gr 2 ( z ) 02/27/2020 ks (1 aw2 ( z )). 2k w ( z ) e , Energy conservation p2 0 rb20 aw e - iY n 1 2 2 [ JJ ] , s rb 2 | as | g

Basic assumption is that fundamental mode dominates (i.e. radiation beam profile remains approximately Gaussian) Ft : trapped electron No./total electron No. Tapered region as (r , z ) as 0 ( z )e - r2 i ( r , z ) rs2 ( z ) as20 ( z z )rs2 ( z z ) - as20 ( z )rs2 ( z ) p20 2 2 rb 0 Ft ( z )( g( z ) - g( z z ) ), Envelope equation s rs K 2 rs 0, Numerical simulation to get the mircobunching term , which contains Ft information Y. Jiao, FLS 2012, Jefferson Lab @19 slides Same with KMR treatment 23 KMR vs. SDE vs. Our Model (cont.) KMR SDE Our Model E-beam Const. radius rb Gaussian beam density profile Transverse dimension Uniform transverse density Same with SDE treatment for the initial distribution, which will deviate from Gaussian with taper.

E-beam A portion (Ft) of electrons oscillate around gr and Yr Longitudinal dimension sin[ Y r ( z )] Electron trapping fraction Ft Ft ( z ) f 0 (r ) N e e - r2 rb20 / (p rb20 ), Numerical simulation to get Y and g Yfor each particle Similar to KMR treatment Mono-energy; since as depends on r, Yr depends on r. Numerical simulation to get the mircobunching term , which contains Ft information Same with KMR treatment local trapping fraction Ft(r, z) depends on r, - gr ( z )gr( z ) . a w ( z ) k s | as ( z ) | Y 2 ( z ) - Y1 ( z ) , 2p Y2, Y1 are the max. and min. Y of the ponderomotive bucket 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides

Ft ( z ) 24 1 Ne rmax 0 Ft (r , z ) f 0 ( r )2p rdr , FEL process in a tapered FEL The decreasing of refractive guiding and trapping fraction is the major cause of the saturation of the radiation intensity and power. Particle Trapping Development Region @I V2 > 1, G ~ 0, rapid rs growth, evident detrapping. 3 -5 s On-axis |a | The tapered undulator system, from the initial saturation location to the end, three successive regions. x 10 2 1 0 60 Radiation Intensity Growth Region @II I II III Radiation Power Growth Region @III V2 < 1, G ~1, rapid rs growth, as0 decreases, power still grows. s r (m) V2 > 1, G ~ 1, slow rs growth, strong as0 increase.

40 20 0 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 0 20 40 60 80 100 120 z (m) Case A 25 Particle Trapping Development Region This region starts from initial saturation and lasts ~ 2 Rayleigh lengths (2zr). At the beginning, rs rb, G ~ 0. The -1 term dominates the expression for K2, leading to strong diffraction of the radiation and an increase in rs. 4 G ( z ) (1 - f ) / (1 f ) 2 , f ( z ) ( rb / rs ) 2 , K 2 2 ( - 1 V 2G ...) rs- 4 . ks The energy conservation has equivalent form 2 p0 2as20 ( z1 )rs ( z1 ) rs ( z ) 2rs2 ( z )as 0 ( z )as 0 ( z ) Er (rad ) Ea (rad ) 2 s rb20 Ft ( z )gr( z ) z, E (e- ) The increase in rs corresponds to a large Er(rad), suggesting a large portion of the ene rgy extracted from the electron beam contributes to the radiation expansion in radial d irection. As rs increases, G also increases. By the end of this region, G is large enough that the V2 G 1, and the optical focusing effects now become strong. At this point the rs growth b ecomes much slower and as0 will begin to increase strongly. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 26 Particle Trapping Development Region (Cont.) gr 2 ( z ) ks (1 aw2 ( z )). 2k w ( z ) sin[ Y r (r , z )] - gr ( z )gr( z ) . aw ( z ) k s | as ( r , z ) |

