Synchrotron radiation: generation, coherence properties and applications for beam diagnostics Gianluca Geloni European XFEL GmbH

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Caveats, Acknowledgments and References My experience with SR was mainly focused on e- beams and undulators Some articles/notes I co-authored on the subject:  ‘Paraxial Green’s function in SR theory’, DESY , G. Geloni, E. Saldin, E. Schneidmiller and M. Yurkov  A study on ‘Transverse coherence properties of X-ray beams in third- generation synchrotron radiation sources’ can be found in G. Geloni, E. Saldin, E. Schneidmiller and M. Yurkov NIM A 588 (2008) 463  Fourier treatment of near-field synchrotron radiation theory, Gianluca Geloni, Evgeni Saldin, Evgeni Schneidmiller, Mikhail Yurkov, Optics Communications 276 (2007) 167–179 I wish to thank E. Saldin for many discussions during the preparation of this talk

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Contents Generation of radiation from a single charged particle  SR in general, approximations and characteristics  Example: BM radiation  Wavefront propagation Coherence properties  Many particles, statistical optics treatment, coherence  Near and Far zone  SR and particle diagnostics (beam size measurements) Application  Parameters for the CERN case of interest Conclusions

I. Generation

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Classical framework From Maxwell eqs we arrive at the wave equations for E and B: Use space-frequency representation Use space-frequency representation to obtain the wave equations in the space-frequency domain: Component by component: inhomogeneous Helmholtz Equation

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Source & geometry specification We first specialize to a single particle. We set the sources in the space- frequency as: s=s(t’) is the curvilinear abscissa and we re- parametrize t’=t’(z); we assume constant speed. z P x y r’(t’) We introduce This can always be done. It is useful in the case when the field envelope is a slowly varying function of z wrt

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Green Function Solution for Helmholtz equation For z o >z’ we can show that   : time slippage photon-to-charge on the z axis in units of 1/   r : difference of fly-time between photon from r’ to r 0 and photon between z’ and z 0 in units of 1/  Here

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Formation Length When the total phase varies of ~1, as we move along z’, the integrand begins to oscillate  zero extra-contribution to the integral. Let us first analyze Definition of formation length L f : interval  z’ such that  0 =1 We now analyze the second phase term At what condition we obtain variations, along  z ~L f, of a quantity ~1? It can be demonstrated that sufficient condition is

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Paraxial Approximation (And additionally ) Whenever the paraxial approximation applies, i.e. when: It can be shown that the total expression for the field boils down to

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Ultrarelativistic Approximation We now assume (always ok in our case) ultrarelativistic charges What can we say about the formation length L f ? Since  z >>1 it follows L f >> /(2  ) ~1 z  z >>1 Condition to have variation in the second phase factor ~1 Implies automatically Ultrarelativistic approximation  Paraxial approximation!

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Ultrarelativistic Approximation and U.R. appr. applies Summing up. Whenever Radiation is confined into a cone The formation length L f >> /(2  ) The paraxial approximation applies The field can be calculated as: Important note: for a single charge far zone simply means z 0 >> L f (Always in our case!)

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Bending Magnet, single particle, far zone z 0 >>L f And redefine z’  z’ + R  x ; then we have: For c /(2  )~R/  3  L f ~R/  Sometimes it is convenient to define

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Bending Magnet, single particle, far zone z 0 >>L f In those units, the above expression is equivalent to I x (AU) I y (AU)  =1  x (rad)  y (rad)

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Synchrotron Radiation Propagation We calculated the field at a given position z (does not matter where!) Then, we can calculate the field at any other position in free-space using We can even propagate at (virtual) positions actually occupied by sources!

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Synchrotron Radiation Propagation Free-space propagation can be easily analyzed in the spatial Fourier domain Field propagation in the spatial Fourier domain is almost trivial. Note that the field can be seen as a superposition of plane waves  angular spectrum u corresponds to a given propagation angle

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Synchrotron Radiation Propagation We can start with the field distribution in the far zone, and calculate the field distribution at the ‘virtual’ source position, in the middle of the magnet! The field at z=0 is “virtual”. That particular field distribution at z=0 reproduces all effects associated with sources Note that free space basically acts as a Fourier Transform

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Bending Magnet virtual source For a BM, the field seen by imaging the middle of the magnet (virtual source) with a lens is

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Synchrotron Radiation Propagation Radiation from a single charge is like a special (non-Gaussian) laser beam Rayleigh length L f

II. Coherence

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH SR and statistical optics Random variables: e - angle + offset: Arrival times t k On top of that: energy spread, which I will not consider here… so, the full phase-space! Shot noise  random position, direction, but also arrival time  Current fluctuations SR is a random process  Statistical Optics!

