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Understanding transverse coherence properties of X-ray beams in third generation Synchrotron Radiation sources

This paper describes a theory of transverse coherence properties of Undulator Radiation. Our study is of very practical relevance, because it yields specific predictions of Undulator Radiation cross-spectral density in various parts of the beamline. On the contrary, usual estimations of coherence properties assume that the undulator source is quasi-homogeneous, like thermal sources, and rely on the application of van Cittert-Zernike (VCZ) theorem, in its original or generalized form, for calculating transverse coherence length in the far-field approximation. The VCZ theorem is derived in the frame of Statistical Optics using a number of restrictive assumptions: in particular, the quasi-homogeneous assumption is demonstrated to be inaccurate in many practical situations regarding undulator sources. We propose a technique to calculate the cross-spectral density from undulator sources in the most general case. Also, we find the region of applicability of the quasi-homogeneous model and we present an analytical expression for the cross-spectral density which is valid up to the exit of the undulator. For the case of more general undulator sources, simple formulas for the transverse coherence length, interpolated from numerical calculations and suitable for beamline design applications are found. Finally, using a simple vertical slit, we show how transverse coherence properties of an X-ray beam can be manipulated to obtain a larger coherent spot-size on a sample. This invention was devised almost entirely on the basis of theoretical ideas developed throughout this paper.


Introduction
In recent years, continuous evolution of third generation light sources has allowed dramatic increase of brilliance with respect to older designs, which has triggered a number of new techniques and experiments unthinkable before. Among the most exciting properties of today third generation facilities is the high flux of coherent X-rays provided. The availability of intense coherent Xray beams has fostered the development of new coherence-based techniques like fluctuation correlation dynamics, phase imaging, coherent X-ray diffraction (CXD) and X-ray holography. In this context, understanding the evolution of transverse coherence properties of Synchrotron Radiation (SR) along the beam line is of fundamental importance.
In general, when dealing with this problem, one should account for the fact that Synchrotron Radiation is a random statistical process. Therefore, the evolution of transverse coherence properties should be treated in terms of probabilistic statements: the shot noise in the electron beam causes fluctuations of the beam density which are random in time and space. As a result, the radiation produced by such a beam has random amplitudes and phases.
Statistical Optics [5,6,10] affords convenient tools to deal with fluctuating electromagnetic fields in an appropriate way. Among the most important quantities needed to describe coherent phenomena in the framework of Statistical Optics is the correlation function of the electric field. In any interference experiment one needs to know the system (second order) correlation function of the signal at a certain time and position with the signal at another time and position. Alternatively, and equivalently, one can describe the same experiment in frequency domain. In this case one is interested in the correlation function of the Fourier transform of the time domain signal at a certain frequency and position with the Fourier transform of the time domain signal at another frequency and position. The signal one is interested to study is, indeed, the Fourier transform of the original signal in time domain. In SR experiments the analysis in frequency domain is much more natural than that in the time domain. In fact, up-to-date detectors are limited to about 100 ps time resolution and they are by no means able to resolve a single X-ray pulse in time domain. They work, instead, by counting the number of photons at a certain frequency over an integration time longer than the radiation pulse. Therefore, in this paper we will deal with signals in the frequency domain and we will often refer to the "Fourier transform of the electric field" simply as "the field".
For some particular experiment one may be interested in higher order correlation functions (for instance, in the correlation between the intensities) which, in general, must be calculated separately. In the particular case when the field fluctuations can be described as a Gaussian process, the field is often said to obey Gaussian statistics. In this case, with the help of the Moment Theorem [5] one can recover correlation functions of any order from the knowledge of the second order one: this constitutes a great simplification to the task of describing coherence properties of light. A practical example of a field obeying Gaussian statistics is constituted by the case of polarized thermal light. This is more than a simple example: in fact, Statistical Optics has largely developed in connection with problems involving optical sources emitting thermal light like the sun, other stars, or incandescent lamps. As a consequence, Gaussian statistics is often taken for granted. Anyway, it is not a priori clear wether Synchrotron Radiation fields obey it or not; our analysis will show that Synchrotron Radiation is indeed a Gaussian random process. Therefore, as is also the case for polarized thermal light and any other signal obeying Gaussian statistics, when we deal with Synchrotron Radiation the basic quantity to consider is the second order correlation function of the field. Moreover, as already discussed, in Synchrotron Radiation experiments it is natural to work in the space-frequency domain, so that we will focus, in particular, on the second order correlation function in the space-frequency domain.
Besides obeying Gaussian statistics, polarized thermal light has two other specific properties allowing simplifications of the theory: the first is stationarity 1 and the second is quasi-homogeneity. Exactly as the property of Gaussian statistics, also stationarity and quasi-homogeneity of the source are usually taken for granted in Statistical Optics problems but, unlike it, they do not belong, in general, to Synchrotron Radiation fields. In fact, as we will show, Synchrotron radiation fields are intrinsically non-stationary and not always quasi-homogeneous. Nevertheless, up to now it has been a widespread practice to assume that undulator sources are completely incoherent (i.e. homogeneous) and to apply the well known van Cittert-Zernike (VCZ) theorem for calculating the degree of transverse coherence in the far-field approximation [1]. Using the VCZ theorem, the electric field cross-correlation function in the far field is usually calculated (aside for a geometrical phase factor) as a Fourier transformation of the intensity distribution of the source, customarily located at the exit of the undulator.
Although the VCZ theorem only deals with completely incoherent sources, there exists an analogous generalized version of it which allows to extend the treat the case of quasi-homogeneous sources as well. Actually there is no unambiguous choice of terminology in literature regarding the scope of the VCZ theorem. For instance, a very well-known textbook [6] reports of a "Zernike-propagation" equation dealing with any distance from the source. Also, sometimes [5], the generalized VCZ theorem is referred to as Schell's theorem (and also used in some paper [2]). In this paper we will refer to the VCZ theorem and its generalized version only in the limit for a large distance from the source and for, respectively, homogeneous and quasi-homogeneous sources. However, irrespectively of different denominations, the fundamental fact holds, that once a cross-correlation function is known on a given source plane it can be propagated through the beamline at any distance from the source. It should be noted that, from this viewpoint, a source simply denotes an initial plane down the beamline from which the cross-correlation function is propagated further. Then, the position of the source down the beamline is suggested only on the ground of opportunity. On the contrary, when dealing with the VCZ theorem, the source must be (quasi)-homogeneous which explains the customary location at the exit of the undulator.
In some cases, the VCZ theorem or its generalized version may provide a convenient method for calculating the degree of transverse coherence in various parts of the beamline once the transverse coherence properties of the photon beam are specified at the exit of the undulator, that is at the source plane. In most SR applications though, such treatment is questionable. First, the source (even at the exit of the undulator) may not be quasi-homogeneous. Second, even for specific sets of problem parameters where the quasi-homogeneous model is accurate, the specification of the far-field zone depends not only on the electron beam sizes, but also on the electron beam divergencies (in both direction) and on the intrinsic divergence of the radiation connected with the undulator device. At the time being, widespread and a-critical use of the VCZ theorem and its generalization shows that there is no understanding of transverse coherence properties of X-ray beams in third generation Synchrotron Radiation sources.
If, on the one hand, the definition of the far-zone and the possible non quasihomogeneity of SR sources constitute serious problems in the description of the coherence properties of Synchrotron light, on the other hand the intrinsic non-stationarity of the SR process does not play a very important role. In particular, as we will show, assumption of a minimal undulator bandwidth much larger than the characteristic inverse bunch duration (which is always verified in practice) allows to separate the correlation function in space-frequency domain in the product of two functions. The first function is a spectral correlation describing correlation in frequency. The second function describes correlation in space and is well-known also in the case of stationary processes as the crossspectral density of the process. Then, the cross-spectral density can be studied independently at any given frequency giving information on the spatial correlation of the field. Subsequently, the knowledge of the spectral correlation function brings back the full expression for the space-frequency correlation.
In this paper we aim at the development of a theory of transverse coherence capable of providing very specific predictions, relevant to practice, regarding the cross-spectral density of undulator radiation at various positions along the beam-line. A fully general study of undulator sources is not a trivial one. Difficulties arise when one tries to include simultaneously the effect of intrinsic divergence of the radiation due to the presence of the undulator, of electron beam size and electron beam divergence into the insertion device. The full problem, including all effects, poses an unsolvable analytical challenge, and numerical calculations are to be preferred. Generally, the cross-spectral density of the undulator radiation is controlled by nine physical parameters which model both the electron beam and the undulator: the horizontal and vertical geometrical emittances of the electron beam ǫ x,y , the horizontal and vertical minimal betatron functions β o x,y , the observation distance down the beamline z o , the observation frequency ω, the undulator resonant frequency ω o , the undulator length L w and the length of the undulator period λ w .
We will make a consistent use of dimensional analysis. Dimensional analysis of any problem, performed prior to analytical or numerical investigations, not only reduces the number of independent terms, but also allows one to classify the grouping of dimensional variables in a way that is most suitable for subsequent study. The algorithm for calculating the cross-spectral density can be formulated as a relation between dimensionless quantities. After appropriate normalization, the radiation cross-spectral density from an undulator device is described by six dimensionless quantities: the normalized emittanceŝ ǫ x,y = ω o ǫ x,y /c, the normalized betatron functionsβ x,y = β o x,y /L w , the normalized observation distanceẑ o = z o /L w and the normalized detuning parameter C = 2πN w (ω−ω o )/ω o , where N w = L w /λ w is the number of undulator periods.
At some point in this work we will find it convenient to poseĈ = 0. In other words we will assume that parameters are tuned at perfect resonance. It is relevant to note that even under this simplifying assumptions, conditions for the undulator source to be quasi-homogeneous still include four parametersǫ x,y andβ x,y . For storage rings that are in operation or planned in theÅngstrom wavelength range, the parameter variation ofǫ x ∼ 10 − 10 3 ,ǫ y ∼ 10 −1 − 10, β x,y ∼ 10 −1 − 10 are possible: these include many practical situations in which the assumption of quasi-homogeneous sources and, therefore the (generalized) VCZ theorem, is not accurate.
In this paper we will first deal with the most general case of non-homogeneous sources. In fact, from a practical viewpoint, it is important to determine the cross-spectral density as a function ofǫ x,y ,β x,y ,Ĉ andẑ o . Once a general expression for the cross-spectral density is found, it can be used as a basis for numerical calculations. A second goal of this work is to find the region of applicability of the quasi-homogeneous source model (i.e. of the generalized VCZ theorem) which will arise automatically from the dimensional analysis of the problem. Finally, we will derive analytical expressions for the cross-spectral density atĈ = 0 in various parts of the beamline.
Results may also be obtained using numerical techniques alone, starting from the Lienard-Wiechert expressions for the electromagnetic field and applying the definition of the field correlation function without any analytical manipulation. Yet, computer codes can calculate properties for a given set of parameters, but can hardly improve physical understanding, which is particularly important in the stage of planning experiments: understanding of correct approximations and their region of applicability with the help of a consistent use of dimensional analysis can simplify many tasks a lot, including practical and non-trivial ones. Moreover, at the time being, no code capable to deal with transverse coherence problems has been developed at all.
It should be noted that some theoretical attempt to follow this path has been proposed in [4]. Among the results of that paper is the fact that van Cittert-Zernike theorem could not be applied unless the electron beam divergence is much smaller than the diffraction angle, which is never verified in practice in the horizontal plane. We will show that this conclusion is incorrect.
We organize our work as follows. After this Introduction, in Section 2 we present a second-order theory of coherence for fields generated by Synchrotron Radiation sources. In Section 3 we give a derivation of the cross-spectral density for undulator-based sources in reduced units. Subsequently, we analyze the evolution of the cross-spectral density function through the beamline in the limit forĈ = 0. A particular case of quasi-homogeneous sources and its applicability region is treated under several simplifying assumptions in Section 4. Effects of the vertical emittance on the cross-spectral density are discussed in detail in Section 5, while a treatment of some non quasi-homogeneous source is given in the following Section 6. Obtained results include approximate design formula capable of describing in very simple terms the evolution of the coherence length along the beamline in many situation of practical interest. A good physical insight is useful to identify possible applications of given phenomena. In particular in Section 7 we selected one practical application to exploit the power of our approach. We show that, by means of a simple vertical slit, it is possible to manipulate transverse coherence properties of an X-ray beam to obtain a convenient coherent spot-size on the sample. This invention was devised almost entirely on the basis of theoretical ideas of rather complex and abstract nature which have been described in this paper. Finally, in Section 8, we come to conclusions. 2 Second-order coherence theory of fields generated by Synchrotron Radiation sources

