research papers
Numerical analysis of
and coherent of segmented undulator radiation based on statistical optics^{a}Engineering and System Science, National Tsing Hua University, No. 101, Section 2, Guangfu Road, East District, Hsinchu City 30013, Taiwan, ^{b}National Synchrotron Radiation Research Center, 101 HsinAnn Road, Hsinchu Science Park, Hsinchu City 30076, Taiwan, and ^{c}Electrophysics, National Chiao Tung University, 1001 University Road, Hsinchu City 30010, Taiwan
^{*}Correspondence email: luo.hw@nsrrc.org.tw, cshwang@nsrrc.org.tw
A method based on wave optics together with electron tracking is used to analyze synchrotron radiation from a segmented undulator in a double or multi miniβ function lattice storage ring. Radiation and transverse coherence features are investigated, where the former is calculated with the Wigner distribution function and the latter is evaluated by integrating the and crossspectral density to exhibit the coherent and overall degree of coherence. To be specific, radiation properties for a single undulator in a typically single miniβ function and a tandem undulator in a double miniβ_{y} lattice are compared in this work. As a result, both, and coherent can be enhanced by a second tandem undulator at the Taiwan Photon Source.
Keywords: Wigner distribution function; brilliance; transverse coherence; double miniβ_{y} lattice; segment undulator.
1. Introduction
There has been significant progress in generating highbrilliant and transverse coherent synchrotron radiation (SR) in storage rings by reducing the electron beam emittance (Eriksson et al., 2016; Willeke, 2015; Kuo et al., 2015) and by installing very long undulators (Hara et al., 2002; Yamamoto et al., 2014). The first method has been encouraged because of progress in synchrotron technologies, such as lattice design, magnets, and the injection schemes. Upgrade projects or new facilities are proposed or are under construction. The second approach mainly relies on insertion device (ID) technology. Even though a continuoussegment ID in a very long straight section is desired, it is convenient to separate a long ID into shorter segments for a number of practical reasons. First, there is the electron lifetime degradation due to with residual gas molecules and the function of β_{y} in the narrow gap of an undulator, which pushes for a shorter length of the undulator. Second, several essential components requiring space, such as vacuum pumps, beam diagnostic elements, steering and quadrupole magnets, must be installed for normal beam operation along the ID.
The technologies of segmented undulators have been applied at SPring8 (Yamamoto et al., 2014), ESRF (Chavanne et al., 1996) and BESSY II (Reichardt et al., 2001). At ESRF, the measured angular from two 1.6 mlong segments compared with a single segment increases by a factor of 3.1 at 3.5 keV and 2.3 at 20 keV (Onuki & Elleaume, 2003). The angular being less than the ideal factor of four, is photonenergydependent and depends on the emittance and energy spread of the electron beam and on perfect matching of the electron and photon phases in the transverse phase space. To observe the interference effect, it is necessary to install a monochromator to improve the spectral resolution. In general, the phase matching in the transverse phase space decreases with increasing photon energy. The phase matching of segmented undulator radiation is a sensitive factor but has never been described rigorously in terms of photonenergydependent degradation of SR. Here we pose one question: how to improve the SR for high photon energies? A straightforward method is to increase the overlapping region of the two transverse phase spaces by minimizing the βfunction at the location of each segmented undulator (Chiu et al., 2010) (see Fig. 1). Installation of quadrupoles, however, raises two flags. First, the electron beam deflection and displacement introduced by the quadrupoles should be kept small to preserve the spatial overlap of the photon beams emitted from both segments. Second, the path length of the electron beam should be adjusted so as to retain temporal matching between radiation fields emitted from both segments. The first case relates to the electron beam quality and beam dynamics control (Baron et al., 2016), yet our work will focus on the SR performance for a perfect transverse alignment. Due to the fact that quadrupoles will affect phase matching, we leave most of the electron deflection issues to the qualitative discussion in Section 5.
The SR et al., 2015). A typical evaluation of the SR emitted from an electron beam passing through an undulator is to apply a convolution of the Wigner distribution function (WDF) of the photon beam from a single electron and the electron bunch beams (Bahrdt, 1997). In the case of a large number of undulator periods, the Gaussian approximation on the photon beam is obtained by comparison with the angular distribution of intensity (a sinc function) (Lindberg & Kim, 2015). The approximation facilitates the convolution process, because of the Gaussian distribution of the electron beam in a storage ring, and gives a simplified expression of the The Gaussian approximation, however, is not appropriate for the case of segmented undulators in a double miniβ function lattice, due to interference effects of the SR generated in each segment. The angular distribution of intensity generated by an electron passing through segmented undulators depends on the incident position and angle of the electron, corresponding to different optical phases. A nonGaussian profile is therefore needed to evaluate the of a segmented undulator. However, it should also be noted that the nonGaussian profile is important if the electron emittance is comparable with the wavelength of the photon beam.
is a figure of merit for a light source, indicating how well SR can be focused (GeloniThe SR wave character becomes noticeable for small electron beam emittances, especially for a diffractionlimited light source. The SR properties in a lowemittance ring have therefore attracted many investigations (Geloni et al., 2008; Bazarov, 2012; Tanaka, 2014). A typical method is to compute the WDF directly from the electron beam providing a general way to analyse the wave properties of SR without a Gaussian approximation (Kim, 1986). It is appropriate to evaluate the SR for segmented undulators. The onaxis WDF is chosen as the definition of the throughout this paper.
