research papers
Elliptical plasmafilled waveguide as a new standard shortperiod undulator
^{a}Department of Physics, Shahid Beheshti University, Tehran, Iran, and ^{b}Iranian Light Source Facility, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
^{*}Correspondence email: bshokri@sbu.ac.ir
Undulators as the sources of highbrilliance synchrotron radiation are of widespread interest in new generations of light sources and freeelectron lasers. Microwave propagation in a plasmafilled elliptical waveguide can be studied as a standard shortperiod undulator. This structure as a lucrative insertion device can be installed in the storage ring of third and fourthgeneration light sources to produce highenergy and highbrilliance synchrotron radiation. In this article, the propagation of the transverse electric modes in a plasmafilled waveguide with an elliptical
is investigated, and the field components, the cutoff frequencies and the electron beam trajectory are calculated. With due consideration of the electron beam dynamics and in order to achieve a standard shortperiod undulator, parameters such as the dimensions of the waveguide elliptical the microwave frequency and the plasma density are calculated.1. Introduction
Synchrotron radiation has many applications in different branches of science such as physics, chemistry, biology, materials science, medicine, etc. (Onuki & Elleaume, 2003). Undulators and wigglers are the foremost sources of synchrotron radiation in light source facilities and freeelectron lasers (FELs). Undulators can generate synchrotron radiation with higher and higher than wigglers (Clarke, 2004). In recent years, many attempts have been made to design new generations of undulators capable of generating photons with higher energy and higher The energy of generated synchrotron radiation photons in an undulator is a function of the undulator period length and deflection parameter as
where n is the harmonic number, E is the electron beam energy, λ_{u} is the period length, K is the deflection parameter, γ is the Lorentz factor, and θ is the observation angle. The synchrotron radiation generated on an undulator axis (θ = 0) depends on the undulator deflection parameter in the form of φ_{n} = 1.43 × 10^{14}N_{p}IQ_{n}(K), where N_{p} is the undulator number of periods, I is the electron beam current in ampere, and Q_{n}(K) is a function of the deflection parameter. When the deflection parameter decreases (K < 1), Q_{n}(K) decreases to zero especially for higher harmonics, and when the deflection parameter increases, Q_{n}(K) increases and reaches its maximum for K ≃ 5 and remains almost constant for larger deflection parameters (Clarke, 2004).
The deflection parameter depends on the undulator period length and magnetic field as K = 0.934B_{u} [T]λ_{u}[cm], which signifies the fact that period length reduction leads to a reduction of the deflection parameter. There are two ways to generate synchrotron radiation with higher energy: increasing the electron beam energy or reducing the undulator period length. Period length reduction, besides its lower cost, provides the ability to generate higher in incorporating more periods in insertion devices (IDs) or installing more IDs within the available space in the straight sections of light source facilities and FELs. So, many attempts have been made to design new undulators with shorter period lengths, such as invacuum undulators (IVUs) ( ≃ 10 mm) (Chavanne et al., 2003; Huang et al., 2014; Kitamura, 1995), cryogenic permanent magnets undulators (CPMUs) ( ≃ 10 mm) (Benabderrahmane et al., 2017; Chavanne et al., 2008; Hara et al., 2004; Tanaka et al., 2006), microwave undulators (MUs) (λ_{u} ≃ 1 mm) (Batchelor, 1986; Kuzikov et al., 2013; Pellegrini, 2005; Tantawi et al., 2014), crystalline undulators (λ_{u} ≃ 100 nm) (Bellucci et al., 2003) and plasma undulators (λ_{u} ≃ 1 mm) (Corde & Ta Phuoc, 2011; Joshi et al., 1987; Rykovanov et al., 2015). However, achieving shorter period length generally comes at a price. Most of these undulators have very small deflection parameters (K < 1) which leads to a reduction of and It also inevitably results in the separation of the harmonics of undulators' tuning curves.
The deflection parameter of an undulator with overlapping radiated harmonics and continuous energy spectrum is larger than two (K > 2). Such an undulator has been called a `standard undulator' (Huang et al., 2017). Separation of the harmonics, due to the period length reduction, gives rise to nonstandard undulators.
