Telescope system

A beam telescope is a tracking detector system that typically comprises four or more planes with silicon pixel sensors. The pixel sensors on each plane must be thin and highly granular. To minimize the effect of multiple Coulomb scattering, the detection area typically includes only a silicon pixel sensor and two layers of Kapton film for light shielding. A schematic of the beam telescope is shown in Fig. 1. The DUT is generally positioned at the center of the beam telescope with high-energy charged particles traversing both the telescope planes and the DUT. Generally, a mechanical structure consisting of two symmetrical telescope arms is used to support telescope planes. Both arms are adjustable along the beam direction, facilitating the easy installation of DUTs of different sizes.

Fig. 1Full-screen View

(Color online) Schematic of the beam telescope with six telescope planes and the DUT

In the data analysis of beam tests, detector alignment is important for calibrating the position of the DUT and each plane of the telescope by minimizing the track residuals through a large number of tracks [24]. After the alignment, the reference tracks can be reconstructed using the minimum χ² matrix algorithm [25]. For low-energy beams, the effect of multiple Coulomb scattering on the measured resolution cannot be ignored, particularly in the case of the non-negligible material budget of the DUT. Algorithms such as the Kalman Filter with a covariance matrix generated by multiple Coulomb scattering [26, 27] or General Broken Lines [28] can be utilized for track fitting and detector alignment to mitigate the effect of multiple Coulomb scattering. In this study, to avoid the differences caused by the algorithms, we used the same alignment algorithm as in [24] and the Kalman filter track fitting as in [26] for both the simulation and beam test data.

Measured resolution

The fitting points of the DUT can be obtained via track fitting. The measured spatial resolution of the DUT, which is the measured residual between the fitting points and the measured points, is unbiased if no DUT information is considered during track fitting. The measured resolution σ_meas includes contributions [29] from the intrinsic resolution of the DUT σ_DUT, telescope resolution σ_tel and the effect of multiple Coulomb scattering σ_MS, as described by Eq. (1): ${\sigma }_{\text{meas}}^2={\sigma }_{\text{DUT}}^2+{\sigma }_{\text{tel}}^2+{\sigma }_{\text{MS}}^2$ (1) Notably, the effect of multiple Coulomb scattering originates not only from the DUT but also from the telescope planes. Therefore, telescope planes are typically designed to be as thin as possible to minimize the effects of multiple Coulomb scattering.

The telescope resolution σ_tel can be determined if all telescope planes have the same intrinsic resolution σ_plane and the DUT is placed at z=0, using Eq. (2) and Eq. (3): ${\sigma }_{\text{tel}}^2=k{\sigma }_{\text{plane}}^2$ (2) $k=\frac{{\displaystyle {\sum }_i^N{z}_i^2}}{N{\displaystyle {\sum }_i^N{z}_i^2}-{\left({\displaystyle {\sum }_i^N{z}_i}\right)}^2}$ (3) where zi and N represent the position in the beam direction and the number of telescope planes, respectively. If the telescope planes are parallel and symmetrically distributed on both sides of the DUT, then k is reduced to 1/N.

When the beam energy is sufficiently high, the effect of multiple Coulomb scattering can be ignored, and it is assumed that the DUT and telescope planes are identical, by combining Eq.(1), Eq.(2) and Eq.(3), σ_DUT and σ_tel can be calculated from measured resolution σ_meas, using Eq. (4) and Eq. (5): ${\sigma }_{\text{DUT}}^2=\frac{{\sigma }_{\text{meas}}^2}{1+k}$ (4) ${\sigma }_{\text{tel}}^2=\frac{k}{1+k}{\sigma }_{\text{meas}}^2$ (5) Based on Eq. (1), the measurement result of the DUT is influenced by both the resolution of the beam telescope and the effect of multiple Coulomb scattering. However, for the silicon pixel chips of a telescope, the chip design and resolution are constrained by multiple factors and cannot be significantly improved. Therefore, evaluating and minimizing the effect of multiple Coulomb scattering is crucial for improving the measurement accuracy of DUT with beam tests.

