# Self-Consistent Computation of Electromagnetic Fields and Phase Space Densities for Particles on Curved Planar Orbits Page: 4 of 5

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and pB(r, s) has its support for r small, we have

PB(z(R,u), x(R);z(R,u) + ),u) ~pB(t; 0,U), where

= MT(,U)(R - R,(3,u)). Thus

PL(R; u) ~PB(r; rv) (19)

JL (R; u) 0,c [pB (F; j3 )t(z + 0hy) +

<rF; ,u)n(z + )hu)], (20)

where the JL approximation is derived similarly to that for

PL-

There is a subtlety in the second transformation caused

by interchanging the roles of a and s as independent and

dependent variables. The phase space density transforma-

tion and the approximations are discussed in detail in [3].

UNPERTURBED SOURCE MODEL (UPS)

In this model we uncouple the Vlasov-Maxwell system

so that it becomes a Liouville-Maxwell system. Here the

source evolves with no self-fields and the fields are calcu-

lated from this unperturbed source. The Vlasov equation

thus becomes a Liouville equation which defines the evo-

lution of the beam frame phase space density.

We have focused on the bunch compressor with an ini-

tial Gaussian PSD density in the beam frame. Because (13)

with out self-forces is a linear system, the unperturbed PSD

is Gaussian at each s and thus the Lab frame charge density

in (19) is Gaussian. This fact speeds up the field calcula-

tion considerably. We define an {s,,} grid along the ref-

erence orbit and a (Z, X) grid at each s, which contains

the bunch and is based on the evolution of the unperturbed

charge density. We then calculate the self-forces (14) on

this grid. We could integrate the Liouville equation using

the method of local characteristics discussed below. To date

we have proceeded as follows. We generate an initial en-

semble of beam frame phase space points. We then move

the points from s, to si using (15) with the self-forces at

s,. We interpolate to determine the self-forces at points off

the (Z, X) grid.

The UPS model is not self-consistent nor is it the first

term in a systematic perturbation expansion in the size of

the self-forces. Nevertheless, it has been helpful in the de-

velopment of our self-consistent code because it is a good

testing ground for our numerical and approximation pro-

cedures and computation with a Gaussian is fast. Fur-

thermore, it may give a good approximation to the self-

consistent case in some parameter range, [4, 5, 6].

Two particular points are worth mentioning. Our study

of the UPS has given us insight into how to construct a

space-time grid for the SCMC algorithm described below.

In addition, we have found that for certain parameters, e.g.,

a small uncorrelated energy spread, a moving grid will be

necessary for the PSD.SELF-CONSISTENT VLASOV-MAXWELL

ALGORITHMS

We have discussed our method for calculating the fields

in the lab frame and the determination of the lab frame

charge and current densities from the beam frame PSD.

Here we discuss two approaches for coupling the numerical

integration of the Vlasov equation with field calculation.

Self-Consistent Monte Carlo (SCMC) Method

Here the basic algorithm is the same as in the UPS case

except the field calculation cannot be done up front and

PL can not be computed analytically. We discuss the ba-

sic algorithm and contrast it with the PIC method used in

Vlasov-Poisson codes.

1. We generate an ensemble of IID phase space points

from the density fB (z, x, Pz, PX; 0) using the rejection

method. As an improvement we investigate a Quasi-

Monte Carlo approach "which seeks to construct a set

of initial points that perform significantly better than

the average of a Monte Carlo approach", see [7]. A

similar procedure could be used in a PIC code.

2. We create a globally smooth lab frame charge den-

sity from the scattered beam frame phase space points.

We fit the data with a finite Fourier series where the

Fourier coefficients are calculated, as in Monte Carlo

integration, from the scattered data. This is a tech-

nique used in statistical estimation, see e.g., [8]. We

have found that a smooth representation is quite im-

portant as Borland found for Elegant before us. Note

that this is a meshless proceedure in contrast to the

charge deposition of a PIC code.

3. We calculate the fields from the history of the Fourier

coefficients using our field formula in (10). In a PIC

code the Poisson equation is solved at this step (of

course, the history of the beam is not needed).

4. We use 3) to advance the particles in the interaction

picture of (15). In a PIC code 3) is also used to ad-

vance the particle positions.

5. The procedure is iterated going back to 2.

We note that our approach can treat a Vlasov-Poisson sys-

tem as a special case. Also, our method is not a macropar-

ticle method in the sense of modeling an N particle bunch

with M < N macroparticles and letting them interact as

point particles. We assume the electron bunch is well ap-

proximated by a continuum evolved by the Vlasov equa-

tion and hope that our algorithm approximates the true

Vlasov dynamics. This is analogous to Monte Carlo inte-

gration where convergence follows from the strong law of

large numbers and the central limit theorem and we hope to

prove convergence. However, even though we expect con-

vergence, the calculation of the PSD is probably beyond

current or near future computing capability.

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Ellison, J.A.; Bassi, G.; Heinemann, K.A.; U., /New Mexico; Venturini, M.; /LBL, Berkeley et al. Self-Consistent Computation of Electromagnetic Fields and Phase Space Densities for Particles on Curved Planar Orbits, article, November 2, 2007; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc888527/m1/4/: accessed September 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.