The transverse coherence F_{coh} is therefore signified by the multiplication of the and the minimum value λ^{2}/4 of the phase space area in the transverse directions, where λ is the radiation wavelength (Kim, 1989). To be specific, F_{coh} as well as the degree of coherence in segmented undulator radiation must be evaluated by considering interference effects.
of SR is enhanced with an increase of the SR The coherentA typical method to determine the degree of coherence of the SR is to integrate the et al., 2008). Specifically, the VCZ theorem can be applied to analyse the coherent properties of quasihomogeneous light, such as thermal light, but not to SR from thirdgeneration light sources with a very small emittance in the vertical direction. The condition of the in statistical optics means that the intensity distribution of the light source varies slowly compared with the width of the coherence (Goodman, 2015). The SR intensity distribution of the light source can be described by the electron beam size σ_{x/y} in the horizontal and vertical direction. The coherent length is determined by the angular divergence of the electron beam σ_{x′/y′} and given by λ/(2σ_{x′/y′}), which can be evaluated from equalizing the phase term of the electric field in equation (30) of Geloni et al. (2008) to be λ/2. The quasihomogeneous condition for a suitable application of the VCZ theorem is therefore the electron beam emittance ∊_{x/y} > λ/2. In the case of the Taiwan Photon Source (TPS) (energy and natural emittance are 3 GeV and 1.6 nm rad, respectively), the vertical emittance ∊_{y} is around 1.6 pm rad, for a coupling constant of 0.001. The homogeneous condition is valid only if the photon energy is higher than 400 keV. To investigate the coherent properties of the desired photon energy at 0.4–20 keV, the direct integral of the crossspectral density (CSD) is used to determine the overall degree of coherence instead of applying the VCZ theorem in this work.
over a coherent area defined by the van Cittert–Zernike (VCZ) theorem. However, the VCZ theorem was studied and proved to be unsuitable in the near diffractionlimited region (GeloniTypical methods to determine the ) and transverse coherence properties (i.e. the VCZ theorem) for segmented undulator radiation are incorrect. The magnets between segments, e.g. quadrupoles and phase shifter, change the focus of the electron beam as well as the trajectories of all electrons, leading to an overburdened evaluation. In this work, numerical analyses were performed to show a complete investigation on both properties for two arrangements, i.e. single and segmented undulators. We organize the paper as follows. In Section 2, the theoretical background for undulator radiation is given. It includes a review and discussion on the definition and properties of WDF and transverse coherence. The corresponding issues of a segmented undulator are then given in Section 3. Several practical issues such as the effects of phase shifter and quadrupoles are discussed. In Section 4, numerical results for the and transverse coherence in segmented undulator radiation at the TPS are shown. In Section 5, we discuss two issues affecting SR from the virtual source point to the endstation, i.e. electron beam displacement and focusing the beam at the sample point. The conclusion is given in Section 6.
(Wiedemann, 20072. Theoretical background of undulator radiation
The WDF was invented by Wigner (Wigner, 1932) to deal with a quantummechanicsbased thermodynamic problem. Walther noticed the relation between the WDF and the radiance and coherence properties of optics (Walther, 1968). Then the concept of WDF was introduced by Coisson & Walker (1985) and Kim (1986) to the SR community about thirty years ago and is based on a phasespace method in wave optics. It follows the same transport rule as the geometrical optics and contains all the information of secondorder correlation of the electric field. The WDF is indeed a fourdimensional distribution which cannot be plotted on paper or on a computer screen. But the twodimensional projection of it provides a visual way to deal with the distribution of light in phase space although it is not a real photon distribution in phase space as in geometrical optics. The onaxis WDF contains information on the transverse coherence of two points placed symmetrically about the optical axis (Lindberg & Kim, 2015). However, the use of twodimensional apertures to define the with coherent features is what the user needs. So, a reasonable parameter to define photon beam coherence should contain all information in the twodimensional area. Although the onaxis WDF contains coherent information for a twodimensional region, it still does not contain all coherent information because the information contained therein is only from symmetrically placed pairs of points. So, it is not easy to obtain the overall degree of coherence by using the WDF. For simplicity, the rootmean square of the degree of coherence is calculated by integrating the crossspectral density in the whole area of the aperture. Note that the coherent that passes through the aperture is in general not equal to the well known definition of coherent (F_{coh}) that is the multiplication of (B) and minimum phase space volume (λ^{2}/4) (Kim, 1989).
2.1. Wigner distribution function for brilliance
The Wigner distribution function for SR is defined by (Coisson & Walker, 1986; Kim, 1986)
where r = (x, y) defines the coordinates in any transverse plane and θ = (θ_{x}, θ_{y}) is the angle coordinate between observer and undulator axis, E and ∊ are the position and angular representation of the electric field in the frequency domain, respectively, k and λ are the corresponding wavenumber and wavelength, respectively. The angle brackets represent the ensemble average (incoherent summation) of the radiation that is generated by all electrons in the beam. Since the electron bunch length is much longer than the wavelength of interest and there is no microbunch structure in the TPS, the WDF of the electron beam is treated as an incoherent summation of the radiation from each electron as described by Geloni et al. (2015) and Bazarov (2012).
Several possible definitions of the ) and the authors chose the onaxis WDF W(0, 0) as the definition of the in their paper, that is
through the WDF were discussed by Bazarov (2012which is, in general, the maximum value of the WDF for onaxis undulator radiation thereby also satisfying the definition given by Geloni et al. (2015). The physical meaning of can be regarded as the maximum concentration of the as discussed by Geloni et al. (2015). In addition, in the limit of geometric optics (the emittance of the electron beam is much larger than the diffractionlimited photon beam emittance), this definition of is the maximum in the defined phase space (Geloni et al., 2015).