Simulation setup

We use Allpix² [30] to simulate the test beam system and study how multiple Coulomb scattering influences the measured resolution. Allpix² is an open-source software framework written in C++, built on Geant4 and ROOT, and specifically designed for simulating silicon pixel detectors. It can easily simulate both individual detectors and more complex systems such as test beam systems, providing detailed information such as pixel charge collection and digitized responses.

Unless otherwise stated, all simulations in this study were conducted with the following settings: the beam telescope consisted of four telescope planes, divided into two upstream and two downstream planes. The DUT was positioned at the center of the telescope. The gap between the telescope planes and the distance from the DUT to the front and rear telescope planes were set to 20 mm. The DUT and telescope planes use the same model, which consists of a MIMOSA28 chip [31] thinned to 50 μm and two layers of 50 μm Kapton film, resulting in a total material budget of 0.088% X₀ per plane. The MIMOSA28 chip contains 928 (row) × 960 (column) pixels with a pitch of 20.7 μm. It has a 15 μm-thick high-resistivity epitaxial layer as a sensitive layer that is not fully depleted, thus allowing charge collection through thermal diffusion. MIMOSA28 chips are also being considered for use in the upcoming proton beam telescopes on CSNS. In this simulation, the electronic noise of the chip was set to 30 e^- based on the threshold scan results, and the threshold was set to 5 σ to mitigate the influence of the electronic noise and achieve optimal resolution.

Beam energy

To study the impact of the beam energy on the multiple Coulomb scattering angle and, consequently, the simulation result of the resolution (σ_meas), represented as the simulated resolution in the following, we simulated different electron beam energies from 0.5 GeV to 30 GeV, and the DUT was the same as the plane model of the telescope with a material budget of 0.088% X₀. Based on Eq. (6), with this system setting, when the beam energy is high enough, for example, 120 GeV, the effect of multiple Coulomb scattering can be ignored. Therefore, a beam energy of 120 GeV can be used as the reference without the effect of multiple Coulomb scattering. Figure 2 shows the simulated resolution as a function of beam energy. Increasing the beam energy can significantly improve resolution in the low-energy range. When the beam energy is sufficiently large, the simulated resolution no longer improves and approaches the value of 120 GeV electrons. Therefore, the effect of multiple Coulomb scattering at different energies can be calculated using Eq. (1). When the beam energy was greater than 12 GeV, σ_MS was less than 0.5 μm. Compared with the value achieved with 120 GeV electrons, the simulated resolution with 12 GeV experiences a degradation of less than 1%.

Fig. 2Full-screen View

Simulated resolution of the DUT as a function of the beam energy from 0.5 to 30 GeV. The dotted line shows the simulated resolution with 120 GeV electrons. The effect of multiple Coulomb scattering with different energies is calculated and shown as the blue line

Material budget

Beam telescopes are typically designed to minimize the material budget, making the material budget of the DUT the primary factor influencing the effects of multiple Coulomb scattering. Thus, to study the impact of the material budget on the simulated resolution of the DUT, the material budget of the DUT is changed from 0.106% X₀ to 0.53% X₀ by increasing the equivalent thickness of silicon from 100 μm to 500 μm without Kapton films, while the material budget of each telescope plane remains 0.088% X₀ unchanged. Figure 3 shows the simulated resolution as a function of the material budget with 5 and 120 GeV electrons. With 120 GeV electrons, as the material budget increased, the simulated resolution remained nearly constant because the effect of multiple Coulomb scattering was sufficiently small to be ignored. However, when using 5 GeV electrons, doubling the material budget results in an increase of approximately 25% in the effect of multiple Coulomb scattering and degradation of the simulated resolution by approximately 5%.

