Beam Emittance and Bunch Length Diagnostics for the MIR-FEL Beamline at Chiang Mai University

Abstract

The generation of high-quality mid-infrared free-electron laser (MIR-FEL) radiation depends critically on precise control of electron beam parameters, including energy, energy spread, transverse emittance, bunch charge, and bunch length. At the PBP-CMU Electron Linac Laboratory (PCELL), effective beam diagnostics are essential for optimizing FEL performance. However, dedicated systems for direct measurement of transverse emittance and bunch length at the undulator entrance have been lacking. This paper addresses this gap by presenting the design, simulation, and analysis of diagnostic stations for accurate characterization of these parameters. A two-quadrupole emittance measurement system was developed, enabling independent control of beam-focusing in both transverse planes. An analytical model was formulated specifically for this configuration to enhance emittance reconstruction accuracy. Systematic error analysis was conducted using ASTRA beam dynamics simulations, incorporating 3D field maps from CST Studio Suite and fully including space-charge effects. Results show that transverse emittance values as low as 0.15 mm·mrad can be measured with less than 20% error when the initial RMS beam size is under 2 mm. Additionally, quadrupole misalignment effects were quantified, showing that alignment within ±0.95 mm limits systematic errors to below 33.3%. For bunch length measurements, a transition radiation (TR) station coupled with a Michelson interferometer was designed. Spectral and interferometric simulations reveal that transverse beam size and beam splitter properties significantly affect measurement accuracy. A 6% error due to transverse size was identified, while Kapton beam splitters introduced additional systematic distortions. In contrast, a 6 mm-thick silicon beam splitter enabled accurate, correction-free measurements. The finite size of the radiator was also found to suppress low-frequency components, resulting in up to 10.6% underestimation of bunch length. This work provides a practical and comprehensive diagnostic framework that accounts for multiple error sources in both transverse emittance and bunch length measurements. These findings contribute valuable insight for the beam diagnostics community and support improved control of beam quality in MIR FEL systems.

1. Introduction

Mid-infrared (MIR) radiation, covering wavelengths from 3–30 $\mathsf{\mu }$m, is interesting for both scientific research and practical applications. This radiation is highly absorbed by many materials, especially in biological materials, which the absorption fingerprints in MIR region mainly associate with the vibration of bending bonds. The absorption peaks in the MIR region can be employed to determine chemical groups in various materials [1,2]. The unique pattern of the MIR absorption spectrum makes it a powerful tool for identifying the composition of biological and chemical molecules. Moreover, the pump–probe experiment based on time-resolved spectroscopy is one of the powerful techniques utilizing short MIR radiation pulses to characterize material properties that depend on time such as transient reflectivity or transmittivity [3]. One of the most effective techniques to generate such high-quality MIR pulses is through an accelerator-based free-electron laser (FEL). FEL light sources possess unique properties, including narrow bandwidth, coherence, and high intensity, enabling superior measurement resolution. Furthermore, FEL pulses have sub-picosecond durations, making them a proper light source for pump–probe experiments. Additionally, the wavelength is conveniently tunable by changing the electron beam energy and the strength of the undulator magnet. With a tunable wavelength range, MIR FELs provide a powerful tool for studying and enhancing material properties at the molecular level. MIR FEL pulses interact with materials through resonant absorption mechanisms, where the laser energy couples selectively to specific molecular vibrations. The generation of coherent radiation in these spectral regions is essential for applications such as spectroscopy, material modification and analysis, and pump–probe experiments. The development of FEL facilities in the infrared (IR) region has driven significant advancements across various scientific and industrial fields. Several laboratories have developed MIR-FEL systems to support these diverse applications [4,5,6].

At the PBP-CMU Electron Linac Laboratory (PCELL), a research unit of the Plasma and Beam Physics (PBP) Research Facility at Chiang Mai University (CMU), we focused on developing an electron accelerator and systems to generate coherent mid-infrared (MIR) and terahertz (THz) radiation. The MIR radiation is produced using oscillator free-electron laser (FEL) technology, while THz radiation is generated from femtosecond electron pulses via transition radiation and super-radiant undulator radiation techniques. The layout of the accelerator system and MIR beamline at PCELL is illustrated in Figure 1. The system consists of an electron beam injector comprising a one-and-a-half-cell standing-wave radio-frequency (RF) electron gun with a thermionic cathode [7], and an alpha magnet used for both electron beam energy filtering and first-stage bunch compression. The alpha magnet is a half-quadrupole magnet equipped with an iron mirror plate to terminate the fringe magnetic field. It generates a constant magnetic field gradient that bends electron trajectories into an alpha-like shape. The path length of electrons inside the magnet’s field depends on their energy. Consequently, an energy slit placed inside the alpha magnet’s vacuum chamber can serve as an energy filter, removing the fraction of the electron beam with energy below a desired threshold. These slits can also be used to measure the energy spectrum of the electron beam, enabling the determination of its mean energy and energy spread. Furthermore, by appropriately adjusting the magnetic field gradient of the alpha magnet, the bunch length can be compressed, resulting in a shorter electron bunch at the desired location after exiting the magnet [8]. Downstream of the alpha magnet, a traveling-wave RF linear accelerator (linac) structure, small dipole magnets for beam steering, and quadrupole magnets for beam-focusing are also included. Four dipole magnets (D1, D2, D4, and D5) form a 180-degree achromat magnetic system for the MIR-FEL beamline, while the D3 dipole magnet bends the electron beam toward the super-radiant THz undulator radiation beamline and serves as part of its achromat magnetic system. The MIR-FEL beamline, which is the focus of this paper, is located in the section following the 180-degree turnaround. This section includes quadrupole magnets, steering magnets, an undulator with an optical cavity for MIR-FEL generation, an electron beam dump, and beam diagnostic devices.

At the gun exit, electron beams have a maximum energy of approximately 2–2.5 MeV. They are then further accelerated in the linac, reaching kinetic energies in the range of 8–25 MeV, depending on the accelerating gradient determined by the supplied RF power [9]. For MIR-FEL generation, a 25 MeV electron beam is directed into an undulator with a periodic magnetic structure, causing the electrons to undergo sinusoidal motion and emit spontaneous radiation in the MIR regime. This radiation is trapped and amplified within an optical cavity consisting of two gold-coated copper mirrors, which reflect the radiation back into the undulator to interact with subsequent electron pulses. The interaction between the emitted radiation and the electron beam induces microbunching at the radiation wavelength, approximately 9.5–12.5 µm [10], leading to coherent amplification of the emitted light. As the gain reaches saturation, a fraction of the MIR-FEL is extracted through a 2.6 mm diameter coupling hole in the upstream mirror.

The performance of MIR-FEL production critically depends on the electron beam quality, particularly parameters such as short pulse duration, high bunch charge, low energy spread, and low transverse emittance. Accurate characterization of these properties is essential for optimizing FEL performance. At our facility, bunch charge is monitored along the accelerator system and beamlines using a current transformer (CT). The combination of the alpha magnet and a CT also enables energy and energy spread measurements after the gun exit. Following acceleration through the linac, the electron energy and energy spread are measured again using a dipole magnet and a screen station with a resolution of 0.1 mm/pixel [11]. Multiple screen stations with the same resolution are distributed along the accelerator and beamlines for beam profile monitoring. While existing setups allow for the measurement of energy, energy spread, and transverse beam profile (i.e., the beam’s spatial distribution across the horizontal and vertical axes), dedicated systems for directly measuring transverse beam emittance and bunch length have not yet been established. This work addresses this gap by designing and evaluating diagnostic stations upstream of the undulator entrance for direct measurement of these parameters. Through detailed simulations and systematic error estimation, we aim to develop reliable diagnostic tools to support beam quality optimization for MIR-FEL generation.

