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Accelera'on II

•  Coupling of wave with a charged par4cle

•  e.m. waves need to be localized in space to maximize interac4on region.

PHYS 790-‐D Special topics in Beam Physics, Fall 2014 1

change in energy

velocity

dEdt

= qvvv.EEEapplied E field

accelera'on with a plane wave 1

•  consider a par4cle moving at an angle w.r.t. a plane wave

•  Change in energy is where we took


kkk✓

EEE = E0 cos

⇣!t� !

cz + '

⌘

z = �ct cos ✓x = �ct sin ✓

dEds

= eE0 sin ✓ cos[!t(1� � cos ✓) + �]

[from A. Chao, USPAS lecture Note (2000)]

accelera'on with a plane wave 2

•  par4cle does not have a net energy gain as in vacuum

•  this is the essence of Woodward-‐Lawson theorem which states that given a free-‐space field the effec4ve accelera4ng field is of the form

where PHYS 790-‐D Special topics in Beam Physics,

Fall 2014 3

dEds

= eE0 sin ✓ cos[!t(1� � cos ✓) + �]

1� � cos ✓ > 0

E0Eeff = ⌘E0

⌘ ⌧ 1

beyond the plane-‐wave model

•  laser beams are actually localized in space •  simplest mode is the Gaussian beam (see L4)

•  main issue is that E-‐field is orthogonal to direc4on of propaga4on…


how to characterize a laser?

•  laser field are o[en wri\en in term of the vector poten4al

•  and the normalized vector poten4al is o[en introduced

•  Amplitude of a transverse field of a linearly polarized laser pulse


EEE = �@AAA

@t, BBB =rrr⇥AAA

aaa =eAAA

mec2

E0[TV/m] =mec2k

ea0 ' 3.21

a0�[µm]

Woodward-‐Lawson’s theorem •  Theorem: A charged par4cle interac4ng with a wave in free space does not acquire a net energy gain

•  Assump4ons: – wave in vacuum and free-‐space i.e. with no boundaries,

–  the charged par4cle is ultra-‐rela4vis4c –  no sta4c fields (either E or B) are present –  the interac4on region is infinite, –  ponderomo4ve (nonlinear forces) effects are neglected


� ' 1

Woodward-‐Lawson’s theorem

•  Consequence: to have a net energy gain communicated by a wave to a charged par4cle we need: – a wave not in free-‐space: •  boundaries (as in an RF cavity), •  dielectric media, gas

– a non-‐ultrarela4vis4c charged par4cle


Reduc'on of laser phase velocity

•  scheme dubbed as inverse Cerenkov accelerator a gas with index of refrac4on is introduced

•  taking yields and a net energy gain


1� n� cos ✓ = 0

n

cos ✓ = n�1� = 1

dEds

= qE0

p1� n�2

⌧ 1

example of Cerenkov accelerator

•  accelera4on by cou-‐ pling a laser to an e-‐ beam in a gas cell,


[from Piot/Tikhoplav/Mellisinos Proc. PAC 05 (2005)]

accelera'on in semi-‐infinite vacuum

•  Consider a Gaus-‐ sian beam

•  net energy gain is

•  our simple model based on plane wave gives a similar result.


[T. Ple\ner et al., Phys. Rev. ST Accel. Beams 8, 121301 (2005)]

laser phase

semi-‐infinite vacuum: experiment

•  carried at SLAC ~2005


semi-‐infinite space

free space

[T. Ple\ner et al., Phys. Rev. ST Accel. Beams 8, 121301 (2005)]

How to make a laser with a strong axial E-‐field?

•  The Helmoltz equa4on has a solu4on known as Hermite-‐Gaussian beams

where are Hermite polynomial of rank n


Hn(x)

Hermite polynomials

•  Recurrence rela4on is •  First few polynomials are


Mul4ply by a Gaussian

Hermite polynominals 2

•  pa\erns


linearly-‐polarized HG01

•  consider HG10

•  from so this type of mode produced an axial E-‐field (along the direc4on of propaga4on)


rrr.EEE = 0

E

x

= E0xe� x

2+y

2

w

2e

�ik

x

2+y

2

2R2f(⇣)

Ez(x = 0) = E0f(⇣)

use radially-‐polarized mode instead

•  asd


Donut “beams” were proposed to serve as an accelera4on mechanism for charged

par4cle beams

accelera'on in free space with a “donut” wave

•  s4ll L-‐W theorem applies •  violate one of the L-‐W assump4on by having a large axial E-‐field that can substan4ally affect the velocity of the par4cle during on laser cycle

•  ultra-‐rela4vis4c assump4on breaks down and LW does not apply


laser accelera'on with radially-‐polarized pulses

•  combine rela4vis4c e-‐ with an intense laser

•  during interac4on e-‐ mo4on is non ultra-‐rela4vis4c


[L. J. Wong et al. Op4cs Express, 18, pp. 25035 (2010)]

a0 ⇠ O(1)

using waveguides

•  recently an op4cal cavity was tested


[E. Peralta et al., Nature 503, 91 (2013)]

laser accelera'on in presence of a magnetosta'c field

•  couple the laser to the beam inside an undulator

•  undulator is a magnet that provide a field

•  velocity acquire a transverse component so that and energy exchange is possible.


vvv.EEE 6= 0

By = B0 sin(kuz)

[from J. P. Duris et al. Phys. Rev. ST Accel & Beams 15, 061301 (2012)]

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