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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
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 2
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…
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 4
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
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 5
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
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 6
� ' 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
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 7
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
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 8
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,
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 9
[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.
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 10
[T. Ple\ner et al., Phys. Rev. ST Accel. Beams 8, 121301 (2005)]
laser phase
semi-‐infinite vacuum: experiment
• carried at SLAC ~2005
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 11
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
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 12
Hn(x)
Hermite polynomials
• Recurrence rela4on is • First few polynomials are
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 13
Mul4ply by a Gaussian
linearly-‐polarized HG01
• consider HG10
• from so this type of mode produced an axial E-‐field (along the direc4on of propaga4on)
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 15
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
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 16
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
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 17
laser accelera'on with radially-‐polarized pulses
• combine rela4vis4c e-‐ with an intense laser
• during interac4on e-‐ mo4on is non ultra-‐rela4vis4c
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 18
[L. J. Wong et al. Op4cs Express, 18, pp. 25035 (2010)]
a0 ⇠ O(1)
using waveguides
• recently an op4cal cavity was tested
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 19
[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.
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 20
vvv.EEE 6= 0
By = B0 sin(kuz)
[from J. P. Duris et al. Phys. Rev. ST Accel & Beams 15, 061301 (2012)]