The distribution of x − θ_{x} or y − θ_{y} planes (Bazarov, 2012),
and in the transverse phase space could be obtained by projecting the fourdimensional WDF on theThese equations can be solved in a simplified form (Bazarov, 2012) by
As an example, a threedimensional and contour map of the WDF for undulator radiation is shown in Fig. 2. In the contour map of Fig. 2, the electric field ∊_{k} of the first harmonic of EPU48 radiation can be obtained (Lindberg & Kim, 2015) from equation (5) at the midpoint of the EPU48. The EPU48 is an elliptically polarizing undulator whose period length is 48 mm. Another undulator that will be discussed in this paper is the invacuum undulator IU22 whose period length is 22 mm. Both the EPU48 and IU22 are operated in horizontal linear mode throughout this paper,
where [JJ] = J_{0}[K^{2}/(4+2K^{2})]  J_{1}[K^{2}/(4+2K^{2})], k_{r} = , = , k_{r} is the wavenumber at resonant energy, and N_{u}, λ_{r}, K and λ_{u} are the period number, resonant wavelength, deflection parameter of the undulator and period length, respectively. The paraxial equation of the electric field ∊_{k}(θ;z = 0) with nonzero initial coordinates of electrons (x_{j}, x_{j}′, η_{j}, t_{j}) is expressed by equation (6) (Lindberg & Kim, 2015). The terms x_{j}, x_{j}′, η_{j} and t_{j} represent the position, angle, energy deviation and time shift, respectively. η_{j} is not directly shown in equations (5) and (6) but it changes the γ term to γ(1 + η_{j}) in the wavenumber at resonant energy in equation (5),
For high odd harmonic radiation, the profile of the radiation electric field ∊(θ_{x}, θ_{y}) (that is the angular terms of the sincfunction) should be modified by Bessel's functions if the number of undulator periods is not large. For the IU22 at TPS, in our experience, the term is needed to be modified for the equation of reference (Wiedemann, 2007) with the replacement of θcosφ with θ_{x} and omission of the γ^{2}θ^{2} terms. The resulting modified equation is
where C_{1} = h[K^{2}/(4+2K^{2})], C_{2} = , C_{3} = and h is the harmonic number. The electric field in the modified equation [equation (7)] was used to calculate the for comparison with SPECTRA results (Tanaka & Kitamura, 2001) to make sure that this formula is avaliable.
2.2. Overall degree of coherence
The cross spectral density Γ with contributions from any two points on a transverse plane with longitudinal coordinate z is defined as (Mandel & Wolf, 1995)
The angle brackets show the ensemble average of the radiation from the entire electron beam. The degree of coherence is 0 ≤ γ_{12}(x_{1}, x_{2}; z) ≤ 1 with
where I is the intensity. The degree of coherence is directly related to the visibility V of interference fringes in Young's interference experiment (Geloni et al., 2008) in equation (10),
For a finite radiation distribution, the overall degree of coherence γ (Luis, 2007) or the coherence mode purity is obtained by integrating the product of intensity and degree of coherence from the entire photon source size,
where γ is the rootmeansquare value of the degree of coherence by integrating any twopoint pairs over the whole aperture region. As the radiation passes through an ideal aperture, the electric field is set to zero outside the aperture. Then γ presents the averaged coherence information of the that passes through the aperture to reach the experimental sample.
The well known VCZ theorem is usually used to estimate the coherent area (the size of the aperture) of SR and thus the coherent et al., 2008). In fact, the derivation of the VCZ theorem is based on the assumption of an incoherent source (Mandel & Wolf, 1995). Instead, the degree of coherence in this work is obtained by direct calculation of the crossspectral density based on the Monte Carlo method. To compare the coherent performance of different configurations of the undulator in different lattices, the coherent is the through an aperture which is defined by equation (12), the socalled coherent length l_{cx,y},
However, it is invalid in the neardiffractionlimited region due to the nonhomogeneous coherence feature of the undulator radiation (Geloniwhere Σ_{x, y} and λz are the effective source size and the photon wavelength, respectively, and
Here, σ_{r} means the size of the photon beam of a single electron. For simplicity, equation (12) is still based on the VCZ theorem following the design report for the TPS beamline (Huang et al., 2015). However, the degree of coherence for each possible configuration is not exactly the same since the aperture size which is defined by the VCZ theorem is indeed unavailable and the coherence properties are different for each configuration. To avoid a biased comparison between single and double undulator radiation, we need to check the difference of the overall degree of coherence to make sure that the degrees of coherence are close to each other for each configuration.
3. Issues of a tandem undulator
Quadrupoles between two collinear undulators focus offaxis electrons and thus change the electron path lengths passing the two midpoints of each undulator as shown in Fig. 3. This effect, which depends on the initial conditions of the electron trajectory at the entrance of the upstream undulator, will result in a phase difference between the two radiation beams that are generated in each of the undulators. In addition, a phase shifter is installed between two undulators to tune the path length difference. Another issue is the fraction of electrons that effectively contribute to the [this is somewhat related to the core emittance as discussed by Bazarov (2012)]. For a single undulator, the onaxis is calculated by the convolution of the WDF of the radiation generated by a single electron and the electron beam distribution in phase space. The calculation method of the WDF for a tandem undulator in a double miniβ_{y} lattice is different from that of a single undulator. In the simulation of a tandem undulator, the trajectories of each electron need to be calculated to know the phase difference of the radiation from two undulators since the quadrupole magnets change the path length difference. In addition, the radiation from two undulators needs to be propagated to the same transverse plane to obtain the total radiation field. These processes do not exist in the calculation of the radiation properties of a single undulator.
