Generation and detection of transition radiation

At the experimental station, an aluminum (Al) foil serves as the radiator. By tilting the radiator 45° relative to the electron-beam direction, backward transition radiation is emitted at the radiator surface and subsequently collimated by a parabolic mirror. The collimated radiation passes through a high-density polyethylene (HDPE) window, as illustrated in Fig. 2. The radiation then enters the measuring system for energy, power, transverse-profile, polarization, and spectral-range measurements. Because TR is directly related to the electron-bunch length, it is also employed to measure this parameter.

Fig. 2Full-screen View

(Color online) Setup for generating backward transition radiation and characterizing radiation properties

The Al-foil radiator has a thickness of 25 μm and diameter of 24 mm. THz radiation is generated at the interface between the vacuum and Al foil. In the THz regime, the relative dielectric constant of Al is approximately 1 × 10⁵, which is significantly higher than that of vacuum ($ϵ=1$). Therefore, Al foil can be assumed to be a perfect conductor. A 90° gold-coated parabolic mirror is positioned at the focal point of the mirror within the vacuum chamber below the center of the radiator. It has a diameter of 25.4 mm and focal length of 76.2 mm. The HDPE window has a diameter of 32 mm and thickness of 1.25 mm. The majority of the radiation in the THz regime can exit the HDPE window with more than 80% transmission.

Consequently, the radiation enters the measuring system and is divided into two parts using a flipable flat mirror. The first part involves the measurement of the radiation profile, energy, and power. The radiation profile is captured using a Pyrocam III camera featuring 124×124 sensor arrays with active dimensions of 1.24 cm ×1.24 cm. A parabolic mirror is employed to collimate the radiation onto the detector, with a diameter of 50.8 mm and focal length of 101.6 mm. Furthermore, the setup allows the measurement of radiation polarization. To achieve this, a wire-grid polarizer (Graseby-Space Model IGP223) with a 4-μm wire-grid photo-lithed onto a polyethylene substrate is used. By placing the polarizer in front of the Pyrocam III and rotating it, the polarization in various directions can be measured. The second part involves measuring the radiation spectrum and electron-bunch length using a Michelson interferometer. The details of the Michelson interferometer are described in Sect. 3.4. Following the characterization, the radiation is guided to the experimental room by another parabolic mirror (M2).

Optimization of electron-beam properties

To optimize the electron-beam properties, A Space Charge Tracking Algorithm (ASTRA) software [24] was used to simulate the electron-beam dynamics throughout the accelerator system, starting from the RF gun to the TR experimental station. In our study, the space-charge effect was included in the simulation. We generated four million macroparticles at the cathode of the RF gun to initiate the simulation. These macroparticles were then separated into groups using a 2856 MHz RF wave, forming an electron bunch. Our simulation specifically focused on studying a single electron bunch, assuming that other electron bunches had similar characteristics. The electron bunch at the gun exit had an average energy of 1.92 MeV and a bunch charge of 160 pC. These electrons then traveled to the alpha magnet, which functioned as a magnetic bunch compressor. The beam had a bunch length of 69.2 ps at the entrance of the alpha magnet. Within the magnet, higher-energy electrons traveled a longer path than lower-energy electrons. Consequently, the lower-energy electrons eventually caught up with the higher-energy electrons at the designated position, which, in this case, was the experimental station. This process resulted in the compression of the electron bunch, leading to a shorter electron-bunch length. The alpha-magnet gradient was varied to obtain the shortest electron-bunch length at the experimental station. Additionally, the energy slit inside the alpha magnet was used to filter out low-energy electrons. This filtering process optimized the energy spread of the beam and enabled selective control of the bunch charge.

After exiting the alpha magnet, the beam traversed the three-meter linac and gained energy of up to 22 MeV. Subsequently, the size of the transverse electron beam at the TR station was measured using two quadrupole magnets. The performance of the alpha magnet was evaluated by varying its gradients, and the results showed that a gradient of 420 g/cm yielded the shortest bunch length of 200 fs at the TR station for an electron-beam kinetic energy of 22 MeV, as shown in Fig. 3. The simulated longitudinal and transverse electron-beam distributions at the TR station are shown in Fig. 4. The histogram provides a representation of the longitudinal shape of the electron beam and exhibits a root-mean square (RMS) value of 352 fs. To further characterize the beam profile, a Gaussian fit was applied to the histogram, yielding a standard deviation that matched the RMS value obtained from the histogram. The electron-beam parameters obtained from the simulations are listed in Table 1.