Thermal light and Synchrotron Radiation: some concepts and definitions
A great majority of optical sources emits thermal light. Such is the case of the sun and the other stars, as well as of incandescent lamps. This kind of radiation consists of a large number of independent contributions (radiating atoms) and is characterized by random amplitudes and phases in space and time. The electromagnetic fields can be then conveniently described in terms of Statistical Optics, a branch of Physics that has been intensively developed during the last few decades. Today one can take advantage of a lot of existing experience and theoretical basis for the descriptions of fluctuating electromagnetic fields [5,6].
Consider the light emitted by a thermal source passing through a polarization analyzer (see Fig. 1). Properties of polarized thermal light are well-known in Statistical Optics, and are referred to as properties of completely chaotic, polarized light [5,6]. Thermal light is a statistical random process and statements about such process are probabilistic statements. Statistical processes are handled using the concept of statistical ensemble, drawn from Statistical Mechanics, and statistical averages are performed indeed, over many ensembles, or realizations, or outcomes of the statistical process under study.
Polarized thermal light is a very particular kind of random process in that it is Gaussian, stationary and ergodic. Let us discuss these characteristics in more detail.
The properties of Gaussian random processes are well-known in Statistical Optics. For instance, the real and imaginary part of the complex amplitudes of the electric field from a polarized thermal source have Gaussian distribution, while the instantaneous radiation power fluctuates in accordance with the negative exponential distribution. Gaussian statistics alone, guarantees that higher-order correlation functions can be expressed in terms of second-order correlation functions. Moreover, it can be shown [5] that a linearly filtered Gaussian process is also a Gaussian random process. As a result, the presence of a spectral filter (monochromator) and a spatial filter as in the system depicted in Fig. 1 do not change the statistics of the signal, because they simply act as linear filters.
Stationarity is a subtle concept. There are different kinds of stationarity. Strict-stationarity means that all ensemble averages are independent on time.
Wide-sense stationarity means that the signal average is independent on time and that the second order correlation function in time depends only on the difference of the observation times. However, for Gaussian processes strict and wide-sense stationarity coincide [5,6]. As a consequence of the definition of stationarity, necessary condition for a certain process to be stationary is that the signal last forever. Yet, if a signal lasts much longer than its coherence time τ c (which fixes the short-scale duration of the field fluctuations) and it is observed for a time much shorter than its duration σ T , but much longer than its coherence time it can be reasonably considered as everlasting and it has a chance to be stationary as well, as in the case of thermal light.
Ergodicity is a stronger requirement than stationarity. Qualitatively, we may state that if, for a given random process all ensemble averages can be substituted by time averages, the process under study is said to be ergodic: all the statistical properties of the process can be derived from one single realization. A process must be strictly stationary in order to be ergodic. There exist stationary processes which are not ergodic. One may consider, for instance, the random constant process: this is trivially strictly stationary, but not ergodic because a single (constant) realization of the process does not allow one to characterize the process from a statistical viewpoint. However, this is a pathologic case when both the coherence time τ c and the duration time of the signal σ T are infinite. On the contrary, a stationary process like the radiation from an incandescent lamp driven by a constant current has, virtually, infinite duration. In this case different ensembles are simply different observations, for given time intervals, of the same, statistically identical phenomenon: then, the concept of ensemble average and time average are equivalent and the process is also ergodic. Statistical Optics was developed starting with signals characterized by Gaussian statistics, stationarity and ergodicity. Let us consider any Synchrotron radiation source. Like thermal light, also Synchrotron Radiation is a random process. In fact, relativistic electrons in a storage ring emit Synchrotron Radiation passing through bending magnets or undulators. The electron beam shot noise causes fluctuations of the beam density which are random in time and space from bunch to bunch. As a result, the radiation produced has random amplitudes and phases. As already declared in the Introduction we will demonstrate that the SR field obeys Gaussian statistics. In contrast with thermal light though, Synchrotron Radiation is intrinsically non-stationary (and, therefore, non-ergodic) because even if its short pulse duration cannot be resolved by detectors working in the time domain, it can nonetheless be resolved by detectors working in frequency domain. For this reason, in what follows the averaging brackets ... will always indicate the ensemble average over bunches. In spite of differences with respect to the simpler case of thermal light, as we will see in this paper, also Synchrotron Radiation fields can be described in terms of Statistical Optics. Fig. 2 shows the geometry of the experiment under consideration. The problem is to describe the statistical properties of Synchrotron Radiation at the detector installed after the spatial and spectral filters. Radiation at the detector consists of a carrier modulation of frequency ω subjected to random amplitude and phase modulation. The Fourier decomposition of the radiation contains frequencies spread about the monochromator bandwidth ∆ω m : it is not possible, in practice, to resolve the oscillations of the radiation fields which occur at the frequency of the carrier modulation. It is therefore appropriate, for comparison with experimental re- sults, to average the theoretical results over a cycle of oscillations of the carrier modulation. Fig. 3 gives a qualitative illustration of the type of fluctuations that occur in cycle-averaged Synchrotron Radiation beam intensity. Within some characteristic time, a given random function appears to be smooth, but when observed at larger scales the same random function exhibits "rough" variations. The time scale of random fluctuations is the coherence time τ c . When τ c ≪ σ T the radiation beyond the monochromator is partially coherent. This case is shown in Fig. 3: there, we can estimate τ c ≃ ∆ω −1 m . If the radiation beyond the monochromator is partially coherent, a spiky spectrum is to be expected. The nature of the spikes is easily described in terms of Fourier transform theory. We can expect that the typical width of the spectrum envelope should be of order of ∆ω/ω ∼ (τ c ω) −1 . Also, the spectrum of the radiation from a bunch with typical duration σ T at the source plane should contain spikes with characteristic width ∆ω/ω ≃ (ωσ T ) −1 , as a consequence of the reciprocal width relations of Fourier transform pairs (see, again, Fig. 3).

Second-order correlations in space-frequency domain
We start our discussion in the most generic way possible, considering a fixed polarization component of the Fourier transform at frequency ω of the electric field produced at location (z o , r ⊥o ), in some cartesian coordinate system, by a given collection of sources. We will denote it withĒ ⊥ (z o , r ⊥o , ω) and it will be linked to the time domain field E ⊥ (z o , r ⊥o , t) through the Fourier transform so that This very general collection of sources includes the case of an ultra relativistic electron beam going through a certain magnetic system and in particular an undulator, which is our case of interest. In this case z o is simply the observation distance along the optical axis of the system and r ⊥o are the transverse coordinates of the observer on the observation plane. The contribution of the k-th electron to the field Fourier transform at the observation point depends on the transverse offset (l xk , l yk ) and deflection angles (η xk , η yk ) that the electron has at the entrance of the system with respect to the optical axis. Moreover, an arrival time t k at the system entrance has the effect of multiplying the field Fourier transform by a phase factor exp (iωt k ) (that is, in time domain the electric field is retarded by a time t k ). At this point we do not need to specify explicitly the dependence on offset and deflection. The total field Fourier transform can be written as where η k , l k and t k are random variables and N is the number of electrons in the beam. It follows from Eq. (3) that the Fourier transform of the Synchrotron Radiation pulse at a fixed frequency and a fixed point in space is a sum of a great many independent contributions, one for each electron, of the formĒ s⊥ ( η k , l k , z o , r ⊥o , ω) exp (iωt k ). For simplicity we make three assumptions about the statistical properties of elementary phasors composing the sum, which are generally satisfied in Synchrotron Radiation problems of interest.
1) We assume that for a beam circulating in a storage ring random variables t n are independent from η n and l n . This is always verified, because the random arrival times of electrons, due to shot noise, do not depend on the electrons offset and deflection with respect to the z-direction. Eq. (3) states that the k-th elementary contribution to the totalĒ ⊥ can be written as a product of the complex phasors exp (iωt k ), andĒ s⊥ that, in its turn, can be written as a product of modulus and phase asĒ s⊥ ( η k , l k , z o , r ⊥o , ω) =|Ē s⊥k | exp (iφ k ).
Under the assumption of statistical independence of t n from η n and l n the complex phasors exp (iωt k ), andĒ s⊥ are statistically independent of each other and of all the other elementary phasors for different values of k. The ensemble average of a given function f of random variables η n , l n and t n is by definition: where P ( η n , l n , t n ) is the probability density distribution in the joint random variables η n , l n , t n . Independence of t n from η n and l n allows us to write where we also assumed that the distribution in the horizontal and vertical planes are not correlated. Since electrons arrival times are completely uncorrelated from transverse coordinates and offsets, the shapes of F ηx,lx , F ηy ,ly and F t are the same for all electrons.
2) We assume that the random variables |Ē s⊥k | (at fixed frequency ω), are identically distributed for all values of k, with a finite mean |Ē s⊥k | and a finite second moment |Ē s⊥k | 2 . This is always the case in practice because electrons are identical particles.
3) We assume that the electron bunch duration σ T is large enough so that ωσ T ≫ 1: under this assumption the phases ωt k can be regarded as uniformly distributed on the interval (0, 2π). The assumption ωσ T ≫ 1 is justified by the fact that ω is the undulator resonant frequency, which is high enough to guarantee that ωσ T ≫ 1 for any practical choice of σ T .
The formal summation of phasors with random lengths and phases is illustrated in Fig. 4. Under the three previously discussed assumptions we can use the central limit theorem to conclude that the real and the imaginary part ofĒ ⊥ are distributed in accordance to a Gaussian law. Detailed proof of this fact is given in Appendix A. As a result, Synchrotron Radiation is a Gaussian random process and second-order field correlation function is all we need in order to specify the field statistical properties. In fact, as already remarked, higher-order correlation functions can be expressed in terms of second-order correlation functions.
In Synchrotron Radiation experiments with third generation light sources detectors are limited to about 100 ps time resolution and are by no means able to resolve a single X-ray pulse in time domain: they work, instead, by counting the number of photons at a certain frequency over an integration time longer than the pulse. Therefore, for Synchrotron Radiation related issues the frequency domain is much more natural a choice than the time domain, and we will deal with signals in the frequency domain throughout this paper. The knowledge of the second-order field correlation function in frequency domain is all we need to completely characterize the signal from a statistical viewpoint. For the sake of completeness it is nonetheless interesting to remark that it is possible (and often done, in Statistical Optics) to give equivalent descriptions of the process in time domain as well. First, note that the time domain process E ⊥ (t) is linked toĒ ⊥ (ω) by Fourier transform, and that a linearly filtered Gaussian process is also a Gaussian process (see [5] 3.6.2). As a result, E ⊥ (t) is a Gaussian process as well. Second, the operation of ensemble average is linear with respect to Fourier transform integration. This guarantees, that the knowledge of Γ ω in frequency domain is completely equivalent to the knowledge of the second-order correlation function between E ⊥ (z o , r ⊥o1 , t 1 ) and E ⊥ (z o , r ⊥o2 , t 2 ). The latter is usually known as mutual coherence function and was first introduced in [9]: For the rest of this paper we will abandon almost entirely any reference to the time domain and work consistently in frequency domain with the help of Eq. (6) because, as has already been said, this is a natural choice for Synchrotron Radiation applications. In particular, as has already been anticipated, under non-restrictive assumptions on characteristic bandwidths of the process, it is possible to break the correlation function Γ ω (ω, ω ′ ) in space-frequency domain in the product of two factors, the spectral correlation function F ω (ω − ω ′ ) , and the cross-spectral density of the process G ω (z o , r ⊥o1 , r ⊥o2 , ω) [6]. The crossspectral density can be studied independently at any given frequency giving information on the spatial correlation of the field. Subsequently, the knowledge of the spectral correlation function brings back the full expression for Γ ω .
Substituting Eq. (3) in Eq. (6) one has With the help of Eq. (4) and Eq. (5), the ensemble average exp (iωt k ) t can be written as the Fourier transform of the bunch longitudinal profile function F t (t k ), that is Using Eq. (10), Eq. (9) can be written as where F * ω (ω ′ ) = F ω (−ω ′ ) because F t is a real function. When the radiation wavelengths of interest are much shorter than the bunch length we can safely neglect the second term on the right hand side of Eq. (11) since the form factor product F ω (ω)F ω (−ω ′ ) goes rapidly to zero for frequencies larger than the characteristic frequency associated with the bunch length: think for instance, at a centimeter long bunch compared with radiation in the Angstrom wavelength range. It should be noted, however, that when the radiation wavelength of interested is longer than the bunch length the second term in Eq. (11) is dominant with respect to the first, because it scales with the number of particles squared : in this case, analysis of the second term leads to a treatment of Coherent Synchrotron Radiation phenomena (CSR). In this paper we will not be concerned with CSR and we will neglect the second term in Eq. (11), assuming that the radiation wavelength of interest is shorter than the bunch length: then, it should be noted that F ω (ω − ω ′ ) depends on the difference between ω and ω ′ , and the first term cannot be neglected. We can therefore write As one can see from Eq. (12) each electron is correlated just with itself: crosscorrelation terms between different electrons was, in fact, included in the second term on the right hand side of Eq. (11), which has been dropped. It is important to note that if the dependence ofĒ s⊥ on ω and ω ′ is slow enough, so thatĒ s⊥ does not vary appreciably on the characteristic scale of F ω , we can substituteĒ * s⊥ ( η, l, z o , r ⊥o2 , ω ′ ) withĒ * s⊥ ( η, l, z o , r ⊥o2 , ω) in Eq. (12). The situation is depicted in Fig. 5. On the one hand, the characteristic scale of F ω is given by 1/σ T , where σ T is the characteristic bunch duration. On the other hand, the bandwidth of single particle undulator radiation at resonance is given by ω o /N w , where ω o is the resonant frequency and N w is the number of undulator periods (of order 10 2 − 10 3 ). In the case of an electron beam the undulator spectrum will exhibit a longer tail, as has been shown in [3], which guarantees that ω o /N w is, indeed, a minimum for the radiation bandwidth, and is the right quantity to be compared with 1/σ T . As an example, for wavelengths of order 1Å, N w ∼ 10 3 and σ T ∼ 30 ps (see [1]), ω o /N w ∼ 2 · 10 16 Hz which is much larger than 1/σ T ∼ 3 · 10 10 Hz. As a result we can simplify Eq. (12) to Fig. 5. Schematic illustration of the relative frequency dependence of the spectral correlation function F ω (ω − ω ′ ) and of the cross-spectral density function (the cross-power spectrum) G ω (z o , r ⊥o1 , r ⊥o2 , ω) of the Synchrotron Radiation at points r ⊥o1 and r ⊥o2 at frequency ω.
Eq. (13) fully characterizes the system under study from a statistical viewpoint. However, in practical situations, the observation plane is behind a monochromator or, equivalently, the detector itself is capable of analyzing the energy of the photons. The presence of a monochromator simply modifies the right hand side Eq. (13) for a factor T (ω)T * (ω ′ ), where T is the monochromator transfer function: Independently on the characteristics (and even on the presence) of the monochromator, it should be noted that both in Eq. (13) and Eq. (15), correlation in frequency and space are expressed by two separate factors. In particular, in both these equation, spatial correlation is expressed by the cross-spectral density function G ω (z o , r ⊥o1 , r ⊥o2 , ω). In other words, we are able to deal separately with spatial and spectral part of the correlation function in space-frequency domain with the only non-restrictive assumption that ω o /N w ≫ 1/σ T . From now on we will be concerned with the calculation of the correlation function G ω (z o , r ⊥o1 , r ⊥o2 , ω), independently on the shape of the remaining factors on the right hand side of Eq. (15) which can have a simple or a complicated structure, accounting for the characteristics of the monochromator.
Before proceeding with the analysis of G ω (z o , r ⊥o1 , r ⊥o2 , ω) though, let us spend some words on these remaining factors; the presence of a monochromator introduces another bandwidth of interest. If we indicate the bandwidth of the monochromator with ∆ω m and the central frequency of interest at which the monochromator is tuned with ω o (typically, the undulator resonant frequency), then T is peaked around ω o and goes rapidly to zero as we move out of the range (ω o − ∆ω m /2, ω o + ∆ω m /2). Now, if the characteristic bandwidth of the monochromator, ∆ω m , is large enough so that T does not vary appreciably on the characteristic scale of F ω , i.e. ∆ω m ≫ 1/σ T , then F ω (ω − ω ′ ) is peaked at ω = ω ′ . In this case the process resembles more and more a stationary process, although it will be still intrinsically non-stationary. Consider a signal observed for a time much shorter than its duration, but much longer than its coherence time, and such that the stationary model applies to it. Now imagine that we extend the observation time to a duration which is still much shorter than the signal duration, but long enough that we need to account for the intrinsical non-stationarity of the process due to finite signal duration. In this case the stationary model does not apply anymore strictly. To describe this situation, we can define a property weaker than stationarity, but nonetheless very interesting from a physical standpoint: quasi-stationarity. The time domain correlation function (that is, the mutual coherence function) can be written as When ∆ω m ≫ 1/σ T , and with the help of new variables ∆ω = ω−ω ′ and ω, we can simplify Eq. (16) accounting for the fact that F ω (ω −ω ′ ) is strongly peaked around ∆ω = 0. In fact we can consider T (ω)T * (ω ′ )G ω (z o , r ⊥o1 , r ⊥o2 , ω) ≃ |T (ω)| 2 G ω (z o , r ⊥o1 , r ⊥o2 , ω), so that we can integrate separately in ∆ω and ω to obtain In other words, in the quasi-stationary case, Γ t (z o , r ⊥o1 , r ⊥o2 , t 1 , t 2 ) is split on the product of two factors, a "reduced mutual coherence function", that is G t (z o , r ⊥o1 , r ⊥o2 , t 1 − t 2 ), and an intensity profile, that is F (t 2 ).
If we now assume ∆ω m N w /ω o ≪ 1 (that is usually true), G ω (z o , r ⊥o1 , r ⊥o2 , ω) = G ω (z o , r ⊥o1 , r ⊥o2 , ω o ) is a constant function of frequency within the monochromator line. In this case, G ω contains all the information about spatial correlations between different point and is, in fact, the quantity of central interest in our study, but it is independent on the frequency ω. As a result we have which means that the mutual coherence function Γ t (z o , r ⊥o1 , r ⊥o2 , t 1 , t 2 ) is reducible, in the sense that it can be factorized as a product of two factors, the first Ng t (t 1 − t 2 )F t (t 2 ), characterizing the temporal coherence and the second G ω (z o , r ⊥o1 , r ⊥o2 , ω o ) describing the spatial coherence of the system 2 . This case is of practical importance. In fact for 1Å radiation we typically have ∆ω m /ω o ≃ 10 −4 ÷ 10 −5 and N w ≃ 10 2 ÷ 10 3 , i.e. ∆ω m N w /ω o ≪ 1. It should be noted that, although Eq. (18) describes the case when ∆ω m N w /ω o ≪ 1 and ∆ω m ≫ 1/σ T , only the former assumption is important for the mutual coherence function to be reducible. In fact, if the former is satisfied but the latter is not, from Eq. (16) one would simply have: that is still reducible.
(b) The intensity distribution of the radiation pulse F t (t 2 ) and the spectral correlation function F ω (∆ω) form a Fourier pair.
The statement (a) can be regarded as an analogue, for quasi-stationary sources, of the well-known Wiener-Khinchin theorem, which applies to stationary sources and states that the temporal correlation function and the spectral density are a Fourier pair. Since there is symmetry between time and frequency domains, a "anti" Wiener-Khinchin theorem must also hold, and can be obtained by the usual Wiener-Khinchin theorem by exchanging frequencies and times. This is simply constituted by the statement (b).
The assumption of quasi-stationarity is not vital for the following of this work, since the cross-spectral density can be studied in any case as a function of frequency. In this respect it should be noted that, although in the large majority of the cases monochromator characteristics are not good enough to allow resolution of the non-stationary process, there are cases when it is not allowed to treat the process as if it were quasi-stationary. For instance, in [8] a particular monochromator is described with a relative resolution of 10 −8 at wavelengths of about 1Å, or ω o ∼ 2 · 10 19 Hz. Let us consider, as in [8], the case of radiation pulses of 32 ps duration. Under the already accepted assumption 1/σ T ≪ ω o /N w , we can identify the radiation pulse duration with σ T . Then we have ∆ω m ∼ 2 · 10 11 Hz which is of order of 2π/σ T ∼ 2 · 10 11 Hz: this means that the monochromator has the capability of resolving the non-stationary processes in the frequency domain. On the contrary, also in this case, the mutual coherence function is reducible, in the sense specified before, because ∆ω m /ω o ≃ 10 −8 and N w ≃ 10 2 ÷ 10 3 , i.e. ∆ω m N w /ω o ≪ 1. However, such accuracy level is not usual in Synchrotron Radiation experiments.
To sum up, condition ω o /N w ≫ 1/σ T alone allows separate treatment of transverse coherence properties at a given frequency through the function G ω (z o , r ⊥o1 , r ⊥o2 , ω). The condition for the monochromator bandwidth ∆ω m ≫ 1/σ T defines a quasi-stationary process. If the monochromator bandwidth is such that ∆ω m > ω o /N w , the ratio N w /(ω o σ T ) gives us a direct measure of the accuracy of the stationary approximation which does not depend, in this case, on the presence of the monochromator. When a monochromator is present, with a bandwidth ∆ω m < ω o /N w , it is the ratio 1/(∆ω m σ T ) which gives such a measure. The condition ∆ω m ≪ ω o /N w ensures, instead, that the mutual coherence function of the signal is reducible, in the sense specified by Eq. (19). In the following we will need only the first of these conditions, ω o /N w ≫ 1/σ T . In fact, this is all we need in order to separate transverse and temporal coherence effects, as shown in Eq. (15).
Once transverse and temporal coherence effects are separated one can focus on the study of transverse coherence through the function G ω (z o , r ⊥o1 , r ⊥o2 , ω).
There exists an important class of sources, called quasi-homogeneous. As we will see, quasi-homogeneity is the spatial analogue of quasi-stationarity.
In general, quasi-homogeneous sources are a particular class of Schell's model sources. Schell's model sources are defined by the condition that their crossspectral density at the source plane (that is for a particular value of z o ) is of the form where g( r ⊥o2 − r ⊥o1 , ω) is the spectral degree of coherence (that is normalized to unity by definition, i.e. g(0, ω) = 1) 3 . Equivalently one may simply define Schell's model sources using the condition that the spectral degree of coherence depends on the positions across the source only through the difference r ⊥o2 − r ⊥o1 (see [6], 5.2.2) from which Eq. (20) follows.
Quasi-homogeneous sources are Schell's sources obeying the following extraassumption: the spectral density G ω ( r ⊥o , r ⊥o , ω) at the source plane, considered as a function of r ⊥o , varies so slowly with the position that it is approximatively constant over distances across the source, which are of the order of the correlation length ∆ (that is the effective width of |G ω ( r ⊥o1 , r ⊥o2 , ω)|, see Fig. 6 for a qualitative illustration in one dimension).
Since for quasi-homogeneous sources the spectral density is assumed to vary slowly we are allowed to make the approximation where is the field intensity distribution. A schematic illustration of the measurement of the cross-spectral density of a undulator source is given in Fig. 7.
We reported definitions and differences between Schell's sources and quasihomogeneous sources as treated in [6] in order to review some conventional language. However, in our paper we will study and classify sources with the help of parameters from dimensional analysis of the problem. In particular, in the following we will introduce quantities that model, in dimensionless units, the electron beam dimension and divergence in the x and y directions (N x,y andD x,y , respectively). When some of these parameters are much larger or much smaller than unity, in certain particular combinations discussed in the following part of this work, we will be able to point out simplifications of analytical expressions. These simplifications do not depend, in general, on the fact that the source is quasi-homogeneous or not, but simply on the fact that some of the above parameters are large or small. It will be possible to describe some parameter combination in terms of Schell's or quasi-homogeneous model, but this will not be the case, in general. In order to link the physical properties of certain kind of sources to a certain range of parameters we will find it convenient to extend the concept of quasi-homogeneity to the concept of "weak quasi-homogeneity". It is better to familiarize with our new definition already here: a given wavefront at fixed positionẑ o will be said to be weakly quasi-homogeneous, by definition, when the modulus of the spectral degree of coherence |g| depends on the position across the source only through the difference r ⊥o2 − r ⊥o1 , i.e. |g| = w( r ⊥o2 − r ⊥o1 , ω). This is equivalent to generalize Eq. (21) to As a remark to both the definitions of quasi-homogeneity and weak quasihomogeneity, it should be noted that they only involve conditions on the crossspectral densities through Eq. (21) or Eq. (23): one can apply these definitions to any wavefront at any position z o . This is consistent with what has been remarked in the Introduction: the choice of a source plane down the beamline is just a convention. Therefore, our definition of weak quasi-homogeneity is completely separated from the concept of source plane. It may seem, at first glance, that the definition of "weak quasi-homogeneity" is somehow artificial but it is, on the contrary, very convenient from a practical viewpoint. In fact, in any coherent experiment, the specimen is illuminated by coherent light from some kind of aperture, or diaphragm. Think, for instance, to the usual process of selection of transversely coherent light through a spatial filter, where a diaphragm is placed downstream a pinhole. Physically, when the modulus of the spectral degree of coherence depends only on the coordinate difference r ⊥o2 − r ⊥o1 the coherence properties of the beam do not depend on the position of the diaphragm with respect to the transverse coordinate of the center of the pinhole, which we may imagine on the z axis: for instance, the coherence length will depend only on r ⊥o2 − r ⊥o1 and not on the average position ( r ⊥o2 + r ⊥o1 )/2. Mathematically, as has already been said, the definition of weak quasi-homogeneity is linked with particular combination of small and large parameters which will lead to simplifications of equations and to analytical treatment of several interesting cases.
After having introduced the definition of "weak quasi-homogeneity" we should go back to the concept of usual quasi-homogeneity to describe a vary particular feature of it: quasi-homogeneity can be regarded as the spatial equivalent of quasi-stationarity, as anticipated before. Exactly as the time domain has a reciprocal description in terms of frequency, the space domain has a reciprocal description in terms of transverse (two-dimensional) wave vectors. However, since the frequency is fixed, the ratio between the horizontal or vertical component of the wave vector and the longitudinal wave number is representative of the propagation angle of a plane wave at fixed frequency. Therefore any given signal on a two-dimensional plane can be represented in terms of superposition of plane waves with the same frequency and different angles of propagation, which goes under the name of angular spectrum. This defines an angular domain which is the reciprocal of the space domain. Intuitively the angular spectrum representation constitutes a picture of the effects of propagation in the far zone whereas the near zone is described by the space domain. In this Section, an analogous of the Wiener-Khinchin theorem and its reciprocal form for quasi-stationary processes has been discussed. Substitution of times with position vectors and frequencies with angular vectors allows to derive similar statements for the near (space domain) and far (angular domain) zone [6]. If we use θ = r ⊥o /z o (and θ 1,2 = r ⊥o1,2 /z o ) as variables to describe radiant intensity and cross-spectral density in the far field, and if we identify points on the source plane with r ⊥ (and r ⊥1,2 ), we have: (a') The cross-spectral density of the field at the source plane g( r ⊥2 − r ⊥1 ) and the angular distribution of the radiant intensity I( θ) are a Fourier Pair.
(b') The cross-spectral density of the far field g( θ 2 − θ 1 ) and the source-intensity distribution I( r ⊥ ) are, apart for a simple geometrical phase factor, a Fourier Pair.
The statement (b') can be regarded as an analogue, for quasi-homogeneous sources, of the far-zone form of the van Cittert-Zernike theorem. The statement (a') instead, is due to the symmetry between space and angle domains, and can be seen as an "anti" VCZ theorem. This discussion underlines the link between the VCZ theorem and the Wiener-Khinchin theorem.
Many undulator radiation sources are quasi-homogeneous sources in the usual way. In the case of a quasi-homogeneous source of typical linear dimension d, the angular spectrum at distance z o ≫ dω∆/c is expected to exhibit speckles with typical linear dimension z o (dω/c) −1 , as illustrated in Fig. 8, which shows an intuitive picture of the propagation of transverse coherence.
They generate fields which are relatively simple to analyze mathematically and still rich in physical features. Although both VCZ and "anti" VCZ theorem are based on the assumption of usual quasi-homogeneity we will see that these are often applicable, at least in some sense, also in the case of "weakly quasihomogeneous" wavefronts.

