eh ty l l an i•F angular spectral fluence takes the form

P. Piot, PHYS 571 – Fall 2007

Case of Circular motion: angular spectral fluence

• Finally the angular spectral fluence takes the form

• ….


Angle-integrated spectrum I

• Last Lesson we noted– High frequency radiation occupies angles θ<γ-1 (<<γ-1 for ω<<ωc)

– Low frequency (ω<<ωc) we have

where the critical angle was defined as


Angle-integrated spectrum II

• But so

• And

• Broad spectrum γ-independent

• Can do a similar asymptotic expansion for the high frequency region of the angle-integrated spectrum – let as an exercise…


Angle-integrated spectrum III

• Derived by Schwinger to be

1.33 (ω/ωc)1/3


Angular distribution (frequency integrated) I

• Need to evaluate

• Change of variable gives:

• Where the identity


Angular distribution II

• So finally we have

• Let’s do a consistency check and consider the total radiated energy

• then!!!!

σσ

ππ


Total power

• So finally we have

with

• In agreeement with the Pcirc we derived at the beginning of chapter 4:


Case of periodic circular motion I

• Up to now we considered the steady case circular motion (no transient) and computed instantaneous spectra.

• If the motion is periodic


Case of periodic circular motion II

• And we can show (following the steps we did for the instantaneous case) that

• The spectrum is now discrete at ω= n ω0

Same general form as for instantaneous motiona factor sqrt(2π) come from the difference in normalization between Fourier transforms and Fourier series…


Multiparticle Coherence I

• In real life a bunch consists of many particle so one may wonder how does this affect all the results previously derived

• It depends on the frequency (wavelength) of observation!

Electric field radiated by two particle at “small” (right) and “long”wavelength (compared to the particle spacing)


Multiparticle Coherence II

• Let’s compute the total field generated by an ensemble of N electrons.

• Let’s assume the single particle field have the same value at the observation P. Then spectral angular fluence is

• Let’s evaluate the multiplicative factor

kti

kkN ePEPE ωδ−∑= )()(

2

1

22

2

|)(| ∑ −

Ω≡∝

Ω k

tiN

N

keddWdPE

ddWd ωδ

ωω


Multiparticle Coherence III

• We have.

• Introducing the line charge density Λ(t) we can write

+=

= ∑∑∑∑∑ −

≠

++−−

j

ti

jk

ti

k

ti

j

ti

j

ti jkkj eeNeee ωδωδωδωδωδ

2

)(~

)(~)1(

2

2

ω

ωωδ

Λ+≈

Λ−+=∑ −

NN

NNNej

ti

Fourier transform of the line charge densityTypically N>>1


Multiparticle coherence IV

BBF measurement (easy!) can provide information on the bunch longitudinal charge distribution

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Multiparticle coherence V

Example of real measurement…

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(a.u

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Low Frequency Extrapolation

Deduced Spectrum

(C)

(A) (B)

(D)

110 mµ


Multi-particle coherence: example of CSR

SR

CSR enhancementBeam pipeinduced frequency cut-off

Coherent Synchrotron Radiation

eh ty l l an i•F angular spectral fluence takes the form

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