In this paper, we present the design, simulation, and analysis of diagnostic systems for characterizing the transverse emittance and electron bunch length at the MIR-FEL beamline at Chiang Mai University. One novelty of this work is the implementation of a two-quadrupole emittance measurement system, which enables independent control of beam-focusing in both transverse planes. This configuration enhances measurement flexibility and significantly reduces systematic errors compared to conventional single-quadrupole methods. To support this setup, we introduce a practical analytical model tailored specifically for the two-quadrupole configuration, improving the accuracy of emittance reconstruction. Comprehensive systematic error analyses were performed using ASTRA beam dynamics simulations, which incorporated realistic 3D electromagnetic field maps from CST Studio Suite and fully included space-charge effects. The results demonstrate that transverse emittance values as low as 0.15 mm·mrad can be reliably measured under well-controlled conditions. Furthermore, the impact of quadrupole magnet misalignment on measurement accuracy was investigated, leading to the establishment of practical alignment tolerances for experimental implementation. Extensive analyses of transverse emittance measurement errors presented in this study also provide a valuable contribution to the accelerator and beam diagnostics community, offering practical insights into minimizing systematic uncertainties and improving measurement reliability. For bunch length measurements, a transition radiation station coupled with a Michelson interferometer was developed, capable of resolving femtosecond-scale bunch lengths. The system was validated through spectral and interferometric simulations, accounting for critical physical effects including transverse beam size, beam splitter interference, and the finite size of the radiator. Together, these outcomes provide a practical framework for high-precision beam diagnostics. By addressing multiple sources of error in both transverse emittance and bunch length measurements, this study contributes significantly to improving beam quality control and advancing the performance of MIR-FEL systems.

2. Electron Beam Dynamics Simulation

The ASTRA (A Space Charge Tracking Algorithm) software [12] was used to perform electron beam dynamics simulations from the cathode of the RF-gun to the undulator entrance, with the objective of optimizing the electron beam properties at this location for MIR-FEL production. Space charge effects were included in the simulations throughout the optimization process. A cylindrical grid algorithm was used from the cathode to the entrance of the alpha magnet, where the beam is expected to be cylindrically symmetric. From the entrance of the alpha magnet to the undulator, a three-dimensional (3D) mesh algorithm in Cartesian coordinates was applied, as the beam becomes transversely asymmetric in this region. The 3D electromagnetic field distributions inside the RF-gun, alpha magnet, quadrupole magnets, and dipole magnets were obtained using the software CST Studio Suite [13]. Based on the extensive study in [14], the simulation results provided the electron beam properties expected from the accelerator system. The simulated electron beam properties at the dipole D5 exit, as indicated in Figure 1, are summarized in

Table 1. Expected electron beam parameters at the dipole D5 exit, obtained from optimization using beam dynamics simulations [14]. Here, RMS denotes the root mean square; FWHM is the full width at half maximum; and ${\sigma }_G$ is the standard deviation of the Gaussian function.

Parameter	Value
Average kinetic energy	25 MeV
Energy spread (RMS)	0.1%
Bunch charge	62.5 pC
Bunch length (RMS)	219 fs
Bunch length (${\sigma }_G$)	208 fs
Bunch length (FWHM)	490 fs
Peak current	127.6 A
Horizontal emittance	0.18 mm.mrad
Vertical emittance	0.16 mm.mrad
Horizontal beam size (RMS)	1.1 mm
Vertical beam size (RMS)	1.1 mm

, while the corresponding transverse and longitudinal electron distributions are shown in Figure 2. Using these results, we designed beam diagnostic setups and measurement procedures for characterizing transverse beam emittance and bunch length. Notably, the bunch length was obtained from Gaussian fitting of the longitudinal distribution.

Several techniques can be used for emittance measurement, including the pepper-pot technique [15,16], the slit-scan method [17,18], and the quadrupole scan technique [19,20]. In this work, the quadrupole scan technique is chosen. This approach is advantageous as it allows us to leverage the existing quadrupole magnets and a screen station already installed in the beamline. The quadrupole scan technique is widely used for measuring the transverse emittance of charged particle beams at various facilities, including Fermilab’s Advanced Superconducting Test Accelerator (ASTA) [21] and the Brookhaven Accelerator Test Facility [22]. In our facility, the transverse beam emittance measurements will be performed before the undulator entrance using a setup consisting of two quadrupole magnets and a screen station. For the beam dynamic simulations, the emittance measurements were modeled with the ASTRA program using both an ideal beam (a Gaussian distribution generated at the dipole D5 exit) and a realistic beam (a distribution obtained from start-to-end simulations starting from the cathode to the dipole D5 exit). The simulated longitudinal and transverse particle distributions at the exit of dipole magnet D5 are presented in Figure 2. The longitudinal phase space ($E_k$-t) shows a clear electron bunch compression, with an RMS bunch length of 219 fs. The transverse distribution (x-y) exhibits asymmetric feature with the same RMS beam sizes of 1.1 mm in both the horizontal and vertical planes. The x-$x^{\prime }$ phase space indicates strong focusing in the horizontal plane, while the y-$y^{\prime }$ phase space shows minimal beam divergence. The color scale in all subfigures represents an energy spread of 0.1%. The results obtained from beam dynamic simulations are essential for designing the emittance measurement system and estimating the systematic error associated with our setup.

Several methods are available for measuring electron bunch length, including streak cameras, electro-optic detection, and RF deflectors. For example, streak cameras have been employed at facilities such as the Diamond Light Source (DLS) [23] and the Beijing Electron Positron Collider (BEPCII) [24] to measure electron bunch lengths from 1 to 60 ps using coherent synchrotron radiation. The FELIX facility [25] used the electro-optic detection to measure bunch lengths in the hundreds of femtoseconds, while the Photo Injector Test Facility at DESY in Zeuthen (PITZ) [26] utilized an RF deflector achieving a resolution of about 0.5 ps. While these techniques are powerful, they often require complex, costly, or specialized equipment. In contrast, the Michelson interferometer offers a practical and effective alternative. It can resolve bunch lengths in the sub-picosecond range, features a relatively simple and cost-effective setup, and works well with broadband detectors for accurate profile reconstruction. For our case, we adopt a transition radiation (TR) station coupled with a Michelson interferometer. TR is a well-established method for generating coherent radiation and has been successfully implemented in numerous laboratories [27,28,29,30,31]. Coherent transition radiation (CTR), generated when ultrashort electron bunches cross an interface between materials with different dielectric constants, has a spectrum that directly reflects the bunch’s longitudinal profile [32,33]. By analyzing the CTR interferogram with a Michelson interferometer, bunch lengths as short as a few hundred femtoseconds can be measured.

At the PCELL facility, a TR station equipped with a Michelson interferometer was chosen for its high temporal resolution, simple setup, and compatibility with broadband CTR detection, making it a practical and effective solution for our measurement requirements for femtosecond electron bunches. This setup has already been successfully used to measure femtosecond-scale electron bunch lengths at the linac exit [34]. In this work, we plan to install a TR station before the undulator entrance to enable direct bunch length measurements at that location, ensuring proper bunch compression for MIR-FEL generation. The longitudinal beam distribution at the dipole D5 exit, obtained from beam dynamics simulations (Figure 2), was used as the input for the analytical calculations in Section 4 to estimate the expected bunch length of the electron beam. Additionally, we investigated how the electron beam’s transverse size, the Michelson interferometer’s beam splitter, and the finite size of the radiator influence the accuracy of the bunch length measurement.

