1. Introduction

When a charged particle goes through the boundary between two media with different refractive index, Transition Radiation (TR) is emitted both in the forward and backward direction (Fig. 1). The radiation results from the interaction of the elecromagnetic (EM) field of the travelling particle with the screen surface. The angular dependence of the TR spectral intensity per unit solid angle for the single particle is given by the Ginzburg-Frank formula [1], showing a peak of emission at θ = 1/γ, being γ the Lorentz factor and θ the angle depicted in Fig. 1:

 

Fig. 1 Transition Radiation generation scheme. The electron beam, going through the boundary between two media with different dielectric constant (ϵ₀ and ϵ₁), generates radiation in both forward and backward directions. In particular, if the source is tilted with respect to the electron propagation, the backward radiation follows the Snell’s law.

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When only part of the EM field interacts with the surface, Diffraction Radiation (DR) is produced. In this case, the particle goes through an aperture of the screen, therefore DR is produced if the extension of the relativistic particle EM field, i.e. (γλ/2π), being λ the emitted wavelength, is comparable with the aperture.

When a bunch of electrons, with finite size in both transverse and longitudinal dimensions, passes through, or nearby, a boundary between two media, the EM field of each particle emits its own radiation field with a relative phase depending on the position within the bunch. Therefore, TR and DR from an electron bunch are characterized upon a longitudinal form factor [2] that, for a wavelength λ and a Gaussian temporal profile described by σ_t, is given by $F_{\parallel }(\lambda )=e^{-{(\frac{2\pi {\sigma }_{t}c}{\lambda })}^2}$ for F_‖(λ) = 1 the radiation is totally coherent, while for F_‖(λ) = 0 is incoherent. Moreover, concerning the plane orthogonal to the propagation direction, the radiation is described also by a transverse form factor [2] $F_{\perp }(\lambda ,\theta )=e^{-{\left(\left(2\pi /\lambda \right)\sin \theta \right)}^2({\sigma }_x^2\cos {\varphi }^2+{\sigma }_y^2\sin {\varphi }^2))}$, where σ_x and σ_y describe the transverse beam size, while ϕ is the azimuthal angle of the radiation plane. This term is responsible of an angular distribution narrower and less intense as the transverse beam size increases.

TR is commonly used as diagnostics tool to retrieve transverse and longitudinal beam parameters. In the optical range, TR is used to measure the beam transverse size by imaging the incoherent source, while the beam divergence is retrieved by analyzing the angular distribution in a lens focal plane [3–9]. In the far infrared and THz range, coherent TR is employed for temporal profile measurements [10–14]. Moreover, the use of sub-ps relativistic electron beams allows to obtain a coherent source [15–17] to be employed for THz spectroscopy in applied sciences, e.g. material characterization [18] and biomedical applications [19]. Nevertheless, all these applications need specific optical lines, therefore it is crucial to understand the system performances to properly retrieve the experimental data.

Recently, it was suggested the use of a well-known tool for optical simulation, Zemax [20], to simulate the behaviour of the angular distribution for a single electron TR and DR field and to analyze the optical system in terms of Point Spread Function (PSF) [21]. We present here the effect of the whole beam in the TR and DR emission mechanism, since collective effects are not visible in a single electron analysis: for example, the angular distribution of the incoherent radiation is sensitive to beam divergence, while the characteristics of the coherent one are strongly affected by the transverse beam size. To our knowledge, this issue, that cannot be neglected, has not been included in any optical code devoted to the simulation of a complete experiment.

In this work, we report about a new algorithm based on Zemax lens design software, able to simulate the radiation coming from a whole electron bunch through a specific optical system. After a brief discussion about how our algorithm works, we show some comparison between simulations and experiments. In detail, we analyzed the results of the optical diffraction radiation interference (ODRI) experiment [6], performed at FLASH (DESY) [22], and a THz imaging experiment [16] installed at the SPARC_LAB Test Facility [23] exploiting coherent TR. In both cases, our simulations are in excellent agreement with the experimental outcomes.

2. Simulation methods

Simulations have been performed by means of Zemax optical design software [20]. Although it is commonly used for industrial purposes, such as illumination device and lens design, it allows to simulate many kinds of optical systems. In detail, this software provides two different analysis modes: geometrical ray tracing and physical optical propagation (POP). Concerning the first one, it represents a good solution to simulate the behaviour of an optical system in the ray approximation, by neglecting any diffraction effects related to the wave nature of the light. Therefore, in order to take into account any effects due to the wave nature of the radiation, e.g. polarization and diffraction, the use of POP mode is mandatory.