3.1. Phase shifter effect
An electron beam with zero emittance and energy spread is used to determine the effect of the phase shifter in a double undulator structure. To calculate the electric field ∊ of a segmented undulator (∊_{total}), the Fresnel diffraction formula equation (14) (Kim, 1986) is used to propagate the radiation field from a virtual source point in each undulator to a common transverse plane (e.g. midpoint between two undulators),
where ∊_{1}(θ; z_{up}) and ∊_{2}(θ; z_{down}) are the electric fields for each undulator [equation (6)] at their virtual source points z_{up/down} and t_{2} − t_{1} represents the time delay for the electron transport from the virtual source point in the upstream undulator to the downstream undulator. l_{1} and l_{2} are the distances from each virtual source point to the common transverse plane, and the diffraction formula is the propagation of the electric field in the angular space with paraxial approximation. It is the same as the integral of Fresnel diffraction in position space.
For example, we consider a single electron passing through the two undulators (each of length L_{u}), the drift space (d) and a phase shifter (with phase ξ) between undulators as shown in Fig. 4. The total electric field is described by equation (16),
Defining as the average longitudinal velocity of the electron in the undulator, then
where
For simplicity, the phase difference term , generated by the travel time of the electron, is merged into the phase ξ for every case throughout this paper. We substitute the electric field of each undulator (∊_{1} and ∊_{2}) by the same symbol ∊ as in
where C_{4} = . The total electric field becomes then
The
from a single particle in a segmented undulator can be expressed byand its numerical . The oscillation of the normalized depends on the phase difference which represents the variation of the total spectral at the observation point. The twodimensional projection of WDF at the centre between the two undulators (i.e. at the central quadrupole) from a single particle is plotted in Fig. 6.
results are shown in Fig. 53.2. Quadrupole magnet effect
The quadrupole magnets between two undulators kick the offaxis electrons and change their trajectories. The corresponding phase difference caused by the quadrupoles between the undulators due to the path length difference is
where ΔL_{in/out of undulator} = L_{offaxis} − L_{onaxis} is the path length difference of the offaxis and onaxis electron that is calculated in or out of the undulator. The phases of the offaxis electrons are the phase of the onaxis electron plus the result of equation (22). The relation of the phase difference between the offaxis electron and the onaxis electron to the electron position in phase space at the middle of the upstream undulator is plotted in Fig. 7. In this figure the vertical emittance and energy spread are assumed to be zero for simplicity. Once the position and velocity of an electron at the upstream undulator are known, the phase difference due to the quadrupole magnets can be obtained. The horizontal and vertical scales of Fig. 7 are ±σ_{x} and ± σ_{x′} of the horizontal electron beam size and angular divergence, respectively, at the upstream undulator in the double miniβ_{y} lattice. It seems that only a small fraction of electrons in the electron beam are in the inphase region (white region in Fig. 7) in which the phase of the offaxis electron is close to that of the onaxis electron. So the phase relation between undulators cannot be preserved even for a soft Xray (a few hundred eV) photon beam as shown in Fig. 7(a). But the above description of the phase relation for is not true. Only the paraxial electrons in the area of the wavelength in the horizontal phase space can effectively contribute to the onaxis Wigner function or because the emittance of the photon beam of a single electron is at the scale of the wavelength. For 3 m EPU48 at 220 eV, the photon distributions in phase space are on the scales of σ_{r} = 15 µm and σ_{r′} = 29 µrad, respectively, where σ_{r′} is the angular divergence of the photon beam of a single electron. Comparing this photon distribution with the inphase region in Fig. 7(a), it can be seen that most of the electrons that can effectively contribute to the are in the inphase region; and the electrons with various phase differences outside the inphase region do not contribute to the The phase relation is therefore preserved for the of the soft Xray photon beam under the influence of quadrupole magnets.
Fig. 7(b) shows that the higher the photon energy, the smaller the inphase region in phase space. It seems that the interference effect should be less significant for a highenergy photon since fewer electrons preserve the phase relation. However, the emittance of the photon beam of a single electron is inversely proportional to the photon energy. Therefore, the phase relation could be preserved for the hard Xray photon beam (a few keV) if the reduction of the inphase region is slower than the emittance of the photon beam as the photon beam increases. It turns out that, from numerical calculation, the inphase region is also almost inversely proportional to the photon energy for the horizontal phase space of the double miniβ_{y} lattice in the TPS. As a result, the interference efficiency of double undulators on the is almost independent of the photon energy under the influence of quadrupole magnets, since the ratio of the inphase region and the emittance of the photon beam is almost kept the same for different photon energies. Nevertheless, considering the contribution of the phase difference from the energy spread of the electron beam, the phase relation decreases as the photon energy increases. The energy spread changes the speed of each electron and causes a phase difference. This effect plays an important role when the photon energy is sufficiently high. For the TPS, the energy spread destroys the phase relation for a photon beam of energy a few keV.
