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 angles 1 1 for c Low frequency c we have where the critical angle was defined as P Piot PHYS 571 Fall 2007 Angle integrated spectrum II But so And Broad spectrum 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 2007 Angle integrated spectrum III Derived by Schwinger to be 1 33 c 1 3 P Piot PHYS 571 Fall 2007 Angular distribution frequency integrated I Need to evaluate Change of variable gives Where the identity 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 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 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 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 Same general form as for instantaneous motion a factor sqrt 2 come from the difference in normalization between Fourier transforms and Fourier series The spectrum is now discrete at n 0 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 P Piot PHYS 571 Fall 2007 Multiparticle Coherence II Let s compute the total field generated by an ensemble of N electrons E N P E k P e i t k k Let s assume the single particle field have the same value at the observation P Then spectral angular fluence is 2 dW d d dW i t k e E N P d d 1 k 2 N 2 Let s evaluate the multiplicative factor P Piot PHYS 571 Fall 2007 2 Multiparticle Coherence III We have 2 e j i t i t j i t k i t k i t j e e e N e j k j k j Introducing the line charge density t we can write 2 e i t N N N 1 j N N 2 Typically N 1 P Piot PHYS 571 Fall 2007 Fourier transform of the line charge density Multiparticle coherence IV BBF measurement easy can provide information on the bunch longitudinal charge distribution 13 100000 10 90000 12 10 11 70000 BFF a u Population 80000 60000 50000 40000 30000 10 10 9 10 8 10 7 10 20000 6 10 10000 0 5 0 100000 10 5 2 5 0 0 2 5 10 5 10 13 10 5 0 90000 BFF a u Population 1 10 0 10 1 10 2 10 11 60000 50000 40000 30000 10 10 10 9 10 8 10 7 10 20000 6 10 10000 5 2 5 0 0 2 5 10 5 10 13 10 5 0 90000 4 10 3 10 2 1 10 10 0 10 1 10 2 10 12 10 80000 11 70000 BFF a u Population 2 10 12 70000 60000 50000 40000 30000 10 10 10 9 10 8 10 7 10 20000 6 10 10000 0 5 0 3 10 10 80000 0 5 0 100000 4 10 5 2 5 0 0 z 2 5 5 0 10 5 10 4 10 3 10 2 1 10 10 z P Piot PHYS 571 Fall 2007 z 0 10 1 10 2 10 Multiparticle coherence V Example of real measurement 4 2 B Autocorrelation a u Interferogram a u A 3 2 1 1000 0 Mirror Position microns 1000 60 110 m 1 0 1 1000 0 MIrror Position microns 2 50 Low Frequency Extrapolation 40 30 Deduced Spectrum 20 10 0 100 Wavenumber 1 cm D Bunch Population a u Power Spectrum a u C 0 1000 1 5 1 0 5 0 0 5 0 1 0 05 P Piot PHYS 571 Fall 2007 0 s mm 0 05 0 1 Multi particle coherence example of CSR Beam pipe induced frequency cut off Coherent Synchrotron Radiation CSR enhancement SR P Piot PHYS 571 Fall 2007