Computation of Transfer Mapsfrom Surface Data withApplications to ILC DampingRing Wigglers

Chad Mitchell and Alex DragtUniversity of Maryland

March 2007 Version

Simulations indicate that the dynamic aperture of proposed ILC DampingRings is dictated primarily by the nonlinear properties of their wigglertransfer maps. Wiggler transfer maps in turn depend sensitively on fringe-field and high-multipole effects. Therefore it is important to have detailedmagnetic field data including knowledge of high spatial derivatives. Thistalk describes how such information can be extracted reliably from 3-dimensional field data on a grid as provided, for example, by various 3-dimensional field codes available from Vector Fields. The key ingredient isthe use of surface data and the smoothing property of the inverse Laplacianoperator.

Abstract

Surface Methods in ILC DR Lattice Design• Developing methods to produce accurate transfer maps for realistic

beam-line elements.

• Such methods use accurate 3-d field data provided by finite elementmodeling (Opera) to incorporate fringe field effects, nonlinearmultipoles.

• Once accurate transfer maps have been found for individual beam-lineelements, one can determine all single-particle properties of the ring:dynamic aperture, tunes, chromaticities, anharmonicities, linear andnonlinear lattice functions, etc.

• Key is the use of surface data to compute interior data. Surface mustenclose design trajectory and lie within all iron or other sources.

Objective

• To obtain an accurate representation of the interior field that isanalytic and satisfies Maxwell equations exactly. We want avector potential that is analytic and .

• Use B-V data to find an accurate series representation ofinterior vector potential through order N in (x,y) deviation fromdesign orbit.

• Use a Hamiltonian expressed as a series of homogeneouspolynomials

• We compute the design orbit and the transfer map about thedesign orbit to some order. We obtain a factorized symplecticmap for single-particle orbits through the beamline element:

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M = R2e: f3 :e: f4 :e: f5 :e: f6 :...

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K = −(pt + qφ)2

c 2− (p⊥ − qA⊥ )

2 − qAz = hs(z)Ks(x, pxy, py,τ, pτ )s=1

S

∑

0=×∇×∇ A

∑=

=L

ll

xlx yxPzazyxA

1

),()(),,(L=27 for N=6

S=923 for N=6

Advantages of Surface Fitting

Maxwell equations are exactly satisfied.

Error is globally controlled. The error must take its extrema onthe boundary, where we have done a controlled fit.

Careful benchmarking against analytic results for arrays ofmagnetic monopoles.

Insensitivity to errors due to inverse Laplace kernel smoothing.Improves accuracy in higher derivatives. Insensitivity to noiseimproves with increased distance from the surface: advantageover circular cylinder fitting.

•Solenoids and multipoles -- circular cylinder (M. Venturini)

•RF cavities -- circular cylinder (D. Abell)

•Wiggler magnets -- elliptical / circular cylinder (C. Mitchell, M. Venturini)

•Bending dipoles -- bent box / bent cylinder (C. Mitchell, P. Walstrom)

Realistic transfer maps can now be computed for all beamline elements of the damping ring using surface methods:

Fitting Wiggler Data Using Elliptical Cylinder

x

Data on regular Cartesian grid

4.8cm in x, dx=0.4cm

2.6cm in y, dy=0.2cm

480cm in z, dz=0.2cm

Field components Bx, By, Bz in onequadrant given to a precision of 0.05G.

Fit data onto elliptic cylindrical surfaceusing bicubic interpolation to obtain thenormal component on the surface.

Compute the interior vector potential andall its desired derivatives from surface data.

y

x4.8cm

2.6cm

11.9cm

3.8cm

Place an imaginary elliptic cylinder betweenpole faces, extending beyond the ends of themagnet far enough that the field at the ends iseffectively zero.

fringe region

Fit to the Proposed ILC Wiggler Field Using Elliptical Cylinder

Note precision of data.

Fit to vertical field By at x=0.4cm, y=0.2cm.

Residuals ~ 0.5G / 17kG

Fit to the Proposed ILC Wiggler Field Using Elliptical Cylinder

No informationabout Bz wasused to createthis plot.

Fit tolongitudinal fieldBz at x=0.4cm,y=0.2cm.

