Radiated power and radiation
reaction forces of coherently
radiating charged particles in
classical electrodynamics
Pardis Nikejadi
John Madey
Jeremy Kowalczyk (jeremymk@hawaii.edu)
University of Hawaii at Manoa
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Why is coherent radiation important?


Coherently radiating electrons

Multiple electrons spaced within a fraction of their
radiation wavelength

Fields constructively interfere, increased radiated
power
Electron beam based light sources
(synchrotrons, FELs, compton sources)

~1010 electrons/bunch

1 mm diameter

100 fs bunch length

424 electrons/(1μm X 1μm X 1μm) => strong
coherence
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Show inconsistency with Maxwell:
model system

Two electrons

Spacing 'r'

Oscillating parallel to 'r'

Same frequency, phase
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Maxwell Energy Integral
Subtract, use vector identity, integrate over all volume
Change in field energy
= 0 for steady state
Work done by
field on charge
per unit time
Radiated
power
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Maxwell Energy Integral
Work done by
field on charge
per unit time
Radiated
power
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Two Charge Model System:
Radiated Power vs. Separation
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What about E∙j ??



Assuming retarded
only fields
(Sommerfeld
condition)
No radiation field
along z direction
What about the
'induction' (1/r2) field
?
7
Induction field diverges:
can't match ExH term
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(ALD radiation reaction field) * j
doesn't match ExH either !!
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ALD radiation reaction:
doesn't account for adjacent
particles

Local field (sometimes call a self field)

Only at the location of the particle itself

EALD is zero everywhere except at the particle


Factor of two difference in ExH vs. E∙j power
for 2 particles.
Think about adding a third particle...

Factor of 3 difference in ExH vs. E∙j power for
closest spacing
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Problem summary:
Retarded only CED does not satisfy
Maxwell's energy integral
for coherent oscillation
One solution:
Time symmetric CED does satisfy
Maxwell's energy integral
for coherent oscillation
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Time symmetric CED



Maxwell's equations has two solutions to
radiation: Liénard–Wiechert fields

Retarded

Advanced
In retarded only CED, apply auxilliary Sommerfeld
condition

Excludes the advanced solutions

Provides uniqueness
In time-symmetric CED, prescribe the field

E = ½ (advanced solution) + ½ (retarded solution)
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Time-symmetric CED (TSCED)



Preserves experimentally verified results of
CED
Takes into account boundary conditions of
the universe to solve the radiation problem

Particles other than those being analyzed lumped
into an 'absorber'

Note that retarded only CED does not account for
these particles, assumes empty space
Is consistent with Maxwell's energy integral
for coherent radiation
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TSCED is consistent with Maxwell's
Energy Integral
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How does TSCED provide the
necessary field?

For one radiating charge

E_radiating_charge


The sum of the advanced fields of all 'absorber'
particles



½ (advanced) + ½ (retarded)
- ½ (advanced) + ½ (retarded)
Total field is just the full retarded field of
experience
For two radiating charges


Advanced fields from particle 1 affect particle 2
Induction fields have opposite sign, cancel
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Proposed test of TSCED



Using a 1/10 wave
antenna
Model current in
antenna with TSCED
oscillators
Measure the field
near the antenna!!
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Proposed test of TSCED
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Questions/Discussion
Reference:
Pardis Niknejadi, John M.J. Madey, and Jeremy M.D. Kowalczyk.
Radiated power and radiation reaction forces of coherently
oscillating charged particles in classical electrodynamics.
Physical Review D, 91(9):1–14, 2015.
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