Coherence properties of hard x-ray synchrotron sources and x-ray free-electron lasers

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T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c s

New Journal of Physics

Coherence properties of hard x-ray synchrotronsources and x-ray free-electron lasers

I A Vartanyants1 and A SingerDeutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg,GermanyE-mail: Ivan.Vartaniants@desy.de

New Journal of Physics 12 (2010) 035004 (23pp)Received 21 September 2009Published 31 March 2010Online at http://www.njp.org/doi:10.1088/1367-2630/12/3/035004

Abstract. A general theoretical approach based on the results of statisticaloptics is used for the analysis of the transverse coherence properties of thirdgeneration hard x-ray synchrotron sources and x-ray free-electron lasers (XFEL).Correlation properties of the wavefields are calculated at different distances froman equivalent Gaussian Schell-model source. This model is used to describecoherence properties of the 5 m undulator source at the synchrotron storage ringPETRA III. In the case of XFEL sources the decomposition of the statisticalfields into a sum of independently propagating transverse modes is used for theanalysis of the coherence properties of these new sources. A detailed calculationis performed for the parameters of the SASE1 undulator at the European XFEL.It is demonstrated that only a few modes contribute significantly to the totalradiation field of that source.

1 Author to whom any correspondence should be addressed.

New Journal of Physics 12 (2010) 0350041367-2630/10/035004+23$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

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Contents

1. Introduction 22. Basic equations 5

2.1. Correlation functions of wavefields . . . . . . . . . . . . . . . . . . . . . . . . 52.2. Coherent-mode representation of correlation functions . . . . . . . . . . . . . 62.3. Propagation of the wavefield correlation functions in the free space . . . . . . . 6

3. Coherence properties of third generation synchrotron sources 73.1. Gaussian Schell-model (GSM) source . . . . . . . . . . . . . . . . . . . . . . 73.2. Transverse coherence properties of the PETRA III source . . . . . . . . . . . . 11

4. Coherence properties of XFEL 144.1. Coherent-mode decomposition for the GSM source . . . . . . . . . . . . . . . 144.2. Transverse coherence properties of the European XFEL source . . . . . . . . . 15

5. Summary 18Acknowledgments 19Appendix 20References 22

1. Introduction

With the construction of third generation, hard x-ray synchrotron sources (ESRF, APS andSPring8) with small source sizes and long distances from source to sample (figure 1(a)),new experiments that exploit the high coherence properties of these x-ray beams havebecome feasible. New, high brilliance, low emittance, hard x-ray synchrotron sources, suchas PETRA III at DESY, are starting their operation [1] and will provide an even highercoherent photon flux for users. Experiments exploiting the coherence properties of thesex-ray beams have become even more important with the availability of fourth generationx-ray sources—the so-called x-ray free-electron lasers (XFEL) (figure 1(b)). These are presentlyin the commissioning phase in the USA [2] and under construction in Japan and Europe [3, 4]).Based on the self-amplified spontaneous emission (SASE) process [5], they will provideultrashort, coherent x-ray pulses of unprecedentedly high brightness.

New areas of research have emerged that exploit the high coherence properties of x-raybeams including x-ray photon correlation spectroscopy (for a review see [6, 7]) and coherentx-ray diffractive imaging [8]–[13]. In the former, the dynamic properties of a system are studiedand in the latter static features are analyzed and a real space image of the sample can beobtained by phase retrieval techniques [14, 15]. Similar ideas can be realized through theuse of newly emerging free-electron laser (FEL) sources. These experiments can be furtherextended by the use of coherent femtosecond pulses of extremely high intensity. The firstdemonstration experiments of the possibility of single pulse [16, 17] and single pulse train [18]coherent diffraction imaging were performed recently at the first FEL for extended ultravioletwavelengths, the free-electron laser in Hamburg (FLASH) [19, 20]. In the future, when x-rayFELs with a unique femtosecond time structure will become available, this approach can evenbe used for different applications in materials science [21], the study of dynamics [22] andbiology including such exciting possibilities as single molecule imaging [23].

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Figure 1. Schematic view of (a) the spontaneous radiation from the undulatorsource, and (b) the coherent radiation from the free-electron laser based on theSASE principle.

From these perspectives it is clear that understanding the coherence properties of beamsemerging from third generation synchrotron sources and new FELs is of vital importance forthe scientific community. This includes beamline scientists designing beamlines for coherentapplications, or experimentalists planning experiments with coherent beams. A beamlinescientist, for example, would like to estimate the performance of optics in the partially coherentbeam emerging from a synchrotron source. Nowadays, several codes are used for simulationof x-ray propagation from a source through the optical system of a beamline towards anexperimental hutch. Most of them are based either on the ray tracing approach (SHADOW [24],RAY [25], etc) which is, in fact, the geometrical optics limit, or on the Fourier optics approach(PHASE [26], etc). It will be shown in our paper that the first approach of ray tracing canbe safely used for synchrotron radiation in the horizontal direction, where the radiation fieldis mostly incoherent, but cannot be used in the vertical direction, where the radiation fromthe undulator sources is highly coherent. Unfortunately, the approach of Fourier optics in thedescription of the scattering of synchrotron radiation in the vertical direction is also limitedwhen the radiation is not fully coherent but rather partially coherent. Simulations performed forthe XFEL sources at their saturation [27] suggest that these sources are not fully coherent,and as a consequence, partial coherence effects must be carefully considered for them aswell. The contribution of partial coherence effects is also important in phase retrieval asa reduced coherence of the incoming beam can produce artifacts in reconstructed images[28]–[30].

Different approaches may be used to analyze the coherence properties of the synchrotronand XFEL sources. One is based on a detailed modeling of the radiation process ofultrarelativistic particles in a storage ring or an insertion device (see e.g. [31, 32]), or adetailed modeling of the SASE process by performing calculations of nonlinear electromagneticequations at different conditions of operation (linear regime, saturation, etc) [27]. Anotherpossible approach is based on the results of statistical optics [33, 34], when the statistics ofthe radiation wavefield is analyzed with the very general assumptions about the origin of theradiators [35]–[38]. An attractiveness of this approach is based on the fact that with just a fewparameters the coherence properties of the beam can be described at different distances fromthe source. A famous example of such an approach is the van Cittert–Zernike theorem [33, 34]that predicts the coherence properties of an incoherent source at any distance from that source.