Fig. 3Full-screen View

Electron-bunch length optimization at TR station by varying alpha-magnet gradients. The results were obtained from an ASTRA simulation

Fig. 4Full-screen View

(Color online) (a) Simulated longitudinal electron-beam distribution (blue dot) including energy histogram (red) and Gaussian fit (black line). (b) Simulated transverse electron-beam distribution

Angular distribution and polarization

In our experimental setup, the incidence angle of the electron beam is 45° (ψ = 45°) with respect to the direction of travel. Backward TR radiation is then emitted perpendicularly. This radiation comprises two polarization components: parallel polarization, in which the electric field lies in the radiation plane, and perpendicular polarization, in which the electric field is perpendicular to the radiation plane. The spectral angular distribution of the TR generated from an electron with a kinetic energy of 22 MeV and its cross-section are shown in Fig. 5. The results indicate that no radiation is emitted at the center of the radiation distribution. The distribution reveals asymmetry in the horizontal angle. This asymmetry is clearly depicted in Fig. 5(b), which illustrates the cross sections of the spectral-angular distribution in the horizontal and vertical directions. The observed asymmetry in the distribution arises from the oblique incidence of electrons in the xz-plane. The radiation emitted near the interface exhibits a higher intensity than that emitted near the z-axis. The transition radiation exhibits radial polarization, as shown in Fig. 6. The polarization is asymmetric in the horizontal direction and symmetric in the vertical direction.

Fig. 5Full-screen View

(Color online) Spectral-angular distribution (a) and the cross section (b) of the backward spectral angular distribution for 45°-incidence transition radiation calculated from an electron with kinetic energy of 22 MeV

Fig. 6Full-screen View

(Color online) Calculated radiation spectral-angular distributions for horizontal polarization (left) and vertical polarization (right)

Effects of electron-bunch distribution

The radiation emitted by several electrons can be considered by adding all the radiation fields emitted by each electron in the bunch. The radiation intensity of each electron, $I_{\text{e}}\left(\omega \right)$, is proportional to the absolute value of the squared electric field $\left(I_{\text{e}}\propto {\left|E\right|}^2\right)$. Without considering the transverse-distribution effect, for radiation with a wavelength that is much longer than the bunch length, the phase difference of the radiation from each electron is small. Thus, the total radiation intensity is proportional to the square of the number of electrons in the bunch, leading to coherent radiation.

The influence of bunch distribution on the emitted radiation is investigated. To illustrate this phenomenon, a histogram representing the longitudinal distribution of the electron bunch is shown in Fig. 4(a), obtained from electron-beam-dynamics simulation. The coherent TR spectrum resulting from this bunch was analyzed and compared with a Gaussian distribution. By performing a Fourier transform on the longitudinal distribution of the simulated electron bunch, the resulting radiation spectrum was compared with that of a Gaussian bunch with a length of 352 fs (represented by its standard deviation, σz). The radiation spectra are shown in Fig. 7.

Fig. 7Full-screen View

Coherent TR spectrum of electron bunch with a length of 352 fs obtained from electron-beam-dynamics simulation and Gaussian distribution

Based on the obtained results, the radiation spectrum of the simulated electron bunch aligns well with that of the Gaussian bunch. The spectral range covers up to approximately 2 THz or a wavenumber of 70 cm^-1 for both the distributions. For simplicity, we assume a Gaussian spectrum for further investigation to explore the additional factors affecting the radiation spectrum. In this section, we focus on a scenario in which the electron beam has a kinetic energy of 22 MeV and bunch charge of 62.5 pC, as obtained from the simulations in Sect. 2.2. The bunch distribution is assumed to be Gaussian with an RMS width of 352 fs. The calculated coherent TR spectra for a 22-MeV electron beam at different bunch lengths are illustrated in Fig. 8. The effect of the transverse beam size was excluded.