3. Transverse Beam Emittance Measurement System

The experimental setup for the quadrupole scan technique used to measure transverse beam emittance consists of two quadrupole magnets (Q17 and Q18) located at the beginning of the MIR beamline and a screen station (SC12), positioned 0.866 m downstream of the scanning quadrupole magnet (Q18). The quadrupole magnet Q17 is used to control the electron beam size, ensuring that the beam distribution remains within the detection range of the screen station. The distance between quadrupole magnets Q17 and Q18 is 0.178 m. An engineering drawing presenting this setup is shown in Figure 3. The quadrupole scan method is based on the relationship between the transverse beam size and the effective focal length of quadrupole magnet, which depends on the quadrupole strength and the magnet’s effective length. By systematically adjusting the strength of quadrupole magnet Q18 and recording the beam size at the screen station, a second-order polynomial fit of beam size versus quadrupole strength can be used to extract the transverse emittance. This technique is widely used in particle accelerators due to its non-destructive characteristics, ability to measure emittance at different beam energies, and compatibility with existing quadrupole magnets along our MIR-FEL beamline. In this work, the quadrupole scan method is implemented before the undulator entrance to characterize and optimize the electron beam for MIR-FEL generation.

3.1. Theoretical Model

To model the beam transport through two quadrupole magnets, drift spaces, and the screen station, a matrix transformation using the thin-lens approximation is applied. Let $L_1$ and $L_2$ denote the distances between quadrupole magnets Q17 and Q18, and between Q18 and the screen station SC12, respectively. Defining $f_1$ and $f_2$ as the focal lengths of Q17 and Q18, respectively, the total transfer matrix R is

$$\begin{array}{cc}\hfill R& =\left[\begin{array}{cc}1& L_2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -1/f_2& 1\end{array}\right]\left[\begin{array}{cc}1& L_1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -1/f_1& 1\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{cc}(f_2{f}_1-L_2{f}_1-L_1{f}_2+L_2{L}_1-L_2{f}_2)/f_2{f}_1& (L_2{f}_2+L_1(f_2-L_2))/f_2\\ (-f_1+L_1-f_2)/f_1{f}_2& (f_2-L_1)/f_2\end{array}\right].\hfill \end{array}$$

(1)

Let ${\sigma }_q$ and ${\sigma }_s$ represent the beam matrices at the quadrupole magnet Q17 and the screen position, respectively. The electron beam matrix at the screen is then

$${\sigma }_s=R{\sigma }_q{R}^T.$$

(2)

The root mean square transverse beam size at the screen, ${\left({\sigma }_s\right)}_{11}$, is related to elements of the transformation matrix and beam matrix by

$${\left({\sigma }_s\right)}_{11}=R_{11}^2{\left({\sigma }_q\right)}_{11}+2R_{11}{R}_{12}{\left({\sigma }_q\right)}_{12}+R_{12}^2{\left({\sigma }_q\right)}_{22}.$$

(3)

In this work, the quadrupole strength of Q17 was optimized to confine the beam within the screen area, and its value was held constant during the scan of quadrupole magnet Q18. Consequently, the beam size at the screen depends solely on the focal length $f_2$ of the quadrupole magnet Q18 and can be described by a second-order polynomial as

$${\left({\sigma }_s\right)}_{11}=A{\displaystyle \frac{1}{f_2^2}}+B{\displaystyle \frac{1}{f_2}}+C.$$

(4)

Here, A, B, and C are the polynomial coefficients, which are used to retrieve the beam matrix elements at the quadrupole magnet Q17 through the following relationships:

$$A=a_1{\left({\sigma }_q\right)}_{11}+2a_2{\left({\sigma }_q\right)}_{12}+a_3{\left({\sigma }_q\right)}_{22},$$

(5)

$$B=b_1{\left({\sigma }_q\right)}_{11}+2b_2{\left({\sigma }_q\right)}_{12}+b_3{\left({\sigma }_q\right)}_{22},$$

(6)

$$C=c_1{\left({\sigma }_q\right)}_{11}+2c_2{\left({\sigma }_q\right)}_{12}+c_3{\left({\sigma }_q\right)}_{22},$$

(7)

with the coefficients $a_1$–$a_3$, $b_1$–$b_3$, and $c_1$–$c_3$ defined as

$$\begin{array}{cc}\hfill a_1& =L_2^2+L_1^2{L}_2^2/f_1^2-2L_2^2{L}_1/f_1,\hfill \\ \hfill a_2& =L_2^2{L}_1-L_2^2{L}_1^2/f_1,\hfill \\ \hfill a_3& =L_1^2{L}_2^2,\hfill \\ \hfill b_1& =-2L_2+2L_2^2/f_1-2L_2{L}_1^2/f_1^2-2L_1{L}_2^2/f_1^2+4L_2{L}_1/f_1,\hfill \\ \hfill b_2& =-L_2^2+2L_2{L}_1^2/f_1+2L_2^2{L}_1/f_1-2L_2{L}_1,\hfill \\ \hfill b_3& =-2L_2^2{L}_1-2L_1^2{L}_2,\hfill \\ \hfill c_1& =1+L_1^2/f_1^2+L_2^2/f_1^2-2L_1/f_1-2L_2/f_1+2L_1{L}_2/f_1^2,\hfill \\ \hfill c_2& =-L_1^2/f_1-L_2^2/f_1+L_2+L_1-2L_1{L}_2/f_1,\hfill \\ \hfill c_3& =L_2^2+2L_1{L}_2+L_1^2.\hfill \end{array}$$

(8)

Let $k_1$ and $k_2$ represent the quadrupole strengths of Q17 and Q18, and $l_{\mathrm{eff},1}$ and $l_{\mathrm{eff},2}$ be their corresponding effective lengths. Then, their focal lengths are related to the quadrupole strength and the effective length as

$$f_1={\displaystyle \frac{k_1}{l_{eff,1}}},f_2={\displaystyle \frac{k_2}{l_{eff,2}}}.$$

(9)

When the effective lengths of the quadrupole magnets are known, the beam size on the screen depends only on the quadrupole strength, which can be determined from the relationship between the quadrupole gradient and the applied electric current. In our case, this relationship is described in Section 3.2.

Once the beam matrix elements ${\left({\sigma }_q\right)}_{11}$, ${\left({\sigma }_q\right)}_{12}$, and ${\left({\sigma }_q\right)}_{22}$ are achieved, the transverse emittance $\epsilon$ can be calculated using

$$\epsilon =\sqrt{{\left({\sigma }_q\right)}_{11}{\left({\sigma }_q\right)}_{22}-{\left({\sigma }_q\right)}_{12}^2},$$

(10)

which defines the area of the phase space ellipse and quantifies the beam quality. The accuracy of this calculation relies on precise beam size measurements and proper utilization of the quadrupole scan technique. Although the quadrupole scan method is well-established, our contribution lies in a novel and practical analytical formulation tailored for two-quadrupole configurations, which offers advantages over a single-quadrupole setup. The analytical expressions derived in this work are generalizable to any emittance measurement setup using two quadrupole magnets, strengthening the theoretical foundation of the quadrupole scan technique. One major benefit of this approach is the independent control of beam-focusing in horizontal and vertical planes, which is particularly valuable when the beam exhibits asymmetry. This allows emittance measurements in each plane with greater precision. Additionally, having two quadrupoles increases the degrees of freedom in the transport matrix, enabling more accurate tailoring of the beam optics and resulting in improved fitting accuracy during emittance reconstruction. Furthermore, while the primary aim of our work is to measure the projected transverse emittance, the analytical framework we present here also provides access to the beam matrix elements, including the off-diagonal term ${\left({\sigma }_q\right)}_{12}$, which represents the correlation between spatial position and angle. This information is valuable for understanding the phase space distribution and for optimizing beam transport and matching conditions.