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The most important assumption when deriving this theorem is that the source is completelyincoherent. This means that there are no correlations between any two points of the source atany separation between these two points. As such, it does not require a detailed descriptionof the physical origin of the radiation process of the source. This theorem is widely appliedfor the estimate of the coherence properties of third generation synchrotron sources. This isbased on the assumption that each electron in the electron bunch is radiating independentlyfrom another electron, which means the radiation should be considered as incoherent. However,this assumption, which is absolutely correct from the point of radiation physics, brings us toa certain contradiction when a radiation field at a large distance from a synchrotron source isanalyzed. The van Cittert–Zernike theorem gives a realistic estimate of the coherence lengthdownstream from the source, but at the same time it predicts that such a source should radiatein the solid angle of 4π as a completely incoherent source. However, it is well known fromelectrodynamics (see e.g. [39]) that ultrarelativistic particles radiate mostly in the forwarddirection in a narrow cone 1θ ∼

√1 − v2/c2 around the direction of the velocity vector v,

where v is the velocity of the particle and c is the speed of light. As a consequence, typical x-raybeams produced by synchrotron radiation sources have a beam character with a narrow cone andnegligible radiation in all other directions. The only way to solve this contradiction in the frameof statistical optics is to assume a certain degree of coherence for a synchrotron source. Withthis approach we will substitute a real synchrotron source by an equivalent source with a certainsource size and, what is especially important, a finite degree of coherence that will producewavefields with the same statistical properties as a radiation field from a real source. Withthese general assumptions, we will use the results of statistical optics to calculate the statisticalproperties of beams emerging from a new source, PETRA III, at different distances from thesource.

We use a different approach for the description of the coherence properties of the FELs.These highly coherent sources can be described with a finite number of transverse modes.We will use a decomposition of the statistical fields into a sum of independently propagatingtransverse modes and calculate correlation fields at different distances from the source. Thesource itself will be described by the same Gaussian functions, as in the case of the synchrotronradiation, but clearly with different values of the source parameters. Recently, we characterizedthe coherence properties of FLASH using this method [40]. For the calculations, in this paper,we will consider the concrete example of the European XFEL SASE1 undulator source [4]. Wewill show that for this source only a few modes contribute substantially to the total radiationfield. In principle, this approach is quite general [34] and can be applied to any statisticalfield (not necessarily a Gaussian source) [41]. For a limited number of contributing modes,it significantly simplifies numerical calculations by reducing the number of variables. This maybe especially important for beamline optics characterization.

The paper is organized as follows. In section 2 the basic equations of statistical optics willbe presented. In section 3 this general approach will be applied for the characterization of thestatistical properties of the beams emerging from the third generation synchrotron sources. Insection 4 the coherent-mode representation of wavefields will be used for the characterizationof the coherence properties of the XFEL beams. The paper ends with the conclusions andan outlook in section 5. In the appendix we compare results obtained in this paper with thetheoretical approach developed in [32] for calculation of coherence properties of radiationemerging from the undulator sources.

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2. Basic equations

2.1. Correlation functions of wavefields

The central concept in the theory of partial coherence is the so-called mutual coherence function(MCF), 0(r1, r2; τ), that describes the correlations between two complex scalar2 values of theelectric field at different points r1 and r2 and at different times. It is defined as [33, 34]

0(r1, r2; τ)= 〈E(r1, t + τ)E∗(r2, t)〉T , (1)

where E(r1, t + τ) and E(r2, t) are the field values at the points r1 and r2, τ is the time delayand brackets 〈. . .〉T mean an averaging over times T much longer than the fluctuation time ofthe x-ray field. It is also assumed that the radiation is ergodic and stationary. From the definitionof the MCF it follows that when two points coincide the average intensity is given by

〈I (r)〉 = 0(r, r; 0)= 〈E(r, t)E∗(r, t)〉T . (2)

It is usual to normalize the MCF as

γ (r1, r2; τ)=0(r1, r2; τ)

√〈I (r1)〉

√〈I (r2)〉

, (3)

which is known as the complex degree of coherence. For all values of the arguments r1, r2 andτ the absolute values of the complex degree of coherence 06 |γ (r1, r2; τ)|6 1.

Following Mandel and Wolf [34] the cross-spectral density (CSD) function, W (r1, r2;ω),can be introduced. According to its definition it forms a Fourier transform pair with the MCF inthe time–frequency domain

W (r1, r2;ω)=

∫∞

−∞

0(r1, r2; τ)eiωτ dτ. (4)

The CSD is a measure of correlation between the spectral amplitudes of any particular frequencycomponent ω of the light vibrations at the points r1 and r2. When two points r1 and r2 coincide,the CSD represents the spectral density, S(r;ω) (or the power spectrum) of the field

S(r;ω)= W (r, r;ω). (5)

Introducing the normalized CSD function, µ(r1, r2;ω), which is called the spectral degree ofcoherence (SDC) at frequency ω, we obtain

µ(r1, r2;ω)=W (r1, r2;ω)

√S(r1;ω)

√S(r2;ω)

, (6)

where again, as for the case of the complex degree of coherence, the following inequality isvalid 06 |µ(r1, r2;ω)|6 1.

In order to characterize the transverse coherence properties of the wavefields by onenumber the normalized degree of the transverse coherence can be introduced as [27]

ζ(ω)=

∫|µ(r1, r2;ω)|

2S(r1;ω)S(r2;ω) dr1 dr2∣∣∫ S(r;ω) dr∣∣2 . (7)

According to its definition the values of the parameter ζ(ω) lie in the range 06 ζ(ω)6 1.For narrow bandwidth light, the complex degree of coherence can be approximated by

γ (r1, r2; τ)≈ γ (r1, r2; 0) exp(−iω̄τ ) leading to a relationship |γ (r1, r2; τ)| = |µ(r1, r2; ω̄)|,

2 In the following, for simplicity, we consider only one polarization of the x-ray field.

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where ω̄ is the average frequency. In the same conditions, similar relationships can be obtainedfor the spectral density and the average intensity S(r;ω)= 〈I (r)〉 δ(ω− ω̄), where δ(ω) is thedelta function. As a consequence, for narrow bandwidth light, the spectral density is equivalentto the average intensity.