3.3. Electron bunch effect on the onaxis WDF
There are two sources of β_{y} lattice. The first one is the amplitude of the electric field which determines the number of photons generated by a single electron being directly proportional to the total length of the undulator. The second one is the matching of the electron and photon beam in the two transverse phase spaces. Comparing Figs. 2(b) and 6(c) with the coherence relation, the angular divergence of the photon beam is smaller in the double undulator than in the single undulator configuration. This means that more electrons will contribute to the for a tandem undulator than a single undulator if σ_{x} ≫ σ_{r} and σ_{r′} ≫ σ_{x′}. This conclusion also applies to single undulators of different lengths. If, on the other hand, two undulators are considered as two independent sources (i.e. incoherence relation), the phasespace matching will take place at the virtual source points of each undulator in the double miniβ_{y} lattice. In this case, the phasespace matching in the double miniβ_{y} lattice is more effective than for a single miniβ_{y} for σ_{y} ≫ σ_{r} and σ_{r′} ≫ σ_{y′}. For these two reasons, the depends nonlinearly on the total length of the tandem undulator.
gain from a tandem undulator in a double mini4. Numerical analysis results using WDF
In this section, the numerical analysis results of both . The electric field generated by two undulators is calculated by equations (5)–(7) and propagated through equation (14) to the centre of the straight section in order to calculate the The propagated electric fields of each undulator are combined together to form the total electric field of the segmented undulator. To calculate the the electric field of each electron is used to find the onaxis Wigner function [equation (2)] and sum up incoherently to obtain the onaxis of the electron beam. For the about 250000 and 1000000 particles are tracked in the soft and hard Xray regions, respectively. First, the numerical results of the normalized as a function of the phase delay of the phase shifter are shown in Fig. 8. Figs. 9 and 10 then show the for other possible undulator configurations of EPU48 and IU22 at the TPS that are listed in Table 1. Five configurations are studied to understand the characteristics of the and coherent a single 3 mlong undulator in (i) a 7 m straight section (3M7MS), (ii) a 12 m straight section (3M12MS), and (iii) one 6 mlong undulator in a 12 m straight section (6M12MS), (iv) a 3 mlong undulator downstream of a 12 m double miniβ_{y} section (3MDMBYS) and finally (v) two 3 mlong undulators in a 12 m double miniβ_{y} section (6MDMBYS). The ratios of and for double and single undulators are calculated to see whether a double undulator can generate more brilliant light and provide more coherent than a single undulator. In all these configurations, the aperture size for the tandem undulators is chosen to be the same as the value for a single downstream undulator. For the numerical results of the the effect of a phase shifter is significant and is more important in the soft Xray than hard Xray region as shown in Fig. 8. The can vary by 2.9 times the value of the different phase in the soft Xray region but there is no difference in the hard Xray region. This is due to more significant interference effects for the tandem undulator in the original phase space for soft Xrays which is not the case in the hard Xray region.
and coherent are shown and discussed. In order to understand the effect on the and coherent due to emittance and energy spread in the tandem undulator, a Monte Carlo simulation with WDF is used. In the simulation, the initial transverse position, velocity and energy of electrons are generated by a Gaussian distribution with the standard values equal to the electron beam size, electron beam divergence and energy spread of the electron beam, respectively, at the centre of the upstream undulator. The distribution in the transverse position–velocity phase space is an inclined Gaussian distribution according to the betatron function. Then the matrix method is used to calculate the trajectories of the electrons from the upstream to the downstream undulator without consideration of the focusing effect of the undulator. The lengths of the trajectories of the offaxis electrons are compared with those of the onaxis electrons to obtain the phase difference in equation (22)

4.1. Undulator configuration in the collinear case
Fig. 10 shows a comparison of the in the soft Xray (220–1600 eV) and in the hard Xray region (5.5–9.3 keV) for different undulator configurations. All values from different configurations are normalized to one 3 m undulator downstream of the double miniβ_{y} section (3MDMBYS). The vertical electron beam size and angular divergence of 3M7MS are close to those of 6MDMBYS and 3MDMBYS. Meanwhile, the horizontal electron beam size and angular divergence of 3M12MS are close to those of 6M12MS, 6MDMBYS and 3MDMBYS. It can be easily understood that the from 3M7MS is higher than that from 3M12MS and 3MDMBYS in the soft Xray region. Meanwhile, the 3MDMBYS and 3M7MS values are higher than that from 3M12MS in the hard Xray region. This is because in the soft Xray region (e.g. σ_{r} and σ_{r′} are 15 µm and 29 µrad at 220 eV) all 3 m undulator cases satisfy the relations σ_{x} ≫ σ_{r}, σ_{x′} ≃ σ_{r′}, and . The variations of σ_{y} and σ_{y′} do not affect the significantly since the vertical size divergences are both dominated by the photon beam. The effect on from σ_{x} is direct, since it dominated the horizontal size, but not so for σ_{x′}, since σ_{x′} does not dominate the horizontal divergence and is even smaller than σ_{r′}. The effect of decreasing σ_{x} is larger than that of increasing σ_{x′}. Considering that the product of σ_{x} and σ_{x′} is almost constant for each configuration, it is better to have a small σ_{x} rather than a small σ_{x′} in the soft Xray region. On the other hand, the brilliances of 3MDMBYS and 3M7MS are higher than from 3M12MS in the hard Xray region. For the hard Xray region (e.g. σ_{r} and σ_{r′} are 3 µm and 6 µrad at 5.5 keV), the beam size and angular divergence relations are σ_{x} ≫ σ_{r}, σ_{x′} > σ_{r′}, σ_{y} ≃ σ_{r} and σ_{y′} < σ_{r′}. The horizontal electron beam size and divergence are not important in terms of influencing the since both σ_{x} and σ_{x′} are larger than σ_{r} and σ_{r′}, respectively. It is more important to have a small σ_{y} than a small σ_{y′} because σ_{r′} dominates the vertical divergence while neither σ_{y} nor σ_{r} dominate the vertical size.