In order to tackle this problem in IDs with permanentmagnet structures such as CPMUs and IVUs, a strong magnetic field and, therefore, permanentmagnet blocks with high magnetic remanence field are required. Permanent magnets in CPMUs are cooled down to cryogenic temperatures and therefore their remanence is improved significantly, but the highest accessible remanence field is bounded up to 1.7 T. To achieve a stronger magnetic field, it is also possible to decrease the undulators' magnetic gap. However, the minimum accessible magnetic gap is bounded to the beam stayclear. In fact, due to these limitations, the period length of a standard permanent magnet undulator (PMU) could not be smaller than 10 mm.
In an MU, the electromagnetic field of the microwave radiation is responsible for the electron's wiggling motion (Tantawi et al., 2014), and to reduce its period length the microwave frequency needs to be increased. Though it is possible to attenuate to some extent the undesirable effect of period length reduction on the deflection parameter through increasing the microwave intensity, a very high microwave intensity may result in an electrical breakdown in the MU's vacuum waveguides. In fact, the deflection parameters of shortperiod MUs are smaller than two (K < 2) and these undulators are nonstandard undulators.
To prevent electrical breakdown, the idea of using plasmafilled waveguides is investigated here. Using plasma makes it possible to amplify the microwave intensity in order to reach a shorter period length while avoiding the electrical breakdown. In other words, to keep K > 2 in shortperiod MUs, microwave undulators with plasmafilled waveguides, hereafter called microwave plasma undulators (MPUs), are proposed and investigated in this work.
In an MPU, highpower and highfrequency microwave radiation propagate in a plasmafilled waveguide, and the electromagnetic fields of the microwave radiation makes the electron beam oscillate and generate synchrotron radiation. In this work, an introduction to MPUs is given and the advantages of using an elliptical plasmafilled waveguide compared with conventional undulators are given. MPUs are similar to MUs except for the plasma in their waveguides. This means that the way of using an MPU is similar to that of an MU, and previous studies concerning different aspects of accelerator physics of MUs are applicable to MPUs. The only substantial difference is plasma in their waveguides and its effect on the electron beam quality through electron–plasma interactions. This effect is investigated here and parameters such as the emittance and the energy spread of an electron beam in a plasmafilled waveguide are calculated. The calculation of other parameters such as the stored energy, shunt impedance, filling time and quality factor which are essential for designing accelerator cavities are not under the scope of this work. These parameters could be investigated in detail in a future study to find their optimum values and to select the best shape and dimensions of a plasmafilled waveguide. For example, a circular corrugated or an elliptical corrugated plasmafilled waveguide could be studied for this purpose (Zhang et al., 2019). However, regarding the use of a plasmafilled waveguide as a shortperiod undulator, it can be mentioned that, after installing the waveguide in a storage ring, an appropriate gas should be injected to produce plasma inside it. Because the plasma generation method depends on the gas density and temperature, the gas should be converted to plasma using a plasma generation method (like microwave heating) that is appropriate for the storage ring condition (Wong & Mongkolnavin, 2016; Smirnov, 2015). The waveguide is then ready for the injection of highintensity microwaves to create a standing wave in it. The electromagnetic fields of the standing wave cause the electron beam to oscillate and produce synchrotron radiation.
Due to the electron–plasma interactions in plasmafilled waveguides, there might be a concern about the beam quality degradation and increased beam emittance and energy spread. Since the beam quality degradation is a function of the electron beam and plasma properties, attempts have been made to limit it by choosing suitable parameters for plasma to maintain the emittance and the energy spread of the electron beam in an acceptable range of the beam dynamics of fourthgeneration light sources.