Fig. 8Full-screen View

(Color online) Coherent TR spectra of electron bunches calculated from different bunch lengths

The plots in Fig. 8 clearly show that a shorter electron-bunch length corresponds to a broader radiation spectrum. In our specific case, the simulated electron-bunch length (σz) is estimated at 105.5 μm (352 fs), which results in an emitted TR of up to 70 cm^-1. Coherent radiation dominates when the electron-bunch length is equal to or shorter than the wavelength of the emitted radiation. Consequently, an increase in the electron-bunch length leads to a reduction in the coherent radiation at shorter wavelengths or higher wavenumbers, resulting in a narrower spectral range. However, by employing a shorter bunch length of 40 μm, we achieve a significantly broader spectral range extending up to 180 cm^-1.

In a realistic scenario, the electron bunches also have a transverse size that contributes to the form factor [21, 27]. The coherent radiation is suppressed when the beam size increases. In this case, the transverse bunch distribution reduces the form factor for a particular radiation wavelength. As a case study, we considered an electron bunch with a Gaussian distribution in both the transverse and longitudinal directions with a form factor that can be expressed as $f(\omega \mathrm{,}\theta )=f_t\left(\omega \mathrm{,}\theta \right)f_l\left(\omega \mathrm{,}\theta \right)$, where σρ represents the effective transverse beam size and θ denotes the observation angle, which is limited by the acceptance angle.

For an electron beam with a kinetic energy of 22 MeV, the radiation spectra, including the effect of the transverse size, are depicted in Fig. 9. Notably, these spectra were calculated within an acceptance angle of ±165 mrad (the angle at which the radiation is collected by the first parabolic mirror in our experimental setup) and for a fixed electron-bunch length of 105.5 μm. The effective transverse beam size, defined as ${\sigma }_{\rho }=\sqrt{{\sigma }_x^2+{\sigma }_y^2}$, is approximately 0.12 mm. The figure shows that the spectral range narrows as the transverse size increases. With a beam size of 2.5 mm, the coherent TR spectrum is suppressed at high frequencies and the spectral range is less than 60 cm^-1. To achieve a broader spectral range, a smaller beam size, ideally smaller than 0.1 mm, is necessary to obtain radiation up to a wavenumber of 70 cm^-1.

Fig. 9Full-screen View

(Color online) Calculated impact of transverse beam size on coherent TR spectra for a 22-MeV electron beam

Effect of finite radiator size

In previous sections, the radiator was assumed to be infinitely large. However, the influence of radiator size must be considered, as highlighted by Castellano et al. [28]. This effect becomes significant when the factor γλ exceeds the transverse dimension of the radiator, where γ is the Lorentz factor and λ is the radiation wavelength. To comprehensively analyze the total radiation spectrum, integration was performed over an acceptance angle of ±0.165 rad. A comparison between the electron bunches with kinetic energies of 8 and 22 MeV is presented in Fig. 10 by considering a radiator radius of 12 mm.

Fig. 10Full-screen View

(Color online) Spectra of TR radiation calculated from finite sizes of radiator collected within the acceptance angle of ±0.165 rad. The considered electron-bunch length in this case is 105.5 μm

The results in Fig. 10 indicate that some parts of the low frequencies are suppressed owing to the effect of the size of the radiator. For an electron beam with a kinetic energy of 22 MeV, the radiation spectrum is suppressed at a frequency lower than 0.3 THz (10 cm^-1). The same result was obtained for an electron kinetic energy of 8 MeV. This suppression is consistent with the experimental findings reported in [20]. More importantly, this frequency is the lower limit for designing radiation-transportation systems.

Effect of mirror in Michelson interferometer

The Michelson interferometer serves as a tool for measuring both the radiation spectrum and electron-bunch length. However, when using an interferometer, the diffraction effect caused by a finite aperture during radiation propagation must be considered. Subsequently, optimizing the size of the mirrors used in the interferometer is essential to achieve the best performance of the system. Fraunhofer diffraction is employed to determine the energy transmission of the radiation passing through the Michelson interferometer, assuming circular apertures [29].

As depicted in Fig. 2, the radiation emitted from the first parabolic mirror (with a diameter of 25.4 mm) travels through the HDPE window before entering the Michelson interferometer. Subsequently, the radiation impinges on the flat mirrors with a diameter of 50.8 mm and is then reflected back to the beam splitter before being collected by another parabolic mirror onto the detector. The HDPE window is positioned at a distance of 76.2 mm away from the first parabolic mirror. A flat mirror M1 is situated inside the Michelson interferometer at a distance of 367.3 mm from the HDPE window. Moreover, the distance between the flat mirror M1 and parabolic mirror P2 is 215.9 mm.

