Measurement of Bunch Length Using Spectral Analysis of Incoherent

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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



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                     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




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  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

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    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.




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   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




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                     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 ⎝           ⎠



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      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
                                  −∞



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              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


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             LEUTL at APS




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             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




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                    Single shot spectrum
Typical single-shot spectrum




     Average spectrum




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             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.


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                    Spectrum correlation

             Cn =   ∑ P(ω i ) P(ω i+n ) /
                     i
                                            ∑i
                                              P (ω i ) 2




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                      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


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             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


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             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




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                        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                  ⎝             ⎠



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             Phase retrieval example

             Calculation of longitudinal distribution for different bunches

                                                                          Calculated shape
                                                                          Exact shape




                                    Time (arb. units)




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                     Bunch profile
Two different measurements (two sets of 100 single-shot spectra)



                                   FWHM≈4ps




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                      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




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                      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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