Even though increasing the microwave frequency can pave the way for reaching undulators with shorter period length, it may also result in the excitation of higherorder harmonics and therefore in the reduction of undulator efficiency. To excite only the basic mode in an undulator while increasing the microwave frequency, the waveguide ). The cutoff frequencies of the basic mode and the next two modes of this elliptical waveguide are 14.64 GHz, 27.19 GHz and 38.71 GHz, respectively.
should be reduced. The minimum accessible of a waveguide is confined to the horizontal and the vertical beam stayclear (BSC) of the beam. The horizontal and the vertical BSC of the Iranian Light Source Facility (ILSF) storage ring are 6 mm and 2.1 mm, respectively. This means that the smallest allowed circular waveguide diameter is 6 mm. For such a circular waveguide, the cutoff frequencies of the basic mode (TE11) and the next two modes (TM01 and TE21) are 14.64 GHz, 19.13 GHz and 24.29 GHz, respectively. So, to excite only the basic mode, the microwave frequency inside the waveguide should not be higher than 19.13 GHz. The period length of the TE11 mode in such a circular waveguide with a cutoff frequency of 19.13 MHz is 9.9 mm. It is possible to use smaller crosssections by using elliptical waveguides. In an elliptical waveguide, it is possible to set one of the semiaxis dimensions to 2.1 mm (vertical beam stay clear) and set the other one to 6 mm (horizontal beam stay clear) (Fig. 1Using plasma, in addition to providing the possibility of applying higher microwave intensities, increases the cutoff frequencies of different modes. For example, the cutoff frequencies of the basic mode and the next two modes of the elliptical waveguide in Fig. 1, filled by cold plasma with a density of n_{0} = 3.1 × 10^{14} cm^{−3}, are 96.61 GHz, 177.5 GHz and 253 GHz, respectively. Therefore, it is possible to increase the applied microwave frequency up to 177.5 GHz without exciting the higher modes and without reducing the quality of the electron beam. This increase in the microwave frequency makes it possible to achieve undulators with shorter period length.
In the following, an elliptical plasmafilled waveguide with the electron beam characteristics of the ILSF storage ring is investigated, and the transverse electric (TE) mode field components of the microwave radiation, the electron wiggling motion, the deflection parameter, the period length, the
and the spectrum of this waveguide are calculated.2. TE mode field components
As defined by the standard hydrodynamic model, the dielectric tensor of a cold collisionless unmagnetized plasma is (Krall & Trivelpiece, 1973)
where ω is the microwave frequency, ω_{p} is the electronplasma frequency, and n_{0} is the plasma density. Considering the elliptical waveguide in Fig. 1 and assuming the direction of propagation to be along the zaxis, the relations between the elliptic coordinates and their rectangular counterparts are given as (McLachlan, 1951)
where 0 ≤ ξ ≤ ξ_{0} = tanh^{−1}(b_{0}/a_{0}), 0 ≤ η ≤ 2π, and d = is the semifocal length of the ellipse. The axis of the elliptical waveguide is considered to be at the origin. To calculate the TE mode field components of this waveguide, the dielectric tensor is inserted into Maxwell's equations, the boundary conditions of a complete plasmafilled elliptical waveguide surrounded by a perfect conduction material are applied, and the fact that E_{z} is equal to zero for the TE mode field components is considered,
The behavior of the electric and the magnetic field components along the waveguide axis is as exp(±iβz). Using equation (5) for the electrical field along the zaxis gives
where k_{c}^{2} = and l = d(cosh^{2}ξ − cos^{2}η)^{1/2}. Equation (6) is a differential equation, known as the Mathieu differential equation, with a well known solution and eigenvalue (McLachlan, 1951) as
where ce_{m}(η, q) and se_{m}(η, q) are the even and the odd solutions of the angular Mathieu equation, Ce_{m}(ξ, q) and Se_{m}(ξ, q) are the even and the odd solutions of the radial Mathieu equation of the first kind, and C_{m} and S_{m} are computable arbitrary constants. For any m, there are even and odd solutions in the following form,
Given the boundary condition = = 0, then = 0 and = 0. Considering q_{m,r} and as the rth root of and , respectively, therefore
and the TE_{mr} field components are
where β is the propagation constant given by β = with k_{c}^{2}(e) = 4(q_{m,r}/d^{2}) and k_{c}^{2}(s) = . The corresponding cutoff frequency and wavenumber are
where (e) and (s) refer to even and odd modes. It can be seen that the cutoff frequency (f_{c}) depends on the waveguide dimensions. The electromagnetic field components perpendicular to the direction of propagation are
where
In Fig. 2, the electric field patterns of the first three modes of the plasmafilled elliptical waveguide are plotted by CST Studio (CST Microwave Studio, 2008) software. In an elliptical waveguide, the cutoff frequency of the even TE_{11} mode is smaller than those of the other modes and therefore this mode is the dominant mode.