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH SR as a Gaussian Process

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Correlation function: space-frequency domain

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Simplifications  =2.5e-10 s   =1/  =4e9 Hz  Much larger relative spectral width

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Relation with intensity and coherence Cross-spectral density and spectral degree of coherence Cross-spectral density and measured intensity Requirement on longitudinal coherence l c =   >> max optical path difference

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH How to calculate G explicitly Far zone Virtual source Call with  0 (virtual source) or exp[i  z  2 / ]  f (far zone) any polarization component of the previously calculated fields and Introduce normalized units We need generalization of the field to account for offset and deflection of particles

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH How to calculate G explicitly New variables: Cross-spectral density at the virtual source Cross-spectral density in the far zone

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH How to calculate G explicitly: models… Assume min  -functions at the virtual source position (z=0) Assume horizontal and vertical motions uncoupled Assume Gaussian particle beam shapes and divergences

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Far zone for G Far zone definition For a single charge on axis:  Spherical wavefront For the cross spectral density we rely on a similar definition.

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Overall General expression for G Cross-spectral density at the virtual source Cross-spectral density in the far zone

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH First asymptote: diffraction limit Suppose first Nx<<1, Ny<<1, Dx<<1, Dy<<1 Cross-spectral density at the virtual source Cross-spectral density in the far zone

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Second asymptote: geometrical optics limit Now consider Nx>>1, Ny>>1, Dx>>1, Dy>>1 Cross-spectral density at the virtual source Cross-spectral density in the far zone

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Second asymptote: geometrical optics limit Let’s have a closer look Nx>>1, Ny>>1, Dx>>1, Dy>>1 Cross-spectral density in the far zone Cross-spectral density at the virtual source

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH The VCZ theorem Let’s have a closer look Nx>>1, Ny>>1, Dx>>1, Dy>>1 Cross-spectral density in the far zone Cross-spectral density at the virtual source I0I0 gfgf FT

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH The VCZ theorem Let’s have a closer look Nx>>1, Ny>>1, Dx>>1, Dy>>1 Cross-spectral density in the far zone Cross-spectral density at the virtual source I0I0 gfgf FT IfIf g0g0

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH The VCZ theorem Cross-spectral density in the far zone Cross-spectral density at the virtual source I0I0 gfgf FT IfIf g0g0 THIS IS JUST THE Van Cittert-Zernike Theorem (and its inverse, because near and far zone are reciprocal)

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH The VCZ theorem Van Cittert-Zernike Theorem Van Cittert-Zernike theorem: For a QH stochastic process, power spectral density (intensity) on source, I0 and gf form a Fourier pair (Coherence length normalized to diffraction size!) S r d

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH The VCZ theorem …we can even follow g as a function of z!

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH How do we treat other cases? Example: study case Nx>>1, Ny>>1, Dx<<1, Dy<<1 Cross-spectral density at the virtual source Cross-spectral density in the far zone

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH How do we treat other cases? Example: study case Nx>>1, Ny>>1, Dx<<1, Dy<<1 Cross-spectral density at the virtual source Cross-spectral density in the far zone I0I0 gfgf FT

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Can we find the beam size? Different alternative methods (Nx>>1, Ny>>1, Dx<<1, Dy<<1): We can measure the intensity at the virtual source using a lens We can measure the fringe visibility in the far zone In principle we can even measure the fringe visibility at any position, but in this case we need a-priori information on the charged beam divergence in order to get back N… …remember:

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Can we find the beam size? Going to Nx~1, Ny~1, Dx<<1, Dy<<1 we can still say something but analysis implies deconvolution: Virtual source Far zone

III. Application

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH CERN parameter case How about our case? Moreover I took min = 200nm; max = 900nm

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Formation length and critical wavelength L f (m)  c [  m]  Caveat: Whether there is enough light in the visible is another matter of concern, and should be considered separately.

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Beam 1 – Main paramaters How about our case? Beam 1 NyNy DyDy  

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Beam 1 – main parameters How about our case? Beam 1 Far zone for ( z[m] ) 2 much larger than  Y Note: for z=26 m, z 2 =676 m 2

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Beam 2 – main parameters How about our case? Beam 2 NyNy DyDy  

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Beam 2 – main parameters How about our case? Beam 2 Far zone for ( z[m] ) 2 much larger than  Y Note: for z=26 m, z 2 =676 m 2

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Where can we use SR for diagnostics? Challenging for N<<1, D<<1 Still, there are “intermediate regions” where N>1, (with D<<1) The reason for the existence of these “intermediate regions” ( even if  <<  ), is that the beta are large

Conclusions

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Conclusions Generation of radiation  Single particle  Green function for free-space  Paraxial and Ultrarelativistic approximations  Formation length, emission angle  Example: BM radiation  SR from a single particle as a laser-like beam

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Conclusions Coherence properties  Charged bunch: enters shot noise and statistical optics  SR from a bunch as a collection of laser-like beams  Cross-spectral density, relation to intensity and coherence  Near and Far zone  Limiting cases  Diffraction limit  Quasi-homogeneous Gaussian source and VCZ  SR and particle beam size  Intensity measurements  Interferometric measurements

Synchrotron radiation: generation, coherence properties and applications for beam diagnostics – CERN, Genf Gianluca Geloni, European XFEL GmbH Conclusions Applications to diagnostics  Our case of interest