2.2. Coherent-mode representation of correlation functions

It has been shown [34], that under very general conditions, one can represent the CSD of apartially coherent, statistically stationary field of any state of coherence as a sum of independentcoherent modes

W (r1, r2;ω)=

∑j

β j(ω)E∗

j (r1;ω)E j(r2;ω), (8)

where β j(ω) and E j(r;ω) are the eigenvalues and eigenfunctions that satisfy the Fredholmintegral equation of the second kind∫

W (r1, r2;ω)E j(r1;ω) dr1 = β j(ω)E j(r2;ω). (9)

The eigenfunctions in (8) and (9) form an orthogonal set.According to the definition of the spectral density, S(r;ω), (5) we have in the case of a

coherent-mode representation of the fields

S(r;ω)=

∑j

β j(ω)|E j(r;ω)|2. (10)

2.3. Propagation of the wavefield correlation functions in the free space

For our purposes it is especially important to calculate correlation functions at differentdistances from the source. Propagation of the CSD W (r1, r2;ω) in the half-space z > 0 fromthe source plane at z = 0 to the plane at distance z in paraxial approximation is determined bythe following expression [34]:

W (r1, r2, z;ω)=

∫6

∫6

WS(s1, s2, 0;ω)P∗

z (r1 − s1;ω)Pz(r2 − s2;ω) ds1 ds2, (11)

where WS(s1, s2, 0;ω) is the value of the CSD at the source plane z = 0, Pz(r − s; ω) is theGreen’s function (or propagator) and the integration is made in the source plane. In (11) andbelow coordinates, s1 and s2 are taken in the plane of the source and coordinates r1 and r2 aretaken in the plane at the distance z. The propagator Pz(r − s;ω) describes the propagation ofradiation in free space and is defined as

Pz(r − s;ω)=1

iλzexp

[i

k

2z(r − s)2

], (12)

where k = 2π/λ and λ is the wavelength of radiation.In the case of the coherent-mode representation of the correlation functions, the values of

the CSD can be obtained at different distances by propagating the individual coherent modes.Propagation of the individual modes in the half-space z > 0 from the source plane at z = 0 tothe plane at distance z in the paraxial approximation can be obtained from

E j(r, z;ω)=

∫6

ESj (s;ω)Pz(r − s;ω) ds, (13)

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where ESj (s;ω) are the values of the field amplitudes at the source plane, z = 0, and Pz(r − s;ω)

is the same propagator as in (12). Due to the statistical independence of the modes [34], theCSD, after propagating a distance z, is given as a sum of propagated modes E j(r, z;ω) with thesame eigenvalues β j(ω) defined in (9):

W (r1, r2, z;ω)=

∑j

β j(ω)E∗

j (r1, z;ω)E j(r2, z;ω). (14)

3. Coherence properties of third generation synchrotron sources

3.1. Gaussian Schell-model (GSM) source

In this section, we apply the general theory of the propagation of Gaussian optical beams froma source of any state of coherence to the case of synchrotron radiation. We assume that a realsynchrotron source (for example an insertion device like an undulator) can be represented byits equivalent model that produces x-ray radiation with statistical properties similar to a realx-ray source. This equivalent model source will be positioned in the center of the real undulatorsource.

We will assume that the x-ray radiation is generated by a planar GSM source3. Such sourcesare described by the following CSD function [34]:

WS(s1, s2)=

√SS(s1)

√SS(s2)µS(s2 − s1), (15)

where the spectral density and SDC of the x-ray beam in the source plane are Gaussianfunctions4

SS(s)= S0x S0y exp

(−

s2x

2σ 2Sx

−s2

y

2σ 2Sy

),

µS(s2 − s1)= exp

(−(s2x − s1x)

2

2ξ 2Sx

−(s2y − s1y)

2

2ξ 2Sy

).

(16)

Here parameters σSx,y define the rms source size in the x- and y-directions and ξSx,y give thecoherence length of the source in the respective directions.

The starting expression for the source CSD function in the form of expression (15), is infact, very general and is based on the definition of the SDC (6). Here, the main approximationsare that the source is modeled as a plane two-dimensional source, that the spectral density, SS(s),and the SDC, µS(s2 − s1), (16) are Gaussian functions, and that the source is spatially uniform(SDC µS(s2 − s1) depends only on the difference of spatial coordinates s1 and s2). The fact thatsynchrotron radiation sources are typically elongated in the horizontal direction is specificallyintroduced in expression (16) by allowing the source size σSx,y and coherence length of thesource ξSx,y to be different in x- and y-directions. What is especially important in this model isthe assumption of a certain degree of coherence of the source expressed by a finite coherencelength of that source ξSx,y . Only with this finite coherence length of the source, it is possible toobtain a reasonable description of the synchrotron radiation with its extremely small divergence.

3 The applicability of the GSM for the description of the synchrotron radiation is discussed in details in [31, 32,36, 37].4 In this equation and below the frequency dependence ω is omitted.

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It can be shown [34] that with a suitable choice of source size, σS, and coherence length, ξS,a GSM source can generate a field whose intensity has appreciable values only within a narrowcone of solid angle. In optics, radiation with a narrow angular divergence is called a beam. Thiswill be a good model for x-ray beams generated by insertion devices, such as undulators, whichproduce x-ray beams with a divergence of a few microradians. In order to generate the beam,the parameters of a GSM source have to satisfy the following inequality in each direction:

1

δ2Sx,y

�2π2

λ2, (17)

where1

δ2Sx,y

=1

(2σSx,y)2+

1

ξ 2Sx,y

. (18)

For x-ray wavelengths of about 0.1 nm, we obtain for the right-hand side of this inequality2 × 109 µm−2. The smallest size of a source that is at the moment available at third generationsynchrotron in the vertical direction is of the order of a few microns. We have not yetestimated the coherence length ξS of the source, but as we will see later it is also of theorder of a few microns. From these estimates, it is seen that the beam condition (17) is verywell satisfied for x-ray wavelengths and third generation synchrotron sources. This gives usconfidence using the beam approach to describe the properties of the x-ray radiation from thesesources.

There are two important limits which we can describe as an incoherent or coherent source.The source will be called incoherent if its coherence length ξS is much smaller than the sourcesize ξS � σS. From the beam condition (17) we find for this source

δS ≈ ξS �1

√2πλ.

This means that to satisfy the beam conditions for a spatially incoherent source, the coherencelength has to be small but at the same time larger than the wavelength σS � ξS � λ. In theopposite limit of a spatially coherent source ξS � σS, we find from the beam condition (17)

δS ≈ 2σS �1

√2πλ.

This means that to satisfy the beam condition for a spatially coherent source, the source sizeshould be larger than the wavelength ξS � σS � λ.