3.2. Simulation Setup and Procedure

In the quadrupole scan method, the electron beam size at the screen station is monitored while varying the quadrupole magnet strength. To measure the transverse beam profile, the screen station comprising a phosphor screen and a CCD camera is used. The phosphor screen is mounted at a 45° angle with respect to the beam axis to efficiently capture the electron beam image. This setup is capable of detecting the electron beam image within a usable detection area of 25.0 mm in the vertical direction and 23.3 mm in the horizontal direction. The spatial resolution of the imaging system is 0.1 mm per pixel, covering the entire screen area. To accurately simulate the beam measurement process, a virtual screen was implemented in the numerical model. The electron beam distribution at the screen position was divided into a grid with a cell width of 0.1 mm in both the horizontal and vertical directions. The beam size was then calculated as the root mean square of the projected transverse beam intensity distribution.

The three-dimensional (3D) quadrupole magnetic field distribution was simulated using the software CST EM Studio. The quadrupole magnet, designed by Chaipattana Saisa-ard, serves the purpose of focusing and controlling the electron beam with a maximum energy of 30 MeV. The quadrupole magnets with this design have been integrated into our accelerator system after the linac acceleration and MIR and THz FEL beamlines for various functions, including achromat magnetic systems, emittance measurement setups, beam size control in TR stations, and beam transport lines. The quadrupole magnet was designed and constructed with a bore radius of 20 mm, considering the vacuum tube outer diameter of approximately 38.1 mm. Each magnetic pole consists of 201 turns of coil windings. A 3D model of the quadrupole magnet and an actual fabricated magnet are shown in Figure 4a and Figure 4b, respectively. Manufacturing errors were controlled to be less than 50 µm based on the design requirement. All quadrupole magnets in our accelerator and beamlines were dimensional inspected and the magnetic field measured to verify their quality before the installation.

In the CST EM Studio simulation, the magnetic properties of locally available low-carbon steel were included. The magnetic behavior of this material was characterized by measuring the relationship between magnetic field and permeability using a split-coil permeameter at Deutsches Elektronen-Synchrotron (DESY), Germany [35]. Both quadrupole magnets (Q17 and Q18) in our emittance measurement setup have the same design and construction standards, ensuring equivalent reliability. The simulated relationship between the quadrupole gradient and coil current (G vs. I), shown in Figure 4c, exhibits linearity for currents up to 8 A. To ensure the operating temperature remains below 60 °C, the coil current is limited to a maximum of 5 A, corresponding to a maximum quadrupole gradient of 6.22 T/m. The linear fit equation for this operating range is given by the following:

$$G\left(\mathrm{T}/\mathrm{m}\right)=1.24I\left(A\right)+0.01.$$

(11)

The effective length of the quadrupole magnet $l_{eff}$ was determined using the following integral formula:

$$l_{\mathrm{eff}}={\displaystyle \frac{\int G\left(z\right)dz}{G_0}},$$

(12)

where $G\left(z\right)$ represents the gradient distribution along the longitudinal axis of the magnet, and $G_0$ is the central gradient. To evaluate the magnetic field characteristics, the coil current was set to half of the maximum operational current. The gradient distribution along the z-axis is illustrated in Figure 5a, from which the effective length was calculated to be 111 mm. The gradient distribution along the x-axis, including its deviation, is shown in Figure 5b. The good field region extends over 16 mm, where the gradient deviation remains below 0.1%, ensuring beam-focusing accuracy.

Preliminary simulation was performed and the results indicate that using a single quadrupole magnet for beam size scanning is not feasible, as it causes beam loss before reaching the screen station. To prevent this and ensure accurate emittance measurement, a dual-quadrupole configuration was implemented. In this investigation, the quadrupole Q18 was used to scan the electron beam size, while the quadrupole Q17 was applied to control the beam size in the orthogonal plane, ensuring that at least 99% of the particles successfully reached the screen station. This configuration offers two key advantages: first, it provides measurement flexibility by enabling a wider range of beam sizes at the observation screen, which improves sampling and leads to a more reliable parabolic fit in emittance determination. Second, controlling the beam size and shape with the quadrupole Q17 reduces systematic errors. As a result, the effects of aberrations and potential beam clipping, which can distort measurement accuracy, are minimized. During the study of measurement process, the quadrupole Q17’s strength was optimized for each beam configuration to achieve the desired beam size conditions on the screen. Then, the quadrupole Q18 was scanned while the quadrupole Q17 was held constant each beam configuration. This approach simplifies beam optics modeling and reduces potential sources of error. Varying both magnets simultaneously would complicate the analysis, introduce fitting degeneracies, and increase systematic uncertainties. Holding the quadrupole Q17’s strength fixed allows for isolation of the quadrupole Q18’s effect and facilitates a straightforward, reliable fitting procedure for emittance evaluation. In our setup, the quadrupoles Q17 and Q18 have the same effective lengths ($l_{eff,1}$ = $l_{eff,2}$) of 110 mm. With known values of $L_1$, $L_2$, and $f_1$, the coefficients $a_1$–$a_3$, $b_1$–$b_3$, and $c_1$–$c_3$ in Equation (8) become constants, resulting in a simplified analysis using Equations (4)–(7).

3.3. Results and Discussion

The emittance measurement simulation was initially performed using a real beam, represented by an electron beam distribution at the dipole D5 exit, obtained from the beam dynamics simulation described in Section 2 with the properties listed in Table 1. The initial scan of the quadrupole Q18 was first conducted in the horizontal direction. The quadrupole magnet Q17 was used to vertically focus the beam by adjusting its gradient from 0 to −3.12 T/m in steps of 0.62 T/m. Simultaneously, the quadrupole Q18 was adjusted to horizontally focus the beam, with its gradient varying from 0 to 6.24 T/m in steps of 0.62 T/m. As an example, Figure 6 illustrates the relationship between bunch charge, horizontal beam size, and quadrupole Q18’s strength at the screen position, with the quadrupole Q17 gradient fixed at −2.49 T/m. The results of the initial quadrupole scan in the horizontal direction are shown in Figure 6a; a turning point was observed at a quadrupole Q18 strength of approximately 22 m⁻². Similarly, the results of the initial quadrupole scan for measuring the vertical emittance are presented in Figure 6b. The results show a turning point in the vertical direction at a quadrupole Q18 strength of approximately 15 m⁻². The bunch charge at the screen position as a function of quadrupole Q18 strength for the scans performed in both the horizontal and vertical directions is illustrated in Figure 6c. During the initial scans, the optimal gradient of quadrupole magnet Q17 for controlling the vertical beam size was found to range from −1.87 T/m to −3.12 T/m. The gradient of −2.49 T/m was chosen and it could maintain the beam on the observation screen during the scan of quadrupole Q18 within a strength range of 0 to 57 m⁻², ensuring the bunch charge remains above 99%. When the quadrupole Q18 strength exceeds 57 m⁻², the bunch charge drops below 99% due to excessive vertical beam size, causing part of the beam to fall outside the screen area and and therefore not be detected.