3. Period length and deflection parameter
The period length, the deflection parameter and the number of periods are the main parameters used to obtain the synchrotron radiation characteristics of an ID, such as energy,
and brilliance. Electromagnetic fields are responsible for the electron undulating trajectory inside waveguides. The period length of the electron undulating trajectory and the maximum transverse deflection angle are regarded as the undulator period length and deflection parameter, respectively.Considering a moving electron along the zaxis with the speed of V_{z}, the Lorentz force and its component along the xdirection are
where E_{x} = and B_{y} = . E_{1x} = E_{⊥}cos(η) and B_{1y} = B_{⊥}cos(η) are the electric and the magnetic field peaks and = . Substituting E_{x} and B_{y} into equation (26) gives
where = and = are the wavelength of microwave in free space and in the waveguide, respectively. The first term in equation (27) is responsible for the counterpropagating mode and the second term is responsible for the copropagating mode. The period length in the counterpropagating and in the copropagating modes for relativistic electrons are
The counterpropagating mode generates the desired highfrequency wiggling motion of electrons which in turn results in the generation of synchrotron radiation with higher energy compared with the copropagating mode. Therefore, the counterpropagating mode is more favorable. For λ_{0} ≅ λ_{g} the effect of the copropagating mode on the electron beam vanishes which means . Using the first term in equation (27) to calculate the deflection parameter gives
so we have
where
Similarly, for the Lorentz force along the ydirection, the deflection parameter of the vertical oscillation along the yaxis is
where E_{1y} = E_{⊥}sin(η).
4. Numerical results and calculations
The plasma density and the microwave frequency are significant in MPUs. MPUs are proposed to reach standard undulators with shorter period length compared with conventional MUs. As mentioned before, to obtain a shorter period length, the microwave frequency needs to be increased, and, in addition to technical limitations, other considerations need to be taken into account. For example, to prevent detrimental effects of higher propagating modes on the electron beam dynamics, the frequency should be chosen so that only the dominant mode will be excited.
Using plasma instead of vacuum makes it possible to increase the cutoff frequencies of propagating modes. There is a direct relationship between the plasma frequency and the cutoff frequencies, i.e. the higher the plasma frequency, the higher the cutoff frequencies will be. However, high plasma density is a prerequisite of high plasma frequency, but high plasma density adversely affects the electron beam dynamics. So there should be a tradeoff between increasing plasma density and degradation of beam quality.
Beam quality degradation results from two main sources: beam instabilities due to the beam–plasma interactions and wake fields. Two significant instabilities are raised, called TwoStream and Weible instabilities. The growth rate of the TwoStream instability is proportional to the electron beam density and the plasma density in the form of δ ≃ , where n_{b} is the density of the electron beam, n_{0} is the density of the plasma, and γ is the Lorentz factor. If the efolding length of this instability is larger than the beam length [], the TwoStream instability will be constant (Joshi et al., 1987). A single bunch of electrons inside the ILSF storage ring with n_{b} ≃ 8.6 × 10^{17} cm^{−3} and γ = 5870 has a length of about 8 mm. Therefore, to prevent the TwoStream instability, the plasma density (n_{0}) should be less than 4.76 × 10^{21} cm^{−3} and the plasma frequency (ω_{p}) should be smaller than 3.89 × 10^{15} rad s^{−1}.
The Weible instability is a purely transverse instability responsible for the filamentation of wide beams in plasma. Although this instability is not suppressed for short bunches, it is suppressed mainly for bunches narrower than the plasma skin depth, i.e. narrower than c/ω_{p}. To avoid the Weible instability for the ILSF storage ring electron beam (an ultralow emittance beam with small horizontal and vertical beam sizes of σ_{x} = 68.9 µm and σ_{y} = 2.96 µm), plasma density (n_{0}) should be less than 6.2 × 10^{16} cm^{−3} and the plasma frequency (ω_{p}) should be smaller than 4.4 × 10^{12} rad s^{−1}.
