He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip Duxbury, Chong-Yu Ruan, Kyoko Makino, Martin Berz. Michigan State University

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1 The Multiple Level Fast Multipole Method in the Differential Algebra Framework for Space Charge Field Calculation and Simulations on femtosecond electron imaging and spectroscopy He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip Duxbury, Chong-Yu Ruan, Kyoko Makino, Martin Berz Michigan State University FEIS13, Key West, FL He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip TheDuxbury, MultipleChong-Yu Level FastRuan, Multipole Kyoko Method Makino, the Martin Differential Berz Alg

2 Outline Multiple level fast multipole algorithm (MLFMA) in the differential algebra framework Simulation results on femetosecond electron imaging and spectroscopy He Zhang

3 I. Multiple Level Fast Multipole Algorithm in DA framework He Zhang

4 Space charge effect is an important effect in the photo-emission process that generates the femtosecond electron bunch To simulate space charge effect, one needs an algorithm that has good efficiency and good accuracy. The MLFMA is a good choice, because it scales linearly with the number of particles for any arbitrary charge distribution (grid free) does NOT artificially smooth the field Using DA, one can calculate not only the field, but also its high order derivatives represent the multipole/local expansions in Cartesian coordinates naturally considerably simply the math He Zhang

5 The DA based MLFMA is developed as a package for COSY Infinity COSY is a scientific computing system DA and TM data types (ODE, flows, PDE, automatic differentiation, range bounding, etc.) beam physics package More information on He Zhang

6 Two operations in COSY: Automatic Taylor expansion of a function In COSY, f (x + δx) = f (x) + f (x)δx + 1 2! f (x)δx ! f (x)δx f (x + da(1)) = f (x) + f (x)da(1) + 1 2! f (x)da(1) ! f (x)da(1) Composition of two maps G(x) = G(F ) F (x), or G(x) = G(F (x)) In COSY, it can be done by the command POLVAL.

7 Fast Multipole Method (FMM), L.Greengard and V.Rokhlin, 1987 Cut the whole region into boxes of hierarchical tree structure. Multipole expansions and local expansions. For any arbitrary distribution, scales linearly with the number of particles. He Zhang

8 b He Zhang, Martin Berz The Fast Mulitpole Algorithm in the Differential Algebra Fr

9 Multipole expansion from charges (for the childless boxes) R(x, y,z) R(x, y,z) R1 R2 φ R ( R i) = φ R ( M) M S(0, 0, 0) S(0, 0, 0) Ri

10 φ = = n i=1 i=1 q i (x xi ) 2 + (y y i ) 2 + (z z i ) 2 n d r q i 1 + (x 2 i + yi 2 + zi 2)d r 2 2x i d x 2y i d y 2z i d z with = d r φ c2m x d x = x 2 + y 2 + z, 2 dy = y x 2 + y 2 + z, 2 z d z = x 2 + y 2 + z, 2 dr = dx 2 + dy 2 + dz 2, n { } φ c2m = q i / 1 + (xi 2 + yi 2 + zi 2)d r 2 2x i d x 2y i d y 2z i d z. i=1 ( a ) p+1 1 Error ɛ C r r a, where C = n q i and r i a for any i. i=1

11 Translate the position of a multipole expansion R(x, y, z) R(x, y, z ) M S (x s, y s, z s) S (0, 0, 0) M S(0, 0, 0) φ R ( M) = φ R ( M ) S( x s, y s, z s)

12 In parent box frame, new DA variables are chosen as d x = x x o r 2 = x r 2, d z = z z o r 2 = z r 2, d y = y y o r 2 = y r 2 Relation between the old and new DA variables. d x = (d x + x o (d 2 x + d 2 y + d 2 z )) R, d y = (d y + y o (d 2 x + d 2 y + d 2 z )) R, Mm2m d z = (d z + z o (d x 2 + d y 2 + d z 2 )) R, with R = (x o 2 + y o 2 + z o 2 )(d x 2 + d y 2 + d z 2 ) + 2x od x + 2y od y + 2z od z.

13 In child box frame φ = d r φ c2m. In the parent box frame φ = d r R φ m2m = d r φ m2m with d r = d x 2 + d y 2 + d z 2, and φ m2m = φ c2m M m2m, where M m2m is the map from the old DA variables into the new DA variables

14 Convert a multipole expansion into a local expansion R(x, y, z) R(x, y, z ) O(xo, yo, zo) L O(0, 0, 0) M φ R ( M) = φ R ( L) S(0, 0, 0) S( x o, y o, z o)

15 New DA variables in the observer frame d x = x x o = x, d y = y y o = y, d z = z z o = z. The relation between the new and the old DA variables d x = (x o + d x) R, d y = (y o + d y ) R, with d z = (z o + d z) R. R = Mm2l 1 (x o + d x) 2 + (y o + d y ) 2 + (z o + d z) 2.

16 The multipole expansion in the source frame φ = d r φ The local expansion in the observer frame φ = R φ m2l = φ m2l where R is converted from d r, φ m2l = φ M m2l, and M m2l is the map between the DA variables. Error ( ) p+1 a ɛ C r o 1 r o a + C ( ) r p+1 b 1 b r.

17 Translate a local expansion from a parent box to its child boxes R(x, y, z) R(x, y, z ) O (x o, y o, z o ) L O(0, 0, 0) φ R ( L) = φ R ( L ) L O (0, 0, 0) O( x o, y o, z o )

18 DA variables in the child box frame d x = x o + d x, d y = y o + d y, d z = z o + d z. Ml2l The local expasion in the parent box frame is φ m2l. The local expansion in the child box frame is φ = φ m2l M l2l = φ l2l, where M l2l is the map between the old and the new DA variables.

