paper review onDipole fringe fieldK IL E A N H WA N G, C H A D M IC H E L L , R O B E R T R Y N E

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Introduction

•We briefly review some portions from the following papers that includes some analytic expression of dipole fringe field effects

[1] Karl Brown, SLAC-75, 1982

[2] Etienne Forest et at, "Second Order Fringe in MAD-X for the Module PTC", 2006

[3] Riley Molloy, Sam Blitz, "Fringe Field Effects on Bending Magnets, Derived for

TRANSPORT/TURTLE", 2013

[4] Kilean Hwang, S. Y. Lee, "Dipole fringe field thin map for compact synchrotrons", 2015

•A little more details for [4]

Karl Brown, SLAC-75

2nd order Taylor map

from design orbit

(hard-edge curvature) with

Frenet-Serret

coordinate system

Entrance Exit

Karl Brown, SLAC-75

Etienne Forest et at, "Second Order Fringe …"

2nd order (of the field strength) symplectic map from

Cartesian coordinate system

They claim that the map agrees with SLAC-75 to the 2nd order of

the field strength.

Lie map calculation is too complex. Instead, they built an effective

generating function to represent the symplectic map.

but calculation is very

complex. (even in the

Cartesian coordinate

system)

at Entrance

Etienne Forest et at, "Second Order Fringe …"

Riley Molloy, Sam Blitz, "Fringe Field ... for TRANSPORT"

On momentum 4D Matrix map + Closed orbit deviation from design orbit with

Frenet-Serret coordinate system

at Exit

On momentum 4D Matrix map + Closed orbit deviation

from design orbit with

Frenet-Serret coordinate

at Exit

Riley Molloy, Sam Blitz, " ... for TRANSPORT"

Kilean Hwang, S. Y. Lee, "Dipole fringe..."

4th order 4D (but include exact dependence on momentum deviation) Lie map from design orbit with Frenet-Serret

coordinate system. (But implementation of the Lie map is difficult. Truncated Taylor map is presented.)

Entrance Exit

Comparision with simulation

with Logistic model:

Comparision with SLAC-75

Recall, the gradient for combined function magnet was included in SLAC-75

Entrance (SLAC75) Entrance

Exit (SLAC75) Exit

Comparision with TRANSPORT

Exit

Exit

(TRANSPORT)

A little more details: Field Model

Hamiltonian

Effective Hamiltonian

Truncation / Approximation

Effective Hamiltonian

Entrance

Conclusion

We extended existing dipole fringe field map theory:

•Agrees with SLAC-75 except linear chromatic term

•Agrees with TRANSPORT closed orbit term at dipole exit.• We also derived the closed orbit and momentum offset at the entrance

•Octupole like potential is derived which diverge at hard-edge limit.