Fig. 3Full-screen View

Simulated resolution of the DUT as a function of the material budget with 5 GeV and 120 GeV electrons. The material budgets of the DUT models change from 0.106% X₀ to 0.53% X₀. The effect of multiple Coulomb scattering with 5 GeV electrons is calculated and represented as the purple line

Based on the above conclusions, the simulated resolution with 120 GeV electrons (approximately 4.18 μm) can be used as a reference for studying the effects of multiple Coulomb scattering with different material budgets and beam energies. This allows the estimation of the minimum energy at which the effect of multiple Coulomb scattering can be ignored for different material budgets of the DUT. We changed the material budget of the DUT from 0.106% X₀ to 1.06% X₀ at energies ranging from 5 to 50 GeV. Figure 4 shows the effect of multiple Coulomb scattering as a function of beam energy for different DUT models. Increasing the beam energy significantly reduced the effect of multiple Coulomb scattering. When σ_MS was less than 0.5 μm, the resolution of the DUT decreased by less than 1%. Therefore, the simulated resolution of the DUT is influenced more by the randomness of the simulation and statistical errors, resulting in fluctuations in the curves of σ_MS at high beam energies, as shown in Figures 2 and 4. Figure 5 shows the minimum energy at which the effect of multiple Coulomb scattering can be ignored as a function of the various DUT material budgets. The minimum energy increases as the material budget of the DUT increases.

Fig. 4Full-screen View

Effect of multiple Coulomb scattering as a function of the beam energy from 5 to 50 GeV. The models of the DUT have different material budgets from 0.106% X₀ to 1.06% X₀, while the material budget of telescope planes remains 0.088% X₀ unchanged

Fig. 5Full-screen View

Minimum energy as a function of the material budget of the DUT. When the beam energy is larger than the minimum energy, the contribution of the effect of multiple Coulomb scattering is less than 0.5 μm, causing a decrease of less than 1% in σ_meas, which can be ignored

Telescope layout

The telescope arms can be adjusted in the beam direction to modify the distance between the DUT and telescope planes, allowing the testing of various device sizes. Modifying the layout influences the effect of multiple Coulomb scattering even if the multiple Coulomb scattering angle θ₀ remains constant. Furthermore, altering the number of beam telescope planes affects the factor k in Eq. (2) and the effect of multiple Coulomb scattering. To study these impacts, we conducted simulations with a 4-layer, 5-layer, and 6-layer telescope with different layouts. For the 5-layer beam telescope, there were three planes before and two layers after the DUT. In the simulation, the DUT was set the same as that in the telescope plane.

The resolution of the DUT is strongly influenced by the distance between the DUT and the nearest telescope planes. As shown in Fig. 6(a), the simulation results indicate that reducing the gap between the DUT and telescope planes within the range of 10–100 mm can improve the simulated resolution. Increasing the track length resulted in a larger measured residual between the fitting and measured points, thereby degrading the simulated resolution. However, the distance between the DUT and the telescope is limited by the size of the DUT. For DUTs of varying sizes, minimizing the gap between the DUT and nearest telescope planes is crucial for achieving a better resolution. Figure 6(b) shows the simulated resolution as a function of the distance between the telescope planes. Minimizing the distance between the telescope planes can reduce the effect of multiple Coulomb scattering. Theoretically, increasing the number of telescope planes can improve telescope resolution, as shown in Eq. (5). However, this increases the total material budget of the beam telescope. With 5 GeV electrons, the simulated resolution with a 4-layer telescope is essentially the same as the resolution with a 6-layer telescope.

Fig. 6Full-screen View

(a) Simulated resolution of the DUT as a function of the distance from the DUT to both the front and rear telescope planes, simulated with 5 GeV electrons. The gap between telescope planes is kept 20 mm. (b) Simulated resolution of the DUT as a function of the distance between telescope planes simulated with 5 GeV electrons. The distance from the DUT to both the front and rear telescope planes is kept 20 mm