Evolution of the cross-spectral density function through the undulator beamline
In our work [3] we presented an expression for the reduced fieldẼ ⊥ (ω) = E ⊥ (ω) exp (−iωz o /c) of a single particle with offset l and deflection η with respect to the optical axis z in an undulator. In order to derive our result, we used a Green's function approach to solve the paraxial Maxwell equations for the Fourier transform of the electric field and we took advantage of a consistent use of the resonance approximation. The fieldẼ ⊥ differs fromĒ ⊥ for a phase factor which depends on the variable z o and on the frequency ω only: therefore, the use of one expression instead of the other in the equation for G does not change the result. In [3], we presented results in normalized units in the far field zone for a particle with offset and deflection. Based on that work we can calculate the field in normalized units for a particle with offset and deflection at any distance from the exit of the undulator, where the center of the undulator is taken at z = 0, as specified in Fig. 9: Eq. (24) is valid for the system tuned at resonance with the fundamental harmonic ω o . This means that we are considering a large number of undulator periods N w ≫ 1 and that we are looking at frequencies near the fundamental and at angles within the main lobe of the directivity diagram. In this situation one can neglect the vertical y-polarization of the field with an accuracy (4πN w ) −1 . This constitutes a great simplification of the problem since, at any position of the observer, we may consider the electric field Fourier transform,Ê s⊥ , as a complex scalar quantity corresponding to the surviving x-polarization component of the original vector quantity. Normalized units were defined as Fig. 9. Illustration of the undulator geometry and of the observation plane.
K being the deflection parameter, L w being the undulator length, being the resonant frequency, J n the Bessel function of the first kind of order n, λ w the undulator period, (−e) the electron charge and γ the relativistic Lorentz factor. l is the normalized offset in the center of the undulator. Finally, the parameterĈ represents the normalized detuning, which accounts for small deviation in frequency from resonance.
As it is shown in Appendix B, Eq. (24) can be rewritten aŝ represents the observation angle and Φ U is given by Starting from the next Section we will restrict our attention to the caseĈ = 0 for simplicity. Therefore, it may be interesting to note that in the particular caseĈ = 0, the function S can be represented in terms of the exponential integral function Ei as: It is easy to show that the expression for the function S(·) reduces to a sinc(·) function asẑ o ≫ 1 . In fact, in this limiting case, the expression for the electric field from a single particle, given in Eq. (24) is simplified tô Eq. (33) can be integrated analytically givinĝ where A comparison between sinc(ζ 2 /4) and the real and imaginary parts of S(0,ẑ o , ζ 2 ) forẑ o = 1 is given in Fig. 10.
Let us now go back to the general case forẑ o 1/2 and use Eq. (28) to calculate the cross-spectral density. The cross-spectral density G ω is given Eq. (14) in dimensional units and as a function of dimensional variables. Since the field in Eq. (28) is given in normalized units and as a function of normalized variablesẑ o , θ x,y andĈ, it is convenient to introduce a version of G ω defined by means of the field in normalized units: Transformation of G ω in Eq. (14) toĜ (and viceversa) can be easily performed shifting from dimensional to normalized variables and multiplying G ω by an inessential factor: As a result, we can always useĜ in stance of G ω . Substituting Eq. (28) in Eq.
(36) we obtain: Expanding the exponent in the exponential factor in the right hand side of Eq. (38), it is easy to see that terms inl 2 x,y cancel out. Terms inθ 2 x,y contribute for a common factor, and only linear terms inl x,y remain inside the ensemble average sign. Substitution of the ensemble average with integration over the beam distribution function leads tô Here integrals d η and in d l are to be intended as integrals over the entire plane spanned by the η and l vectors. Eq. (39) is very general and can be used as a starting point for computer simulations.
We already assumed that the distribution in the horizontal and vertical planes are not correlated, so that F η, l = Fη x ,lx Fη y ,ly . If the transverse phase space is specified at positionẑ o = 0 corresponding to the minimal values of the βfunctions, we can write Fη x ,lx = Fη x Fl x and Fη y ,ly = Fη y Fl y with From Eq. (25) and Eq. (29) it is easy to see that where σ x,y and σ x ′ ,y ′ are the rms transverse bunch dimension and angular spread. ParametersN x,y will be indicated as the beam diffraction parameters and are, in fact, analogous to Fresnel numbers and correspond to the normalized square of the electron beam sizes, whereasD x,y represent the normalized square of the electron beam divergences.
Substitution of relations (40) in Eq. (39) yields, at perfect resonance (Ĉ = 0): For notational simplicity, in Eq. (43) we have substituted the proper notation Undulator radiation as a quasi-homogeneous source When describing physical principles it is always important to find a model which provides the possibility of an analytical description without loss of es-sential information about the feature of the random process.
In order to get a feeling for some realistic magnitude of parameters we start noting that the geometrical emittances of the electron beam are simply given by ǫ x,y = σ x,y σ x ′ ,y ′ . Here they will be normalized asǫ x,y = 2πǫ x,y /λ. Then σ 2 x,y = β o x,y ǫ x,y , where β o x,y are the minimal values of the horizontal and vertical betatron functions. In this paper we will assume that the betatron functions will have their minimal value at the undulator center. Therefore we have This is always the case in situations of practical interest, withN x which may range from values much smaller to much larger thanD x . AssumingD x ≫ 1 andN x ≫ 1, independently on the values ofD y andN y , introduces simplifications in the expression for the cross-correlation function and allows further analytical investigations. As we will see, in particular, a model of the electron beam based on these assumptions contains (but it is not limited to) the class of quasi-homogeneous sources discussed in Section 2.2.
In the next Sections 4.1 and 4.2 and later in Section 5, we will see what are the conditions in terms of the dimensionless parametersN x,y andD x,y for some undulator radiation wavefront at positionẑ o , to be quasi-homogeneous in the usual and in the weak sense (according to the definition in Section 2.2), we will justify the introduction of the concept of weak quasi-homogeneity itself and we will discuss the applicability regions of the VCZ (and "anti" VCZ) theorem. Then, in Section 6, we will also discuss some case characterized by non weakly quasi-homogenous fields.