The energy transmissions calculated based on the aforementioned setup illustrated in Fig. 11 indicates that radiation-energy density is suppressed at low frequencies when the effect of the mirrors in the Michelson interferometer is considered. In this figure, the diameters of the flat and parabolic mirrors are 50.8 mm and 25.4 mm for diffraction set 1 and 76.2 mm and 50.8 mm for diffraction set 2, respectively. The figure shows that doubling the size of the mirrors does not significantly decrease the energy density.

Fig. 11Full-screen View

(Color online) Coherent transition-radiation spectra demonstrating the influence of mirror diffraction in a Michelson interferometer, calculated for an electron beam with a kinetic energy of 22 MeV and a bunch length of 105 m

Effects of beam splitters

In a Michelson interferometer, a beam splitter divides the radiation into two paths, which are subsequently recombined to form an interferogram pattern. Conventionally, the reflectance and transmittance of a beam splitter were assumed to be constant across all frequencies. However, in the THz region, an appropriate beam splitter exhibits variations in reflectance and transmittance owing to multiple reflections within the beam-splitter materials [30]. This phenomenon can be predicted as follows. The total amplitudes of the reflection (R) and transmission (T) coefficients are defined as follows (Hecht, 2002): $R=-\left|r\right|\frac{1-e^{i\psi }}{1-r^2{e}^{i\psi }}\mathrm{,}$ (1) $T=-\left(1-r^2\right)\frac{e^{i\psi /2}}{1-r^2{e}^{i\psi }}\mathrm{.}$ (2) For an incident angle of 45°, the phase difference (ψ) between two parallel surfaces with thickness d is given by $(2\omega d/c)\sqrt{n^2-1/2}$, where n is the refractive index of the beam splitter, ω is the angular frequency, and c is the speed of light. The following holds true for all frequencies: ${\left|R\right|}^2+{\left|T\right|}^2=1$. However, the amplitude of the reflection (r) varies with the polarized radiation. Fresnel’s equations describe this difference for parallel ($r_{\parallel }$) and perpendicular ($r_{\perp }$) polarization (Hecht, 2002): $r_{\parallel }=\frac{n_1\cos \left({\theta }_1\right)-n_2\cos \left({\theta }_2\right)}{n_1\cos \left({\theta }_1\right)+n_2\cos \left({\theta }_2\right)}\mathrm{,}$ (3) $r_{\perp }=\frac{n_1\cos \left({\theta }_2\right)-n_2\cos \left({\theta }_1\right)}{n_1\cos \left({\theta }_2\right)+n_2\cos \left({\theta }_1\right)}\mathrm{.}$ (4) In this context, θ₁ represents the incident angle and θ₂ denotes the refractive angle in the beam splitter, functioning as the second medium. The efficiency of the beam splitter is defined as $|RT{|}^2$, and for combined polarization, the efficiency becomes $\left({\left|R_{\parallel }{T}_{\parallel }\right|}^2+{\left|R_{\perp }{T}_{\perp }\right|}^2\right)/2$. In our case, Kapton (polyimide film) and Si beam splitters were chosen because of their low absorption coefficients in the THz region [19, 31]. The following sections discuss the effects of these two beam-splitter types. The chosen thicknesses of the beam splitters were based on their availability in our laboratory.

3.5.1

Beam splitter and radiation spectrum

The beam-splitter efficiencies of Kapton at various thicknesses are shown in Fig. 12 (left) and the corresponding effects on the radiation spectra are shown in Fig. 12 (right). The efficiency fluctuates across the frequency range, reaching zero at specific frequencies at which the interfering radiation from both surfaces of the beam splitter becomes destructive. For a 25.4-μm Kapton beam splitter (with $n\sim 1.67$), the efficiency spans a wide frequency range up to 115 cm^-1 before the first zero. However, thinner beam splitters exhibit strong suppression at low frequencies. Increasing the beam-splitter thickness can recover the low-frequency spectrum, for example, reaching 5 cm^-1 with a thickness of 127 μm. Moreover, the significant difference between the p-polarization and s-polarization leads to a low efficiency for the Kapton beam splitter of approximately 0.18. This issue can be addressed by utilizing a beam splitter with a higher refractive index, such as Si (with $n\sim 3.4$).