In addition to the beam–plasma instabilities, the wake field can also degrade the electron beam quality. The longitudinal and the radial wake forces affect the energy spread and the emittance of the electron beam, respectively. The radial wake force exerts a pinching force on the beam. This force might be used to focus the electron beam in some cases, such as a plasma lens, but in MPUs it leads to undesirable angular variation and emittance growth. In vacuum, the radial repulsion of a relativistic electron beam space charge is canceled by the beam's own azimuthal magnetic field, and therefore the radial repulsion does not increase the beam emittance. In plasma, the plasma electrons are expelled by the beam space charge and therefore the beam space charge radial repulsion is neutralized by the plasma ions. However, due to the V_{z} × B_{θ} force and since the plasma does not completely neutralize the beam current, the net result is the pinching of the beam. As stated by Joshi et al. (1987), the angular spread of a beam can be estimated quantitatively from the betatron motion of individual particles so that, if the betatron wavelength is approximately equal to β ≃ , the beam angular spread of = results from the betatron oscillations. For the ILSF storage ring beam parameters in Table 1, the angular spread of the beam in the presence of a plasma with n_{0} = 6.2 × 10^{16} cm^{−3} is (Δθ)_{β} = 6.28 × 10^{−6} rad. This value is too small to affect the beam emittance.

In the new generation of light sources and FELs, the electron beam emittance and the energy spread are smaller than 400 pm and 0.2%, respectively. Degradation of the electron beam quality increases the energy spread and the emittance and decreases the photon beam
In conclusion, the plasma density should be chosen so that, despite the detrimental effects of the beamplasma interactions on the beam quality, the values of the beam emittance and energy spread remain within their acceptable ranges.In addition to the instabilities and wake forces, the period length of an undulator can also affect the beam emittance and energy spread (Clarke, 2004). In MUs and MPUs, the period length is inversely proportional to the microwave frequency. In Fig. 3, the energy spread is plotted as a function of the microwave frequency for different plasma densities while considering the effects of TwoStream and Weible instabilities, longitudinal wake force and undulator period length. The deflection parameter and the length of the undulator are k = 2.06 and L = 100λ_{u}, respectively, and the natural energy spread is equal to the natural energy spread of the ILSF storage ring beam, i.e. 6.79 × 10^{−4}. In Fig. 4, the emittance versus microwave frequency is plotted for different plasma densities while considering the effects of TwoStream and Weible instability, transverse wake force and undulator period length. From these two figures, it can be seen that, as the microwave frequency increases, the energy spread and the emittance increase. As mentioned before, increasing the microwave frequency decreases the period length, which in turn reduces the deflection parameter. To keep k = 2.06, it is essential to increase the microwave intensity. As the microwave intensity increases, the strength of the electromagnetic fields increases too. In spite of the fact that a stronger electromagnetic field can compensate the effect of period length reduction on the deflection parameter, it may degrade the beam quality. Considering Figs. 3 and 4, in order to keep the energy spread smaller than 0.002 and the emittance smaller than 0.4 nm rad, the maximum value of the microwave frequency for n_{0} = 3.1 × 10^{14} cm^{−3} is 10^{12} rad s^{−1}.
Another parameter to be considered in the study of beam dynamics in a storage ring is the damping time, τ. The fraction of the damping times with and without an MPU in a storage ring, i.e. , for the MPU with the parameters in Table 2, is equal to 0.93, which indicates the minor effect of the MPU on the damping time.

In the following, the elliptical MPU with parameters listed in Table 2 is investigated, and its period length, synchrotron radiation and are calculated. Besides considerations about the beam quality degradation, the microwave frequency and the plasma density are chosen so that only the dominant mode of the waveguide is excited.
In Fig. 5, the electric field pattern of the microwave radiation along the elliptical plasmafilled waveguide axis is shown, in which the only present electrical field is along the yaxis. Therefore, the main oscillation of the electron beam is in the y–z plane, and the synchrotron radiation from this undulator has linear polarization.
