Radiation Spectrum I Starting from the radiation field, we have - - PowerPoint PPT Presentation

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Radiation Spectrum I Starting from the radiation field, we have - - PowerPoint PPT Presentation

Feb 12, 2023 437 likes •596 views

Radiation Spectrum I Starting from the radiation field, we have Where A is defined as To obtain the power spectrum we need to work in the frequency domain P. Piot, PHYS 571 Fall 2007 Radiation Spectrum II Lets define

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  • P. Piot, PHYS 571 – Fall 2007

Radiation Spectrum I

  • Starting from the radiation field, we have
  • Where A is defined as
  • To obtain the power spectrum we need to work in the frequency

domain

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SLIDE 2
  • P. Piot, PHYS 571 – Fall 2007

Radiation Spectrum II

  • Let’s define the “symmetrized Fourier transform”
  • Parseval’s theorem states that
  • Since A is a real function
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SLIDE 3
  • P. Piot, PHYS 571 – Fall 2007

Radiation Spectrum III

  • The radiation spectrum is therefore
  • Starting from A(t)
  • A(ω) is
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SLIDE 4
  • P. Piot, PHYS 571 – Fall 2007

Radiation Spectrum IV

  • This must be evaluated at the retarded time t’
  • Note that in the far-field regime

is constant in time and argument of the exponential in the far-field is

ignore

⇒

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  • P. Piot, PHYS 571 – Fall 2007

Radiation Spectrum V

  • We finally have
  • And the corresponding angular spectral fluence distribution
  • This is the most general formula for computing the angular spectral

fluence.

  • JDJ re-write the vector part of the integrand as a total time derivative
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SLIDE 6
  • P. Piot, PHYS 571 – Fall 2007

Radiation Spectrum VI

  • Consider
  • Then the time derivative is
  • We have
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SLIDE 7
  • P. Piot, PHYS 571 – Fall 2007

Radiation Spectrum VII

  • So we can do an integration-by-part

=0 in principle but care must be taken to verify this in practice

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SLIDE 8
  • P. Piot, PHYS 571 – Fall 2007

Radiation Spectrum VIII

  • So finally the angular spectral fluence is

note that

  • Here we could also start introducing the polarization, but we will do

this for the special case of circular motion

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SLIDE 9
  • P. Piot, PHYS 571 – Fall 2007

Case of Circular motion I

  • Introduce the polarization unit

vectors ε’s

  • Then

q r O

ω0t

P x z y n

ε⊥ ε|| θ

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SLIDE 10
  • P. Piot, PHYS 571 – Fall 2007

Case of Circular motion II

  • The argument of the exponential writes
  • If an observer catches an impulse from the charge q: θ is small and

the pulse originated close to t = 0, so under these approximations and

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SLIDE 11
  • P. Piot, PHYS 571 – Fall 2007

Case of Circular motion III

  • The angular spectral fluence is
  • This displays the two polarizations.

σ π

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SLIDE 12
  • P. Piot, PHYS 571 – Fall 2007

Case of Circular motion IV

  • We have to compute the integrals
  • We have
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SLIDE 13
  • P. Piot, PHYS 571 – Fall 2007

Case of Circular motion V

  • Finally the angular spectral fluence takes the form
  • ….
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SLIDE 14
  • P. Piot, PHYS 571 – Fall 2007

Case of Circular motion VI

σ π σ π

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SLIDE 15
  • P. Piot, PHYS 571 – Fall 2007

Case of Circular motion VII

σ+π