Fig. 12Full-screen View

(Color online) Calculated assessing efficiency and radiation-spectrum impact of Kapton beam splitters with thicknesses of 25.4, 50.8, and 127 μm and Si beam splitter with thickness of 2 mm

The efficiency of the Si beam splitter with a thickness of 2 mm is presented in Fig. 12. The calculated results indicate that the Si beam splitter offers higher efficiency for both p-polarized and s-polarized radiation, reaching a maximum value of 0.25 and an average value of approximately 0.21. Additionally, the Si beam splitter demonstrates good coverage at low frequencies, particularly below 0.5 cm^-1 for a 2-mm thickness. However, because of the higher refractive index of Si and thickness of the beam splitter, zero efficiency is observed at closely spaced frequencies, as shown in Fig. 12. For the 2-mm Si beam splitter, the frequency spacing between zeros is at 0.75 cm^-1, resulting in numerous sharp dips in the radiation spectrum. These are particularly noticeable when the spectral resolution is lower than the zero-frequency spacing of the Si beam splitter. In such cases, the radiation spectra contain an average beam-splitter response.

Following the investigation, the results showed that the Kapton beam splitter with a thickness of 25.4 μm is highly suitable for accurately measuring the radiation spectrum. This conclusion is based on the significant separation observed between the dips in the beam splitter. This considerable separation indicates that the Kapton beam splitter allows for more precise measurement of the radiation spectrum using a Michelson interferometer.

3.5.2

Beam splitter and bunch-length measurement

As mentioned previously, TR can be utilized to measure the electron-bunch length through autocorrelation using a Michelson interferometer. The interferogram obtained directly from the Michelson interferometer reflects the electron bunch-length characteristics. However, the effect of beam-splitter efficiency on the accuracy of these measurements must be considered. The efficiency of the beam splitter in the interferometer can introduce potential errors during the measurement process. According to the results presented in the previous section, the width of the interferogram cannot be directly used to determine the electron-bunch length, owing to the presence of beam-splitter interference. The intensity-suppression effect of the beam splitter leads to negative valleys in the interferogram, resulting in an artificially smaller interferogram width, as depicted in Fig. 13(a)–(c).

Fig. 13Full-screen View

Calculated interferograms including the effect of beam-splitter efficiency for Kapton beam splitter with thicknesses of (a) 25.4 μm, (b) 50.8 μm, and (c) 127 μm, and (d) Si beam splitter with a thickness of 2 mm

To investigate this effect further, the simulated interferogram is compared with a reference Gaussian interferogram. Simulated interferograms are obtained by applying the inverse Fourier transform to the radiation spectra. The reference interferogram is constructed using a Gaussian bunch distribution with a full width at half maximum (FWHM) of 334 μm, corresponding to an electron-bunch length of 105.5 μm.

By increasing the thickness of the beam splitter, the negative valleys move further away from the center burst (central positive intensity) of the interferogram. As a result, measuring the bunch length with a Kapton beam splitter requires a correction of the FWHM values. The FWHM of the measured interferograms corrected with the beam-splitter efficiency are shown in Fig. 14. When the beam-splitter thickness is equal to or exceeds twice the electron-bunch length, the measured FWHM of the interferogram accurately reflects the actual bunch length. The SUNSHINE facility also conducted a study on the impact of a beam splitter within the Michelson interferometer, with the aim of measuring the bunch length of 26-MeV electrons. This investigation is documented in [32]. The experimental results exhibit a trend similar to that observed in our calculations. In this study, a Kapton beam splitter with a thickness of 127 μm yields a measured bunch length that is close to the real value for a bunch length of 105.5 μm (FWHM = 334 μm). In the case of the Si beam splitter, the interferograms exhibit negative valleys far from the center burst. Because these negative valleys do not alter the appearance of the center burst, the width of the interferogram can be used to extract the bunch-length value directly. Regardless of the light absorption by the beam splitter, we can conclude that thick beam splitters are more suitable for bunch-length measurements than thin beams, as discussed above.

Fig. 14Full-screen View

(Color online) FWHM correction of calculated interferogram for Kapton beam splitters with thicknesses of 25.4, 50.8, and 127 μm. The reference case for the Gaussian electron bunch with a length of 105.5 μm is shown by the dashed curve