Now, a comparison of the σ_{y} is smaller in the former configuration, as described in Section 3.3.
in the 12 m sections of 6M12MS, 3MDMBYS and 6MDMBYS is made and the results are shown in Fig. 10. The of 6MDMBYS is about 2.5 and 2 times higher compared with 3MDMBYS in the soft Xray region and hard Xray region, respectively. This result shows that an interference exists between the radiation pulses from the two undulators in the soft Xray region but not in the hard Xray region. The of 6MDMBYS is higher than that of 6M12MS. This is due to the fact that the phasespace matching between electron and photon beam in the vertical phase space is better for 6MDMBYS than for 6M12MS sinceNext, we consider the et al. (2015), the sample position is located 59 m away from the centre of the straight section and the aperture size (i.e. the coherence length) is determined by this distance. For 3MDMBYS, the longitudinal position z in equation (12) is chosen as 59 m minus the distance between the middle of the straight section and the middle of the downstream undulator, and the effective beam sizes Σ_{x,y} are the values at the virtual source point in the downstream undulator. For the case of 6MDMBYS, the coherent area (i.e. the aperture) is chosen to be the same as that of 3MDMBYS for simplicity. Unlike the effect of the phase shifter on the coherent shown in Fig. 11, is not significant even in the soft Xray region. This is because more electrons contribute to the coherent than the The coherent is defined as the that passes through the aperture. Since the aperture includes the optical axis, all the electrons that have a nonzero onaxis contribute to the coherent However, since the photon beam divergence is larger than the electron beam divergence in the horizontal axis for the soft Xray region, the radiation from almost all the electrons would contribute to the onaxis and thus to the coherent The distribution of the electron beam is larger (especially in the horizontal size) than the inphase region in Fig. 7(a), so the photon beam with any phase contributes to the coherent and thus the phase relation disappears. As a result, the interference between the upstream and downstream undulators almost disappears and the integration in the aperture region is the incoherent summation for each undulator. The incoherent summation of 6MDMBYS is not double that of 3MDMBYS but only about 1.8 times (see Figs. 12 and 13) due to the distance difference of the virtual source points in the upstream and downstream undulator to the aperture location in the case of 6MDMBYS. The ratio of the coherent between 3MDMBYS and 6MDMBYS could be tuned by using a focusing mirror in the beamline, but a more detailed treatment of this problem is not part of this paper.
for the coherent radiation as requested from users in different configurations of the undulators. From HuangThese results show a different behaviour for ; Thompson et al., 2009) the coherent is directly proportional to the following Kim's equation (Kim, 1989),
and coherent for an electron beam of finite emittance and energy spread. In many articles (Bazarov, 2012where B and (λ/2)^{2} are the and minimum possible phase space area, respectively. The coherent is different for detailed calculation in the aperture area from the definition in equation (23). The coherent depends weakly on the phase shifter at any photon energy, but the is very sensitive to its settings in the soft Xray region. However, as mentioned previously, the determines how well SR can be focused. Therefore, the phase shifter can be used to enhance the of the tandem undulator in the soft Xray region.
4.2. Overall degree of coherence (ODOC) of the constraint in the aperture
The aperture size for collecting the coherent β_{y} lattice should be checked by numerical methods that define the degree of coherence [equation (9)].
of a tandem undulator is set to be the same as that of a single downstream undulator. We have not discussed the behaviour of the ODOC in these configurations so far. Whether the ODOC will be decreased in our case of a double miniTo calculate the ODOC of the electric field in the aperture area, a twodimensional area with equal space is defined in the aperture plane. The ODOC is also obtained by a Monte Carlo based simulation for the degree of coherence of all pairs of elements of the area. In the simulation, the electric field of the segmented undulator is propagated to the aperture, i.e. 59 m away from the centre. A threedimensional array of the electric field is generated, and each element E_{ijn} in the array represents the electric field that is contributed by the nth electron at a transverse position in the aperture corresponding to indices i and j for horizontal and vertical position, respectively. The cross spectral density of any two points in the aperture is the dot product of the corresponding vectors E_{ij} which contains the contribution of all the electrons at a transverse position. The dot products of all pairs of vectors E_{ij} in the array are summed up to obtain the numerator of equation (11) to obtain the ODOC. The result shows that there is no significant (at least lower than our simulation error) difference between 3MDMBYS and 6MDMBYS in both the soft and hard Xray region. The ODOCs of IU22 at 9.3 keV for 3MDMBYS and 6MDMBYS are both around 0.97. This is due to the fact that the distance between two undulators is much smaller than the distance between the aperture and downstream undulator.
5. Electron deviation between the tandem undulator and focusing effect
The discussion in the previous section considered ideal cases where both undulators share the same optical axis. However, the electron beam position may deviate from the ideal orbit in practical cases. Therefore, the effects of the electron trajectory displacement are analysed by a twodimensional projection of the WDF in phase space and the ODOC. The focusing problem for the
passing through the aperture of the tandem undulator is described briefly in this section.The deviation of the electron position in each undulator also affects the performance of the light source. Two examples are given to deal with this issue for both ) where the electron beam is displaced (without a kick angle) in the horizontal direction in the second undulator. If this beam displacement is larger than the size of the photon beam of the single electron, the is defined only by a single undulator. However, if the deviation is smaller than the electron beam size, there exist some electrons (the blue trajectory in Fig. 14) that pass through the upstream undulator that can also contribute to the photon coherence in the downstream undulator. These electrons contribute to the but there may be no interference at the origin of the phase space since the WDF of each electron in the two individual undulators are separated too far in phase space (see Fig. 15 at 220 eV) and the WDF of different electrons in the two undulators add incoherently. In this case, the phase shifter will be useless for Next, the ODOC is calculated for different electron displacements at an energy of 9.3 keV and is shown in Fig. 16 for a tandem undulator as a function of the electron displacement. The electron displacement is measured in units of the effective source size Σ_{x,y} for a single undulator. Fig. 16 shows that the ODOC is almost the same for displacements in both the horizontal and vertical plane. In order to maintain a reasonable and constant ODOC, the electron displacement should be well controlled within one unit of effective source size. It is much more difficult to keep the same ODOC in the vertical direction than in the horizontal since the coupling constant is very small and is less than 0.1%.