19 Calculate the local expansion from charges. S (xo, yo, zo) S (0, 0, 0) φ( R i) φ ( L ) R1 R2 R 1 R 2 S(0,0, 0) S( xo, yo, zo) R2 R 2

20 In the observer (small box) frame, the new DA variables are d x = x x o = x, d y = y y o = y, d z = z z o = z. Then the local expansion is Error φ L = = n i=1 n i=1 q i (x xi ) 2 + (y y i ) 2 + (z z i ) 2 q i (x o x i + d x) 2 + (y o y i + d y ) 2 + (z o z i + d z) 2 ( ) r p+1 1 ɛ C b b r

21 Now we have the potential expressed as a polynomial of coordinates up to order p. Take the derivative of a coordinates to get the field expression in a polynomial of coordinates up to order p 1. Submit the charge positions into the expression to calculate the potential/field.

22 Calculate the field from the multipole expansion. The multipole expansion is φ = d r φ, then E x = { φ d x (d 2 r 2d 2 x ) + 2 φ d y d x d y + 2 φ d z d x d z + φ d x } d r E y = {2 φ d x d y d x φ d y (d 2 r 2d 2 y ) + 2 φ d z d y d z + φ d y } d r E z = {2 φ d z d x + 2 φ d z d y φ (dr 2 dz 2 ) + d x d y d φ d z } d r z with d r = dx 2 + dy 2 + dz 2.

23 Description of the MLFMA Construct the hierarchical box structure (partial tree). Upwards: Calculate the multipole expansions for all the boxes. Downwards: For each box, check its the relation with other boxes and operate according to the above table. Then translate the local expansion from parent boxes to the child boxes. Calculate the potential/field, which comes from direct calculation and multipole or local expansions.

24 Numerical experiments Compare the MLFMA with direct calculation o oe o ( C Atn CDttt Ctt Ct C trc trtc r r o arttt Crtdl trttc CDttt C Atn C Atg C Atk ooooo o o o

25 Numerical experiments Accuracy and computation time txtw r txttw txtttw w otr w ot w ot w ot w ot w wt wtt wbttt r ri n

26 Numerical experiments m m mnm m me f m m

27 Simulation II. Simulation results on femetosecond imaging and spectroscopy He Zhang

28 Simulation Experiment setup: Simulation model: Three-step model of photoemission Extraction field F a, space charge field, surface image charges Up to several millions of macro-particles in simulation He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip TheDuxbury, MultipleChong-Yu Level FastRuan, Multipole Kyoko Method Makino, the Martin Differential Berz Alg

29 Simulation (a) Electron bunch profiles (b) Bunch length vs. Ne emit (c) Ne emit vs. Ne 0 showing virtual cathode effect 1.Z. Tao, H. Zhang, P. M. Duxbury, M. Berz, C.-Y. Ruan, Journal of Applied Physics 111 (2012) J. Portman, H. Zhang, Z. Tao, etc., Applied Physics Letter, in press, 2013 He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip TheDuxbury, MultipleChong-Yu Level FastRuan, Multipole Kyoko Method Makino, the Martin Differential Berz Alg

30 Simulation (a) Initial electron bunch profiles in x-z plane (b) Electron bunch profiles in x-z plane at 100ps under strong F a (10 MV/m) (c) Electron bunch profiles in x-z plane at 100ps under weak F a (0.32 MV/m) (d) Electron bunch profiles in x-z plane at 100ps under weak F a (0.32 MV/m) with a pinned image field (e) Electron charge distributions under different fields He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip TheDuxbury, MultipleChong-Yu Level FastRuan, Multipole Kyoko Method Makino, the Martin Differential Berz Alg

31 Simulation (a) ε x for different field and different Ne emit. Affected by the virtual cathode effect. (b) ε y for different field and different Ne emit. Not affected by the virtual cathode effect. Driven by internal field and nonlinearities. He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip TheDuxbury, MultipleChong-Yu Level FastRuan, Multipole Kyoko Method Makino, the Martin Differential Berz Alg

32 Simulation Figure: 6D emittance ε xε y ε z versus Ne emit for the extended electron sources with sizes σr l (100 µm, 1mm), thermionic guns, Schottky, cold (CFEG) and heated (HFEG) field-emission guns 1 Degeneracy (η = B 6D ε 0, B 6D = N e/(ε 2 x εz)) can improved with flat photoemission cathode until the virtual cathode threshold. Increasing extraction field is helpful, because it defers the onset of virtual cathode effect. Large emitting area increases the Ne emit, but not necessary degeneracy (brightness). No gain by using sharp emitters (FEGs or atom-sized emitters) due to their poor emittance scaling with N e. 1 Science of Microscopy, Eds. P.W. Hawkes, and J.C.H. Spence (Springer, New York, 2008) He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip TheDuxbury, MultipleChong-Yu Level FastRuan, Multipole Kyoko Method Makino, the Martin Differential Berz Alg

33 Summary 1.The multiple Level Fast Multipole Algorithm Grid-free, works for any arbitrary charge distribution with an efficiency O(N). Calculate both the potential/field and its derivatives 2. Studied the virtual cathode effect quantitatively in femtosecond electron generation Ne emit, ε x, η Increase F a can defer the onset of virtual cathod effect 3. Transition into virtual cathode regime might be described by two fluids 4. η is affected by different factors. He Zhang (now Jefferson Lab), Zhensheng Tao, Jenni Portman, Phillip TheDuxbury, MultipleChong-Yu Level FastRuan, Multipole Kyoko Method Makino, the Martin Differential Berz Alg

34 He Zhang