Case of Circular motion: angular spectral fluence Finally the - - PowerPoint PPT Presentation

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Case of Circular motion: angular spectral fluence Finally the angular spectral fluence takes the form . P. Piot, PHYS 571 Fall 2007 Angle-integrated spectrum I Last Lesson we noted High frequency radiation occupies

SLIDE 1

P. Piot, PHYS 571 – Fall 2007

Case of Circular motion: angular spectral fluence

Finally the angular spectral fluence takes the form
….

SLIDE 2

P. Piot, PHYS 571 – Fall 2007

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

SLIDE 3

P. Piot, PHYS 571 – Fall 2007

Angle-integrated spectrum II

But so
And
Broad spectrum γ-independent
Can do a similar asymptotic expansion for the high frequency region
f the angle-integrated spectrum – let as an exercise…

SLIDE 4

P. Piot, PHYS 571 – Fall 2007

Angle-integrated spectrum III

Derived by Schwinger to be

1.33 (ω/ωc)1/3

SLIDE 5

P. Piot, PHYS 571 – Fall 2007

Angular distribution (frequency integrated) I

Need to evaluate
Change of variable gives:
Where the identity

SLIDE 6

P. Piot, PHYS 571 – Fall 2007

Angular distribution II

So finally we have
Let’s do a consistency check and consider the total radiated energy
then

!!!! σ σ π π

SLIDE 7

P. Piot, PHYS 571 – Fall 2007

Total power

So finally we have

with

In agreeement with the Pcirc we derived at the beginning of chapter

4:

SLIDE 8

P. Piot, PHYS 571 – Fall 2007

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

SLIDE 9

P. Piot, PHYS 571 – Fall 2007

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 motion a factor sqrt(2π) come from the difference in normalization between Fourier transforms and Fourier series…

SLIDE 10

P. Piot, PHYS 571 – Fall 2007

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)

SLIDE 11

P. Piot, PHYS 571 – Fall 2007

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

t i k k N

e P E P E

ωδ −

∑

= ) ( ) (

2 1 2 2 2

| ) ( |

∑

−

Ω ≡ ∝ Ω

k t i N N

e d d W d P E d d W d

ωδ

ω ω

SLIDE 12

P. Piot, PHYS 571 – Fall 2007

Multiparticle Coherence III

We have.
Introducing the line charge density Λ(t) we can write

        + =               =

∑∑ ∑ ∑ ∑

− ≠ + + − − j t i j k t i k t i j t i j t i

j k k j

e e N e e e

ωδ ωδ ωδ ωδ ωδ 2

) ( ~ ) ( ~ ) 1 (

2 2

ω ω

ωδ

Λ + ≈ Λ − + =

∑

−

N N N N N e

j t i

Fourier transform of the line charge density Typically N>>1

SLIDE 13

P. Piot, PHYS 571 – Fall 2007

Multiparticle coherence IV

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

0.0 2.5 5.0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Population

10-5 10-4 10-3 10-2 10-1 100 101 102 105 106 107 108 109 1010 1011 1012 1013

BFF (a.u.)

0.0 2.5 5.0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Population

10-5 10-4 10-3 10-2 10-1 100 101 102 105 106 107 108 109 1010 1011 1012 1013

BFF (a.u.)

0.0 2.5 5.0

10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Population

10-5 10-4 10-3 10-2 10-1 100 101 102

105 106 107 108 109 1010 1011 1012 1013

BFF (a.u.)

SLIDE 14

P. Piot, PHYS 571 – Fall 2007

Multiparticle coherence V

Example of real measurement…

100 Wavenumber (1/cm) 10 20 30 40 50 60 Power Spectrum (a.u.) −1000 1000 Mirror Position (microns) 1 2 3 4 Interferogram (a.u.) −0.1 −0.05 0.05 0.1 s (mm) −0.5 0.5 1 1.5 2 Bunch Population (a.u.) −1000 1000 MIrror Position (microns) −1 1 2 Autocorrelation (a.u.) Low Frequency Extrapolation Deduced Spectrum (C) (A) (B) (D) 110 m µ

SLIDE 15

P. Piot, PHYS 571 – Fall 2007

Case of Circular motion: angular spectral fluence

Angle-integrated spectrum I

– 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

Angle-integrated spectrum III

1.33 (ω/ωc)1/3

Angular distribution (frequency integrated) I

Angular distribution II

!!!! σ σ π π

Total power

with

4:

Case of periodic circular motion I

transient) and computed instantaneous spectra.

Case of periodic circular motion II

case) that

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

Multiparticle Coherence I

wonder how does this affect all the results previously derived

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

Multiparticle Coherence II

electrons.

the observation P. Then spectral angular fluence is

e P E P E

∑

= ) ( ) (

| ) ( |

∑

Ω ≡ ∝ Ω

e d d W d P E d d W d

ω ω

Multiparticle Coherence III

        + =               =

∑∑ ∑ ∑ ∑

e e N e e e

) ( ~ ) ( ~ ) 1 (

ω ω

Λ + ≈ Λ − + =

∑

N N N N N e

Multiparticle coherence IV

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

Multiparticle coherence V

Example of real measurement…

Multi-particle coherence: example of CSR

SR CSR enhancement Beam pipe induced frequency cut-off Coherent Synchrotron Radiation