research papers
Spatialfrequency features of radiation produced by a stepwise tapered undulator
^{a}Budker Institute of Nuclear Physics, Novosibirsk, Russia, ^{b}Novosibirsk State University, Novosibirsk, Russia, ^{c}European XFEL, Hamburg, Germany, and ^{d}Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
^{*}Correspondence email: andrei.trebushinin@xfel.eu
A scheme to generate widebandwidth radiation using a stepwise tapered undulator with a segmented structure is proposed. This magnetic field configuration allows to broaden the undulator harmonic spectrum by two orders of magnitude, providing 1 keV bandwidth with spectral ^{16} photons s^{−1} mm^{−2} (0.1% bandwidth)^{−1} at 5 keV on the sample. Such a magnetic setup is applicable to superconducting devices where magnetic tapering cannot be arranged mechanically. The resulting radiation with broadband spectrum and flattop shape may be exploited at a multipurpose beamline for scanning over the spectrum at time scales of 10–100 ms. The radiation from a segmented undulator is described analytically and derivations with numerical simulations are verified. In addition, a starttoend simulation of an optical beamline is performed and issues related to the longitudinally distributed radiation source and its image upon focusing on the sample are addressed.
exceeding 101. Introduction
Synchrotron radiation (SR) serves as a powerful tool for investigating materials with Xrays. High ). This requires fast scanning over 1 keV bandwidth at a subsecond time scale with a sufficient on the sample. In this work, we propose an advanced superconducting undulator scheme for the experiments where fast scanning over the spectrum along with micrometrescale focusing is required. We suggest that this source may be installed for future beamlines specialized in micro and nanoprobing, e.g. I18 at DIAMOND (Mosselmans et al., 2009), P06 and P11 at PETRA III (Schroer et al., 2010; Burkhardt et al., 2016) and ID13 at ESRF (Flot et al., 2010).
and photon energy tuning capabilities provide these sources uncontested benefits over laboratory Xray sources. SR sources facilitate spectroscopy techniques such as Xray absorption nearedge structure (XANES), as well as Xray diffraction techniques. Moreover, fast monochromators allow to speed up data acquisition, for example for extended Xray absorption fine structure (EXAFS) spectroscopy resulting in quickEXAFS (Frahm, 1988The number of periods in an undulator defines a resonance bandwidth for the nth harmonic, expressed as ≃ 1/nN, where denotes a resonant frequency and N stands for the number of periods. In absolute units, this bandwidth is only of the order of 10 eV for an undulator with 100 periods at the resonance of 1000 eV. One way to perform scanning experiments is to set the magnetic field of the undulator in a tracking mode and scan it together with a monochromator. Scanning, however, raises a challenging technical issue connected with onthefly magnetic field tuning, especially at subsecond scales. Another way consists of obtaining the radiation with a bandwidth that covers the desired scan range – about 1 keV – and performing the scan with a fast monochromator. The SR community already exploits undulator tapering around a given harmonic in order to provide the required bandwidth for the experiments. In the tapering mode the magnetic field gradually changes along an electron trajectory in the range from B_{min} to B_{max}. The resonance effectively widens satisfying the resonance conditions for all values of the magnetic field range. The tapering mode for undulators has been studied and used at SR sources (Walker, 1988; Bosco & Colson, 1983; Boyanov et al., 1994).
Here we propose to generate a magnetic field with a stepwise magnetic profile employing a segmented undulator. The radiation from the stepwise tapered magnetic field creates a series of consecutive resonances, summing up to a broad spectrum. Depending on the number of segments, this setup may provide 1 keV spectral bandwidth. The stepwise tapered technique is particularly suitable for setups utilizing superconducting undulators, where the undulator gap is fixed and the only way to create the field gradient is to change the value of the current in the superconducting coils.
In this work, we present our calculations with design parameters for a superconducting undulator project for the SR source SKIF (the Russian acronym for Siberian Circular Photon Source) in Novosibirsk. We design this superconducting undulator in collaboration with the undulator group in Budker Institute of Nuclear Physics (Shkaruba et al., 2018; Ivanyushenkov et al., 2015; Mezentsev & Perevedentsev, 2005).
2. Concept
We propose to divide an undulator with constant period length λ_{w} into n_{s} segments with N_{s} periods each. Thus, we create a multisegment structure as depicted in Fig. 1. With this we aim to obtain a sequence of overlapping resonances, resulting in a flattop spectrum with minimum ontop fluctuation.
