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Measurement of Incoherent Radiation Fluctuations and Bunch Profile Recovery Vadim Sajaev Advanced Photon Source Argonne National Laboratory 07/27/2004 XFEL 2004 Theory • Each particle in the bunch radiates an electromagnetic pulse e(t) N • Total radiated field is E ( t ) = ∑ e( t − t k ) k =1 N • Fourier transform of the field is E (ω ) = e(ω )∑ eiωtk ˆ ˆ k =1 • Power spectrum of the radiation is N N N P (ω ) = e(ω )e* (ω )∑ eiωtk ∑ e −iωtm = e(ω ) ∑ eiω ( tk −tm ) 2 ˆ ˆ ˆ k =1 m =1 k ,m =1 07/27/2004 XFEL 2004 Average power ⎛ N + N 2 f (ω ) 2 ⎞ P(ω ) = e(ω ) ⎜ 2 ˆ Average power is ˆ ⎟ ⎝ ⎠ incoherent coherent radiation radiation The coherent radiation term carries information about the distribution of the beam at low frequencies of the order of ω≈σ -1 07/27/2004 XFEL 2004 Difference between coherent and incoherent power is huge 2 • Coherent to incoherent ratio R(ω ) = N f (ω ) ˆ • consider a Gaussian beam with σt=1 ps and total charge of 1 nC (approximately 1010 electrons) t2 ω 2σ t2 − f (t ) = 1 e 2σ t2 and ˆ (ω ) = e − f 2 2π σ t At 1 THz: R≈1010 At 10 THz: R≈10-34 07/27/2004 XFEL 2004 High frequencies still contain information N P (ω ) = e(ω ) ∑ eiω ( tk −tm ) 2 • Power spectrum before averaging: ˆ k ,m =1 • Each separate term of the summation oscillates with the period ∆ω=2π/(tk-tm)~2π/σt • Because of the random distribution of particles in the bunch, the summation fluctuates randomly as a function of frequency ω. 07/27/2004 XFEL 2004 Example: incoherent radiation in a wiggler Single electron Electron bunch Spectrometer N N e(t) E ( t ) = ∑ e( t − t k ) E (ω ) = e(ω ) ⋅ ∑ e iωtk k =1 k =1 Spike width is inversely proportional to the bunch ~10 fs ~1 ps length 07/27/2004 XFEL 2004 Bunch profile measurements using fluctuations of incoherent radiation • The method was proposed by M. Zolotorev and G. Stupakov and also by E. Saldin, E. Schneidmiller and M. Yurkov • Emission can be produced by any kind of incoherent radiation: synchrotron radiation in a bend or wiggler, transition, Cerenkov, etc. • The method does not set any conditions on the bandwidth of the radiation 07/27/2004 XFEL 2004 Quantitative analysis We can calculate autocorrelation of the spectrum: N ∑e iω ( t k − t m ) + iω ′( t p − t q ) P (ω ) P (ω ′) = e(ω ) e(ω ′) 2 2 ˆ ˆ k , m , p , q =1 ⎛1 + f (ω − ω ′) 2 ⎞ P (ω ) P (ω ′) = N e(ω ) e(ω ′) ⎜ 2 ˆ2 ⎟ 2 ˆ ˆ ⎝ ⎠ 1⎛ ˆ (Ω ) ⎞ 2 g (Ω) = ⎜1 + f ⎟ 2 ⎝ ⎠ 07/27/2004 XFEL 2004 Variance of the Fourier transform of the spectrum Fourier transform of the spectrum: ∞ G (τ ) = ∫ −∞ P (ω )e iωτ dω Its variance: 2 D(τ ) = G (τ ) − G (τ ) It can be shown, that the variance is related to the convolution function of the particle distribution: ∞ D (τ ) = A ∫ f (t ) f (t − τ )dt −∞ 07/27/2004 XFEL 2004 Some of the limitations • Bandwidth of the radiation has to be larger than the spike width • In order to neglect quantum fluctuations, number of photons has to be large 1 ∆ω n ph ≈ αN e 2 ω • Transverse bunch size – radiation has to be fully coherent