A simple model
To provide a first analysis of the problem we adopt some simplifying assumptions that are only occasionally met in practice.
As already assumed vertical emittance is much smaller than horizontal emittance. For notational simplicity we will make the assumptionsN y ≪ 1 and D y ≪ 1. This means that we theoretically assumeη y ≪ 1 andl y ≪ 1. As a result, the terms inη y andl y can be neglected in the S(·) term on the right hand side of Eq. (43). Although this model includes obvious schematization it is still close to reality in many situations, and it is only to be considered as a provisory model for physical understanding to be followed, below, by more comprehensive generalizations. In this Section we will restrict our attention at the correlation function forθ y1 =θ y2 that is on any horizontal plane. Here again, for notational simplicity, we will substitute the proper notation G(ẑ o ,θ x1 , 0,θ x2 , 0) withĜ(ẑ o ,θ x1 ,θ x2 ). Eq. (43) can be greatly simplified leading toĜ Let us now introduce With this variables redefinition we obtain where we have posedD =D x andN =N x for notational simplicity. Eq. (48) can be also written as The integral inη can be performed analytically thus leading tô It is important to remember again that an asymptotic formula forẑ o ≫ 1 can be obtained from Eq.
Then, it is easy to understand that S is bound to go to zero for values of (φ±∆θ) 2 larger than unity, exactly as the asymptotic terms in sinc(·) would do. In fact, onceĈ is set to zero, S depends parametrically on the normalized distanceẑ o alone, that is S = S[ẑ o , (φ ± ∆θ) 2 ], and gives the previously found asymptotic expression of sinc[(φ ± ∆θ) 2 /4] in the limit forẑ o ≫ 1. Since herê z o is supposed to be at least of order unity (ẑ o > 1/2), we can conclude that S must be different from zero only for values of (φ ± ∆θ) 2 of order unity (to be more precise, for values (φ ± ∆θ) 2 ≃ 4, (φ ± ∆θ) 2 /4 being the arguments of the sinc function) as it can be seen, for instance, from Fig. 10 for a particular case. Thus Eq. (50) and its asymptotic equivalent forẑ o ≫ 1 share the same mathematical structure.
Let us now introduce the non-restrictive assumptions: and definê The physical interpretation ofÂ follows from that ofσ/ẑ o :Â is the dimen-sionless square of the apparent angular size of the source at the observer point position, calculated as if the source was positioned atẑ o = 0. IfN ≫ 1 and D ≫ 1 we have (2Âẑ 2 oD )/(Â +D) ≫ 1 for any value ofẑ o and any choice of N andD. As a result, from the exponential factor exp [−2Âẑ 2 o ∆θ 2D /(Â +D)] outside the integral sign in Eq. (50) we have that G ω (ẑ o ,θ, ∆θ) is different from zero only for ∆θ ≪ 1. Then we can neglect terms in ∆θ in the factors S(·) within the integral sign thus gettinĝ The maximal value ofθ is related with the width of the exponential function exp [−θ 2 /(2Â +2D)] outside the integral sign in Eq. (53). It follows that in the limit forD ≫ 1 we can neglect the exponential factor exp [−(φ 2 + 2φθ)/(2Â + 2D)] within the integral sign: in fact, its argument assumes values of order unity for φ ≫ 1, but the factor | S[ẑ o ,φ 2 ] | 2 cuts off the integrand forφ 1. Therefore Eq. (53) can be simplified as follows: The integral in Eq. (54) is simply the Fourier transform of the function Since the function f (φ) has values sensibly different from zero only asφ is of order unity or smaller, its Fourier Transform will also be suppressed for values of 2Âẑ o |∆θ|/(Â +D) larger than unity, by virtue of the Bandwidth Theorem. This means that the integral in Eq. (54) gives non-negligible contributions only up to some maximal value of |∆θ|: On the other hand, the exponential factor outside the integral in Eq. (54) will cut off the functionĜ around some other value It is easy to see that, for any value ofẑ o , |∆θ| max ≫ |∆θ| max2 . In fact we have in the limit forD ≫ 1. As a result the Fourier transform in Eq. (54) is significant only for values of the variable −2Âẑ o ∆θ/(Â +D) near to zero and contributes toĜ only by the inessential factor In order to use the correlation functionĜ for calculation of coherence length and other statistical properties, one has to use the spectral degree of coherence g, which can be presented as a function ofθ and ∆θ instead of x ⊥o2 and x ⊥o1 : From Eq. (54) we obtain : In the asymptotic limit for a large value ofẑ o ,Â ≪ 1, Eq. (60) can be simplified to From Eq. (54) it is easy to see that the region of interest where the field intensity is not negligible is whenθ D . Therefore, Eq. (61) can be further approximated to It is interesting to calculate the transverse coherence lengthξ c as a function of the observation distanceẑ o . For any experiment, complete information on the coherence properties of light are given by the function g. When calculating the coherence length one applies a certain algorithm to g thus extracting a single number. This number does not include all information about the coherence properties of light, and the algorithm applied to g is simply a convenient definition. Then, in order to calculate a coherence length one has, first, to choose a definition among all the possible convenient ones. In this paper we will simply follow the approach by Mandel, originally developed for the time domain, but trivially extensible to any domain of interest, in our case the angular domain. The coherence length, naturally normalized to the diffraction length L w c/ω is defined aŝ where the factor 2 in front of the integral on the right hand side is due to the fact that we chose Mandel's approach and that our definition of ∆θ differs of a factor 1/2 from his definition. Performing the integration in Eq. (63) with the help of Eq. (60) yields:

Discussion
The coherence length in Eq. (64) exhibits linear dependence onẑ o , that iŝ ξ c −→ π/Nẑ o while forẑ o −→ 1/2 that is at the end of the undulator, it converges to a constantξ c −→ [π/(4N)+π/D] 1/2 . Eq. (64) and its asymptotes are presented in Fig. 11 and Fig. 12 for the caseN = 10 3 ,D = 10. It is evident that at the exit of the undulator,ξ c ∼ 1/ D , becauseN ≫D. On the other hand, horizontal dimension of the light spot is simply proportional to N as it is evident from Eq. (60). This means that the horizontal dimension of the light spot is determined by the electron beam size, as is intuitive, while the beam angular distribution is printed in the fine structures of the intensity function, that are of the dimension of the coherence length. In the limit for z o ≫ 1 the situation is reversed. The radiation field at the source can be presented as a superposition of plane waves, all at the same frequency ω o , but with different propagation angles with respect to the z-direction. Since the radiation at the exit of the undulator is partially coherent, a spiky angular spectrum is to be expected. The nature of the spikes is easily described in terms of Fourier transform theory, in perfect analogy with what has been said about the frequency spectrum in Section 2.1. From Fourier transform theorem or, directly, from Eq. (54) or from geometrical optics arguments we can expect an angular spectrum envelope with Gaussian distribution and rms width of D .
Also, the angular spectrum should contain spikes with characteristic width 1/ N , as a consequence of the reciprocal width relations of Fourier transform pairs (see Fig. 13). This can be seen realized in mathematical form from the expression for the cross-spectral density, Eq. (60) and from the equation for the coherence length, Eq. (64). SinceN ≫ 1, the horizontal width of the coherence spot is much smaller than the vertical one.
It is also important to remark that the asymptotic behavior forÂ ≪ 1 of g in Eq. (62) andξ ĉ are direct application of van Cittert-Zernike theorem. In fact, the last expo- nential factor on the right hand side of Eq. (61) is simply linked with the Fourier transform of Fl x (l x ). We derived Eq. (61) forN ≫ 1 andD ≫ 1, witĥ z 2 oD ≫N : in non-normalized units these conditions mean that the VCZ theorem is applicable when the electron beam divergence is much larger than the diffraction angle, i.e. σ 2 x ′ ≫ λ/(2πL w ), the electron beam dimensions are much larger than the diffraction size, i.e. σ 2 x ≫ λL w /2π, and (σ x ′ z o ) 2 ≫ σ 2 x . On the contrary, authors of [4] state that, in order for the van Cittert-Zernike theorem to be applicable, "the electron-beam divergence must be much smaller than the photon divergence", that is our diffraction angle, i.e. σ x ′ ≪ λ/(2πL w ) (reference [4], page 571, Eq. (57)). Our derivation shows that this conclusion is incorrect.
In [5] (paragraph 5.6.4) a rule of thumb is given for the applicability region of the generalization of the VCZ theorem to quasi-homogeneous sources. The rule of thumb requires z o > 2d∆/λ where d is "the maximum linear dimension of the source", that is the diameter of a source with uniform intensity and ∆ "represents the maximum linear dimension of a coherence area of the source". In our case d ≃ 2σ x , since σ x is the rms source dimension, and from Eq. (64) we have ∆ = ξ c ≃ λ/(2 √ πσ x ′ ). The rule of thumb then requires z o > 2σ x /( √ πσ x ′ ): in dimensionless this readsẑ o N /D. This is parametrically in agreement with our limiting conditionẑ 2 oD ≫N , even though these two conditions are obviously different when it come to actual estimations: our condition is, in fact, only an asymptotic one. To see how well condition z o N /D works in reality we might consider the plot in Fig. 11. Therê N = 10 3 andD = 10 so that, following [5] we may conclude that a good condition for the applicability of the VCZ theorem should beẑ o 10. However as it is seen from the figure, the linear asymptotic behavior is not yet a good approximation atẑ o ≃ 10. This may be ascribed to the fact that the derivation in [5] is not generally valid, but has been carried out for sources which drop to zero very rapidly outside the maximum linear dimension d and whose correlation function also drops rapidly to zero very rapidly outside maximum linear dimension ∆.
However, at least parametrically, the applicability of the VCZ theorem in the asymptotic limitẑ 2 oD ≫N can be also expected from the condition z o > 2d∆/λ in [5]. In other words, with the help of our approach we were able to specify an asymptotic region where the VCZ theorem holds. Such a region overlaps with predictions from Statistical Optics. Statistical Optics can describe propagation of the cross-spectral density only once it is known at some source plane position. Our treatment allows us to specify the crossspectral density at the exit of the undulator, but it should be noted that we do not need to use customary results of Statistical Optics and propagate the cross-spectral density from the exit of the undulator in order to obtain the cross-spectral density at some distance along the beamline. In fact our approach, which consists in taking advantage of the system Green's function in paraxial approximation and, subsequently, of the resonance approximation, allows us to calculate the cross-spectral density directly at any distance from the exit of the undulator.
Let us now consider the structure of Eq. (60) and discuss the meaning of the phase terms inθ∆θ. These are important in relation with the condition for quasi-homogeneous source: their presence couples the two variablesθ and ∆θ and prevents the source to be quasi-homogeneous at any given value ofẑ o 4 , unless they compensate each other in some parameter region.
Let us discuss the limit,Â ≫ 1. We may consider two subcases. First, consider A ≫D ≫ 1. In this case, inspection of Eq. (60) shows that the two phase terms compensate and the source is quasi-homogeneous, because the crossspectral density is factorized in a function ofθ and a function of ∆θ. It should be noted that if conditionÂ ≫ 1 is not satisfied at the exit of the undulator, whereẑ o ∼ 1, then it is never satisfied. IfN ≫D ≫ 1 we have a quasihomogeneous source at the exit of the undulator.
Second, considerD ≫N ≫ 1. This correspond to a situation with a low value of the normalized betatron function in the horizontal direction. Figure  14 shows a numerical example withǫ x = 100 andβ x = 0.3 that isN = 30 and D = 300: the value for the horizontal betatron function is similar to the low-β case reported at page 12, Table 2.2.2 in [1], where β x = 1.3 m for a 5m-long insertion device. The valueǫ x = 100 corresponds to a wavelength of about 0.6Å for the PETRA III case. WhenD ≫N ≫ 1 no compensation of the phase terms in Eq. (60) is possible, not even at the exit of the undulator. In this case, whatever the value ofẑ o we can never have a quasi-homogeneous wavefront. This constitutes no problem. Simply, the wavefront is not-quasi-homogeneous in this case. However, we may interpret the situation by saying that a "virtual" quasi-homogeneous source placed in the center of the undulator would result in the non-homogeneous source described by Eq. (60) at the exit of the undulator. Although it physically makes no sense to discuss about Eq. (60) inside the undulator, the "virtual" source analogy is suggested by the fact that settinĝ z o = 0 in Eq. (60), both phase terms become zero. Note that in general, whatever the values ofN ≫ 1 andD ≫ 1, one never has quasi-homogeneous sources in the limit forÂ ≪ 1. In fact, in the asymptotiĉ A ≪ 1 only the phase factor exp(i2θ∆θẑ o ) contributes, which couplesθ and ∆θ. Such a factor is connected with phase of the field from a single electron in an undulator in the far zone, ω(x 2 o + y 2 o )/(2cz o ), which represents, in paraxial approximation, the phase difference between the point (x o , y o , z o ) and the point (0, 0, z o ): in the asymptotic for large values ofẑ o , the electric field generated by a single electron with offset and deflection in an undulator has a spherical wavefront (see [3]). When one calculates the field correlation function at two different points, he ends up with a contribution equal to the difference (due to complex conjugation) between ω(x 2 o2 + y 2 o2 )/(2cz o ) and ω(x 2 o1 + y 2 o1 )/(2cz o ), that for the vertical (and separately, the horizontal) direction gives exactly the shift 2θ∆θẑ o in normalized units. It should be noted that such a reasoning is not limited to Synchrotron Radiation sources, but it is quite general since, as already discussed, it relies on the fact that the wavefront of a single radiator (in our case, an electron) produces a spherical wavefront in the far field. This is, for instance, the case of thermal sources as well. In other words, if the far field radiation of a quasi-homogeneous source is taken as a new source, that new source will never be quasi-homogeneous.
A common property of all situations withN ≫ 1 andD ≫ 1 is that, for any value ofẑ o , the modulus of g, i.e. |g|, is always independent onθ. Moreover, it is always possible to apply the VCZ theorem starting either from a virtual quasi-homogeneous source placed at center of the undulator when D ≫N ≫ 1, or otherwise from a real one placed at the exit of the undulator (or at any other positionẑ o close enough to the exit of the undulator to guarantee a quasi-homogeneous wavefront). These observations suggest to extend the concept of quasi-homogeneity, and introduce the new concept of "weak quasi-homogeneity" as discussed before. With respect to the new coordinates θ and ∆θ, a given wavefront at fixed positionẑ o is said to be weakly quasihomogeneous when |g| is independent ofθ. With this new definition at hand we can restate some of our conclusions in a slightly different language. We have seen that in the far field, when the VCZ theorem holds, the wavefronts are weakly quasi-homogeneous, but never quasi-homogeneous in the usual sense. In the caseN ≫D ≫ 1 we pass from quasi-homogeneous wavefronts (in the usual sense) in the near field to weakly-quasi homogeneous wavefronts (but not quasi-homogenous in the usual sense) in the far field. Note that the wavefronts are always weakly quasi-homogeneous, even during the transition from near to far zone. In the caseD ≫N ≫ 1 instead, the VCZ theorem is applicable already from the exit of the undulator, as it can be seen from Fig. 14, and the wavefront is not quasi-homogeneous in the usual sense, but still weakly quasi-homogeneous from the very beginning.
The weak quasi-homogeneity of the wavefronts at any value ofẑ o , i.e. the fact that |g| is independent ofθ for any value ofẑ o guarantees that the plot in Fig. 14 is universal. It should be noted that this fact depends on the choice of large parametersN ≫ 1 andD ≫ 1, but it is also strictly related with the Gaussian nature of the electron distribution in angles and offsets, that is a well-established fact for storage-ring beams. If angles or offsets were obeying different distribution laws, in general, one could not perform the integral inη in Eq. (48) and, in general, |g| would have shown a dependence onθ: our noticeable result is linked with the properties of the exponential elementary function. However, it should be clear that even in the case when angles or offsets were obeying different distribution laws, i.e when the plot in Fig. 14 is not universal, we could have situations when wavefronts are quasi-homogeneous in the usual sense near the exit of the undulator and are weakly quasi-homogeneous in the far field limit, but not along the transition between these two zones. A more detailed discussion of this issue will be given in Section 5, where we will be discussing conditions for the source to be quasi-homogeneous.
Another remark to be made pertains the applicability of the VCZ theorem. As we deal with a quasi-homogenous source (in the usual sense) the knowledge of I(θ) and g(∆θ) in the far zone allow, respectively, the calculation at the source plane of g(∆x) through the "anti" VCZ theorem and of I(x) through the VCZ theorem (here we consider only one dimension, the horizontal one x). Viceversa the knowledge of I and g at the source allow calculation of I and g in the far field. In terms of intensity, all information regarding wavefront evolution (assuming a quasi-homogeneous source, in the usual sense) is included in I(θ) and I(x). For instance, the knowledge of I(θ) allows calculation of g(∆x) at the source plane through the "anti" VCZ theorem. Then, the knowledge of g(∆x) and I(x) allow the calculation of the cross-spectral density, which can be propagated at any distance, and allow to recover g(∆θ). So, complete characterization of the undulator source is given when I(θ) and I(x), when the source is assumed quasi-homogenous.
Yet we have seen that, when the electron distribution in angles and offsets are Gaussian andN ≫ 1 andD ≫ 1, the VCZ theorem holds also in the caseD ≫ N ≫ 1, when the source is non quasi-homogeneous in the usual sense. We have seen that the situation can be equivalently described with the help of a virtual quasi-homogeneous source in the middle of the undulator. However, such an interpretation is only valid a posteriori. For the caseN ≫ 1 andD ≫ 1 there is a non quasi-homogeneous wavefront at the undulator exit; before our approach was presented one would have concluded that the VCZ theorem cannot be applied, since the spectral degree of coherence does not form a Fourier pair with the intensity distribution at the undulator exit. Our approach is based on the simplification of mathematical results through the use of small and large parameters and subsequent understanding and interpretation of these simplified results: in our analysis we were never limited to the treatment of quasi-homogenous cases alone.
As a closing remark about the coherence length we like to draw the reader's attention on the fact that the dimensional form ξ c of the coherence length, given in normalized units byξ c in Eq. (64), does not include the undulator length. This is to be expected since, in the limitN ≫ 1 andD ≫ 1, the typical size and divergence of the electron beam are much larger than the diffraction size cL w /ω and angle c/(ωL w ), which are intrinsic properties of the undulator radiation. As a result, in this limit, the evolution of the radiation beam is a function of the electron beam parameters only, and does not depend on the undulator length. In the following Section 5, where we will extend our model to a two-dimensional case, we will see that the quasihomogeneous approximation is valid in many practical situations, but we will have to account for diffraction of undulator radiation in the vertical direction. In this case the dimensional coherence ξ c length will be a function of the undulator length L w as well.