as 0 ( z ) sin[Y r (r 0, z )] | aw | . 2k w At initial saturation At taper start-point, aw = 0; then |aw| will be increase d, the ponderomotive bucket evolves with the on-axis Yr increasing from 0 to a positive value, and with Ft de creasing markedly. Electrons gradually bunch themselves to match the ch anged ponderomotive bucket, during which evident de trapping occurs. For case A (right), zr = 4.7 m, 2zr after initial saturation, the trapped electrons have a visible reduction in the a verage beam energy, and Ft decreases by about 17%. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 1zr after initial saturation 2zr after initial saturation 27 Radiation Intensity Growth Region This region follows the particle trapping development region and ends wher e as0 reaches a maximum value. In this region, G ~1, V2 > 1, (-1+ V2G) dominates the expression of K2. Radiation focusin g is relatively strong, rs increases slowly and as0 increases rapidly. Meanwhile, the increase in as0 causes a reduction in V2, and hence a decrease in K2, i.e. reduced optical focusing. rs increases more and more rapidly. The as0 growth gradually slows down, in general eventually reaching an asymptotic valu e (Er(rad) = E(e-) 2, at2 V2~1, K2 ~0). p 0 rb 0 aw0 [ JJ ] as 0,max 2 cos Y , c g0 The length of this radiation intensity growth region can be estimated by Lrigr (as0, maxas0, sat)/, with being the average as0 growth rate in this region. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 28 Radiation Intensity Growth Region (Cont.) In general, it requires a moderate taper to obtain both large on-axis intensit y and large power. 1.5 120 1 s

r (m) Power(TW) 100 0.5 80 60 40 20 0 x 10 20 40 60 80 z (m) 100 0 120 0 20 40 60 80 z (m) 100 -5 Red, moderate taper = 0.12 1 2 0.8 1.5 0.6 F t

s On-axis |a | 2.5 0 Black, strong taper, = 0.16 120 1 0.4 0.5 0.2 0 0 02/27/2020 20 40 60 80 z (m) 100 120 0 Blue, gentle taper = 0.02 0 20 40 60 80 z (m) Y. Jiao, FLS 2012, Jefferson Lab @19 slides 100 120 29 Radiation Power Growth Region This region following the radiation intensity growth region, where the on-axi

s intensity does not increase but the total radiation power still grows. Note that in the case that as0 has not reached the predicted as0,max at the end of th e undulator, there will not be such a region. V2 < 1, rs increases rapidly, as0 decreases with z (in some cases, the as0 decrease rate ca n be small). However, the radiation power will continue to increase as long as there ex ist electrons trapped in the ponderomotive bucket. Associated with the weakened refractive guiding, the energy extraction becomes less and less efficient, independent of the actual taper profile. ---Case I: KMR-based self-design taper algorithm, a design particle at a preset radius remains at a constant Yr, Ft will decease slowly, but will reach power saturation within a long enough ndulator. --- Case II: A taper with smaller |aw| compared with Case I, it will needs even longer undulator le ngth to reach power saturation. ---Case III: A taper with larger |aw| compared with Case I, can produces relatively slower decreas e in as0 within a short distance from the beginning of this region. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 30 Optimization of a finite-length tapered FEL Optical guiding physics limits the on-axis radiation intensity and the overall energy extraction efficiency. Generally, to maximize the overall energy extraction efficiency usually require s a moderate taper within a long enough undulator length, let us say Lw Lsat + 4zr, which results in a relatively large as0, max before and close to the exit of th e undulator. The optical focusing and hence the evolution of the radiation beam is related to both the longitudinal dynamics (through |gr| and Yr) and the transverse d imension parameters (through G), it is interesting to consider if allowing a con trolled variation in the electron beam radius (transverse focusing) can be hel pful in improving the overall energy extraction efficiency. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 31 Contribution of Variation in Transverse Focusing We consider a gradually decreased rb (increasing transverse focusing) in the latter part of a tapered undulator, and compare with the case with the same taper profile but constant rb. G ( z ) (1 - f ) / (1 f ) 2 , f ( z ) (rb / rs ) 2 , K 2 4 2 -4 ( 1