Each segment of the superconducting undulator will be equipped with a separate power supply or correction coils (or auxiliary coils) to control the value of the magnetic field. The magnetic field varies as
where i_{s} = 0,…, n_{s} − 1 is the segment number, B_{0} and dB are, respectively, the field in the first segment and the field leap to the next segment in sequence. We propose to use two windings: a main one with the reference field B_{0} and an auxiliary one which shapes this stepwise magnetic structure with dB increments. Overall, the auxiliary coil in the last segment should provide the additional field n_{s}dB which is less than 20% of the main field for the third harmonic at the energy around 4.5 eV. The magnetic field sets the corresponding value of current in the auxiliary coils. Dividing the undulator into two windings seems to be more efficient in the sense of thermal stability of a cryostat and in terms of cost when one needs to reduce the number of highcurrent power suppliers.^{1}
Each segment has a resonance at the nth harmonic with relative bandwidth ≃ resulting in ≃ , where and are the resonance frequencies for the first harmonic and the nth harmonic. We found that to effectively generate a widebandwidth flattop spectrum the individual resonances of each segment should be shifted by their full width at halfmaximum (FWHM). We justify this criterion in Appendix B. Hence, the resulting combined undulator bandwidth for all odd harmonics becomes
It is worth noting that this undulator can also be used in uniform mode (dB = 0). So, this device is multifunctional.
3. Analytical description of the segmented undulator
In our derivation, we exploit the resonance approximation. This approximation requires a large number of periods (N_{s} ≫ 1). We also use a filament electron beam approximation in our analytical derivation. In other words, the emittance of the electron beam is much smaller than the `emittance' of radiation () and we neglect energy spread effects. These assumptions will be justified later on by comparison with numerical calculations. We present a list of the undulator parameters in Table 1.

We denote the total length and the period of the insertion device as L and λ_{w}, respectively. For the sake of clarity, we provide SKIF electron beam parameters in Table 2.

E and dE denote electron beam energy and its spread, respectively, while β_{x} and β_{y} are the (almost constant) horizontal and vertical beta functions in the straight section, and ε_{n} and κ refer to the natural emittance and the coupling factor, respectively.
3.1. Segmented undulator spectrum: theoretical model and its verification with a numerical simulation
3.1.1. Analytically calculated onaxis spectrum
We provide all derivations in a Gaussian unit system. The derivation of undulator radiation in the frequency domain may be found in textbooks on SR although we follow the approach and notation of Geloni et al. (2007). We derive the radiation distribution emitted by a single electron onaxis from a segmented undulator as a sum of the fields from n_{s} separate segments the centres of which are located at the distance z_{0}(i_{s}) from an observer,
where γ is the Lorentz factor, e is the c is the speed of light and k_{w} = 2π/λ_{w},
where J_{n} is a Bessel function of the first kind of order n. is the fundamental resonant frequency of the i_{s}th segment, L_{s} = N_{s}λ_{w}. is the undulator parameter of the i_{s} segment. is the slowly varying envelope of the Fourier transform of the electric field. Here we consider a fixed polarization component. We give an analytical expression for the radiation from a single segment in Appendix A. By taking the integral in equation (3) we obtain
The radiation from each segment has a resonance with a relative bandwidth ∼1/nN_{s} and shifted from the reference resonance by . Also, the radiation from each segment has a phase shift
Within the filament beam model, the resulting spectral
in units of number of photons per second per unit surface area per unit relative bandwidth is given bywhere ℏ = h/2π is Planck's constant. The spectral is depicted in practical units (i.e. in photons per second per squared millimetre per 0.1% relative bandwidth) in Fig. 2.
The blue solid line in Fig. 2 represents the effect of of the segments, while the black dashed line shows the shape of the spectrum without the interference term in equation (7).
Our analytical model is relatively rough as we exploited the filament beam approximation and the resonance approximation, assuming a large number of undulator periods, while here we only consider 18 periods per segments. Nevertheless, this approach shows the main effect of
that could impede implementing this scheme. In the following subsections, we provide a more thorough investigation of the stepwise tapered magnetic field configuration using a numerical simulations approach.3.1.2. Numerical simulations of the onaxis spectrum
We performed our simulation using a multiphysics software for numerical calculation of electron beam dynamics, freeelectron lasers and SR sources performance – OCELOT (Agapov et al., 2014, https://github.com/ocelotcollab). The OCELOT software allows to calculate SR from a filament beam in an arbitrary magnetic field and to propagate this radiation through an optical beamline.
We provide simulation results for the onaxis spectrum emitted in the segmented undulator, Fig. 3, with the device parameters listed in Table 1. We consider a range of energies from the Ti Kedge, 4.5 keV, up to the edges of heavier elements, 20 keV. As we showed in equation (2), it is possible to obtain 1 keV spectral bandwidth for all desired photon energies. The values of dB/B_{0} for these simulations are listed in Table 3. We crosschecked our OCELOT results with SPECTRA (Tanaka & Kitamura, 2001) that are depicted in Fig. 4. The red line shows the effect of the finite electron beam emittance and energy spread (SKIF electron beam parameters). One can see that assuming an electron beam of finite size results in a less modulated spectral shape without notable loss. Given that the modulated spectrum is undesired for users, we use the filament beam as a conservative approximation.