For a more precise investigation on the emittance measurement, the gradient of quadrupole Q18 was rescanned with finer steps of 0.12 T/m, corresponding to an applied coil current step increment of 0.1 A, to measure the transverse beam emittance under the optimal strength settings of the quadrupole Q17. The electron beam sizes at the screen were calculated using the root mean square (RMS) method, taking into account the screen resolution. Subsequently, the relationship between the square of the electron beam size and the quadrupole strength was determined. This relationship was then fitted using Equations (4)–(9) to obtain the polynomial coefficients and related parameters, which were used to compute the transverse emittance using Equation (10). Figure 7 illustrates examples of RMS beam sizes observed at the screen for varied strengths of the quadrupole magnet Q18, along with the corresponding quadratic fitting for the quadrupole magnet Q17 gradient of −2.49 T/m. The fitted polynomial coefficients show slight variations, with the fitting quadrupole strength ranging from 10.3 to 29.3 m⁻² (Figure 7a) and from 16.1 to 22.0 m⁻² (Figure 7b). However, despite these differences, the calculated transverse emittance remains consistent at 0.16 mm·mrad, indicating that small deviations in the quadratic fitting do not significantly affect the precision of emittance measurement to two decimal places. This consistency suggests that the quadratic fitting can be reliably applied within a quadrupole strength range of ±10 m⁻² around the turning point of the parabola. The horizontal transverse emittance obtained from the results shown in Figure 7 exhibits an error of 11.1% compared to the beam dynamic simulation when considering three decimal places of precision. For the vertical direction, the emittance measurement investigation was conducted following the same procedure, yielding a vertical transverse emittance of 0.18 mm·mrad, with an error of 12.5% at three decimal places. However, when considering values rounded to two decimal places, the results from our approach are highly precise.

To investigate systematic errors in emittance measurement, two factors were considered: the influence of the initial beam size and the misalignment of the quadrupole magnets. The effect of initial beam size was studied by generating an ideal beam at the dipole D5 exit under various conditions. In all cases, the energy spread was kept at 0.1%, while the initial transverse RMS beam size was varied from 1 to 4 mm, with the transverse beam emittance ranging from 0 to 1 mm·mrad. The emittance measurement simulation was then performed, and the results are presented in Figure 8. For initial emittances below 0.4 mm·mrad, the emittance values are close to the ideal linear response for smaller beam sizes (1 and 2 mm), while larger beam sizes (3 and 4 mm) show noticeable deviations. From these results, it was determined that initial emittance values ranging from 0.15 to 1 mm·mrad can be measured with a systematic error of less than 20%, when the incoming beam size remains below 2 mm. The estimated systematic error arises from a few factors. First, the quadrupole scan model relies on beam-matrix transport calculations using the thin-lens approximation, which does not fully account for higher-order optical effects, introducing minor discrepancies in the extracted emittance values. Additionally, the quadrupole scan model does not incorporate space-charge effects because it is based on the matrix transport approach. This omission may slightly affect the precision of the measured emittance, especially at higher beam intensities. Nevertheless, all beam dynamics simulations reported here include space-charge effects, ensuring that the estimated emittance measurements are highly accurate for small beam sizes below 2 mm. This establishes a criterion for the incoming beam size to reliably measure emittance in our setup. Another contributing factor is the energy spread, where the quadrupole focal length is dependent on electron energy. Variations in the energy of individual electrons in the whole beam lead to beam aberrations at the screen station, causing deviations in measured beam sizes. Despite these sources of error, the overall measurement accuracy remains within acceptable limits, particularly when the incoming beam size is controlled.

The accuracy of emittance measurements using the quadrupole scan technique is highly sensitive to the alignment of the quadrupole magnets. Any misalignment can introduce systematic errors that affect the reliability of the measured emittance values. In this study, we investigated how horizontal and vertical alignment errors of quadrupole magnets influence the accuracy of emittance measurements, with the aim of identifying acceptable tolerances for practical implementation. For the analysis, it was assumed that the electron beam travels along the central axis of the beam transport line within the vacuum chamber. The beam position was monitored at two downstream observation screens: SC11 and SC12. The quadrupole magnets used in our setup (Q17 and Q18) have a bore radius of 20 mm, while the vacuum beam pipe has an internal diameter of 38.1 mm, allowing a maximum radial offset of 1.9 mm for the physical installation of each quadrupole.

To consider the influence of alignment, we generated an ideal electron beam with a transverse Gaussian distribution and a beam size of 2 mm (rms) at the entrance of the quadrupole scan section. This beam size was chosen based on prior studies indicating that the incoming beam size with this value minimizes the sensitivity of emittance measurements to other error sources. In the simulations, horizontal and vertical alignment offsets were introduced independently to the two quadrupole magnets. Each magnet’s horizontal position was varied systematically from −1.9 mm to +1.9 mm. The beams with different initial emittance values and the beam size of 2 mm were guided through the misaligned quadrupole magnets during the quadrupole scan, and the beam size at the screen positions was recorded for each scan configuration. The corresponding emittance was then calculated using a parabolic fit to the beam size squared versus quadrupole strength, following standard quadrupole scan methodology. Initial simulation results confirmed that both horizontal and vertical misalignments produced similar levels of emittance measurement error, due to the symmetric field nature of quadrupole magnets. Therefore, the detailed analysis focused on horizontal misalignment as a representative case, with the understanding that the same results can be applied to vertical misalignment. Figure 9 illustrates the systematic errors introduced by different alignment configurations. The errors are most pronounced when the two quadrupole magnets were misaligned in opposite directions (e.g., +1.9 mm for Q17 and −1.9 mm for Q18). The degree of impact is found to strongly depend on the emittance of the input beam. For low-emittance beams (e.g., 0.15 mm·mrad), the relative error introduced by misalignment is more significant, as small distortions have a greater effect on the smaller phase space. In contrast, higher emittance beams exhibit more tolerance to alignment errors due to their larger phase space. These simulation outcomes reveal that if the off-axis displacement of each quadrupole magnet is kept within ±0.95 mm, the beam with the smallest emittance value of 0.15 mm·mrad can be measured with the systematic error below 33.3%. This result provides a practical guideline for alignment tolerances in experimental setups where high-precision emittance measurements are required.

This analysis provides significant insights into practical strategies for optimizing the accuracy of emittance measurements using the quadrupole scan method. The results underscore the critical role of precise quadrupole alignment, particularly for low-emittance beams. Furthermore, to minimize the measurement error, one effective strategy is to precondition the beam by adjusting the incoming beam size prior to entering the quadrupole scan section. This can be achieved by carefully tuning the upstream quadrupole magnets located before the dipole magnet D5, which helps to shape the beam and minimize initial envelope distortions. Additionally, continuous monitoring of the beam trajectory using screen stations SC11 and SC12 is essential for verifying beam centering throughout the measurement process. Together, these integrated optimization strategies contribute to enhancing both the precision and reliability of emittance measurements.

4. Electron Bunch Length Measurement System

The measurement of the electron bunch length will be conducted at the entrance of the undulator magnet using the TR-2 station, as shown in Figure 3. The bunch length measurement setup is illustrated in Figure 10. This setup utilizes an aluminum (Al) foil radiator, which has a thickness of 25 μm and a diameter of 22 mm. The Al foil is tilted at an angle of 45° relative to the electron beam direction, enabling backward transition radiation (TR) to be emitted from the radiator surface toward a high-density polyethylene (HDPE) window. The THz radiation is generated at the interface between the vacuum and Al, where the relative dielectric constant of Al in the THz range is approximately 1 × 10⁵ which is significantly higher than that of the vacuum ($ϵ$ = 1). Consequently, the Al radiator behaves as a perfect conductor, efficiently producing transition radiation. The HDPE window, positioned below the beamline, serves to separate the vacuum section from ambient air. This HDPE window, with a diameter of 36 mm and a thickness of 1.25 mm, is capable of transmitting over 80% of THz radiation [36]. A 90° gold-coated parabolic mirror, located immediately below the HDPE window, collects the backward TR and directs it into a Michelson interferometer for further analysis.