Effect of the vertical emittance on the cross-spectral density
Up to now we were dealing with the field correlation function g within the framework of a one-dimensional model. In fact we considered the limit forǫ yβy ≪ 1 andǫ y /β y ≪ 1 and we calculated g forθ y1 =θ y2 = 0 andĈ = 0, so that our attention was focused on coherent effects in the horizontal direction. We will now extend our considerations to a two-dimensional model always forĈ = 0. This can be done by a straightforward generalization of Eq. (50) which can be obtained from Eq. (43) following the same steps which lead to Eq. (50), but this time without assumptions on N y ,D y ,θ y1 andθ y2 . Finally, at the end of calculations, our final expression forĜ should be normalized tô As has already been seen in Section 4, after normalization toŴ we will obtain the spectral degree of coherence g. With this in mind we will neglect, step after step, unnecessary multiplicative factors that, in any case, would be finally disposed after normalization of the final result. Retaining indexes x and y in our notation we obtain We will still assumeN x ≫ 1 andD x ≫ 1. This allows to factorize the right hand side of Eq. (67) in the product of contribution depending on horizontal (θ x , ∆θ x ) coordinates only with a second depending on vertical (θ y , ∆θ y ) coordinates only. In fact, from the exponential factor outside the integral sign in Eq. (67) it is possible to see that the maximum value of ∆θ 2 x is of order (Â x +D x )/(Â xDxẑ 2 o ) ≪ 1. As a result, ∆θ x can be neglected inside the S func-tions in Eq. (67). Moreover, sinceD x ≫ 1 one can also neglect the exponential factor inφ 2 x + 2φ xθx inside the integral sign. This leads tô Based on the same reasoning in Section 4.1, that we repeat here for completeness, we can also neglect the phase factor inφ x under the integral in dφ x in Eq. (68). Such integral in dφ x is simply the Fourier transform of the function with respect to the variable −2Â xẑo ∆θ x /(Â x +D x ). In the argument of the S functions on the right hand side of Eq. (69),φ 2 x is always summed to positively defined quantities. This remark allows one to conclude that f (φ x ) has values sensibly different from zero only asφ x is of order unity or smaller. Therefore, its Fourier Transform will also be suppressed for values of 2Â xẑo |∆θ x |/(Â x +D x ) larger than unity, by virtue of the Bandwidth Theorem. This means that the integral in dφ x in Eq. (68) gives non-negligible contributions only up to some maximal value of |∆θ x |: On the other hand, the exponential factor outside the integral in Eq. (68) will cut off the functionĜ around some other value It is easy to see that, for any value ofẑ o , |∆θ| xmax ≫ |∆θ| xmax2 . In fact we in the limit forD x ≫ 1. As a result the Fourier transform in Eq. (68) is significant only for values of the variable −2Â xẑo ∆θ x /(Â x +D x ) near to zero. As a result we obtain the following equation forĜ: where horizontal and vertical coordinates are obviously factorized.
Eq. (73) has been derived assumingN x ≫ 1 andD x ≫ 1. Note that assuming setting ∆θ y =θ y = 0 one can obtain Eq. (60) from Eq. (73). This proves that Eq. (60) has a wider range of validity than that forN y ≪ 1 andD y ≪ 1 (as the reader will remember, these assumptions were made just for notational simplicity). In fact, asN x ≫ 1 andD x ≫ 1 horizontal and vertical direction factorize and the horizontal factor is always that in Eq. (60), independently onN y andD y .
Under one of the two extra assumptionsN y ≫ 1 orD y ≫ 1, Eq. (73) can be further simplified and often describes a weakly quasi-homogeneous wavefront according to the definition given at the end of Section 4.1. At the end of the Section we will show that asD y ≫ 1 we always have a weakly quasihomogeneous wavefront, for any value ofN y . It will also be seen that the same applies whenN y ≫ 1 andÂ y ≪ 1 (far field) orÂ y ≫ 1 (near field) for any value ofD y . However, asN y ≫ 1,D y 1 andÂ y ∼ 1 we have an intermediate region between the near and far region were, in general, wavefronts are not quasi-homogeneous, not even in the weak case.
For simplicity of discussion we will setθ x =θ y = 0 thus obtaining from Eq.
which is easier to manipulate. It should be reminded that, if one is interested in ascertaining the weak quasi-homogeneity of a wavefront, one has to deal with the full Eq. (73). Moreover, it should be noted that on the one hand, within the weak quasi-homogeneous case, Eq. (74) is quite general and we can extract from it all important information on the transverse coherence independently on the values ofθ x andθ y . On the other hand though, in the case the weakly quasi-homogeneous assumption fails, Eq. (74), e.g. whenN y ≫ 1,D y 1 and A y ∼ 1 as we will see, |g| depends onθ y and the study of Eq. (74) has a more restricted range of validity, namely for the particular value ofθ y = 0.
In Section 5.1 we will assumeN y ≫ 1 and arbitraryD y , while in Section 5.2 we will study the case with arbitraryN y andD y ≫ 1.
In general, the coherence length in theŷ direction (calculated at ∆θ x = 0, but trivially extendible to the case ∆θ x = 0),ξ cy is a function ofD y ,N y andẑ o , as it can be concluded by inspection of the general expression forĜ in Eq. (67).
AsN y ≫ 1 andD y is arbitrary we will demonstrate thatξ cy can be approximated aŝ where Φ is a universal function of the dimensionless parametersD y andÂ y . We will first study the asymptotic casesD y ≫ 1 andD y ≪ 1, which are useful for physical understanding. Then we will generalize our results accounting for the influence of a finite divergence of the electron beam on the cross-spectral density. An analytical approximation for the function Φ will be proposed. This will be chosen to match in a very simple way both asymptotic expressions for D y ≫ 1 andD y ≪ 1: we will demonstrate that for any value ofD y , discrepancies between the approximated and the actual (numerically calculated) expression are less than 10%, though there is no theoretical reason to assume, a priori, this relatively good accuracy. Also, atÂ y ≪ 1 (andN y ≫ 1) we will see that the VCZ theorem always hold.
AsN y is arbitrary andD y ≫ 1 we will show, instead, that the coherence length (calculated again at ∆θ x = 0, but trivially extendible to the case ∆θ x = 0) can be approximated aŝ N eff (N y ) being a universal function of the Fresnel numberN y . Ψ is a universal function of the dimensionless parametersD y ,N y andẑ o . As usual in this paper, we will first study the asymptotic casesN y ≫ 1 andN y ≪ 1. Then we will generalize our results accounting for any value ofN y . As before, an analytical approximation for Ψ will be proposed. This will be chosen to match in a very simple way both asymptotic expressions forN y ≫ 1 andN y ≪ 1: again, we will demonstrate that for any value ofN y , discrepancies between the approximated and the actual (numerically calculated) expression are less than 13% though there is no theoretical reason to assume, a priori, this relatively good accuracy.
The case for a large Fresnel number with a finite electron beam divergence, or viceversa of a large beam divergence and a finite Fresnel number, is of practical importance. To give a numerical example, let us put λ = 1Å, and consider a typical vertical emittance (for third generation sources in operation) ǫ y ≃ 3 · 10 −11 m, that isǫ y ≃ 2. On the one hand, ifβ y = 3, we haveD y ≃ 0.6 andN y ≃ 6. On the other hand, ifβ y = 0.3 we have, viceversa,D y ≃ 6 and N y ≃ 0.6.

Very large Fresnel numberN y ≫ 1, arbitrary divergence parameterD y
As a matter of fact, the only important assumptions used to derive results Section 4.1 were thatN x ≫ 1 andD x ≫ 1 that are valid here as well. When these assumptions are granted, results can be factorized as a product of factors dependent separately on the x and on the y directions. Then the spectral degree of coherence in the horizontal direction will always be the same as in Section 4.1. Differences will arise here, of course, due toθ y1 =θ y2 .

Case with divergence parameterD y ≫ 1.
This case is the easiest to analyze, because one can follow step by step the derivation for the one dimensional model given in the previous Section 4.1.
Calculations in the vertical direction y simply follow the derivation for the horizontal direction x. As a result, one can start from Eq. (67) and perform, separately for the x and the y directions, the same simplifications which hold for the one-dimensional model. By comparison with Eq. (60), one obtains directly the result Normalization of Eq. (77) according to Eq.(66) has been obtained simply settingĜ(ẑ o , 0, 0) = 1. In this case as well, two-dimensional Fourier Transform of |S[ẑ o ,φ 2 x +φ 2 y ]| 2 calculated with respect the variables −2Â xẑo ∆θ x /(Â x +D x ) and −2Â yẑo ∆θ y /(Â y +D y ) gives, in analogy with Eq. (58), an unessential factor which has been included in the normalization. Again, similarly as before one can calculate the coherence areaΩ c (ẑ o ) defined in analogy withξ c (ẑ o ) aŝ Performing the integration yields: In the limit for a large value ofẑ o the coherence area exhibits quadratic dependence onẑ o , that isΩ c −→ πẑ 2 o /(N xNy ) 1/2 while forẑ o −→ 1/2, that is at the end of the undulator, it converges to the constant valueΩ c −→ π 1/(4N x ) + 1/D x 1/2 1/(4N y ) + 1/D y 1/2 .
It should be noted as before that the asymptotic behavior forẑ o ≫ 1 of g are direct application of van Cittert-Zernike theorem, as it must be sincê A x,y ≪ 1. In fact, Eq. (81) is simply linked with the two-dimensional Fourier transform of Fl x (l x )Fl y (l y ).