Though the final design of this undulator does not contain free space gaps or phase shifters between the segments (V. Shkaruba & N. Mezentzev, private communication), we provide further analysis of the spectral shape including the influence of free space gaps in Appendix B.
4. Wavefront propagation simulation: focusing
Beamlines for the energy range between 5 keV and 40 keV typically consist of an entrance aperture, a doublecrystal monochromator and a focusing system, e.g. a pair of Kirkpatrick–Baez mirrors. We present a simplified scheme of the SKIF beamline in Fig. 4. In this section we study imaging of the undulator radiation on the sample using the and radiation propagation modules of OCELOT. In particular, we model the third harmonic radiation of an intersectionless segmented undulator with optimized magnetic field increments. We consequently propagate the emitted radiation through the beamline up to the sample location. We will not account for the monochromator in our simulations as it does not affect the image formation process.
As discussed above, in our simulations we use the filament beam model; partially because calculations of radiation by a finite electron beam (Chubar et al., 2011) are computationally expensive, and partially by aspiration to show features related to the insertion device itself, on which we focus in this contribution. One can estimate the radiation beam size and divergence for a given electron beam on each optical element using raytracing codes, for example SHADOW (Sanchez del Rio et al., 2011).
4.1. Radiation on the sample in the case when no focusing is applied
In the case of no focusing, the transverse size of the radiation on the sample is about 1 mm FWHM after 25 m of freespace propagation. In this case the calculated spectral ^{14} photons s^{−1} mm^{−2} (0.1% bandwidth)^{−1}. Owing to the angular dependence of the undulator resonance condition, the spectral integrated over both transverse directions exhibits minor modulations, shown in Fig. 5.
is of the order of 2 × 104.2. The field distribution on the sample after focusing
In order to increase the spectral C. In fact, the longitudinal source scale in the image is reduced by the demagnification factor 1/D^{2} = b^{2}/a^{2}, where a is the distance from the source to the centre of the optical system and b is the distance from the centre of the optical system to the sample. The longitudinal offset is then in the order of centimetres. In Fig. 6 we plot the space–frequency distribution of the radiation on the sample. The onaxis is extremely dispersed across the energy range and the spot position varies longitudinally.
one may introduce focusing elements to concentrate the radiation on a smaller spot on the sample. When we form an image in the focal plane, we reconstruct a demagnified version of the object plane. However, the segmented undulator is a nonuniform source and has chromatic aberrations, which we discuss in AppendixAs we already noted, the segmented undulator is a longitudinally distributed source, therefore the radiation from each segment converges at different distances from the focusing element. To mitigate this effect we introduce an aperture with size 0.3 mm × 0.3 mm in the far zone of the undulator. The aperture increases the Rayleigh length of the image waist as well as the imaginary source, effectively reducing chromatic abberration, see Fig. 14 in Appendix C. Thus, the full spectrum of the radiation is uniformly concentrated in one transverse spot, Fig. 7.
The notable disadvantage of reducing the aperture size is a significant where we applied an aperture with a size of 2 mm × 2 mm (only 36% loss). In this simulation, we change the longitudinal position of the sample and place it at 1.1 m from the second mirror although, on the actual beamline, the same effect can be reached by changing the angle of incidence on the mirror.
loss (over 98%). Therefore, we propose to place the sample slightly out of focus after a much milder collimation with a wider aperture. In this case, we enhance the density converging the radiation on a smaller spot but we do not significantly lose photons on the aperture to reduce chromatic aberration. This is because we do not reimage the source and gain the benefits of spatialfrequency properties of radiation in the far zone. We present simulation results of the outoffocus image on Fig. 8Additionally, we present the transverse intensity distributions for the calculated fields in Fig. 9.
5. Discussion
Undulator tapering is a known method to broaden the spectrum of the undulator radiation (Nonaka et al., 2016; Caliebe et al., 2019) and it is carried out as a gradual incremental change of the K parameter over the undulator length. While we propose to implement a qualitatively different stepwise tapering, the resulting performance is very similar compared with that of the linear tapering, as illustrated in Fig. 10
Stepwise tapering, being easier to implement using an electromagnetic undulator, yields comparable amplitude of spectrum modulation. As already discussed above (see Fig. 3), the latter is further reduced due to the finite electron beam emittance.