Dipole Field Test for Elliptical Cylinder

Pole location: d=4.7008cm

Pole strength: g=0.3Tcm2

Semimajor axis: 1.543cm/4.0cm

Semiminor axis: 1.175cm/0.8cm

Boundary to pole: 3.526cm/3.9cm

Focal length: f=1.0cm/3.919cm

Bounding ellipse: u=1.0/0.2027

y

xd

z

x

y

+g

-g

160cm

• Simple field configuration in which allderivatives of the field and on-axis gradientscan be determined analytically.

• Tested for two different aspect ratios: 4:3and 5:1.• Direct solution for interior scalar potentialaccurate to 3x10-10: set by convergence androundoff.• Computation of on-axis gradients C1, C3,C5 accurate to 2x10-10 before final Fouriertransform accurate to 2.6x10-9 after finalFourier transform.

Remarks about Geometry

For the circular and elliptical cylinder, only the normal component of on the surface was used to determine interior fields.

The circular and elliptical cylinder are special in that Laplace’s equationis separable for these domains.

Laplace’s equation is not separable for more general domains, e.g. a bentbox.

Surface data for general domains can again be used to fit interior dataprovided both and are available on the surface. Analogoussmoothing behavior is again expected to occur.

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Bnormal

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ψ

Laplace equation separates fully in11 orthogonal coordinate systems.

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∇2Ψkl = 0 S1Ψkl =k 2Ψkl S2Ψkl = l2Ψkl

straight-axis geometries

Separated solutions aresimultaneous eigenstates of twosecond-order symmetry operators:

Pro: Solutions have known, optimalconvergence properties.

Con: Difficult functions, techniquesstrongly dependent on particularsurface geometry.

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P32

The Bent Box and Other Geometries

The previous techniques can be used effectively for straight-axis magneticelements.

For elements with significant sagitta, such as dipoles with large bending angles, we must generalize to more complicated domains in which Laplace’s equation is not separable.

Given both and on a surface enclosing the beam, the magneticvector potential in the interior can be determined by the integration of surfacedata against a geometry-independent kernel.

ψ

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Bnormal

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A(r) = −14π

n( ′ r ) × F( ′ r )| r − ′ r |

d ′ S +14π

′ ∇ × F( ′ r )| r − ′ r |

dV 'V∫

S∫

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Φ(r) = −14π

n( ′ r ) ⋅ F( ′ r )| r − ′ r |

d ′ S − 14π

′ ∇ ⋅ F( ′ r )| r − ′ r |

dV 'V∫

S∫

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F =∇ × A +∇Φ

Application of Helmholtz Theorem

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At (r) = −14π

n( ′ r ) × B( ′ r )| r − ′ r |

d ′ S S∫

Φn (r) =14π

n( ′ r ) ⋅ B( ′ r )| r − ′ r |

d ′ S S∫

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∇ × An =∇Φn

To compute a Lie map from the Hamiltonian we require alone.

Decompose the interior field into a divergence and curl

of the form:

In the absence of sources, we may write potentials in terms of surface data:

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B =∇ ×A

?

normal component

tangential components

Dirac monopole construction

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Am(r) = −g4π

d ′ r ×∇ 1| r − ′ r |

L∫

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Am(r) =gm × (r − ′ r )

4π | r − ′ r | (|r − ′ r |−m ⋅ (r − ′ r ))

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Fm(r) =∇ × Am (r) = −g4π

∇1

| r − ′ r |

We exploit the vector potential of a Dirac monopole of charge g:

where L is a curve extending from the monopole source to infinity.We choose L to be a ray in the direction defined by . Then

This vector potential has the desired property that

at all points away from the ray L. Furthermore, .

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g€

L

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m

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m

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∇ ⋅ Am = 0

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An (r) = [n( ′ r ) ⋅ B( ′ r )]G n (r; ′ r ,m( ′ r ))d ′ S S∫

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G n (r; ′ r ,m) =m × (r − ′ r )

4π | r − ′ r | | r − ′ r |−m ⋅ (r − ′ r )[ ]

We define a vector field

where the kernel is given by the vector potential of a monopole at a point :

Here is allowed to vary over the surface. This vector potential will satisfy

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∇ × An (r) = [n( ′ r ) ⋅ B( ′ r )]∇ ×Gn (r; ′ r ,m( ′ r ))d ′ S S∫

=[n( ′ r ) ⋅ B( ′ r )]

4π∇

1| r − ′ r |

S∫ d ′ S =∇Φn€

m

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A = An + At

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At (r) =14π

ψ( ′ r )G t (r; ′ r )d ′ S S∫

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′ r

We also rewrite in terms of a scalar potential on the surface :

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At

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B =∇ψ

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G t (r; ′ r ) =n( ′ r ) × (r − ′ r )| r − ′ r |3

The kernels and are analytic in the interior. Given a point along the design orbit, we may construct a power series for about by integrating against the power series for the ’s . Writing ,

We have the kernels:

where m is a unit vector pointing along some line that does not intersectthe interior (a Dirac string), and n is the unit normal to the surface at .