The Michelson interferometer system comprises two optical paths for analyzing the radiation. All optical components, numbered 2 through 11 in Figure 10b, are positioned at the same height on the optical table. The first flat mirror (no. 3) directs the radiation to a movable flat mirror (no. 4), which allows for beam path selection. When mirror no. 4 is positioned in the radiation path, the total radiation intensity is measured using a parabolic mirror (no. 11) to focus the radiation onto a pyroelectric detector (no. 9). Conversely, when mirror no. 4 is removed from the radiation path, the radiation travels through the second optical path, which is used to generate an interferogram providing information about the electron bunch length. This interferometer path includes a beam splitter (no. 7), a fixed flat mirror (no. 5), a movable mirror (no. 6), another parabolic mirror (no. 10), and a pyroelectric detector (no. 8). To detect the THz transition radiation, two pyroelectric detectors (Model P1-series, Molectron Corporation, Sunnyvale, CA, USA) are used in the setups for both the energy measurement and the Michelson interferometer. These detectors, made of lithium tantalate (LiTaO₃), offer a uniform response across a broad spectral range (0.1 to 1000 μm). They are capable of measuring high-power radiation at room temperature, with an operating temperature ranging from −55 °C to +85 °C. To enhance absorption efficiency, their surfaces are coated with a thin film of black organic material. These detectors were calibrated using a Joulemeter (Model SPJ-A-8-OB, Spectrum Detector Inc., Lake Oswego, OR, USA).

The radiation entering the Michelson interferometer is split into two paths by a beam splitter. One part is reflected toward a fixed mirror, while the other part transmits through the beam splitter to a movable mirror. Both partial beams are then reflected back to the beam splitter, where they are recombined and directed toward the detector. The detector measures the intensity of the recombined beam. As the movable mirror is translated, the detector records intensity variations, establishing the relationship between intensity and the optical path difference, $\delta$, providing a result known as the “interferogram”. For an electron bunch distribution that follows a Gaussian profile in the longitudinal direction, the full width at half maximum (FWHM) of the interferogram can be used to determine the electron bunch length (${\sigma }_z$), given by

$$\mathrm{FWHM}=4\sqrt{\ln 2}{\sigma }_z.$$

(13)

This relationship provides a method to extract the temporal length of the electron bunch. By accurately characterizing the bunch length, the setup enables precise control of the electron beam properties, thereby enhancing the performance of the MIR-FEL system.

4.1. Transition Radiation from 25 MeV Electron Beam

The transition radiation of the 25 MeV electron beam was investigated using beam properties in front of the undulator entrance. When a charged particle crosses the interface between two media with different dielectric constants, radiation is emitted due to the sudden change in the particle’s velocity at the boundary [37]. In the case of oblique incidence, an electron traveling through a vacuum ($ϵ$ = 1) transitions into a perfect conductor ($ϵ$ = ∞) at an incident angle $\psi$ relative to the electron trajectory. The resulting transition radiation consists of two polarization components: parallel polarization ($W^{\Vert }$), where the electric field lies in the radiation plane, and perpendicular polarization ($W^{\perp }$), where the electric field is perpendicular to the radiation plane. The total radiation spectral–angular distribution of the backward radiation is given by

$${\displaystyle \frac{d^{2}W}{d\Omega d\omega }}={\displaystyle \frac{d^2{W}^{\Vert }}{d\Omega d\omega }}+{\displaystyle \frac{d^2{W}^{\perp }}{d\Omega d\omega }},$$

(14)

where $\omega$ is the radiation angular frequency, and $\Omega$ is the solid angle. The spectral–angular distributions of individual electron for both polarization components are expressed as follows:

$${\displaystyle \frac{d^2{W}^{\Vert }}{d\Omega d\omega }}={\displaystyle \frac{e^2{\beta }^2{\cos }^2\psi }{{\pi }^{2}c}}{\left[{\displaystyle \frac{\sin \theta -\beta \cos \varphi \sin \psi }{{(1-\beta \sin \theta \cos \varphi \sin \psi )}^2-{\beta }^2{\cos }^2\theta {\cos }^2\psi }}\right]}^2,$$

(15)

$${\displaystyle \frac{d^2{W}^{\perp }}{d\Omega d\omega }}={\displaystyle \frac{e^2{\beta }^2{\cos }^2\psi }{{\pi }^{2}c}}{\left[{\displaystyle \frac{\beta \cos \theta \sin \varphi \sin \psi }{{(1-\beta \sin \theta \cos \varphi \sin \psi )}^2-{\beta }^2{\cos }^2\theta {\cos }^2\psi }}\right]}^2,$$

(16)

where e is the electron charge, c is the speed of light, and $\beta$ is the relativistic velocity factor ($\beta =v/c$). The angle $\theta$ represents the emission angle relative to the negative z-axis, while $\varphi$ is the azimuthal angle in the $x-y$ plane, measured with respect to the negative x-axis.

The transition radiation is emitted in both the forward and backward directions. In our experimental setup, the electron beam strikes the interface of the Al foil at an incidence angle of 45° ($\theta$ = 45°). To collect the backward radiation, a parabolic mirror is positioned below the beamline. The intensity of TR within the acceptance angle (${\theta }_a$) is determined by integrating the solid angle of Equation (14). The parabolic mirror, responsible for collecting radiation, has a diameter of 50.8 mm and a focal length of 127 mm. It is placed 127 mm from the center of the Al radiator, providing an acceptance angle of 11.3° (197 mrad). The calculation of TR using Equations (14)–(16) was performed considering an electron beam with the properties listed in Table 1. The resulting radiation distributions are presented in Figure 11. The radiation distribution clearly shows no emission at the center (with the zero angle defined as the direction at 45° between the $x^{\prime }$ and $-z^{\prime }$ axes), with intensity peaking around $\theta \sim 1/\gamma$ or approximately 0.02 rad, where $\gamma =1/\sqrt{1-{\beta }^2}$ is the Lorentz factor. The distribution is symmetric in the vertical angle but displays asymmetry in the horizontal angle. Figure 11b illustrates cross-sections of the spectral–angular distribution in both horizontal and vertical directions, clearly demonstrating the horizontal asymmetry. This asymmetry arises due to the oblique incidence of electrons in the $x^{\prime }-z^{\prime }$ plane, where radiation emitted closer to the interface exhibits higher intensity compared to that emitted near the $z^{\prime }$-axis. The collection efficiency of the transition radiation within the acceptance angle was determined to be approximately 49%, ensuring that a significant portion of the emitted radiation is effectively captured and analyzed.