6. Conclusion
In this contribution, we proposed a scheme for generating broadband undulator radiation and delivering it to the sample. The scheme relies on a segmented superconducting undulator where the undulator segments are detuned by the FWHM of their individual spectral bandwidth with respect to each other. This segmented structure emits partially overlapping narrowbandwidth fragments of the final spectrum. The resulting onaxis spectrum has a broadband modulated onaxis distribution, while transversely integrated radiation from a finite electron beam exhibits a smooth flattop spectrum shape. The spectral density is modulated due to interference between the segments. If undulator segments are spatially separated, they should be accompanied by phaseshifters. Numerical simulations indicate the tradeoff between the photon density on the sample and the flatness of the resulting broadband spectrum due to both modulated spectrum and a longitudinally distributed photon source. For the undulator with 11 segments and 18 periods in each segment, we obtained 1 keV spectral bandwidth with ^{14} photons s^{−1} (0.1% bandwidth)^{−1} for filament electron beam calculations. This contribution serves as a conceptual design for the beamline dedicated to microprobe experiments at SKIF.
exceeding 2 × 10APPENDIX A
Radiation of a single electron in a separate segment
The electric field emitted by a single electron moving in a planar undulator is linearly polarized within the resonance approximation, and its magnitude can be calculated as an integral over the particle trajectory in the undulator (for example, Onuki & Elleaume, 2003),
where = , γ is the Lorentz factor, e is the c is the speed of light and k_{w} = 2π/λ_{w},
with J_{n} the Bessel function of the first kind of order n. The field is calculated at an observation point z_{0}, θ = (x/z_{0}, y/z_{0}), z_{0} is measured from the undulator centre, located at z = 0. The resonance frequency is denoted as ω. The polarization lies in the e_{x} plane. The notation of the field representation and the coordinate system used in our derivation coincides with one established by Geloni et al. (2007). Taking the integral in equation (8) we obtain the usual expression for the field distribution from the single electron in the far zone,
We illustrate the corresponding intensity distribution as a function of photon energy and of the transverse coordinate x while y = 0, Fig. 11.
APPENDIX B
Phase advance in the drift between the segments
Heretofore, both analytical Fig. 2 and numerical Fig. 3 results assumed zero drift distance between the segments. However, if the undulator segments are separated by intersections of length L_{d}, an additional phase term appears in the field expression equation (5) (onaxis),
where = is the first harmonic resonant wavelength of the upstream undulator cell. The segments interfere and, if destructively phased, introduce fluctuations in the spectrum. In Fig. 12 we show the influence of the phase shifts between radiation pulses emitted in adjacent segments on the spectral shape.
One can compensate these phase shifts by installing phase shifters in the intersections. Thus, each phase shifter provides a 2nπ phase advance, where n is an integer number, for every photon energy the undulator operates. However, the undulator without the drifts and phase shifters constitutes a more costeffective solution (V. Shkaruba & N. Mezentzev, private communication).
In Fig. 13 we show the dependence of spectrum modulation standard deviation on two undulator parameters: radiation phase advance between undulator segments in radians and the relative difference between the magnetic field of two neighbouring undulator segments. The spectrum modulation is minimized by shifting radiation spectra of adjacent cells by approximately one FWHM of their bandwidth and by introducing a phase advance between segments as the smallest multiple of 2π. This corresponds well with our assumptions made in Section 2.
APPENDIX C
Radiation at the imaginary source
It is convenient to study the radiation properties on the sample by examining those at the imaginary radiation source. Once we have calculated the radiation field in the far zone we propagate it back to the undulator centre by means of the free space propagator (Voelz, 2011; Schmidt & Daniel, 2010). The resulting field we call the field distribution at the imaginary source. The shape of the field distribution at the imaginary source facilitates interpretation and understanding of the frequency and spatial structure of the radiation in the image plane after focusing. The field at the imaginary source consists of the contributions from all the segments, with their minimum waists located in the centres of these segments. Since the central frequency of each contribution also depends on the segment, we deal with a chromatic aberration. To illustrate this we depict the onaxis field distribution at the different locations within the undulator (i.e. the imaginary source) with respect to the photon energies, Fig. 14.
Upon imaging, similar dependence would be present at the sample location. To mitigate this effect we introduce an aperture in the far zone. The aperture, 0.3 mm × 0.3 mm, at 25 m from the source smooths the onaxis spectrum at the imaginary source, Fig. 14(b). In this way, we effectively increase the Rayleigh length of the imaginary source.
Footnotes
^{1}This scheme with auxiliary coil initially is proposed by N. Mezentsev and V. Shkaruba and will be discussed in more technical details in future papers.
Acknowledgements
We thank N. Mezentsev and V. Shkaruba for valuable consultations on the possible undulator constructions, S. Tomin for helping with the OCELOT code and patient consultations. We thank S. Molodtsov for his interest in this work. Open access funding enabled and organized by Projekt DEAL.
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