Integration in (11) with the CSD WS(s1, s2) (15, 16) can be done independently for eachdimension. This gives the following expression for the CSD W (x1, x2, z)5 at distance z from thesource [42, 43] (see also [34])

W (x1, x2, z)=I0x

1x(z)eiψx (z) exp

(−(x1 + x2)

2

8σ 2Sx1

2x(z)

)exp

(−(x2 − x1)

2

2δ2Sx1

2x(z)

). (19)

Here

1x(z)=

[1 +

(z

zeffx

)2]1/2

(20)

5 Below we present results only in the x-direction. The same equations are valid in the y-direction.

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is called an expansion coefficient and

ψx(z)=k(x2

2 − x21)

2Rx(z), Rx(z)= z

[1 +

(zeff

x

z

)2]

(21)

are the phase and the radius of curvature of a Gaussian beam. In (20, 21) an effective distancezeff

x is introduced that is defined as

zeffx = kσSxδSx . (22)

At that distance the expansion coefficient 1(zeffx )=

√2. In the limit of a spatially coherent

source δS ≈ 2σS and an effective distance zeffx coincides with the so-called Rayleigh length

zR = 2kσ 2Sx , which is often introduced in the theory of optical Gaussian beams [44]. According

to (20) an effective distance zeffx can serve as a measure of the distances, where nonlinear effects

in the propagation of the beams are still strong. Distances z � zeffx can be considered as a far-

field limit where the expansion parameter 1x(z)→ z/zeffx and the radius Rx(z)→ z change

linearly with the distance z.Setting x1 = x2 = x in (19), we obtain for the spectral density

S(x, z)=I0x

1x(z)exp

(−

x2

262x(z)

), (23)

where

6x(z)= σSx1x(z)=(σ 2

Sx + θ 26x z2

)1/2(24)

is an rms size of the x-ray beam at a distance z from the source. In (24) θ6x is the angulardivergence of the beam

θ6x =1

2kξSx

(4 + q2

Sx

)1/2. (25)

Here a ratio of the coherence length to the source size

qSx =ξSx

σSx(26)

is introduced. It can be considered as a measure of the degree of coherence of the source.According to its definition (6) and (19, 23), the SDC at a distance z from the source is given

by

µ(x1, x2, z)= eiψx (z) exp

(−(x2 − x1)

2

242x(z)

), (27)

where

4x(z)= ξSx1x(z)=(ξ 2

Sx + θ 24x z2

)1/2(28)

is the effective coherence length of an x-ray beam at the same distance. In (28) θ4x is the angularwidth of the coherent part of the beam

θ4x =1

2kσSx

(4 + q2

Sx

)1/2. (29)

For the incoherent source (qSx � 1), we have from (25, 29)

θ6x =1

kξSx, θ4x =

1

kσSx. (30)

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It is seen immediately that in this limit equation (28) predicts the same values for the coherencelength 4x(z) at large distances z as given by the van Cittert–Zernike theorem. At the same timeexpression (30) gives an estimate for the divergence of the beam from an incoherent source,which is determined by the coherence length ξSx of the source. So, directly from (30), we havean estimate of the coherence length of the incoherent source

ξSx =λ

2πθ6x. (31)

In another limit of a coherent source, when parameter qSx � 1, we obtain from (25, 29)

θ6x =1

2kσSx, θ4x =

1

2kσSxqSx . (32)

In this coherent limit, the angular width of the coherent part of the beam exceeds the angulardivergence of the beam, which is determined now only by the size of the source, and we areapproaching here the limit of a so-called diffraction limited source.

In the frame of the GSM, the coherence length of the source of any state of coherencecan be expressed conveniently through its emittance εSx = σSxσ

′

Sx , where σ ′

Sx is the rms of theangular divergence of the source. It can be obtained by inverting the full expression of theangular divergence of the beam (25)

ξSx =2σSx√

4k2ε2Sx − 1

(33)

and the substitution of the angular divergence θ6x by σ ′

Sx .One important property of the beams generated by the GSM sources is that at any distance

from the source the ratio of the coherence length 4x(z) to the beam size 6x(z) is a constantvalue and is equal to the same ratio at the source. From (24, 28) we have for the parameter qx

qx =ξSx

σSx=4x(z)

6x(z). (34)

The normalized degree of the transverse coherence ζx introduced in (7) can be directlycalculated for the GSM source and related to the values of qx by the following expression:

ζx =qx√

q2x + 4

. (35)

According to relationship (34), the values of the degree of transverse coherence ζx for the GSMare preserved for any distance z from the source.

The emittance of a GSM source εSx can be expressed through the degree of the transversecoherence ζx . From (25) and (35) we have for the emittance of a GSM source

εSx =1

2kζx. (36)

Taking into account that for a source of any degree of coherence the values of ζx lie in the range06 ζx 6 1, the values of the emittance should satisfy an inequality εSx > 1/2k = λ/4π . For afully coherent source ζx → 1 and the emittance εcoh

Sx = λ/4π . This value can be considered asthe emittance of a diffraction limited source. For an incoherent source ζx → 0 and according to(36), εSx � λ/4π .

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Table 1. Parameters of the high brilliance synchrotron radiation sourcePETRA III for a 5 m undulator [1] (energy E = 12 keV, distance from thesource z = 60 m).

High-β Low-βx y x y

Source size σS, µm 141 5.5 36 6Source divergence σ ′

S, µrad 7.7 3.8 28 3.7Transverse coherence length

at the source ξS, µm 2.07 4.53 0.57 4.65Degree of coherence q 0.015 0.82 0.016 0.77Normalized degree of coherence ζ 0.008 0.38 0.008 0.36Effective length zeff, m 18.33 1.48 1.29 1.63Beam size at distance z 6(z), mm 0.48 0.23 1.68 0.22Transverse coherence length

at distance z 4(z), µm 7.08 187.8 26.5 170.5

3.2. Transverse coherence properties of the PETRA III source

Our previous analysis can be effectively used to estimate the coherence properties of the beamsproduced by third generation x-ray sources if source parameters (source size and divergence) areknown. We will make this calculation for the high brilliance synchrotron source PETRA III atDESY [1]. This storage ring is planned to produce εx = 1 nm emittance beams in the horizontaldirection and, due to 1% coupling, the emittance in the vertical direction will be two orders ofmagnitude lower.