4.2. Effect of Transverse Beam Size on Bunch Length Measurement

The radiation intensity from a single electron, $I_e\left(\omega \right)$, is proportional to the square of the absolute value of the electric field ($I_e\propto {\left|E\right|}^2$). In the case of radiation wavelengths equal to or longer than the electron bunch length, the total radiation intensity ($I\left(\omega \right)$) is proportional to the square of the number of electrons in the bunch (N), expressed as $I(\omega ,\theta )\approx N^2{I}_e\left(\omega \right)f(w,\theta )$, where $f(w,\theta )$ is the bunch form factor. This emitted radiation is referred to as “coherent radiation”. Conversely, for radiation wavelengths shorter than the bunch length, the radiation is incoherent, meaning the total radiation intensity is proportional to the number of electrons: $I\left(\omega \right)\approx N{I}_e\left(\omega \right)$. In the case of a Gaussian distribution in both the transverse and longitudinal directions, the bunch form factor can be written as

$$f(\omega ,\theta )=e^{-{\left(\omega {\sigma }_{\rho }sin\theta /c\right)}^2}{e}^{-{\left(\omega {\sigma }_{z}cos\theta /c\right)}^2},$$

(17)

where ${\sigma }_{\rho }=\sqrt{x^2+y^2}$ is the standard radial size, $\theta$ is the observation angle ($\theta$ = 0 is defined along the direction that lies at 45° between the $x^{\prime }$ and $-z^{\prime }$ axes), and ${\sigma }_z$ is the electron bunch length.

From Equation (17), the first exponential term becomes significant when the observation angle is not zero ($\theta \ne 0$). This reduces the bunch form factor, which affects the accuracy of the bunch length measurement. To reduce the transverse size, quadrupole magnets Q17 and Q18 were used together for focusing the electron beam at the radiator to be as small as possible. With the magnets Q17 and Q18 set to gradients of 3.62 T/m and −2.37 T/m, respectively, the standard deviations from the Gaussian fitting are 0.17 mm and 0.20 mm in the horizontal and vertical directions, respectively. The corresponding R² values are 0.94 and 0.96, indicating good agreement between the data and the fit. Based on these results, the standard radial size is calculated to be 0.26 mm. The electron bunch length, determined from Gaussian fitting of the longitudinal beam distribution, is 208 fs, with an R² value of 0.97. Figure 12 shows the longitudinal and transverse distributions of the electron beam at the TR-2 station. By considering the bunch form factor, the spectrum was calculated within the acceptance angle. The interferogram was obtained by applying the inverse Fourier transform. The results shown in Figure 13 indicate that the TR spectra exhibit a reduction in spectral intensity at higher wavenumbers when the transverse beam size is taken into account. The expected wavenumber range of the TR spectrum was defined by comparing the measured spectral intensity with that of the incoherent case. Wavenumbers where the measured intensity exceeded the incoherent level were identified as part of the coherent TR range When the beam size is zero, the spectral range extends from 0 to 114 cm⁻¹, whereas for a standard radial size of 0.26 mm, the spectral range is from 0 to 110 cm⁻¹. This leads to slight differences in the interferogram width, where the calculated electron bunch length was found to be 220 fs, which is 6% different from the case without considering the transverse beam size.

4.3. Effect of Beam Splitter on Bunch Length Measurement

A beam splitter in the Michelson interferometer can affect the accuracy of the bunch length measurement [34,38] due to beam splitter interference. Consequently, its properties, including the refractive index ($n_2$) and thickness (d), must be investigated. For two polarizations, the beam splitter efficiency is

$${\eta }_b=\left(\right|R_{\Vert }{T}_{\Vert }{|}^2+|R_{\perp }{T}_{\perp }{|}^2)/2,$$

(18)

where R and T are the reflectance and transmittance, which can be written as follows [39]:

$$R=-\left|r\right|{\displaystyle \frac{1-e^{i\psi }}{1-r^2{e}^{i\psi }}},$$

(19)

$$T=-(1-r^2){\displaystyle \frac{e^{i\psi /2}}{1-r^2{e}^{i\psi }}}.$$

(20)

From the Fresnel’s equations, the reflection coefficients for parallel ($r_{\Vert }$) and perpendicular ($r_{\perp }$) polarizations can be written as follows [40]:

$$r_{\Vert }={\displaystyle \frac{n_{1}cos{\theta }_1-n_{2}cos{\theta }_2}{n_{1}cos{\theta }_1+n_{2}cos{\theta }_2}},$$

(21)

$$r_{\perp }={\displaystyle \frac{n_{1}cos{\theta }_2-n_{2}cos{\theta }_1}{n_{1}cos{\theta }_2+n_{2}cos{\theta }_1}},$$

(22)

where ${\theta }_1$ represents the angle of incidence, and ${\theta }_2$ denotes the angle of refraction within the beam splitter as the second medium.

In the case of a ${45}^{\circ }$ incident angle, the phase difference between two parallel surfaces with thickness d is given by $(2\omega d/c)\sqrt{n_2^2-1/2}$, where $\omega$ is the angular frequency and c is the speed of light. In this study, Kapton (polyimide film) and silicon (Si) beam splitters were considered due to their low absorption coefficients in the THz frequency range [41]. The selected thicknesses were based on the availability of the beam splitters in our laboratory and their accessibility for online purchase. For the Kapton beam splitter, the thickness are 25.4, 50.8, and 127 µm. The reflective index is approximately 1.67. For the Si beam splitter, the considered thicknesses are 0.5, 1.5, 3.0, 3.5, 5.0, 6.0, and 10.0 mm with a refractive index of about 3.4. An example of the efficiency calculation result in Figure 14 shows that all beam splitters exhibit periodic variations in efficiency. The efficiency drops to zero at certain wavenumbers where the radiation interferes destructively between the two surfaces of the beam splitter. Thinner beam splitters provide broader spectra compared to thicker ones. However, the regions where efficiency drops are less significant for thinner splitters. In the case of the Si beam splitter, it shows much stronger fluctuations in efficiency compared to the Kapton beam splitters. Furthermore, the maximum efficiency of the Si beam splitter is higher than the Kapton ones.

The calculated interferogram was obtained using a Gaussian distribution in the longitudinal direction of the electron beam with a bunch length of 208 fs. The results including the beam splitter efficiency are presented in Figure 15. To examine the impact on bunch length measurement, simulated interferograms are generated by applying the inverse Fourier transform to the radiation spectrum. Figure 16a shows examples of interferograms resulting from beam splitter efficiency. In the case of Kapton beam splitters, negative valleys appear near the main peak of the interferograms, leading to a measured bunch length smaller than the actual value. The valley moves away from the main peak as the beam splitter thickness increases. In the case of the Si beam splitter, the valley is located away from the main peak. Therefore, this beam splitter does not affect the bunch length measurement.

The relationship between the estimated FWHM bunch lengths—derived from interferograms affected by the beam splitter—and the corrected FWHM bunch lengths (i.e., the bunch lengths prior to the introduction of the beam splitter effect) is shown in Figure 16b. The correction was performed by modeling the spectral distortion caused by the beam splitter and applying this model to retrieve the original bunch length values. The plots are calculated for electron bunch lengths ranging from 50 to 500 fs, corresponding to interferogram FWHM values from 50 to 500 µm, which cover the expected electron bunch lengths at the undulator entrance. For Kapton beam splitters, the measured FWHM bunch length is underestimated when it exceeds twice the thickness of the beam splitter. In contrast, for Si beam splitters, the negative valleys are not located near the main peak, allowing the actual electron bunch length to be directly measured from the interferogram width. These graphs provide a correction approach for bunch length measurements across all studied beam splitters. However, the Si beam splitter is more suitable than the Kapton beam splitter for straightforward electron bunch length measurements. Most optical components used in the design bunch length measurement setup have a diameter of 76.2 mm. Among commercially available options, the Si beam splitter with this diameter has a thickness of only 6 mm. Therefore, a 6 mm thick Si beam splitter has been selected for our future bunch length measurement station. The key devices required for this setup are listed in

Table 2. Specifications and details of components in bunch length measurement station.