V G ...) r . s 2 ks The factor G and K2 will be slightly larger, leading to increased optical focusing and to s maller rs & smaller Er(rad). Thus a larger portion of energy contributes to as0 growth. A larger as0, results in smaller Yr(r=0) and higher Ft(r=0). On the other hand, a smaller r s leads to a smaller rmax, causing higher Yr(r) and smaller Ft(r) for electrons at large r. M ore electrons around axis will be trapped, resulting in stronger on-axis optical guiding will tend to lead to a more rapidly growing as0. If squeezing rb too much, diffraction effect dominates. Thus, there is an optimal value for the decreased rb. The change in G is small, thus improvement from an rb-variation is relatively small. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 32 Multi-Dimensional Optimization Scheme Taper profile aw ( z ) aw ( z0 ) [1 - c ( z - z0 )d ], z0 the taper start-point, d the taper profile order, c is the scale coefficient which is relat ed to the taper ratio by c = /(Lw-z0)d. Empirically it is best to start the taper slightly before initial saturation and use a mod erate taper profile order with d 2. The optimal for the maximum radiation power v aries with the undulator length and various initial electron and radiation beam parame ters. Quadrupole strength 3-segment variation K q ( z1 ), with 0 z z1 K q ( z ) K q ( z1 ) [1 - f ( z - z1 )], with z1 z z2 , K ( z ) [1 - g ( z - z )], with z z L , 2 2 w q 2 z1, z2 indicate the Kq variation points, z1 usually around initial saturation and z2 around where as0 reach its maximum value. The coefficients f can be either positive or negative, while g is usually negative. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides

33 Multi-Dimensional Optimization We perform multi-dimensional scans with GENESIS single-frequency simula tions over the following eight parameters, z0, d, , z1, z2, Kq(z1), Kq(z2) and Kq(L w) to obtain the maximum radiation power for a tapered FEL with specific el ectron beam, radiation seed and undulator properties. Matlab code run in Windows system. The code generates input file, invokes GENESIS simulation via Cygwin, and reads GENESIS output. The parameter scan is performed by varying one parameter while fixing the other seven. The variation of the radiation power with respect to a specific parameter can be obtained and the optimal value is then found by means of data analysis. Two Applications 200 m, hard X-ray FEL, LCLS-II like parameters, without break sections. Clarify the contribution of variation in rb. 200 m, hard X-ray FEL, LCLS-II like parameters, with break sections. Clarify the effect of the finite-length break sections. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 34 200 m Hard X-ray FEL w/o Break Sections Optimization with rb-variation (red) vs. with constant rb (green). 100 1 4 80 0.8 s 1 0 100 z (m) 150 t F 0.2

rb 0 50 100 z (m) 0.4 150 0 200 0 50 100 z (m) 150 200 0 50 100 z (m) 150 200 -5 3 2 1 0 02/27/2020 50 100 z (m) 150 200 3.5 3 2.5 2 1.5 1

0.5 0.8 Bunching factor &K 4 0 20 0 200 q x 10 50 0.6 rs 40 w | r &r 2 5 s 60 b 3 0 On-axis |a (m) 5 Normalized a Power (TW) Higher power, 4.9 TW vs. 4.4 TW, increase by 11%, Higher on-axis radiation field, small radiation radius; higher F t. Kq aw

0 50 100 z (m) 150 200 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 0.6 0.4 0.2 0 35 200 m Hard X-ray FEL w/Break Sections, 1m/4.4m 100 1 4 80 0.8 rs 0.6 t 60 40 0.4 s 2 F b 3 1 x 10 50 100 z (m) 150

w s 3 2 1 0 0 50 100 z (m) 150 0 200 0 Increase in rs and reduction in as0, cascade into increase in Yr and decrease in Ft. 50 100 z (m) 150 200 50 100 z (m) 150 200 -5 4 0 0 200 0.2 rb 50