Source parameters for a 5 m long undulator and a photon energy of 12 keV are summarizedin table 1. Two cases of high-β and low-β operation of PETRA III are considered. Thevalues of the coherence length of the source calculated according to (33) vary from 0.6 to2µm in the horizontal direction and are about 5µm in the vertical. We can estimate thevalues of the parameter qS (34) and the normalized degree of the transverse coherence ζS ofthat source. Using tabulated values of the source size, we obtain for the horizontal directionqSx ∼ 0.02 and for the vertical qSy ∼ 0.8. For the normalized degree of the transverse coherenceζS (35) we have in the horizontal direction ζSx ∼ 0.01 and in the vertical ζSy ∼ 0.4. Theseestimates immediately show that in the horizontal direction PETRA III source is a ratherincoherent source with the degree of coherence about 1%; however, in the vertical directionthe coherence length of the source is about the size of the source itself, and it can beconsidered as a rather coherent source with a degree of coherence about 40%. Substitutingthese numbers into (24, 28), we can obtain the values of the intensity distribution and thetransverse coherence length at any distance downstream from the source. These values arelisted in table 1 for a distance z = 60 m, where the first experimental hutches are planned.We see that for this distance the coherence length is varying from 7 to 25µm in thehorizontal direction and is in the range from 170 to 190µm in the vertical one. This definesthe coherence area across the beam within which one can plan experiments with coherentbeams.

The absolute value of the SDC, |µ(1x)|, (27) as a function of the separation of two pointsacross the beam and the spectral density, S(x), (23) as a function of the position across the

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High-β vertical(a) (b) High-β horizontal

Figure 2. Absolute value of the SDC |µ(1x)| as a function of separation of twopoints across the beam at a distance of 60 m downstream from the source forhigh-β operation of PETRA III in the vertical (a) and horizontal (b) directions.The spectral density S(x) as a function of the position across the beam calculatedat the same distance from the source is shown in the insets. The rms values of thebeam size 6x,y(z) and transverse coherence length 4x,y(z) at that distance weretaken from table 1.

beam at a distance 60 m downstream from the source for high-β operation of PETRA III arepresented in figure 2. The rms values of the beam size 6x,y(z) and transverse coherence length4x,y(z) were taken from table 1. The full width at half maximum (FWHM) of the beam isabout 500µm in the vertical direction (figure 2(a)). For separations of up to 100µm the beamis highly coherent (with the degree of coherence higher than 80%). In the horizontal direction(figure 2(b)) the situation is quite different. The FWHM of the beam is about 1 mm. It is wellseen in figure 2(b) that the beam is rather incoherent in the horizontal direction and the degreeof coherence is higher than 80% for separations of up to 5µm.

Calculations of the beam size, 6x,y(z), and the transverse coherence length, 4x,y(z), atdifferent distances z from the source are made according to (24, 28). They are presented forhigh-β operation of PETRA in figure 3. These calculations show that in the vertical directionthe rms values of the coherence length, 4y(z) (black, solid line) are slightly smaller than therms values of the beam size, 6y(z) (red, dash line) along the beamline. At distances larger thanzeff

y ' 1.5 m in the vertical direction, all z-dependencies can be considered to be linear. It meansthat for all practical cases all parameters scale linearly with the distance z. In the horizontaldirection the situation is quite different. Firstly as expected and clearly seen in figure 3, thebeam is quite incoherent in this direction. It is interesting to note that the linear z-dependence ofparameters 6x(z) and 4x(z) for high-β operation starts from further distances from the source.An effective distance zeff

y is about 20 m in this case. Similar results for the low-β operation ofthe PETRA III source can be obtained using parameters listed in table 1.

We compared the results obtained by our approach with the results of different calculationsperformed for the PETRA III 5 m undulator source by the ESRF simulation code SRW [45] aswell as by the analytical results obtained in [32] (see the appendix for details). Our calculationsshow that the divergence of the beam both in the vertical and in the horizontal directions is welldescribed by our model as compared with the SRW calculations (see figure 3). The comparisonwith the results of [32] was performed only in the vertical direction as in the horizontal directionthe source can be described as a quasi-homogeneous source (coherence length of such a sourceis much smaller than the size of the source (ξSx � σSx )). In this limit the analytical results of [32]

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Figure 3. Beam size 6x,y(z) and the transverse coherence length 4x,y(z) atdifferent distances z from the source for a high-β operation of the PETRA IIIstorage ring. Parameters of the source are taken from table 1. In all figuresthe dashed (red) line is the beam size 6x,y(z) and the solid (black) line is thetransverse coherence length 4x,y(z). Open circles correspond to calculationsperformed by the ESRF simulation code SRW [45], open triangles are the beamsize and squares are the transverse coherence length obtained from the analyticalresults of [32]. (a, c) Vertical direction of the beam. (b, d) Horizontal directionof the beam. The vertical dashed line in (c) and (d) corresponds to an effectivedistance zeff. Note, different range for the coherence length comparing to that ofthe beam size in (a, b).

completely coincide with our description of the source in the frame of the GSM. However,in the vertical direction a more careful analysis is required. Using the approach of [32] wecalculated the SDC and spectral density for a 5 m undulator of the PETRA III source in thevertical direction at different distances z from the source. From these calculations, we obtainedthe rms values of the source size 6x(z) and the coherence length 4x(z) at different distancesfrom the source and compared them with the results obtained from the GSM (see figures 3(a) and(c)). This comparison shows very good agreement between two approaches for these energies.However, the analytical approach of [32] gives slightly lower values of the beam size and thecoherence length at larger distances. The lower values of the beam size predicted by Geloniet al [32] can be attributed to effects of a finite energy spread of electrons in the bunch thatwere neglected in calculations. From this comparison, we see that an approach based on theGSM and simulations performed by different methods at a wavelength of 0.1 nm give similarresults6.6 We performed a similar analysis for different energies and came to the conclusion that for PETRA III sourceparameters the GSM model can be safely used at energies higher than 6 keV (see the appendix for details).

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4. Coherence properties of XFEL

4.1. Coherent-mode decomposition for the GSM source

We apply a general approach of coherent-mode decomposition, described in section 2, forthe analysis of the correlation properties of wavefields originating from XFEL sources. Wesubstitute a real XFEL source by an equivalent planar GSM source (15, 16). Coherent modesand eigenvalues obtained as a solution of the Fredholm integral equation (9) for such aGSM source are well known [46, 47] and can be decomposed for each transverse directionEl,m(sx , sy)= El(sx)Em(sy) and βl,m = βlβm . The eigenvalues β j for the GSM source have apower law dependence and the eigenfunctions E j(sx) at such a source are described by theGaussian Hermite-modes7 [48]

β j/β0 = κ j , (37)

E j(sx)=k1/4(

π zeffx

)1/4

1(2 j j!

