L1 intro math - Northern Illinois Universitynicadd.niu.edu/~piot/phys_790_fall2014/Slides/L1_intro_math.pptx.pdfL1_intro_math.pptx Author: Philipe Piot Created Date: 8/26/2014 12:26:55

PHYS 790-‐D: Special topics, Fall 2014:

Charged-‐par*cle beams and waves

(fields) interac*ons

P.P. 08/26/2014

PHYS 690-‐D Special topics in Beam Physics, Fall 2014 1

Charged-‐par*cle beams and waves (fields) interac*ons


electron, posiBon muons, protons, ions (e.g. Carbon)

directed energy ensemble of parBcle w pz>>(px,py)

propagaBng e.m field

e.m. field produced by a parBcle “velocity” or “radiaBon” fields

transfer of energy or momentum

Introduc*on

•  no textbook required note (slides, papers, or short notes) will be provided most of the Bme,

•  grading: •  Homework: 50 % of overall grade, •  Midterm: 20 % of overall grade, •  Final: 30% of overall grade.

•  biweekly homework, midterm on Tues. 10/21 and final will be a small project (read, understand, and summarize a paper of your choice; more details to come).


Course descrip*on This course will discuss basics of charged-‐parBcle beams and wave interacBons and their use in a variety of applicaBons: radiofrequency parBcle accelerators and electron sources, radiofrequency power generators, free-‐electron lasers, laser-‐based and self-‐field acceleraBon techniques, and other assorted "exoBc" topics. Some knowledge of electromagneBsm, electrodynamics, and classical mechanics is desired and will be reviewed as necessary (all within 1st of graduate studies). Some formalism on charged-‐parBcle beams (phase space, staBsBcal descripBons, etc...) and electromagneBc wave and laser (Wigner funcBon, Gaussian and Fourier opBcs) descripBon will also be introduced. One of the goals of this course is to make a connecBon between parBcle and photon beams formalism and their interplay when discussing the interacBon between these two classes of beams. The class is not intended to be a comprehensive beam physics class in the sense that only beam-‐physics concepts required will be introduced. The course will provide an overview of forefront researches being carried in beam physics and connect them with classical mechanics and electromagneBsm formalisms.


Course descrip*on


Math refresher

•  derivaBons of some concepts will be outlined, some of the details, digressions, or extensions will be let as homework or for fun.

•  mathemaBcal tools needed: – coordinate systems (mostly Cartesian + cylindrical), vector and matrix manipulaBons,

– complex analysis, – Fourier transformaBons.


coordinate systems

•  will mostly use cartesian system ( is generally choosen as the direcBon of propagaBon)

•  for some topics we will switch to cylindrical coord.


x

y

zz

x = r cos(✓)

y = r sin(✓)z = z

Cylindrical and Cartersian coordinates


operators in cylindrical coordinate


trajectory of a single par*cle

•  classical mechanics use where posiBon

canonical momentum. •  form a set of canonical-‐conjugate variables

•  alternaBve descripBon use divergence but are not canonical conjugates.


(x,p)x ⌘ (x, y, z)p ⌘ (p

x

, py

, pz

)(x,p)

x

0 ⌘ p

x

/p

z

y0 ⌘ py/pz(x, x0)

trajectory of a single par*cle

•  are pracBcal and are use as a basis of “ray tracing” in magneBc and photonic opBcs

•  the same is valid for the other degrees of freedom and .

•  We implicitly assume that the three degree of freedom are decoupled – can consider the parBcle moBon in each degree of freedom independently from the others.


(x, x0)

(y, y0) (z, z0)

ABCD formalism

•  in a given d.o.f. a single parBcle can be “advanced” via a matrix mulBplicaBon


accelerator beamline

X0 = (x0, x00) Xf = (xf , x

0f )

Xf = RX0

transfer matrix

valid in the “paraxial” approximaBon and assume system is linear

ABCD formalism (2)

•  for a beamline with many component one can mulBply each transfer matrix

•  only works for lumped elements (in sequence)


accelerator or opBcal beamline with n components

1

X0 = (x0, x00) Xf = (xf , x

0f )

2 n

Xf = RnRn�1...R3R3R1X0

example driE (free) space

•  consider moBon in free space


x0x

00

x

0f

xfdirecBon of moBon L

xf = x0 + Lx

00

x

0f = x

00

sta*s*cal descrip*on

•  nth moment of a funcBon

•  1st order is averaging •  2nd order gives variance •  “root-‐mean-‐square” is oben defined as the centered 2nd order moment:


huni =Z +1

�1f(u)undu

f(u)

�u ⌘ [h(u� hui)2i]1/2

sta*s*cal descrip*on (mul*ple dimensions)

•  nth-‐mth coupled moment of a funcBon

•  extensively used to describe the staBsBcal property of a beam

•  somewhat use to describe laser pulse (opBcian like to use full-‐width half max instead) we will use staBsBcal (e.g. RMS) quanBBes


f(u, v)

hunvmi =Z +1

�1umvnf(u, v)dudv

example 1-‐D Gaussian

•  relaBon between rms & FWHM

•  note a Gaussian bunch w. charge :


s/�

e�s2/(2

�2)

rms

FWHM fwhm = 2

p2 log 2⇥ �

I(t) =Qp2⇡�t

e� t2

2�2t

instantaneous current

bunch rms duraCon

Q

Fourier transform

•  consider a temporal signal •  this signal can be seen as a where is the Fourier transform of


S(t)

S(t) =1

2⇡

Z +1

�1S(!)ei!td!

S(!) =

Z +1

�1S(t)e�i!tdt

S(!) S(t)


Some useful rela*ons

example of Fourier transforms


Documents

L1 intro math - Northern Illinois Universitynicadd.niu.edu/~piot/phys_790_fall2014/Slides/L1_intro_math.pptx.pdfL1_intro_math.pptx Author: Philipe Piot Created Date: 8/26/2014 12:26:55