Case with divergence parameterD y ≪ 1.
With respect to the situation treated in Paragraph 5.1.1, this case requires a more careful analysis of the relations between different parameters. In fact, on the one handD y ≪ 1 implies that the electron beam divergence drops out of the problem parameters, but on the other hand in this case the divergence of the radiation is described by the intrinsic divergence of undulator radiation, that is described by a more complicate mathematical function, compared with a Gaussian. In relation with this, it should be noted that simplifications in Section 4.1 pr 5.1.1 were based on the very specific properties of the Gaussian function representing the electron distributions in offset and deflection. Luckily, this is a realistic description in storage ring beam physics.
Let us consider Eq.(74). In order to derive it we only used the assumptionŝ N x ≫ 1 andD x ≫ 1, forθ x =θ y = 0. The result of operations on the right hand side of Eq. (74) depend on howN y scales with respect toẑ 2 o . The cases for A y > 1 cannot be dealt with fully analytically. In the following we will analyze the asymptotic situationÂ y ≪ 1 and then we will treat semi-analytically the generic situation for all values ofÂ y . As we will see, as soon asÂ y < 1 we start to be in the applicability region of the VCZ theorem.
It should be noted that the dependence in ∆θ x and ∆θ y in Eq. (74) are already separated. Therefore, what has been said in Section 4.1 regarding the behavior of coherence properties in the horizontal direction hold independently of the behavior of coherence properties in the vertical direction.
(A) Far zone case:Â y ≪ 1. -SinceN y ≫ 1 it must beẑ 2 o ≫N y ≫ 1 in order to allow forÂ y ≪ 1. Since we are working in quasi-homogeneous source condition (N x,y ≫ 1) in the limiting situationÂ y ≪ 1 we should recover VCZ theorem: this is the far field case. Even in the presence of the extra parameter D y we can treat the generic caseÂ y ≪ 1 independently on howÂ y compares with respect toD y .
Simultaneously, from the oscillating factor, always inside the integral in dφ y , we have a condition for the maximal value of ∆θ 2 y ∼ 1/ẑ 2 o ≪ 1: in fact, if this condition is not fulfilled the integrand will be highly oscillatory. Alternatively, we can obtain a similar condition from the Gaussian exponential factor in ∆θ y outside the integral, since ∆θ 2 As a result, the dependence of S on (φ + ∆θ y ) 2 can be dropped givinĝ The integral in dφ y can be performed givinĝ NormalizingĜ in such a way thatĜ(ẑ o , 0, 0) = 1 we obtain the following expression for the spectral degree of coherence g: Finally, combination of the second and the third exponential functions yields the result Let us first considerẑ o ≫ 1. Looking at the integral inφ y in Eq. (74) it is easy to recognize that its integrand is highly oscillatory inφ y when 2φ yẑo ∆θ y ≫ 1, sinceÂ y /(Â y +D y ) < 1. Therefore, the integrand will contribute to the integral significatively only up to values 2φ yẑo ∆θ y 1. On the other hand, the terms in S give non negligible contributions only for values ofφ y up to order unity. As a result, it must be 2ẑ o ∆θ y 1. Asẑ o ∼ 1 the width of g in ∆θ y is then of order unity. Whenẑ o becomes larger than unity, the angular width ∆θ y will decrease and asymptotically, asẑ o ≫ 1, one will have a rapidly oscillating integrand for ∆θ y ∼ 1/ẑ o ≪ 1. However note that ∆θ y must be multiplied byẑ o in order to obtain the correlation length which means that this remains constant and comparable with the diffraction length cL w /ω asẑ o increases.
As a result, under the assumptionẑ o ≫ 1, terms in ∆θ y can be dropped in the functions S(·) of Eq. (74), and the functions S can be substituted with their limiting form sinc. Moreover, in the limit forÂ y ≫ 1 andD y ≪ 1 we havê where the simplification in the phase under the integral in dφ y is possible forẑ o ∆θ yφyDy /Â y ≪ 1 and the exponential function exp [−φ 2 y /2(Â y +D y )] under the integral in dφ y can be neglected because the sinc(·) function has characteristic length inφ y of order unity. Eq. (88) is therefore valid in the limit forÂ y ≫ 1,D y ≪ 1 andẑ o ≫ 1. The integral in dφ y in Eq. (88) is simply the Fourier transform of the universal function done with respect to the variable 2ẑ o ∆θ y , conjugate toφ y , that iŝ It is not difficult to see that the width of this Fourier transform in ∆θ yẑo is much smaller than the characteristic width imposed by the Gaussian exponentials outside the integration sign, their ratio being of orderD y ≪ 1. This means that the Gaussian in ∆θ 2 y outside the integral sign is almost constant with respect to the behavior of the Fourier transform, and can be neglected, to obtain The assumptionẑ o ≫ 1 was vital for the derivation of Eq. (91). However, for all values ofẑ o such thatÂ y ≫ 1 (and, therefore, up to the exit of the undulator atẑ o = 1/2), it is easy to see from Eq. (67) that the source is quasihomogeneous, so that the "anti" VCZ theorem applies and the cross-spectral density of the field at the source plane forms a Fourier couple with the angular distribution of the radiant intensity, which is simply given by I S as defined in Eq. (89).
It is interesting to justify the fact that I S is in fact the angular distribution of the radiant intensity. To this purpose, it is sufficient to note that I S is simply, normalization factors aside, Eq. (176) of [3], which represents the intensity from a beam withǫ x → ∞ andǫ y → 0. A representation of I S (φ y ) is given in Fig. 15.
As a result, one may therefore conclude that Eq. (91) is valid in general, for any value ofẑ o such thatÂ y ≫ 1, in the asymptotic limitD y ≪ 1.
It is important to note that Eq. (91) can be written as where γ(ẑ o ∆θ y ), given by is a universal function normalized to unity. It is possible to calculate Eq. (93) analytically. To this purpose, it is sufficient to note that the Fourier Transform can be evaluated with the help of the Bessel-Fourier formula as is the sine integral function and Ci(·) is the cosine integral function. As a result one has The function γ(ẑ o ∆θ y ) is illustrated in Fig. 17. It should be noted that γ(ẑ o ∆θ y ) is the spectral degree of coherence g calculated for ∆θ x = 0 in the limit forÂ y ≫ 1 andD y ≪ 1, in agreement with Eq. (92).
(C) CaseÂ y ∼ 1. -In case (B) we have shown that terms in ∆θ y can be dropped in the functions S(·) of Eq. (74), for every value ofẑ o , and the functions S can be substituted with their limiting form sinc. In the case of A y ∼ 1 andD y ≪ 1 we havê The integral in dφ y in Eq. (97) is the Fourier transform of done with respect to the variable 2ẑ o ∆θ y , conjugate toφ y . Similarly as before, it is not difficult to see that the width of this Fourier transform in ∆θ yẑo is much smaller than the characteristic width imposed by the Gaussian exponentials outside the integration sign, their ratio being of orderD y ≪ 1. This means that the Gaussian in ∆θ 2 y outside the integral sign is almost constant with respect to the behavior of the Fourier transform, and can be neglected. Using the definition of F S in Eq. (98) we havê At this point, no simplification is possible and numerical analysis of the problem should be undertaken in order to calculateĜ, followed by normalization according tô as already said, in order to find an expression for the complex degree of coherence g. This discussion underlines again how the correlation angle ∆θ y in the y direction behaves. It starts from a constant value equal to the diffraction angle c/(ωL w ) whenẑ o ∼ 1, which corresponds to the maximal possible value of A y onceN y is fixed. Then it decreases asẑ o grows. Asymptotically in limit for z o ≫ 1 (but still such thatẑ 2 o N y ), it behaves as ∼ 1/ẑ o , as it is clear from the fact that the function F S , which is Fourier Transformed in Eq. (99), does not depend on any parameter. However, it should be noted that ∆θ y must be multiplied byẑ o in order to obtain the correlation length, which means that this remains constant and comparable with the diffraction length cL w /ω aŝ z o increases. Correlation length in the x direction is governed instead by the Gaussian exponential function in ∆θ x outside the integral sign in Eq. (99), exactly as in the simplified model treated in Section 4.1.
We performed some numerical calculation with the aim of giving the reader an exemplification. We set ∆θ x = 0 andN y = 10. Assuming that we are in the asymptotic limitD ≪ 1 we can rely on what has been said in this paragraph to calculate g and on Eq. (63) to calculate the correlation length ξ cy in the vertical direction for any value ofẑ o . These numerical results must agree with the Van Cittert-Zernike limit forÂ y ≪ 1 treated in paragraph 5.1.1 (A): in fact, Eq. (86) and Eq. (63) yield immediately a linear dependence of ξ cy on z o asÂ y ≪ 1. Then, using paragraph 5. 1.1 (B) we can extend the function ξ cy (ẑ o ) to all values ofẑ o . In this way we obtain ξ cy (ẑ o ) for every value ofẑ o . In Fig. 16 we plot Eq. (98) for two particular values ofÂ y :Â y = 10 (solid line) andÂ y = 1 (dashed line). Note that the solid line Fig. 16 is still similar to Fig. 15, although the tails are changing and the width is already smaller.
The following step is to calculate the Fourier Transform of F S according to Eq. (99) after normalization procedure according to Eq. (100). This gives g calculated for ∆θ x = 0. Fig. 18 illustrates | g | forẑ o = 1/2 as a function of z o ∆θ y at ∆θ x = 0. Note that for this parameter choice we are still in the limit forÂ y ≫ 1, which explains why Fig. 18 is practically identical to the universal plot Fig. 17.
One can calculate the coherence lengthξ cy (ẑ o ) straightforwardly by means of Eq. (63). The curve obtained can be then compared, in the limit forÂ y ≪ 1, with the van Cittert-Zernike behavior illustrated in Paragraph 5.1.2 (A). Fig.  19 shows our results. The black circles represent actual numerical calculations. The asymptotic limit forÂ y ≪ 1 (VCZ theorem) is shown with a dotted line. WhenN y ≫ 1 andD y ≪ 1, the coherence lengthξ cy (ẑ o ) can be calculated with the approximated formulâ Under the approximation of negligibly small electron beam divergence in the vertical direction, the normalized coherence length is thus a universal function of one dimensionless parameter,Â y . On the one hand, Eq. (101) accounts for the asymptotic behavior asÂ y ≪ 1 (VCZ theorem). On the other hand, the value ofξ cy in Eq. (101) approaches the constant valueξ cy −→ a for asymptotically large values ofÂ y . Beside accurately reproducing the asymptotes for small and large values ofÂ y , Eq. (101) provides an accuracy of several per cent with respect to the result of numerical calculations (whenÂ y is within the limits (0, ∞)), in the whole range of the parameterÂ y . The solid line in Fig. 19 is calculated with the approximated formula (101), where calculated a numerically using Eq. (91) to calculateξ cy and obtaining a ≃ 1.12. Therefore, according to Eq. (101), whenN y ≫ 1 andD y ≪ 1, we haveξ cy = const = 1.12 at the undulator exit (with accuracyD y ≪ 1 and 1/N y ≪ 1) and, in dimensional units, ξ cy = 1.12 L w c/ω. In this case the coherence length is a function of undulator length and wavelength due to the intrinsic divergence of the undulator radiation and a = 1.12 is a universal constant.

Case with finite divergence parameterD y .
We will first discuss, in Paragraph 5.1.3 (A), the limit forÂ y ≪ 1: in this case, whatever the value ofD y , we will recover the VCZ theorem. Note that, although we are discussing the limitN y ≫ 1 there always be values ofẑ o large enough so thatÂ y ≪ 1. Further on, in Paragraph 5.1.3 (B), we will discuss the caseÂ y ≫ 1: note that, since we are discussing the limit forN y ≫ 1, this will always be the case near the exit of the undulator atẑ o = 1/2. We will be particularly interested to this situation; in fact, the study of the caseÂ y ≫ 1 near the exit of the undulator will allow us to give an explicit representation of Eq. (75).
(A) Far zone caseÂ y ≪ 1. -Eq. (74) is still valid and can be calculated numerically, in principle, for any value ofẑ o . In the caseÂ y ≪ 1 it can be further simplified. In fact, in this situation, the second exponential function of the right hand side of Eq. (74) limits the possible values of ∆θ y to ∆θ y ≪ 1. Moreoverẑ o ≫ 1 so that Eq. (97) is valid in this case. In particular, remembering the definition of F S (φ y ) in Eq. (98) we havê It is not difficult to see that the ratio between the characteristic width in ∆θ y of the exponential function outside the integral sign and the exponential function in inside the integral sign in Eq. (102) is of order Â y /D y and it is always much smaller than unity unlessD y ≪ 1: such a case has already been treated before in Paragraph 5.1.1 (A), and has been shown to obey the VCZ theorem 5 . For all other values ofD y we have, automatically,D y ≫Â y , so that we can neglect the integral in dφ y and we get back once more the Van Cittert-Zernike regime. To sum up, we obtain the following expression for g, which is valid forÂ y ≪ 1 with no restrictions onD y : Calculation of the coherence length from Eq. (103) at ∆θ x = 0 gives once more the behavior which is consistent with the partial result in Paragraph 5.1.1 (A).
(B) Near zone,Â y ≫ 1. -Equation (97) is still valid in this case and following the same line of reasoning as paragraph 5.1.2 (B) one gets valid forÂ y ≫ 1 and arbitraryD y . Note that since we are working in the limit forN y ≫ 1, forẑ o = 1/2 we haveÂ y ≫ 1. We can see from Eq. (105) that, for finite values ofD y , the cross-spectral density g, evaluated at the exit of the undulator for ∆θ x = 0, is given by the product of the exponential function exp [−2D yẑ 2 o ∆θ 2 y ] with the function illustrated in Fig. 18. This remark is intuitively sound. SinceN y ≫ 1 in fact we have weakly quasi-homogeneous wavefronts near the exit of the undulator, and we can use the "anti" VCZ theorem to conclude that g must for a Fourier couple with the intensity distribution in the far zone. This will simply be, for any arbitraryD y , a convolution between a Gaussian distribution with rms width equal to D y and I S , which is the angular distribution of radiant intensity for a beam withǫ x −→ ∞ and ǫ y −→ 0. Finally, the Fourier transform of such a convolution between two function is simply given by the product of the Fourier transforms of the two functions.
Within the assumptionN y ≫ 1, we have seen that ifD y ≫ 1 Eq. (80) is valid with relative accuracy 1/D y . This means that, in this limit,D eff (D y ) =D y . Then we have seen that ifD y ≪ 1 Eq. (101) holds with accuracy 1/N y . This means thatD eff (0) = 2.50. Moreover we have seen that in the far zone case, in the limit forÂ y ≪ 1, the VCZ theorem holds, in agreement with Eq. (104).
We are now in position to calculateD eff = f (D y ) for any value ofD y . Evaluation of Eq. (105) atẑ o = 1/2, followed by normalization according tô G(ẑ o , 0, 0) = 1 gives the function g. Further integration of |g| 2 to calculate the correlation function allows to recoverD eff = f (D y ) as plotted in Fig. 20.
One may choose to calculateD eff = f (D y ) numerically, but it is also possible to use, with reasonable accuracy, the following analytical interpolation ofD eff : There is of some interest to compare the exact and interpolatedD eff (D y ) functions. Fig. 21 shows the function There is seen to be good agreement between the interpolated and exactD eff functions for small and large value ofD y . Noticeable discrepancies forD y close to unity are, anyway, less than 10%.
Our conclusive result is, therefore, the following: whenN y ≫ 1 andD y assumes arbitrary values we have: Also, it is important to remember that under conditionsN y ≫ 1 and arbitrarŷ D y the spectral degree of coherence is given by Eq. (105) in the near zone, and by Eq. (103) in the far zone.
The final step is to check that our main work hypothesis, i.e. that the coherence length has the form in Eq. (106) is correct. This can be done comparing Eq. (109) with numerical calculations for any given value ofN y ≫ 1 and a finitê D y , which give a good agreement.