)1/2 H j

(√k

zeffx

sx

)exp

[−

(k

2zeffx

)s2

x

], (38)

where H j(x) are the Hermite polynomials of order j , β0 =√

8π S0xσSxδSx/(2σSx + δSx) andκ = (2σSx − δSx)/(2σSx + δSx). The parameters σSx , δSx and zeff

x have the same meaning as in(16, 18, 22).

Equation (37) for the eigenvalues of the GSM source gives, in fact, the relative weightswith which the different modes contribute to the CSD of the source. It can also be expressedthrough the values of the parameter qx (34) [34], or through the values of the normalized degreeof the transverse coherence ζx (35)

β j

β0=

[1(

q2x/2

)+ 1 + qx

[(qx/2)2 + 1

]1/2

] j

=

(1 − ζx

1 + ζx

) j

. (39)

For a spatially coherent source (ξSx � σSx ) we have from (39)β j

β0≈ q−2 j

x . (40)

According to this equation β j � β0 for all j 6= 0 which means that in the coherent limit thesource can be well characterized by its lowest mode. In the opposite limit of an incoherentsource (ξSx � σSx ), we have from (39)

β j

β0≈ 1 − jqx . (41)

According to this equation many modes are necessary for a sufficient description of the source.Correlation properties of the fields in the space-frequency domain at any distance z from

the source can be calculated with the help of expression (14) by propagating individual modesE j(x, z). In the case of the GSM source, the propagated modes E j(x, z) at a distance z fromthe source are described by the following expression [48]:

E j(x, z)=k1/4(

π zeffx 1

2x(z)

)1/4

1(2 j j!

)1/2 H j

[√k

zeffx

(x

1x(z)

)]exp

[−

k

2zeffx

(x

1x(z)

)2]

× exp

{i[kz − ( j + 1)φx(z)] +

ikx2

2Rx(z)

}, (42)

7 Due to the symmetry of the GSM we consider below only one transverse direction.

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Table 2. Parameters of the SASE1 undulator of the European XFEL [4].

SASE1undulator

Wavelength λ, nm 0.1Source size σS, µm 29.7Source divergence σ ′

S, µrad 0.43Transverse coherence length

at the source ξS, µm 48.3Degree of coherence q 1.63Normalized degree of coherence ζ 0.63Effective length zeff, m 70

where φx(z)= arctan(z/zeffx ). Parameters 1x(z) and Rx(z) have the same meaning as in

(20, 21). For j = 0 these modes coincide with an expression for a monochromatic Gaussianbeam propagating from a Gaussian source.

4.2. Transverse coherence properties of the European XFEL source

We used this approach to make a realistic and simple estimate of the coherence propertiesof the upcoming XFEL sources. For detailed calculations we took parameters of the SASE1undulator at the European XFEL reported in [4] (see also [27]) and summarized in table 2.Simulations were made for a GSM source (15, 16) with an rms source size σS = 29.7µmand a transverse coherence length at the source of ξS = 48.3µm. The latter parameter wasobtained from (33) using the values of the source size and angular divergence listed in table 2.With these parameters the CSD, W (x1, x2, z), (14) was calculated at a distance of 500 m fromthe source with the eigenvalues β j and eigenfunctions E j(x, z) evaluated from (37, 42). Adistance of 500 m was considered because at this distance the first optical elements of theEuropean XFEL are planned [4]. In figure 4 the results of these calculations are presented.An analysis of the results shows that for the parameters of the SASE1 undulator at the EuropeanXFEL a small number of transverse modes contribute to the total field (figure 4(c)). Parameterκ = 0.22 in these conditions, which means that the contribution of the first mode is about 20%of the fundamental and the contribution of the fourth mode is below 1% of the fundamentalβ4/β0 = κ4

= 2.3 × 10−3. Finally, five modes (including the fundamental) were used in (14)for calculations of the CSD W (x1, x2, z) (figure 4(a)). From the obtained values of the CSD,the modulus of the SDC, |µ(x1, x2, z)|, (figures 4(b) and (d)) and the spectral density, S(x), atthat distance were evaluated. As our source is described as a Gaussian source, these functionsare Gaussian as well. At a distance z = 500 m from the source we obtained a coherence length4(z)= 348µm and a beam size 6(z)= 214µm.

Analysis of figure 4(d) shows that our model source, though being highly coherent, cannotbe described as a fully coherent source. The problem lies in the contribution of the higher modesto the fundamental. This is illustrated in more detail in figure 5 where the SDC, |µ(1x)|, iscalculated with a different number of contributing modes at separation distances of up to 1 mmwhere the spectral density S(x) is significant. It is readily seen from this figure that only in thecase of a single mode contribution will an XFEL beam be fully coherent (figure 5(a)). As soonas the first transverse mode contributes to the fundamental, the SDC, |µ(1x)|, drops quickly

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(a) (b)

(c) (d)

Figure 4. Calculations of the coherence properties of the SASE1 undulator at theEuropean XFEL (see table 2) 500 m downstream from the source in the frameof a GSM source: (a) absolute value of the CSD |W (x1, x2)|, (b) absolute valueof the SDC |µ(x1, x2)|, (c) ratio β j/β0 of the eigenvalue β j to the lowest ordereigenvalue β0 as a function of mode number j and (d) absolute value of thespectral degree of transverse coherence |µ(1x)| taken along the white line in(b). In the inset spectral density S(x) is shown that is taken along the white linein (a).

and reaches zero at a separation distance of 1x ≈ 700µm (figure 5(b)). It again increasesup for higher separation distances and reaches the value |µ(1x)| = 0.3 at 1x ≈ 1 mm. Thisincrease in the correlation function is due to the fact that at these distances the contribution ofthe lowest mode (fundamental in this particular case) is negligible and the correlation propertiesare determined again by a single mode (the first in this case). As it is well seen in figure 5 in thisparticular case the spectral density can be described by three modes.