02/27/2020 100 z (m) 150 200 3.5 0.8 3 2.5 Bunc hing fac tor 5 0 20 q 0 On-ax is |a | r & r (m) 5 Normalized a & K P ow er (T W) Optimization for FEL w/o break-section (red) vs. with/break section (black). Power reduction, 46% (from 4.9 TW to 2.7 TW), partially due to less magnetic length (23% reduction), partially due to vacuum diffraction of radiation in break sections. Kq 2 1.5 1 aw 0.5 0 50 100 z (m) 150

200 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 0.6 0.4 0.2 0 0 36 Dependence of the Available Maximum Radiation Power on Various Parameters In Ab se nc e of S ide ba nd Gr ow th Initial electron beam parameters Approximately quadratically with the initial beam current, Linearly with the initial beam energy, The smaller the emittance and energy spread of the electron beam, the larger the max imum extractable radiation power. Input radiation field parameters The impact of the radiation seed parameters, such as the radiation input power and th e initial spot size, is relatively small, especially when the seed power exceeds a specifi c value (e.g., 1 MW for LCLS-II). Undulator system parameters Helical undulator produces higher radiation power than linearly polarized undulator, b y a factor of about 1/[JJ]2; An undulator with shorter break sections produces higher radiation power; A longer undulator can be optimized for higher maximum radiation power and perfor ms best with a taper with smaller |aw| relative to what is true for shorter undulators. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 37 Impact of Sideband Growth on Optimization Sideband growth generally occurs after initial saturation and will be excited by the SASE components which grow in the exponential gain regime and ori ginate from the shot noise on the electron beam. From the viewpoint of energy conservation, the sidebands power accumulat es along the undulator associated with the slowly varying |as| in the tapered

region, generally reducing the energy gain of the primary wave and causing a more rapid detrapping due to the resultant decrease of longitudinal coherenc e. Eventually, these all lead to saturation of the radiation power at a much earlie r point in z than would be true in their absence. A systematic theoretical and simulation analysis of these effects is under way. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 38 Impact of Sideband Growth in Hard X-ray FEL 200 m Hard X-ray FEL w/o break sections with time-dependent effects (blue) vs. w/o time-dependent effects (red). ---time-dependent effects lead to significant detrapping by around 100m, and the average radiation power and on-axis |as| reach saturation much earlier in z and, most importantly, at much smaller values than is true for the single-frequency run. 5 4 4.5 3.5 4 1 0.9 0.8 0.7 2.5 2 1.5 0.6 2 t 3 2.5 F s On-axis |a | P ow er (T W) -5 3 3.5 0.3

0.2 0.5 0.5 0 02/27/2020 50 100 z (m) 150 200 0 0.5 0.4 1.5 1 1 0 x 10 0.1 0 50 100 z (m) 150 200 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 0 0 50 39 100 z (m) 150 200 Conclusions

A new physical model of tapered FEL, has good success in predicting the beh avior with z of the electron and radiation beams; A general description of FEL process and saturation mechanism; Multi-dimensional optimization based on GENESIS single-frequency simulati ons; A reasonable variation of the transverse focusing can to some extent further enhance energy extraction efficiency; Dependence of the available max. radiation power on various parameters; Preliminary study for the impact of the sideband growth. 02/27/2020 Y. Jiao, FLS 2012, Jefferson Lab @19 slides 40 Averaging Ft over Thousands of Slices 1 0.9 0.8 0.7 F t 0.6 0.5 0.4 0.3 0.2 0.1 0 0 02/27/2020 50 100 z (m) Y. Jiao, FLS 2012, Jefferson Lab @19 slides 150 200 41