Very large divergenceD y ≫ 1, arbitrary Fresnel numberN y
We now move on to treat the case with arbitrary beam transverse size compared with the diffraction size (i.e. arbitrary Fresnel numberN y ) and large divergence compared with the diffraction angle (i.e.D y ≫ 1). The particular case forN y ≫ 1 andD y ≫ 1 overlaps with the previous Section 5. 1 and has already been treated in Section 5.1.1. The conclusion was that ξ cy = (π/D y + π/Â y ) 1/2 . In all the other remaining cases Eq. (74) can be simplified as follows:Ĝ that is wherẽ As is shown in Appendix C, having defined we have the important result for every choice ofẑ o . β is defined in such a way to be normalized to unity. If we account for Eq. (114) we obtain the following expression for the spectral degree of coherence g: Fig. 22. Plot of |β| as a function of ∆θ y . This universal plot illustrates the absolute value of the spectral degree of coherence |g| calculated for ∆θ x = 0 in the limit for N y ≪ 1 andD y ≫ 1.
It should be noted that, asN y ≫ 1, the width of the gaussian function in ∆θ y in Eq. (115) becomes much smaller than unity and the function β can be considered constant and drops out of the normalized expression for g. So, even if we did not analyze here the limit forN y ≫ 1 (we did it in Paragraph 5.1.1), we see that the limit of Eq. (115) forN y ≫ 1 restitutes the results found in Paragraph 5.1.1. Therefore we conclude that Eq. (115) is valid for any value ofN y .
As is shown in Appendix C, the function β(∆θ y ) can also be calculated as where Si indicates the sine integral function. The representation of β in Eq.
(116) is easier to deal with numerically, because it involves a one-dimensional integration only. Performing the integral, one can tabulate |β| to obtain the plot in Fig. 22. This is an universal plot. It should be noted that β(∆θ y ) is the spectral degree of coherence g calculated for ∆θ x = 0 in the limit for N y ≪ 1 andD y ≫ 1. Its generalization for arbitraryN y is given by Eq. (115). Using Eq. (115) and the tabulated values for the universal function β we can therefore calculate g numerically for any choice ofN y and subsequently, we can calculate the coherence length ξ cy (ẑ o ). For instance, in the particular casê N y = 10 and ∆θ x = 0 we obtain the simple linear behavior For a generic value ofN y , one can introduce an effective functionN eff (N y ) so that On the one hand, the functionN eff (N y ) can be computed numerically, similarly as we did for the particular caseN y = 10.N eff (N y ) is represented by the solid line in Fig. 23.
On the other hand, one may also use an interpolation forN eff (N y ). First, numerical calculations tell us that, in the particular caseN y −→ 0, we have Second, asD y ≫ 1 andN y ≫D y we have The simpler interpolated formula which satisfies both asymptotes is therefore: The interpolation ofN eff (N y ) is represented by black circles line in Fig. 23.
There is of some interest to compare the exact and interpolatedN eff (N y ) functions. Fig. 24 shows the function There is seen to be good agreement between the interpolated and exactN eff functions for small and large value ofN y . Noticeable discrepancies forN y close to unity are, anyway, less than 13%.
Since in the caseD y ≫ 1 andN y ≫ 1 we concluded that ξ cy = (π/D y + π/Â y ) 1/2 , we can formulate the hypothesis that, forD y ≫ 1 and generic value ofN y one has The final step is to check that such hypothesis is correct. This can be done comparing Eq. (123) with numerical calculations for any given value ofD y ≫ 1 and finiteN y , which give a good agreement.

Conditions for the source to be quasi-homogeneous
Up to this moment we discussed, for simplicity, the caseθ x =θ y = 0. We will now treat the generic case with arbitraryθ x andθ y . This discussion will reduce to the relation between the weakly quasi-homogeneous condition and our assumptionsN x ≫ 1,D x ≫ 1 and eitherN y ≫ 1 orD y ≫ 1. When we deal with weakly quasi-homogeneous wavefront, the results found forθ x = θ y = 0 have extended validity for generic values ofθ x andθ y . As already said before though, we will find that the wavefronts are not always weakly quasihomogeneous in the vertical y direction. In this case, previously found results are only valid in the particular caseθ y = 0. Let us consider the situation in more detail. We will start with Eq. (73), that may also be written aŝ Because of the properties of the S functionφ y can only change of a quantity ∆φ y ∼ 1, otherwise the S functions will drop to zero. Then, sinceD y ≫ 1 we have from Eq. (124) that is obviously weakly quasi-homogeneous.
(A) CaseÂ y ≫ 1.-In the caseÂ y ≫ 1 we can follow the same line of reasoning in Section 5.3.1 simply replacing the roles ofD y withÂ y , obtaininĝ that is obviously weakly quasi-homogeneous.
(B) CaseÂ y ≪ 1.-In this situation it must beẑ o ≫ 1. Therefore we can substitute all S functions with sinc functions. The caseD y ≫ 1 has been already treated. Let us, therefore, first assumeD y ∼ 1. Using the fact that that is obviously weakly quasi-homogeneous. Now let us consider the caseD y ≪ 1. Eq. (124) can be simplified on the assumptionsÂ y ≪ 1 andD y ≪ 1. In fact, the Gaussian exponential factor inside the integral in dφ y in Eq. (124) imposes a maximal value (θ y +φ y ) 2 ∼ A y +D y ≪ 1.
Simultaneously, from the Gaussian exponential factor in ∆θ y outside the integral, we have a condition for the maximal value of ∆θ 2 y ∼ 1/ẑ 2 o ≪ 1 since As a result, the dependence of S on (φ + ∆θ y ) 2 can be substituted by a dependence onθ y and, as has already been said, the S functions can be substituted with sinc functions, givinĝ The integral in dφ y can be performed givinĝ NormalizingĜ in such a way thatĜ(ẑ o ,θ x ,θ y , 0, 0) = 1 we obtain the following expression for the spectral degree of coherence g: which generalizes Eq. (85) for any value ofθ x andθ y , and shows weak quasihomogeneity of the wavefronts.

Discussion.
Let us discuss the results obtained in this analysis of quasi-homogeneity.
In Section 4.1 we have seen that weakly quasi-homogeneous wavefronts which are not quasi-homogeneous in the usual sense are present in the far field, when the VCZ theorem holds. From our analysis in the one-dimensional framework, the notion of far zone arises in the x direction, when the apparent angular dimensionÂ x of the source is much smaller than the divergence of the radiation beam that, in this case, can be identified with the electron beam divergencê D x ≫ 1. BothN x ≫ 1 andD x ≫ 1. IfÂ x ≪D x the wavefront is quasihomogenous but only in the weak sense, one is in the far zone and the VCZ theorem applies. IfÂ x ≫D x one is in the near field zone and the wavefront is quasi-homogeneous in the usual sense. Transition from the near to the far zone always involves, in this case, weakly quasi-homogeneous wavefront.
In the two-dimensional framework studied in the present Section, as for the one-dimensional model, bothN x andD x ≫ 1. The x and the y coordinates appear factorized and the far zone applies now separately to both x and y directions, meaning thatÂ x,y of the source is much smaller than the divergence of the radiation beam. In other words, we are in the far zone as soon as the (square of the) beam size at a givenẑ o begins to be much larger than the (square of the) initial size of radiation (i.e. much larger than the Fresnel numbersN x,y ).
In the caseD y ≫ 1, independently on the value ofN y , we will always have weakly quasi-homogeneous (but not always quasi-homogeneous in the usual sense!) wavefronts at any distanceẑ o . This is exactly the situation discussed for the x direction, where transitions from the near (Â y ≫D y ) to the far field (Â y ≪D y ) involve only weakly quasi-homogeneous wavefronts. Moreover, in the far field zone, the VCZ theorem is valid.
In the case withD y 1,N y ≫ 1 we will have quasi-homogeneous wavefronts in the usual sense atẑ o ∼ 1 (near zone) and weak quasi-homogeneous wavefronts in the far zone, atẑ o ≫ 1.
Finally, ifN y ≫ 1,D y 1 andÂ y ∼ 1, i.e. for distancesẑ o ∼ N y , we have seen that the wavefront is non quasi-homogenous, not even in the weak sense.
These results can be seen in terms of convolution between the Gaussian distribution of intensity associated to the electron beam emittance and the distribution of intensity due to intrinsic properties of undulator radiation. The only case, among those treated up to now, when such a convolution does not simplify into the product of separate factors is whenN y ≫ 1,D y not much larger than unity andÂ y ∼ 1: in this case we do not have quasi-homogeneous wavefronts, not even in the weak sense.

6
Radiation from some non-homogeneous undulator sources In Section 4.1 we treated a simplified situation withN x ≫ 1,D x ≫ 1 and θ y1 =θ y2 . Moreover, for simplicity of calculations we assumedN y ≪ 1 and D y ≪ 1. In Section 5, instead, we treated the case of electron beams witĥ N x ≫ 1,D x ≫ 1 and eitherN y ≫ 1 orD y ≫ 1 (or both). It is important to note that, in the x direction, the results obtained in Section 4.1 are the same as the one in Section 5. In fact asN x ≫ 1 andD x ≫ 1, the cross-spectral density factorizes in the product of two contributions depending separately on the x and y coordinates, and underN x ≫ 1 andD x ≫ 1 the derivation of the x-dependent factor is always the same.
We have seen that in some of the cases discussed in Section 5.3, the assump-tionsN x ≫ 1,D x ≫ 1 and eitherN y ≫ 1 orD y ≫ 1 (or both) were enough to guarantee that the wavefront is weakly quasi-homogeneous in the sense specified by Eq. (23). In this Section we will extend our analytical investigations to some cases outside the range of parameters treated before, where the weakly quasi-homogeneous assumption is not fulfilled in the far zone. In particular, we will demonstrate that, under conditionsN x ≫ 1,D x ≫ 1 and bothN y ≪ 1 andD y ≪ 1, wavefronts are not weakly quasi-homogenous in the vertical y direction (although they are in the horizontal x direction).
First, in Section 6.1 we will analyze the caseN x ∼ 1,D x ≪ 1. Assuming a vertical emittance of the electron beam much smaller than the horizontal emittanceǫ y ≪ǫ x , we have, automaticallyN y ≪ 1 andD y ≪ 1. This corresponds to a practically important situation. For instance, consider a VUV beamline at a third generation light source with λ = 30 nm, ǫ x = 3 · 10 −9 m, and ǫ y = 0.03 · 10 −9 m, i.e.ǫ x = 0.6 andǫ y = 6 · 10 −3 . Ifβ x = 3 we would haveN x = 2 andD x = 0.2. Second, in Section 6.2 we will study the situation N y 1 andD y ≪ 1 withN x ≫ 1 andD x ≫ 1, that will give us back also the limiting case forN y ≪ 1 andD y ≪ 1 withN x ≫ 1 andD x ≫ 1 (already discussed in Section 4.1). The situation with finite vertical Fresnel number and negligible vertical divergence (compared with the diffraction angle) is a very practical one: for instance, given a third generation light source with λ = 1Å and ǫ y = 10 −11 m, i.e.ǫ y = 0.6, a valueβ y = 6 corresponds toD y = 0.1 and N y = 3.6.
Although in these two cases, the weakly quasi-homogeneous assumption is not fulfilled we will see that the choiceẑ o ≫ 1 will allow us to treat these situations in analogy with respect to some weakly quasi-homogeneous case we already dealt with.
Assuming a vertical emittance of the ring much smaller than the horizontal emittanceǫ y ≪ǫ x , we have, automatically,N y ≪ 1 andD y ≪ 1. We start with Eq. (67), that can be specialized to an equation dependent on the x coordinates only and, in the case forẑ o ≫ 1, can be written aŝ where we systematically omitted x subscripts. SinceÂ +D ≪ 1, the exponential factor in (φ +θ) 2 inside the integral sign in Eq. (133) behaves like a δ-Dirac function with respect to the sinc(·) functions inside the same integral: as a result we can substitute φ withθ in the sinc(·) functions, which drop out of the integral sign. Then, the integral in dφ can be calculated analytically so that we have: Combination of the second and the third exponential function yields Finally, in order to obtain the degree of spectral coherence g we should nor-malizeĜ according to Eq. (66) 6 thus obtaining According to Eq. (136) the spectral degree of coherence g is such that |g| is only function of ∆θ. Moreover, the dependence of g on the phase 2θ∆θẑ o is a feature for radiation from Schell's model sources in the far field, that we have already encountered many times in the study of weakly quasi-homogenous cases. As a result we can conclude that the radiation of the undulator source atN x ∼ 1 andD x ≪ 1 represents the far field radiation of a Schell's model source. Finally, it should be noted that in a two-pinhole experiment, for any vertical position of the pinholes, the fringe visibility, i.e. the modulus of the spectral degree of coherence depends only on the separation along the horizontal x direction. As a result, while in Section 4.1 we putθ y1 =θ y2 , thus selecting from the very beginning a horizontal plane, the present case can be fully described by a one-dimensional model, independently on the choice of transverse coordinates of the pinholes.
6.2 CaseN y 1 andD y ≪ 1 withN x ≫ 1,D x ≫ 1 In this situation we go back to a two-dimensional model. Whenẑ o ≫ 1 we haveÂ y ≪ 1, which is the limiting case treated in Paragraph 5.1.2 (A): the difference is that, nowN y ∼ 1. Let us start with Eq. (73) written aŝ A y ≪ 1 andD y ≪ 1 impose a maximal value of (θ y +φ y ) 2 ∼Â y +D y ≪ 1. Moreover the S functions can be substituted with sinc functions since we are working in the limit forẑ o ≫ 1. Then, from Eq. (137) we obtain As done before, the integral in dφ y can be performed givinĝ that isĜ Finally, normalization according to Eq. (66) yields Obviously |g| is a function of bothθ y and ∆θ, so that in this case we have neither weak quasi-homogeneity, neither wavefronts which can be described by Schell's model. However it should be noted that the integrals in Eq. (141) do not contain any parametric dependence.
It is interesting to have some final comment on Eq. (141). In the limit for N y ≪ 1, we recover the case discussed in Section 4.1, the only difference being that we did not setθ y1 =θ y2 = 0: in Section 4.1 we chose to deal with a one-dimensional model putting ourselves on the horizontal plane. Now, Eq. (141) allows to study the full two-dimensional situation in the limit for N y ≪ 1. In this case we see that the exponential function exp [−2Â yẑ 2 o ∆θ 2 y ] can be neglected because we have a maximum value of ∆θ y ∼ 1. As a result we obtain: where It should be noted that Eq. (143) does not depend on parameters and is, in fact, a universal function. Besides a geometrical factor exp [i2θ yẑo ∆θ y ], the function χ represents the spectral degree of coherence in the vertical direction, once the horizontal coordinates are fixed. The fact that it is a universal function means that even in the case of zero vertical emittance we never have full coherence in the vertical direction. On the one hand, this phenomenon can be seen to be an influence of the presence of horizontal emittance on the vertical coherence properties of the photon beam, as the integral in dφ x in χ comes from an integration over the horizontal electron beam distribution. On the other hand, being χ a universal function, the influence of the horizontal emittance on the vertical coherence does not depend, in the limit forN x ≫ 1 andD x ≫ 1, on the actual values ofN x andD x . It is straightforward to see that χ is symmetric with respect to ∆θ y and with respect to the exchange of ∆θ y withθ y . When θ y = 0, i.e.θ y1 = −θ y2 , we obviously obtain χ(0, ∆θ y ) = 1 that corresponds to complete coherence. In Fig. 25 we plot the three-dimensional representation of χ(θ y , ∆θ y ). In order to get a feeling for the behavior of χ we also plot, in Fig. 26 and Fig. 27, two cuts of Fig. 25 illustrating, respectively, the behavior of χ for a fixedθ y = 0.5 (or fixed ∆θ y = 0.5) and atθ y = ∆θ y .
As it is evident from Fig. 25, χ exhibits, for any fixed value of ∆θ y , many different zeros inθ y . In Fig. 28 we illustrate some of these zeros as a function of ∆θ y ,θ y,Z (∆θ y ). The interest of this plot is that, once a certain distancê z o ∆θ y between two pinholes is fixed, it illustrates at what position of the pinhole system,θ y,Z , the spectral degree of coherence drops from unity to zero for the first time.
It is interesting to compare Fig. 28, with the directivity diagram of the radiant intensity I S (θ y ). This comparison is shown in Fig. 29. In the limit forN y ≪ 1 andD y ≪ 1, one may increase the degree of coherence of the beam by spatially filtering the radiation in the far field. If a vertical slit is used with aperture d much larger than the horizontal coherence length, i.e. d ≫ξ cx , one would have poor coherence. Decreasing the aperture of the slit will increase the coherence of the X-ray beam up to some value d smaller thanξ c . Within our assumption N x ≫ 1 one has the far field approximationξ cx = (π/N x ) 1/2ẑ o . When d becomes smaller and smaller with respect to (π/N x ) 1/2ẑ o the spectral degree of coherence g can be identified with the universal function χ, as once can see by inspecting Eq. (142). As a result, as d becomes smaller one loses photons, but the X-ray beam transverse coherence ceases to improve because, as is seen in Fig. 29, the transverse degree of coherence g = χ drops to zero along the vertical radiation pattern of the filtered X-ray beam: for instance, from Fig.  29 one can see that χ drops to zero for the first time at ∆θ ∼ 1θ y ∼ 2, where the X-ray flux is still intense. This behavior of the degree of coherence should be taken into account at the stage of planning experiments. To give an example, after spatial filtering, one may conduct a two-pinhole experiment (like the one illustrated in Fig. 7) and find, surprisingly, that for some vertical positionθ y of the pinholes (at fixed ∆θ y ) well within the radiation pattern diagram he will have no fringes, but for some other vertical position he can find perfect visibility. So, without the knowledge of the function χ a user would not even have the possibility to predict the outcomes of a simple two-pinhole experiment.
From the definitions of χ, β, γ and I S it can be seen that all universal functions introduced in this work are partial cases of the more generic The knowledge of M is all one needs to calculate coherence properties out of many experimental setups, in very practical situations. It is therefore worth to tabulate M. We present a 3D plot of M, obtained from such tabulation, in Fig. 30.
Finally, it is interesting to sum up and compare results for the far field region obtained in this Section (non-homogeneous undulator source) with results obtained in Section 5. Many users performing coherent experiments with X-ray beams are interested in the beam coherence properties in the far field. We have seen that in the most general situation for third generation light sources one is interested in the caseN x ≫ 1 andD x ≫ 1, which guarantees factorization of results in the x and y direction, with arbitraryN y andD y . In this case we will not have, in general, weakly quasi-homogeneous radiation in the vertical direction. The spectral degree of coherence can be found by simplifying Eq. (73) in the mathematical limitẑ o −→ ∞: We can see that for any value ofN y andD y , in the far field limit we obtain a contribution to the cross-spectral density for the x and for the y direction. The contribution for the y direction can be expressed in terms of the product of an exponential function and convolutions between the (Gaussian) electron beam divergence and universal functions.
On the one hand, asD y ≪ 1 we have On the other hand, asD y ≫ 1 we have weakly quasi-homogeneous wavefronts and where β(∆θ y ) is given in Eq. (116).
Moreover, asN y ≫ 1, and for arbitraryD y we have: It should be noted that Eq. (149) is simply a consequence of the application of the VCZ theorem in both horizontal and vertical directions.