and coherent We consider a simple example (Fig. 14Here, we consider another case: Fig. 17 shows a case where the electron trajectory has an angle but still passes through the axis in the middle of the second undulator. Then the radiation generated in the second undulator may not contribute to the The results can be seen in the contour map of the WDF in Fig. 18 which shows that the distributions of the two WDFs in phase space are separated. Unless the electron beam displacement and angle are both larger than the electron beam size and angular divergence, the from the second undulator will not be significantly reduced. However, the phase shifter has still no effect on the if there exists a kick angle larger than the photon angular divergence.
If the electron deviation is well controlled at the level of the effective beam size Σ_{x,y} and divergence Σ_{x′,y′}, the second undulator can contribute to the The of a double undulators is at least two times larger than from a single undulator. If the electron path distortion is much smaller than the photon beam size and divergence, then the photon phase relation between the two undulators will be preserved and the gain could be more than twofold. But this condition is difficult to satisfy in the hard Xray region, since the electron path distortion needs to be controlled to less than a micrometre or microradian.
The
radiated from the upstream and downstream undulator and passing the aperture can be considered as the incoherent summation of the The tandem undulator can be considered as two independent sources at different longitudinal position. The focusing efficiency in the tandem undulator case will become worse, since the distance from the virtual source points of each undulator to the focusing elements are different. Therefore, the focal points for each undulator will be different. However, if the ODOC of the in the aperture is close to unity, the wavefront of the radiation from both undulators will be similar in the aperture and one focusing mirror behind the aperture can focus the radiation beam well in the tandem undulator structure. It could also be expected that the focusing efficiency decreases as the size of the aperture increases.6. Conclusions
In this paper, wave optics based on the WDF method are used to analyse the performance of the tandem undulator in the double miniβ_{y} section in the TPS. The of a tandem undulator can be more than twice that of a single undulator in the soft Xray region since the phase relations of paraxial electrons are preserved. However, the phase relation in the longitudinal axis cannot be kept in the hard Xray region. Numerical results show that the is the summation of the incoherent of two undulators in the hard Xray region. On the other hand, the phase relation is not important for obtaining the coherent that passes through the aperture defined in equation (12) even in the soft Xray region.
Phasespace matching of the electron and the photon beam diffractionlimited emittance in each undulator plays an important role in the double miniβ_{y} lattice especially when the tandem undulator is considered as being two independent sources. The analysis show that the performance of the tandem undulator in the double miniβ_{y} section is superior to a single undulator with the same number of periods in a single miniβ_{y} section. In this case, the performance especially will be more obvious in the hard Xray region.
The coherent i.e. the in the aperture) contribution from the upstream undulator of the tandem undulator is 80% of that of the downstream undulator, which could be improved by increasing the distance between the undulator and the aperture, or by focusing the two photon beams to the same spot size at the aperture location.
calculation shows that the ODOC is the same for the single and the double undulator when the distance from the undulator to the aperture is much larger than the distance between the tandem undulators. Meanwhile, the coherent (In summary, both β_{y} lattice, the phase shifter is important in the soft Xray region but is not necessary in the hard Xray region. In addition, any electron displacement between the tandem undulators must be carefully controlled. For the coherent the phase shifter is not important either in the soft Xray or in the hard Xray region. There are gains in and coherent in the TPS tandem undulator. However, there may not be a gain in other light source facilities depending on the accelerator lattice design and undulator parameters. The phase relation between undulators and phasespace matching of each undulator should be carefully evaluated while any transverse electron beam displacement needs to be taken into account to evaluate the gain performance.
and coherent can be enhanced by the second undulator of the tandem undulator at the TPS. To enhance the in the double miniAcknowledgements
The authors would like to thank Professor Helmut Wiedemann for providing good comments and correcting this article. The authors also thank Ms S. T. Chang for her opinions on programming computer code. NSRRC staff members Dr M. S. Chiu and Dr C. C. Kuo are commended for their TPS double miniβ_{y} lattice assistance.
Funding information
The following funding is acknowledged: Ministry of Science and Technology of Taiwan (contract No. MOST 1062112M213004).
