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ZHS and EP theory. C. W. James, Columbus, Ohio, Feb 23 rd , 2012. Step 1: Liénard-Weichert Potentials. Begin with Maxwell’s equations Add a single ( monopolar ) particle as a source Allow for finite light propagation speed Use Lorentz gauge. Step 2:. Apply to get:. Nearfield Term

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zhs and ep theory

ZHS and EP theory

C. W. James, Columbus, Ohio, Feb 23rd, 2012

step 1 li nard weichert potentials
Step 1: Liénard-Weichert Potentials
  • Begin with Maxwell’s equations
  • Add a single (monopolar) particle as a source
  • Allow for finite light propagation speed
  • Use Lorentz gauge
step 2
Step 2:
  • Apply to get:

Nearfield Term

Energy/area as R-4

Energy decreases with distance

Radiation term

Energy/area as R-2

Energy transport to infinity

“Accelerating charged particles radiate”

Get rid of this

(no Frank-Tamm VC)

do some maths
Do some maths…




  • Endpoints
  • ZHS formula
  • Endpoints -> ZHS:

(the far-field approximation)

toy experiments
Toy experiments
  • Take a straight particle track:
  • Place an observer in x-z plane
  • Calculate emission via…
      • Endpoints
      • ZHS (single track)
      • ZHS (very many sub-tracks)




what do we expect to see
What do we expect to see?
  • Afanasiev, Kartavenko, Stepanovsky J Phys D, 32 (1999)

Vavilov-Cherenkov radiation important

Bremsstrahlung from endpoints dominates

Bremsstrahlung from endpoints dominates

far field far from theta c

1 m

Far-field, far from theta_C
  • Endpoints, ZHS agree perfectly.
  • No ZHS track sub-division needed (1m source at 1 km unresolved)

1000 m

low frequency limit
  • Endpoints reduce to:
  • ZHS low-phase limit:
  • Tends towards a constant term at low frequencies
  • Tends towards zero at low frequencies
difference in the near field

1 m

Difference in the near-field

1 m

  • Observer much closer to track start than track end
  • Endpoints accounts for this, ZHS can not
  • ZHS formula:
      • Accounts for distance difference in phase, but not magnitude
      • true no matter how tracks are subdivided
  • Endpoints:
      • Distance affects both magnitude and phase
  • Clearly, an observer in the nearfield should see a monopolar component to the pulse
      • [total net change in potential]
  • Important for:
      • Lunar Cherenkov? No! (very far field)
      • Important for air-showers? Perhaps (REAS3 vsZHAires).
      • Important for dense media?...
what about near the cherenkov angle
What about near the Cherenkov angle?
  • Endpoint formulation:
  • In ZHS:
  • Result can be arbitrarily large (it blows up)
  • Result is always finite (more sensible)
behaviour near the cherenkov angle

1 m

Behaviour near the Cherenkov angle
  • Endpoints produce a larger contribution (can be arbitrarily large)

1000 m

why do endpoints blow up
Why do endpoints blow up?
  • Endpoints allow:
      • Infinitely small acceleration zone
      • Infinitely small source particle
      • Infinitely small detector
      • [time-domain only] constant refractive index
  • Result: potentially infinite field
  • This should not be unexpected!
      • Very common to see infinities in the literature
      • This is why textbooks always derive the total radiated power and not the field strengths.
  • This is small consolation.
what happens in the near field in the cherenkov regime
What happens in the near-field in the Cherenkov regime?


When good techniques go bad

C. W. James, Columbus, Ohio, Feb 23rd 2012

toy experimental set up
Toy experimental set-up
  • Place the observer firmly in the Cherenkov regime

10 m

1 m

Cherenkov zone

spectrum n 2
Spectrum: n=2
  • Now we see differences…
time domain no band limit
Time-domain (no band limit)
  • Time-domain output (ZHS vs EP) (n=2):
  • Large contribution from ZHS NOT in endpoints!
  • Could this be a ‘true’ Vavilov-Cherenkov emission? (or a numerical artefact?)

(note different y-axis scales)

quick check in vacuum
Quick check: in vacuum
  • We do not expect and Cherenkov shock
  • But we do expect two bremsstrahlung shocks…
  • I do not understand this ZHS behaviour
1cm from the vacuum track
1cm from the vacuum track
  • Large ZHS pulse… in a vacuum.
  • This is not V-C radiation!
  • It is a numerical artefact OR a static term.
summary from toy experiments
Summary from toy experiments
  • Theoretical expectation:
      • EP theory models only bremsstrahlung
        • Handles near-field
        • Breaks down near theta_C
      • ZHS models only bremsstrahlung + far-field approx
        • Breaks down in near-field
        • Handles theta_C
  • What we see:
      • EP theory matches expectation
      • ZHS: some strange results…
        • Produces phantom Vavilov-Cherenkov-like pulse
        • Somehow misses bremsstrahlung
main conclusion
Main conclusion
  • Neither endpoints nor ZHS get it completely right



EP & ZHS agree

(probably correct)

EP theory is better

(probably correct)

Far from θC

ZHS crazy

EP misses VC (main)

(probably both crap)

ZHS is better

(probably not correct)

Near θC

philosophical aside
Philosophical aside
  • What about smooth particle motion?
  • Radiation is emitted constantly
  • Limit (description -> perfection) [inf points]:
      • Endpoints have contributions equal-and-opposite sides of the Cherenkov angle
      • Divergences are expected to cancel
      • Hence tendency towards ZHS treatment in REAS3
what does the zhs formula produce
What does the ZHS formula produce
  • ZHS formula approximates:
  • This approximation can not be made near the Cherenkov angle
      • Same approximation as Tamm (1939)
      • Shown to exclude Frank-Tamm Cherenkov
  • And yet…
      • ZHS formula produces something sensible.
      • Endpoints do not.
  • We do not know what ZHS produces at the Cherenkov angle
is the divergence physical
Is the divergence physical?
  • If:
      • n is constant
      • The acceleration event is truly instantaneous
      • The particle and detector are both infinitely small
  • Then yes!
  • Divergence/magnification at the Cherenkov angle does NOT necessarily mean Vavilov-Cherenkov radiation!
  • Q: Why do we often see total radiated power calculated, but not the fields?
  • A: Because this can hide nasty divergences (integrate away this divergence over finite spatial angles)