One of the main parameters to consider in designing insertion devices is the field rolloff in their good field region. The smaller the field rolloff, the smaller the values of the higherorder integrated multipoles will be. The higherorder integrated multipoles negatively affect the quality of the beam. In the ILSF storage ring, the horizontal good field region (GFRx) at the straight section (where the IDs are installed and dispersion is zero) is ±0.7 mm. In Fig. 6, the horizontal rolloff is plotted. Field variations over the maximum field () over the ±0.7 mm is 2.42% which is an acceptable value for the field rolloff at the GFR.
Other essential parameters in designing IDs are the first and the second field integrals. Ideally, these integrals should be zero, which means the field's net effect on the beam is null. Nevertheless, in practice, these integrals are not zero. One of the main approaches to correct for the nonzero field integrals of an ID is to use corrector magnets at the end of it. These magnets could be used with MPUs as well.
It is worth mentioning that in conventional undulators, in which permanent magnets are used to create the required magnetic field, various factors such as the imperfections of the permanent magnets' block size and their magnetism as well as undulators construction imperfections result in deficiencies in the generated magnetic field. This can negatively affect the quality of generated synchrotron radiation and might result in σ_{φ}) is used by the IDs experts to investigate the effect of magnetic field deficiencies on the reduction. The reduction on the axis of IDs is shown by R = in which n represents the radiation harmonic number (Walker, 2013). This reduction due to the RMS phase error is more drastic for higher harmonics that make it practically impossible to use harmonics higher than the seventh harmonic for a phase error larger than 5° (Clarke, 2004). Therefore, in light sources in which a wide synchrotron radiation energy range (e.g. 1 keV up to 40 keV) is covered with high harmonics, it is necessary to reduce the phase error as much as possible.
reduction. The RMS phase error (In Fig. 7, the and the of the elliptical plasmafilled waveguide are plotted. It can be seen that the first to seventh harmonics are covering the energy spectra ranging from 17 to 150 keV with higher than 10^{14}, and the energy spectra range from 17 to 180 keV with higher than 10^{20}. Comparing the results of Fig. 7 with the results of conventional undulators used to generate hard Xrays shows that the energy spectrum of the generated synchrotron radiation has increased significantly, and the and the are increased by at least one order of magnitude. The ability of MPUs to generate synchrotron radiation with higher and is an evident and significant advantage over conventional undulators. This ability provides researchers with a chance to conduct more diverse experiments.
5. Summary and conclusion
In this paper, microwave propagation in a plasmafilled elliptical waveguide was studied. It was shown that this structure could work as a standard shortperiod undulator, i.e. an elliptical microwave plasma undulator (MPU). The propagation of TE modes in the waveguide was investigated, and the even and odd field components, cutoff frequencies and electron beam trajectory under the influence of electromagnetic fields of microwave radiation were calculated. With consideration of the electron beam dynamics and achieving a standard shortperiod elliptical MPU, the microwave frequency, the plasma density, and the undulator period length and deflection parameter were chosen. It was shown that the elliptical MPU eliminates the limitation of standard MUs in generating hard Xray synchrotron radiation with overlapping harmonics, and it can generate higher with a wider energy spectrum compared with conventional MUs. The effects of beam–plasma interactions on the electron beam dynamics were studied, and the variations of the electron beam emittance and energy spread versus microwave frequency for different plasma densities were plotted. The parameters of the electron beam inside the ILSF storage ring were used for radiation calculations.
It was also shown that MPUs have two significant advantages over conventional MUs, making their use in the storage ring of the new generation of light sources more efficient and more attractive. The first advantage of utilizing plasma, as an ionized medium, is that it does not undergo electrical breakdown and as a result it allows to increase the intensity of microwave radiation. In contrast, in vacuum waveguides, increasing the radiation power leads to electrical breakdown and system malfunctions. The possibility of increasing the radiation power gives the ability to maximize the i.e. K > 2. Besides, the cutoff frequencies of the propagating modes of the microwave radiation in a plasmafilled waveguide are larger than the cutoff frequencies of the vacuum waveguides. So, as a result, it is possible to use microwaves with higher frequencies to reduce an undulators' period length without the necessity of exciting higherorder modes. Exciting higher modes negatively affects the electron beam dynamics and degrades the electron beam quality.
to maintain standard values for the deflection parameter,References
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