∑=

′=′+L

dd yxP1

),(),;(),;(α

αα δδδ nrrGnrrrG

yx yx eer ˆˆ δδδ +=

r′

nG tG

G

Adr

This has been implemented numerically.

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G n (r; ′ r ,m) =m × (r − ′ r )

4π | r − ′ r | | r − ′ r |−m ⋅ (r − ′ r )[ ] normal component

tangential components

1) Vector potential A at any interior point (gauge specified by orientationof strings)

2) Taylor coefficients of A about any design point through degree N

which in turn are used to compute…3) Interior field B at any point4) Taylor coefficients of B about any point5) as Taylor series through degree N-1

Code accepts as input 3d data of the form on a mesh and willproduce as output:

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∇ ⋅ A, ∇ × A, ∇ ⋅ B, ∇ × Bbenchmark tools

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(B,ψ)

Code produces interior fits that satisfy Maxwell’s equations exactly and even if therequired surface integrals are performed approximately.

Transfer maps are then computed from the Taylor expansion of along the designorbit.

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A

Benchmarks for various domains

The procedure has been benchmarked for the above domains using arrays of magnetic monopoles to produce test fields. Power series for the components of at a given point are computed from the power series for . These results can be compared to the known Taylor coefficients of the field. We find, using a surface fit that is accurate to 10-4 , that all computed coefficients are accurate to 10-6.

Similarly, we verify that , , and to machine precision.

drB

A

0=⋅∇ B 0=×∇ B

torus

bent box

straight box

sphere

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∇ ⋅ A = 0

Box Fit to ILC Wiggler Field

As an additional test, data provided by Cornell of the form at grid points was fit onto the surface of a nearly-straight box filling the domain covered by the data. Box: 10 x 5 x 480cm Mesh: 0.2 x 0.4 x 0.2cm

Using this surface data, the power series for the vector potential was computedabout several points in the interior. From this, the value of was computedat various interior points and compared to the initial data.

0.9160.6260.161Bz (G)0.0542.5270.299By (G)0.2300.1870.0417Bx (G)(0, 1.4, 31.2) cm(2, 2, 1) cm(0.4, 0.2, 31.2) cmDifference

Peak field=17kG Largest error/peak ~ 10-4

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B

),,,( ψyxz BBB

interior points

Properties of solution:

• Guaranteed to satisfy Maxwell’s equations exactly.• Sources of error are controlled: surface interpolation andmultivariable integration--can be treated as error in the surfacefunctions and .• Error is globally bounded.• High-frequency random numerical errors on the surface aredamped in the interior (smoothing).

Challenges:

Reduce runtime. Operation count ~ LMNp. Computationcan be parallelized.

L - number of points evaluated along design orbitM - number points on surface meshN - degree of coefficients required by Lie mapp(N) - operations required by TPSA expansion of kernel

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Bnormal

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ψ

Smoothing of surface errors: an illustration

x

yz

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z =1

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z = 0

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δBn ,δψ

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g(k, ′ k ) ≤ Cexp(−λ ⋅k)exp(− ′ λ ⋅ ′ k )

Surface harmonic added.

Kernels are analytic in , soFourier coefficients decay as

Error in the plane is suppressed:

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δAα = π 2 gk∑ (k, ′ k )sin(kxπx)sin(kyπy)

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′ k = ( ′ k x, ′ k y )

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(x, y, ′ x , ′ y )€

x

€

y

€

kx = 3, ky = 2

€

x

€

y€

δAx

€

x

€

y

€

δAy

€

5 ×10−4

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1×10−3

Contributions to ILC Damping Ring Studies

• Characterize all beamline elements by realistic symplectic transfermaps in Lie form

• Potential to compute all single-particle properties of the DR from asingle combined one-turn map, using only real field data for the entirering.

• Includes all fringe-field effects on dynamic aperture, tunes,chromaticities, anharmonicities, linear and nonlinear lattice functions,etc.

• Can be used to check models of beamline elements, but use of modelsis no longer required.

• Could produce hybrid code using both Lie maps and PIC-computedspace charge effects.

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