Number	Component	Specification
1	radiator	material: Aluminium-foil thickness: 25 µm diameter: 22 mm
2	parabolic mirror	coating surface: gold diameter: 50.8 mm focal length: 127 mm
3–6	flat mirror	coating surface: Gold diameter: 76.2 mm
7	beam splitter	material: Silicon thickness: 6 mm diameter: 76.2 mm refractive index: 3.4
8–9	pyroelectric detector	material: LiTaO3 detected wavelength: 0.1–1000 µm operating temp.: −55 °C to +85 °C
10–11	parabolic mirrors	coating surface: Gold diameter: 76.2 mm focal length: 127 mm

.

4.4. Effect of Radiator Finite Size

The analyses presented in the previous sections assume that the radiator has an infinite transverse size. In reality, however, the coherent transition radiation (TR) spectrum is influenced by the finite dimensions of the radiator [42]. This influence becomes significant when the product of the Lorentz factor ($\gamma$) and the radiation wavelength ($\lambda$) exceeds the transverse size of the radiator. In such cases, the spatial coherence of the emitted radiation is altered, particularly at lower frequencies, leading to a suppression in spectral intensity. The effect of the radiator’s finite size on the intensity of the coherent TR spectrum, $I_f$, can be described as follows:

$$I_f\left(\omega \right)=N_e^2{\int }_0^{2\pi }{\int }_0^{{\theta }_a}{I}_e(\theta ,\omega )f\left(\omega \right){\left[1-J_0(krsin\theta )\right]}^{2}sin\theta d\theta d\varphi ,$$

(23)

where the factor ${\left[1-J_0(krsin\theta )\right]}^2$ accounts for the influence of the radiator’s finite radius r, with $k=2\pi /\lambda$. Here, $I_e\left(\theta \right)$ represents the spectral intensity emitted by a single electron at angle $\theta$, assuming normal incidence. The electron bunch is modeled as a longitudinal Gaussian distribution with form factor $f\left(\omega \right)=e^{-{(\omega {\sigma }_z/c)}^2}$, where ${\sigma }_z$ is the RMS bunch length and $N_e$ is the number of electrons in the bunch. The collected radiation is integrated up to an acceptance angle ${\theta }_a$.

Figure 17a shows the total radiation spectrum for a bunch length of 208 fs and a beam energy of 25 MeV, assuming an 11 mm radiator radius and a 197 mrad acceptance angle. As illustrated, the finite size of the radiator leads to suppression of spectral components in the lower frequency range due to destructive interference effects. To evaluate the impact of this suppression on bunch length measurements, Figure 17b presents the interferograms obtained via inverse Fourier transform of the spectra. The FWHM of the interferogram for the finite radiator case is slightly smaller than that for the infinite case, due to the presence of secondary valleys near the main peak. In this example, the electron bunch length inferred from the finite radiator case is 186 fs, which underestimates the actual value by approximately 10.6%.

Beyond the transverse beam size, the effect of the beam splitter, and the radiator finite size, several other factors may influence the TR spectrum and impact the accuracy of bunch length measurements. These include the electron beam’s energy spread, beam divergence, diffraction caused by finite apertures in the optical transport system, and optical misalignment. Electrons with varying energies emit radiation with different spectral characteristics, which broadens the measured spectrum and introduces uncertainty in the inferred bunch length. Similarly, beam divergence affects the angular distribution of the emitted radiation, as the emission angle depends on each electron’s trajectory, potentially leading to spectral shifts or distortions. Additionally, diffraction effects arising from finite mirror apertures and optical misalignments can further alter the collected spectrum by introducing interference patterns or misdirected light. Although these effects are not explicitly modeled in the present study, they are important sources of error and should be considered in future work aimed at improving the precision of bunch length measurements.

5. Conclusions

The coherent mid-infrared free electron laser (MIR-FEL) is one type of radiation that can be generated at the PCELL facility. The expected electron beam properties at the undulator entrance for MIR-FEL production were simulated using the ASTRA program. The characterization of the electron beam properties is essential. In this study, two measurement stations were designed to characterize electron beam properties, including transverse beam emittance and electron bunch length, and to estimate systematic errors. These stations were designed to measure beam properties at the undulator entrance. The ASTRA simulation indicates that an electron with a kinetic energy of 25 MeV should have transverse emittances of 0.18 and 0.16 mm·mrad in the horizontal and vertical directions, respectively, and an RMS bunch length of 219 fs (equivalent to 65.6 $\mu$m).

The quadrupole scan technique was chosen to measure the transverse beam emittance. The setup consists of two quadrupole magnets and a screen station. One quadrupole magnet is used for controlling the beam size, while the other one scans. To estimate the systematic error, the ASTRA program was employed. The positions of the two quadrupole magnets and the screen station are set according to the current installation positions. In the simulation, the resolution of the screen station, which is 0.1 mm/pixel, was considered. The result shows that, for measuring the transverse emittance in a sub mm.mrad, the incoming transverse electron beam size should be equal or less than 2 mm. The initial emittance values ranging from 0.15 to 1 mm·mrad can be measured with a systematic error of less than 20%. To ensure that the incoming electron beams have a beam size of 2 mm or less, the quadrupole magnets located before the dipole magnet (D5) should be adjusted accordingly. Misalignment between the two quadrupole magnets can introduce significant errors in the emittance measurement. Simulation results indicate that by maintaining the transverse (off-axis) alignment error below 0.95 mm, the lowest emittance value of 0.15 mm·mrad can be obtained with a corresponding systematic error of less than 33.3%. This highlights the importance of precise magnet alignment in minimizing measurement uncertainty and ensuring reliable characterization of the beam’s transverse phase space.

The electron bunch length was measured at the undulator entrance via the TR2 station. An Al foil, tilted at an angle of 45° with respect to the electron beam trajectory, was used as the radiator. A two-inch parabolic mirror with a focal length of 127 mm was positioned below the beamline, with its focal point at the center of the radiator, to collect the backward radiation to the Michelson interferometer with an efficiency of about 49%. The Michelson interferometer was employed to measure the electron bunch length. The FWHM of the interferogram can be used to determine the electron bunch length. Two quadrupole magnets in front of the TR-2 station were used to focus the electron beam at the TR-2 station, and the standard radial size was found to be 0.26 mm. This led to a bunch length measurement error of 6%. The beam splitter in the Michelson interferometer can introduce an error in the bunch length measurement. Kapton and Si beam splitters with varying thicknesses were used to investigate this error, and it was found that the 6 mm Si beam splitter can be used to measure the electron bunch length without introducing an error. The finite size of the radiator influences the characteristics of the transition radiation, particularly by suppressing low-frequency components in the spectrum. This spectral distortion leads to an underestimation of the electron bunch length when reconstructed from the TR interferogram. In the case studied, the measured bunch length was approximately 10.6% shorter than the ideal value obtained under the assumption of an infinitely large radiator.

This study contributes to developing beam diagnostics for accelerator-based light sources, particularly in optimizing MIR-FEL performance. Despite these achievements, limitations remain. The quadrupole scan method assumes a thin-lens approximation, which does not fully account for higher-order optical effects. Space-charge effects were included in the simulations, which may introduce small deviations in simulation results. For the bunch length measurement, further optimization of the interferometer design and calibration procedures will be necessary for improved precision. Future work will involve the experimental implementation of these diagnostic stations, verification of simulation results with real beam measurements, and refinement of the measurement techniques. The developed system will support ongoing MIR-FEL research and facilitate applications such as ultrafast spectroscopy and material characterization using MIR-FEL radiation.