The values of the beam size 6(z) and the transverse coherence length 4(z) at differentdistances z from our GSM source are presented in figure 6. Calculations were performedusing a coherent-mode decomposition (14) of the CSD W (x1, x2, z) at different distances fromthe GSM source. It can be seen from figure 6 that contrary to the analysis performed fora synchrotron source, here, in the case of the European XFEL, the values of the transversecoherence length4(z) are higher than the values of the beam size6(z) at all distances from thesource downstream. An effective distance zeff (22) is about 70 m in this case, which means thatfor distances z � 70 m all z-dependencies of parameters, such as the coherence length and thebeam size, can be considered to be linear.

Using the previously introduced values of the normalized degree of the transversecoherence ζ (35) and the parameters obtained for an equivalent GSM source (see table 2), we

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(a) (b)

(c) (d)

Figure 5. Contribution of the higher transverse modes to the absolute valueof the SDC |µ(1x)|. The same for the spectral density S(x) is shown in theinsets. (a) Fundamental mode contribution, (b) fundamental plus first modecontribution, (c) fundamental plus two modes contribution and (d) fundamentalplus three modes contribution. In all figures, the dashed line corresponds to anactual number of modes contributing to |µ(1x)| and S(x). In all figures solidline corresponds to a full calculation of |µ(1x)| and S(x) with the five modes.Calculations were made for the same parameters as in figure 4.

find that ζ = 0.63 for that source. This means we can expect a transverse coherence of about60% at the European XFEL. This number is in good agreement with the value ζ = 0.65 obtainedby the ensemble average of the wavefields produced by the SASE1 undulator of the EuropeanXFEL calculated by the code FAST using the actual number of electrons in the beam [27]. Thisgood agreement obtained for the value of the degree of the transverse coherence ζ by differentapproaches gives good fidelity for the analysis proposed in this work based on the results ofstatistical optics and a simple characterization of the source with a GSM.

Here, for a sufficient description of the transverse coherence properties of FELs, we useda coherent-mode decomposition approach. In principle, the same approach can be used forthe description of the third generation synchrotron sources, however being mostly incoherentsources, especially in the horizontal direction, they would require a large number of modes for asufficient description. This is illustrated in figure 7 where the ratio β j/β0 of the eigenvalue β j tothe lowest order eigenvalue β0 as a function of a mode number j for the PETRA III synchrotronsource is presented. For comparison, results of the calculations for the SASE1 undulator of theEuropean XFEL are also shown in the same figure. In the calculations we considered the samePETRA III parameters as in the previous section (see table 1, high-β operation of the PETRAIII storage ring). Our results demonstrate that in the vertical direction correlation functions canbe properly described by the contribution of eight modes (including the fundamental) and in the

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Figure 6. Beam size6(z) (dashed line) and the transverse coherence length4(z)(solid line) at different distances z from the SASE1 undulator of the EuropeanXFEL source. Parameters of the source are the same as in figure 3. The verticaldashed line corresponds to an effective distance zeff.

Figure 7. Ratio β j/β0 of the eigenvalue β j to the lowest order eigenvalue β0 as afunction of mode number j . Results of the calculations for the parameters of theSASE1 undulator of the European XFEL (open circles), high-β operation of the5 m undulator of the PETRA III source in the vertical direction (triangles) and inthe horizontal direction (squares).

horizontal direction a large number of modes (about 300) is necessary to describe the coherenceproperties of the undulator source. This is in good agreement with the behavior of the modesdescribed by (40, 41) for a coherent and an incoherent source. Our results indicate that in thevertical direction the undulator source is highly coherent, however in the horizontal direction itbehaves as an incoherent source.

5. Summary

In summary, we have demonstrated how a general theoretical approach based on the results ofstatistical optics can be applied to give a sufficient description of the correlation properties of thefields generated by third generation synchrotron sources and FELs. We have substituted a realsource by an equivalent planar GSM source with the same source size and divergence as a realsource. This phenomenological approach gives us the opportunity to characterize this source

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19

with just two parameters, source size and transverse coherence length. What is more importantis that this approach can be used as a tool to calculate correlation properties at different distancesfrom the source with simple analytic functions. In this way, realistic estimates of the size of anx-ray beam and its coherence length can be obtained at any distance from the source. Thisalso gives a beam profile and a complex degree of coherence in the transverse direction at anydistance from the source.

We applied this general approach to the concrete case of the PETRA III source that is underconstruction. Our calculations have shown that the coherence properties of this synchrotronsource are quite different in the vertical and the horizontal directions (this is typical of all thirdgeneration synchrotron sources). In the vertical direction, a beam produced with the parametersof PETRA III is highly coherent with a degree of coherence of about 40%, however, in thehorizontal direction it is a rather incoherent source with a degree of coherence of about 1%.Sixty meters downstream from that source, where the first experimental hutches are planned,the transverse coherence length in the vertical direction reaches a value of 190µm and in thehorizontal direction can be about 25µm. This has to be compared with the size of the beam at thesame distance with the FWHM values about 0.5 mm in the vertical and 1 mm in the horizontaldirections. The values of the transverse coherence length and the beam size scale linearly atthese large distances from the source and can be easily estimated at any other distance from thesource.

In the case of the XFELs, we used a decomposition of the statistical fields into a sumof independently propagating transverse modes for the analysis of the coherence properties ofthese fields at different distances from the source. Calculations were performed for the concretecase of the SASE1 undulator at the European XFEL, which is presently under construction.Our analysis has shown that only a few transverse modes (five in the case of European XFEL)contribute significantly to the total radiation field of the XFEL. It was demonstrated that dueto the contribution of a few transverse modes, the SASE1 undulator source while being highlycoherent (with the degree of coherence about 60%), cannot be considered as fully coherent.One essential difference between the radiation field from the XFEL compared with that of asynchrotron source is that its coherence properties are expected to be of the same order ofmagnitude in the vertical as well as in the horizontal direction (compare with the results of themeasurements performed at FLASH [40]). The transverse coherence length 500 m downstreamfrom the source, where the first optical elements will be located, is expected to be of the orderof 350µm compared to the FWHM of the beam that is expected to be about 500µm.

The approach used in this paper for the analysis of the transverse coherence properties isquite general and can be applied as an effective and useful tool for describing the coherenceproperties of undulator radiation at third generation synchrotron sources and of SASE FELs. Inour future work, we plan to extend this approach to calculate the coherence properties of x-raybeams passing through different optical elements.

Acknowledgments

We acknowledge the help of M Tischer in calculation of the radiation properties of the 5 mPETRA III undulator using SRW code and the interest and support of E Weckert during thework on this project. We also acknowledge a fruitful discussion with E Saldin and M Yurkovand careful reading of the manuscript by A Mancuso.