to observe 100% intensity fluctuations λrad 2πσ θ 07/27/2004 XFEL 2004 LEUTL at APS 07/27/2004 XFEL 2004 Spectrometer Grooves/mm 600 Grating Curv. radius [mm] 1000 Blaze wavelength[nm] 482 CCD Number of pixels 1100×330 camera Pixel size [µm] 24 Concave mirror curv. radius [mm] 4000 Spectral resolution [Å] 0.4 Bandpass [nm] 44 Resolving power at 530 nm 10000 Wavelength range [nm] 250 – 1100 07/27/2004 XFEL 2004 Single shot spectrum Typical single-shot spectrum Average spectrum 07/27/2004 XFEL 2004 Spectrum for different bunch length Long 2-ps rms bunch Short 0.4-ps rms bunch Note: Total spectrum width (defined by the number of poles in the wiggler) is barely enough for the short bunch. 07/27/2004 XFEL 2004 Spectrum correlation Cn = ∑ P(ω i ) P(ω i+n ) / i ∑i P (ω i ) 2 07/27/2004 XFEL 2004 Bunch length From the plot the correlation width is 2 pixels. Frequency step corresponding to one pixel is 2.4·1011 rad/s. Assuming the beam to be Gaussian, from equation 1⎛ ˆ (Ω ) ⎞ 2 g (Ω) = ⎜1 + f ⎟ 2 ⎝ ⎠ we get 1 τb ≈ ≈ 2 ps n ⋅ δω 07/27/2004 XFEL 2004 Convolution of the bunch profile 2 N ch Np Np 1 Gk ,n = ∑ Pm ,n e 2πimk / N ch Dk = ∑ Gk ,m − ∑G k ,n m =1 m =1 Np n =1 Bunch profile 1 0 2 0 2 Step function Gaussian 07/27/2004 XFEL 2004 Convolution recovered from the measurements Convolution of the Gaussian is also a Gaussian with σ = 2 ⋅σ t The Gaussian fit gives us τ b = 1.8 ps 07/27/2004 XFEL 2004 Phase retrieval The amplitude and the phase information of the radiation source can be recovered by applying a Kramers-Kronig relation to the convolution function in combination with the minimal phase approach. ln[ρ ( x) / ρ (ω )] ∞ 2ω ψ m (ω ) = − P ∫ dx π 0 x2 −ω 2 ∞ 1 ⎛ ωz ⎞ S ( z) = ∫ dω ⋅ ρ (ω ) ⋅ cos⎜ψ m (ω ) − c ⎟ πc 0 ⎝ ⎠ 07/27/2004 XFEL 2004 Phase retrieval example Calculation of longitudinal distribution for different bunches Calculated shape Exact shape Time (arb. units) 07/27/2004 XFEL 2004 Bunch profile Two different measurements (two sets of 100 single-shot spectra) FWHM≈4ps 07/27/2004 XFEL 2004 Conclusions • Measurements of incoherent radiation spectrum showing intensity fluctuations were done. • A technique for recovering a longitudinal bunch profile from spectral fluctuations of incoherent radiation has been implemented. Although we used synchrotron radiation, the nature of the radiation is not important. • Typically, analysis of many single shots is required, however one can perform statistical analysis over wide spectral intervals in a single pulse 07/27/2004 XFEL 2004 Conclusions • An important feature of the method is that it can be used for bunches with lengths varying from a centimeter to tens of microns (30 ps – 30 fs) • There are several important conditions for this technique. In order to be able to measure a bunch of length σt, the spectral resolution of the spectrometer should be comparable with 1/σt. Also, the spectral width of the radiation and the spectrometer must be larger than the inverse bunch length 07/27/2004 XFEL 2004

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