References
Bahrdt, J. (1997). Appl. Opt. 36, 4367–4381. CrossRef PubMed CAS Web of Science Google Scholar
Baron, A. Q. R., Tanaka, T., Soutome, K., Takao, M., Nakamura, T., Kobayashi, K., Fujita, T., Takahashi, S., Aoyagi, H., Shimosaki, Y., Seike, T., Uchiyama, H., Ishikawa, D., Chuang, T.H., Kimura, H., Tanaka, H., Kitamura, H. & Ishikawa, T. (2016). AIP Conf. Proc. 1741, 020033. Google Scholar
Bazarov, I. (2012). Phys. Rev. ST Accel. Beams, 15, 050703. Web of Science CrossRef Google Scholar
Chavanne, J., Van Vaerenbergh, P. & Elleaume, P. (1996). J. Synchrotron Rad. 3, 93–96. CrossRef IUCr Journals Google Scholar
Chiu, M. S., Yang, C. H., Chao, H. C., Chang, H. P., Tsai, H. J. & Kuo, C. C. (2010). Proceedings of the First International Particle Accelerator Conference (IPAC2010), Kyoto, Japan, 23–28 May 2010, pp. 4581–4583. THPE030. Google Scholar
Chung, T. Y., Yang, C. S., Chu, Y. L., Lin, F. Y., Jan, J. C. & Hwang, C. S. (2017). Nucl. Instrum. Methods Phys. Res. A, 850, 72–77. CrossRef Google Scholar
Coisson, R. & Walker, R. P. (1985). International Conference on Insertion Devices for Synchrotron Sources, Stanford, CA, USA. (Proceedings published in 1986, SPIE, Vol. 0582, pp. 24–29.) Google Scholar
Eriksson, M., AlDmour, E., Andersson, Å., Johansson, M., Leemann, S. C., Malmgren, L., Tavares, P. F. & Thorin, S. (2016). Proceedings of the 7th International Particle Accelerator Conference (IPAC2016), 8–13 May 2016, Busan, Korea, pp. 11–15. MOYAA01. Google Scholar
Geloni, G., Kocharyan, V. & Saldin, E. (2015). J. Synchrotron Rad. 22, 288–316. CrossRef IUCr Journals Google Scholar
Geloni, G., Saldin, E., Schneidmiller, E. & Yurkov, M. (2008). Nucl. Instrum. Methods Phys. Res. A, 588, 463–493. Web of Science CrossRef CAS Google Scholar
Goodman, J. (2015). Statistical Optics, p. 201. New York: John Wiley and Sons. Google Scholar
Hara, T., Yabashi, M., Tanaka, T., Bizen, T., Goto, S., Maréchal, X., Seike, T., Tamasaku, K., Ishikawa, T. & Kitamura, H. (2002). Rev. Sci. Instrum. 73, 1125–1128. CrossRef Google Scholar
Huang, D.J., Chung, S.C., Jean, Y.C., Fung, H.S., Lin, H.J., Huang, Y.S., Lee, H.Y., Tang, M.T., Hsu, C.H., Lee, K., Pan, K.Y. & Su, H.R. (2015). Technical Design Report of the PhaseI TPS Beamlines, National Synchrotron Radiation Research Center, Hsinchu City, Taiwan. Google Scholar
Kim, K.J. (1986). Nucl. Instrum. Methods Phys. Res. A, 246, 71–76. CrossRef Web of Science Google Scholar
Kim, K.J. (1989). AIP Conf. Proc. 184, 565–632. CrossRef Google Scholar
Kuo, C. C., Chen, J. Y., Chiu, M. S., Chou, P. J., Liu, Y. C., Tsai, H. J., Tseng, F. H., Hsu, K. T., Luo, G. H. & Chen, C. T. (2015). Proceedings of the Sixth International Particle Accelerator Conference (IPAC2015), 3–8 May 2015, Richmond, VA, USA, pp. 1314–1318. TUXC3. Google Scholar
Lindberg, R. & Kim, K.J. (2015). Phys. Rev. ST Accel. Beams, 18, 090702. CrossRef Google Scholar
Luis, A. (2007). JEOS RP, 2, 07030. Google Scholar
Mandel, L. & Wolf, E. (1995). Optical Coherence and Quantum Optics, p. 188. Cambridge University Press. Google Scholar
Onuki, H. & Elleaume, P. (2003). Undulators, Wigglers and their Applications, pp. 192–193. London, New York: Taylor & Francis. Google Scholar
Reichardt, G., Bahrdt, J., Schmidt, J.S., Gudat, W., Ehresmann, A., MüllerAlbrecht, R., Molter, H., Schmoranzer, H., Martins, M., Schwentner, N. & Sasaki, S. (2001). Nucl. Instrum. Methods Phys. Res. A, 467–468, 462–465. CrossRef Google Scholar
Tanaka, T. (2014). Phys. Rev. ST Accel. Beams, 17, 060702. Web of Science CrossRef Google Scholar
Tanaka, T. & Kitamura, H. (2001). J. Synchrotron Rad. 8, 1221–1228. Web of Science CrossRef CAS IUCr Journals Google Scholar
Thompson, A., Vaughan, D., Kirz, J., Attwood, D., Gullikson, E., Howells, M., Kim, K.J., Scofield, J., Liu, Y., Kortright, J., Lindau, I., Pianetta, P., Robinson, A., Underwood, J., Williams, G. & Winick, H. (2009). Xray Data Booklet, https://xdb.lbl.gov/. Google Scholar
Walther, A. (1968). J. Opt. Soc. Am. 58, 1256–1259. CrossRef Google Scholar
Wiedemann, H. (2007). Particle Accelerator Physics. New York, Berlin, Heidelberg: Springer. Google Scholar
Wigner, E. (1932). Phys. Rev. 40, 749–759. CrossRef CAS Google Scholar
Willeke, F. (2015). Proceedings of the Sixth International Particle Accelerator Conference (IPAC2015), 3–8 May 2015, Richmond, VA, USA, pp. 11–16. MOYGB3. Google Scholar
Yamamoto, S., Senba, Y., Tanaka, T., Ohashi, H., Hirono, T., Kimura, H., Fujisawa, M., Miyawaki, J., Harasawa, A., Seike, T., Takahashi, S., Nariyama, N., Matsushita, T., Takeuchi, M., Ohata, T., Furukawa, Y., Takeshita, K., Goto, S., Harada, Y., Shin, S., Kitamura, H., Kakizaki, A., Oshima, M. & Matsuda, I. (2014). J. Synchrotron Rad. 21, 352–365. Web of Science CrossRef CAS IUCr Journals Google Scholar
© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.