P. Piot, PHYS 571 – Fall 2007Case of Circular motion: angular spectral fluence•Finally the angular spectral fluence takes the form•….P. Piot, PHYS 571 – Fall 2007Angle-integrated spectrum I • Last Lesson we noted– High frequency radiation occupies angles θ<γ-1(<<γ-1for ω<<ωc)– Low frequency (ω<<ωc) we have where the critical angle was defined asP. Piot, PHYS 571 – Fall 2007Angle-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…P. Piot, PHYS 571 – Fall 2007Angle-integrated spectrum III• Derived by Schwinger to be1.33 (ω/ωc)1/3P. Piot, PHYS 571 – Fall 2007Angular distribution (frequency integrated) I• Need to evaluate• Change of variable gives:• Where the identityP. Piot, PHYS 571 – Fall 2007Angular distribution II• So finally we have• Let’s do a consistency check and consider the total radiated energy• then!!!!σσππP. Piot, PHYS 571 – Fall 2007Total power• So finally we havewith• In agreeement with the Pcirc we derived at the beginning of chapter 4:P. Piot, PHYS 571 – Fall 2007Case 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 periodicP. Piot, PHYS 571 – Fall 2007Case 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 ω0Same general form as for instantaneous motiona factor sqrt(2π) come from the difference in normalization between Fourier transforms and Fourier series…P. Piot, PHYS 571 – Fall 2007Multiparticle 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)P. Piot, PHYS 571 – Fall 2007Multiparticle 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 factorktikkNePEPEωδ−∑= )()(21222|)(|∑−Ω≡∝ΩktiNNkeddWdPEddWdωδωωP. Piot, PHYS 571 – Fall 2007Multiparticle Coherence III• We have.• Introducing the line charge density Λ(t) we can write+==∑∑∑∑∑−≠++−−jtijktiktijtijtijkkjeeNeeeωδωδωδωδωδ2)(~)(~)1(22ωωωδΛ+≈Λ−+=∑−NNNNNejtiFourier transform of the line charge densityTypically N>>1P. Piot, PHYS 571 – Fall 2007Multiparticle coherence IVBBF measurement (easy!) can provide information on the bunch longitudinal charge distribution-5.0 -2.5 0.0 2.5 5.00100002000030000400005000060000700008000090000100000Population10-510-410-310-210-11001011021051061071081091010101110121013BFF (a.u.)-5.0 -2.5 0.0 2.5 5.00100002000030000400005000060000700008000090000100000Population10-510-410-310-210-11001011021051061071081091010101110121013BFF (a.u.)-5.0 -2.5 0.0 2.5 5.0z/z0100002000030000400005000060000700008000090000100000Population10-510-410-310-210-1100101102/z1051061071081091010101110121013BFF (a.u.)P. Piot, PHYS 571 – Fall 2007Multiparticle coherence VExample of real measurement…0 100Wavenumber (1/cm)0102030405060Power Spectrum (a.u.)−1000 0 1000Mirror Position (microns)1234Interferogram (a.u.)−0.1 −0.05 0 0.05 0.1s (mm)−0.500.511.52Bunch Population (a.u.)−1000 0 1000MIrror Position (microns)−1012Autocorrelation (a.u.)Low Frequency ExtrapolationDeduced Spectrum(C)(A) (B)(D)110 mµP. Piot, PHYS 571 – Fall 2007Multi-particle coherence: example of CSRSRCSR enhancementBeam pipeinduced frequency cut-offCoherent Synchrotron