7
Application: Coherent X-ray beam expander scheme In this Section we show how transverse coherence properties of an X-ray beam can be manipulated to obtain a larger coherent spot-size on a sample.
The idea of increasing the horizontal width of the coherence spot is based on the use of a downstream slit for selection of the transversely coherent fraction of undulator radiation. Imagine a slit very close to the exit of the undulator with an aperture d comparable with the coherent length of the radiation at the exit of the undulatorξ cx = π/D x , as illustrated in Fig. 31.
The new radiation source after the slit is now coherent and characterized by a horizontal dimension of the light spot equal to π/D x . In the far field one can take advantage of the reciprocal width relations of Fourier transform pairs or, equivalently, the expression for the Fraunhoffer diffraction pattern from a slit, i.e. a sinc function, to calculate the magnitude of the coherence spot. There is, of course, some arbitrary convention to agree upon when it comes to the definition of the width of the sinc function but, numerical factors aside, this reasoning shows qualitatively that the coherence spot is of order D xẑo which isǫ x times larger than the spot size dimension in the case of freespace propagation, of orderẑ o / N x . The radiation beyond the slit must be then spectrally filtered by a monochromator (not shown in Fig. 31) to further narrow the spectral bandwidth. Here we assume that the radiation frequency ω is equal to the fundamental frequency ω o . The radiation beyond the slit is transversely coherent when the aperture d is equal (at most) to the coherence length ξ cx .
Let us present a numerical example illustrating the improvement of the horizontal coherence length obtained by slit application. Let us consider the case when the electron horizontal emittance is largeǫ x ≫ 1 and the vertical emittance is smallǫ y ≪ 1, withN x ≫D x . SinceN x =ǫ xβx andD x =ǫ x /β x , this is the case, for instance whenβ x ≃ 10. Then, forǫ x = 100 we haveN x = 1000 andD x = 10. This particular numerical example has been considered in Section 4.1 to illustrate, in free space, the behavior of the coherence length as a function of the position along the beamline, given in Eq. (64). Suppose we install the slit atẑ o = 2. From Eq. (64) we haveξ cx|ẑo=2 ∼ π/D x ∼ 0.6. Atẑ o = 12 we have a situation close to the asymptotic behavior where the field diffracted by the slit can be treated in the Fraunhofer approximation and ξ cx|ẑo=12 ∼ D xẑo ∼ 30. Comparison of the asymptotic behaviors after spatial filtering with respect to the free-space propagation case is given in Fig. 32. Note that only the asymptotic behaviors near the slit and at large distancê z o ≫ 1 are plotted for the spatially filtered radiation. What is important is, in fact, the comparison between the coherent distance at large values ofẑ o with and without spatial filtering in the near zone.
This is an example in which evolution of transverse coherence through the beam line plays an important role. In fact, the ability of spatially filter radiation by a slit requires the knowledge of the transverse coherence length variation along the beamline.
Let us calculate the number of coherent photons observed beyond the aperture.
In the region of parameters whereN x ≫ 1 andD x ≫ 1, the number of transversely coherent photons into the slit aperture d = ξ cx can be calculated as In the near-zone limit the slit is positioned at a position down the beamline z s ≃ β x so that, from Eq. (64) and Eq. (50) we have: and Therefore we can write At the opposite extreme, with the distance z s much larger than β x we find that, always from Eq. (64) and Eq. (50) and Therefore we can write Thus, the number of transversely coherent photons into the slit aperture d = ξ cx will be independent of the distance z s . This means that operation of spatial filtering in the near field region, as proposed by us, will not diminish the number of coherent photons with respect to the usual practice in which spatial filtering to obtain coherent radiation is performed in the far field: its effect will only be that of increasing the horizontal dimension of the coherent spot size. This possibility to create transversely coherent radiation with large divergence in the horizontal direction is important for many experiments. The distance of the slit from the exit of the undulator sets a limit to achievable linear dimension of coherence area. If we build a coherent X-ray beam line, we want to have large linear dimension of coherence area at the specimen position. Therefore, in order to have a largest linear dimension of coherence area at the specimen position ξ cx (z o ) ≃ σ x ′ z o we must have a slit aperture of at most of the size d ≃ λ/(2πσ x ′ ) installed in the near zone at z s ≃ β x . In order not to loose coherent photons instead, the slit aperture must be at least of the size d ≃ λ/(2πσ x ′ ). The right compromise is thus a slit aperture of the size d ≃ λ/(2πσ x ′ ) installed in the near zone at z s ≃ β x .

Conclusions
Before this work, no satisfactory theory describing spatial coherence from undulator radiation sources has been built. In this paper we developed such a theory of transverse coherence dealing with X-ray beams, with particular attention to third generation light sources.
First we studied Synchrotron Radiation as a random statistical process using the language of Statistical Optics. Statistical Optics developed around Gaussian, stationary processes characterized by quasi-homogeneous sources; under these assumptions, the characterization of statistical properties of the process are greatly simplified and the van Cittert-Zernike theorem (or its generalized version) can be used in order to describe the X-ray beam partial coherence properties in the far field region. However, for Synchrotron Radiation, there is no a priori reason to hold these assumptions satisfied.
We showed that Synchrotron Radiation is a Gaussian random process. As a result, statistical properties of Synchrotron Radiation are described satisfactory by second-order field correlation functions. We used a frequency domain analysis to describe them from a mathematical viewpoint. This choice is very natural. In fact, up-to-date detectors are limited to about 100 ps time resolution: therefore, in real-life experiments with third generation light sources, detectors are by no means able to resolve a single X-ray pulse in time domain and work, instead, by counting the number of photons at a certain frequency over an integration time longer than the radiation pulse. As a consequence of the frequency domain analysis we could study the spatial correlation for a given frequency content using the cross-spectral density of the system which, independently of the spectral correlation function, can be used to extract useful information even if the process is not stationary.
We gave an expression for the process cross-spectral density dependent on six dimensionless parameters. Subsequently we tuned parameters at perfect resonance, thus obtaining a simplified expression.
First we studied the limit of applicability of the quasi-homogeneous model from an analytical viewpoint, within the framework of simplifying assumptions, namely in the limit of small electron beam divergence and Fresnel number in the vertical direction, of large electron beam divergence and Fresnel number in the horizontal direction, and performing calculations for the crossspectral density on the horizontal plane only. This simplified study allowed us to introduce the concept of weakly quasi-homogeneous radiation and virtual quasi-homogeneous source while discussing the applicability region of the VCZ theorem.
Second, we studied the effect of the vertical emittance on the cross-spectral density. This study led us to analyze both cases in which the sources are weakly quasi-homogeneous and cases when they are not quasi-homogeneous at all. In the limit for large horizontal beam divergence and Fresnel number, which is always satisfied for third generation light sources, we found that the spectral degree of coherence factorizes in the product of factors depending separately on the horizontal and on the vertical coordinates. In the far field limit the vertical part of the spectral degree of coherence can be expressed in terms of the product of an exponential function (which, alone, would simply satisfy the VCZ theorem) and convolutions between the electron beam divergence in the vertical direction and a universal functions, that we introduced in our work. The universality of such a function implied that even for zero vertical emittance we never have full coherence in the vertical direction. This unexpected result is due to the influence of the horizontal emittance on the vertical coherence properties of the photon beam. Because of this, the degree of coherence changes between zero and unity within the diffraction angle. We also studied the near field zone. When one is interested in the evolution of the degree of coherence along the beamline back up to the exit of the undulator the situation becomes much more complicated with respect to the far zone case, as the observation distance is one of the problem parameters. There are many more asymptotic situations which can be studied, and a large part of our paper is devoted to the calculation of these asymptotic situations. We provided approximate estimations for the vertical coherence length that are valid from the far zone and back, up to the exit of the undulator in the case when either the vertical Fresnel number or the vertical electron beam divergence are much larger than unity. These can be used at the stage of planning experiments.
It should be noted that, throughout this work, we did not discuss the accuracy of the approximation of small and large parameters. In order to do so, one needs to develop a perturbation theory for each asymptotic case studied here, which would considerably increase the size of this paper. As a result we leave this issue for future work.
Finally, we selected an application to show the power of our approach. We discussed how the transverse coherence properties of an X-ray beam can be manipulated to obtain a convenient coherent spot-size on the sample with the help of a simple vertical slit; this invention was predicted almost entirely on the basis of theoretical ideas of rather complex and abstract nature discussed in the previous parts of the paper.
We have straightforwardlȳ because all averages over trigonometric functions are zero.
Finally, we show that ri = 0. In fact ri = 1 N N k,n=1 | α k || α n | cos φ k cos φ n cos ψ k sin ψ n +| α k || α n | cos φ k sin φ n cos ψ k cos ψ n −| α k || α n | sin φ k cos φ n sin ψ k sin ψ n −| α k || α n | sin φ k sin φ n sin ψ k cos ψ n = 1 2N k |α k | 2 (cos φ k sin φ k − sin φ k cos φ k ) = 0 . (161) As a result we have that real and imaginary parts have zero means, equal variances and are uncorrelated. Use of the central limit theorem allows to conclude that the resulting phasor sum is a circular complex Gaussian random variable.
where we changed variables from ∆θ x,y to ∆θ ′ x,y = 2∆θ x,y Then we can apply the autocorrelation theorem in Eq. (178) to the functionf thus obtaining the following relation: where α x,y are now conjugated variables with respect to the anglesφ x,y on which S depends. We will denote withS the two-dimensional Fourier Transform of S, that is: The relation between the function S and the undulator field is given by Eq. Now, on the one hand in the limit forẑ o −→ ∞ the functionf transforms into β, because the S functions inf tend asymptotically to the sinc functions in β. On the other hand, if the Fourier transform off is independent ofẑ o , alsof is independent ofẑ o . As a result it can only bef (∆θ ′ y ) = β(∆θ ′ y ), and f (∆θ y ) = β(∆θ y ) that is Eq. (175) holds, quantum erat demonstrandum.
Note that, based on the autocorrelation theorem, it is also possible to give an analytic expression for the Fourier transform of β. After definition of β = β(ẑ o , ∆θ ′ x , ∆θ ′ y ) as in Eq. (179), application of the autocorrelation theorem simply states that the two-dimensional Fourier Transform of β, that will be indicated withβ can be written as Introducing α 2 = α 2 x +α 2 y and representing the two-dimensional Fourier Transform of β in terms of Fourier-Bessel transform we obtain where Si indicates the sine integral function.