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Figure A.1. Absolute value of the SDC |µ(1y)| in the vertical direction at thedistance 60 m downstream from the source for a high-β operation of PETRA IIIcalculated for different photon energies ((a) 20 keV, (b) 12 keV, (c) 6 keV and(d) 3 keV) using results of [32] (dotted line). The spectral density S(y) (dottedline) calculated in the same conditions is shown in the insets. For comparison,calculations performed in the frame of the GSM are also shown in this figure(solid lines).

Appendix

We compared results obtained by the GSM with the results of [32] for the PETRA IIIsynchrotron source and different photon energies ranging from 3 to 20 keV (see figure A.1).There are two critical dimensionless parameters of the theory [32]

Dx,y = kσ ′2x,y Lu and Nx,y =

kσ 2x,y

Lu, (A.1)

where σx,y and σ ′

x,y are the electron bunch sizes and divergences in the horizontal and in thevertical directions and Lu is the undulator length. The electron beam sizes σx,y and divergencesσ ′

x,y can be calculated from the values of the emittance εx,y and known β-function of thesynchrotron source according to σx,y =

√εx,yβx,y, σ

′

x,y =√εx,y/βx,y (see e.g. [36]). As it was

shown in [32], for large parameter values Dx,y � 1 and Nx,y � 1 the undulator source canbe described in the frame of the GSM (see (15, 16)) with the source size σSx,y =

√Nx,y Lu/k

and parameter δSx,y (18) δSx,y =√

Lu/k Dx,y . Using the values of Dx,y and Nx,y (A.1) and thedefinition of δSx,y (18) we obtain for the source size σSx,y = σx,y and the coherence length ξSx,y

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Table A.1. Parameters of the synchrotron radiation source PETRA III for a5 m undulator, high-β operation and different photon energies. Parameters Nx,y ,Dx,y are defined in (A.1), σTx,y and σ ′

Tx,y are the total photon source sizesand divergences, σr and σ ′

r are the intrinsic radiation characteristics of a singleelectron. The following electron beam sizes σx = 141µm, σy = 4.9µm anddivergences σ ′

x = 7.1µrad, σ ′

y = 2.0µrad were used in these calculations.

20 keV 12 keV 6 keV 3 keV

Nx 405 243 122 61Dx 25 15 7.6 3.8Ny 0.5 0.30 0.15 0.07Dy 2.0 1.3 0.63 0.32σTx , µm 141 141 141 142σ ′

Tx , µrad 7.5 7.7 8.4 9.6σTy, µm 5.3 5.5 6.1 7.1σ ′

Ty, µrad 3.2 3.8 5.0 6.7σr , µm 2.0 2.6 3.6 5.1σ ′

r , µrad 2.5 3.2 4.5 6.4

at the source ξSx,y = 2σx,y/√

4k2ε2x,y − 1. These expressions are exactly the same as discussed

earlier in the paper (compare e.g. (33) for the coherence length ξSx,y) with the only exceptionthat here the electron beam parameters are used instead of the photon beam parameters. For thePETRA III parameters at the photon energy of E = 12 keV and high-β operation (see table 1and PETRA III TDR [1]), we obtain Dx = 15, Dy = 1.3 and Nx = 243, Ny = 0.3. We can seefrom these estimates that in the horizontal direction parameters Dx � 1, Nx � 1 that means thatin this direction GSM can be safely used. However, in the vertical direction at the same energyDy 6 1, Ny 6 1 and a more careful analysis has to be applied.

To compare predictions of the GSM theory with the theoretical results of [32] in the verticaldirection, we used the far-field expressions for the CSD function W (y,1y) ((65) from [32])and spectral density S(y) ((71) from [32]) obtained in the limit Dx � 1 and Nx � 1. The SDCµ(y,1y) was calculated according to (6)

µ(y,1y)=W (y,1y)

√S(y +1y/2)

√S(y −1y/2)

, (A.2)

where y = (y1 + y2)/2,1y = y2 − y1 and y1, y2 are two positions in the vertical direction.Calculations were performed for a 5 m undulator of the PETRA III source, high-β operation,60 m downstream from the source at the central position of the beam (y = 0). In figure A.1results of the calculations of the SDC and spectral density for the energy range from 3 upto 20 keV are presented (see for the parameters used in these simulations table A.1). Theseresults are compared with the GSM theory described in this paper. For the GSM calculations atdifferent photon energies the total photon source size σTy and divergence σ ′

Ty were used. Theyare determined from a convolution of the sizes and divergences of the electron beam (σy , σ ′

y)with the intrinsic radiation characteristics of a single electron (σr , σ ′

r ). The latter are given byHowells and Kincaid [36] σr =

√2λLu/4π, σ ′

r =√λ/2Lu.

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22

As we can see from figure A.1, in spite of the fact that parameters Dy 6 1, Ny 6 1 (seetable A.1), the difference between two approaches is negligible down to an energy of 6 keV. Itbecomes more pronounced only at energies of about 3 keV for large separation distances 1y.It is also interesting to note that the coherence area, defined as the area where the degree ofcoherence drops to 80%, is the same in both approaches down to a lowest energy of 3 keV. Atthe same time, for this very low energy the effects of the single electron radiation (at the photonenergy E = 3 keV σr > σy and σ ′

r > σ′

y) are becoming more pronounced and reveal themselvesin the form of oscillations at large separations 1y. An inspection of figure A.1 shows that theGSM slightly overestimates the values of the SDC compared to the results of [32]. We relatethis to the fact that at low energies, at the source position, the intensity distribution obtained inthe frame of the model [32] contains long tails that effectively produce a larger source size incomparison to a source size obtained by the GSM approach. We note here as well that accordingto table A.1 the approximation Dx � 1 and Nx � 1 is no longer valid at very low energies below3 keV. Consequently, at these low energies equations (65, 71) from [32] cannot be applied forcalculation of the coherence properties of the 5 m undulator source at PETRA III. A more carefultreatment using general expressions for the correlation functions should be used in this case.

In conclusion, our analysis shows that for the high brilliance source PETRA III the GSMcan be safely used for the 5 m undulator at x-ray energies higher than 6 keV. It will also givea reasonable upper limit estimate of the coherence length for the energies as low as 3 keV.Our analysis has also shown that for a shorter undulator of 2 m length, which is typical for thePETRA III source, both approaches give similar results even for lower energies.

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