The beam is represented by an array of discretely sampled points. The entire array is then propagated through the free space between optical surfaces. At each optical surface, a transfer function is computed and applied to the beam. In particular, the software can choose automatically the propagator to be used by checking the Fresnel number defined as $N_F=\frac{a^2}{\lambda L}$, where a is the radial size of the beam, λ is the wavelength and L the distance between initial and final surfaces. In general, for N_F < 1 the Fraunhofer diffraction is computed, while for N_F ≥ 1 the Fresnel propagator is used.

In order to simulate the optical propagation of TR, a DLL has been compiled to provide the right electric field to Zemax POP analysis tool. The library contains the instructions to let the software build the input electric field. In detail, the real and imaginary parts of both the horizontal (h) and vertical (v) polarization components of the single electron TR electric field have been defined as follows [24]

 

Fig. 2 a) Single electron TR total intensity at the source, b) Horizontal and c) Vertical polarization. The field parameters are: γ = 250 and λ = 500nm.

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In order to take into account the finite size of the beam and its divergence, a Zemax macro has been developed exploiting the Zemax Programming Language (ZPL) provided together with the software suite. The ZPL is a language specifically designed to offer the power of user-extensibility. It is similar to the BASIC programming language, with exclusive capabilities and functions for ray tracing and physical optics analysis.

Our custom code contains a specific algorithm, able to propagate the TR field of each electron in the bunch, defined by transverse size and divergence, and sum all of them at the end. In our simulations, the single terms are weighted by a Gaussian spatial distribution. Furthermore, the possibility offered by Zemax to start with an angle with respect to the optical axis has been exploited in order to take into account the electron momentum.

This algorithm allows to define every optical system offered by Zemax itself and analyze TR and DR taking into account the whole source electron beam. Indeed, it acts on the layout of Zemax lens data editor and it is totally uncoupled from that: nothing has to be modified in the code in order to adapt it to a custom set-up. Only electron beam characteristics, in terms of spatial and angular distributions, are needed to properly simulate the propagation.

In next sections, comparison between simulations performed by means of Zemax and experimental data will be shown, for both diffraction and transition radiation.

3. Simulation results

In this section, the comparison between simulations and experimental results will be shown. The results of two experiments have been analyzed: the measurement of Optical Diffraction Radiation Interference (ODRI) angular distribution performed at FLASH (DESY) and an imaging system in the THz spectral range from coherent TR (CTR) at SPARC_LAB.

3.1. Diffraction radiation: ODRI experiment

Experimental data have been taken at FLASH (DESY, Hamburg) [22]. Here, an experiment concerning the characterization of the interference pattern coming from the radiation produced by two radiators with different aperture sizes has been performed to retrieve the transverse emittance [6, 7]. In detail, incoherent radiation in the optical spectral range is generated through ps-scale, relativistic electron bunches, accelerated up to 1 GeV. The total charge was 200 pC per microbunch (in a train with up to 30 microbunches in our experiment). The beam transverse size and angular divergence have been retrieved by measuring the OTR both the image plane and focal plane, respectively, produced by an aluminum-coated silicon screen placed on the same DR target holder, resulting in a σ = 90 µm and σ′ = 50 µrad. For ps-long electron beams, F_‖(λ) ~ 0 in the visible spectrum and the TR can be considered fully incoherent. Therefore, our algorithm combines the single electron intensities.

Figure 3 shows the ODRI experimental setup: two aluminum-coated silicon screens, 25 mm far from each other, placed in ultra high vacuum environment (10⁻¹⁰mbar), are made with two different aperture sizes: 1 mm and 0.5 mm, respectively. The second screen is placed at 45°. The total radiation, i.e. the interference between the forward DR coming from the first slit and the backward DR coming from the second slit, reaches a custom apochromatic convex lens (f=527 mm) [6] and it is collected by an air-cooled, high sensitivity, 16-bit CCD camera (Hamamatsu ORCA II-BT-512G model type C4742-98-26LAG2), placed in the lens focal plane. Here, the ODRI angular distribution in the far field turns out, as depicted in Fig 3 with its central line profile shown in Fig. 4. In addition, several narrow band interferential filters and a polarizer can be inserted to select the observation wavelength and polarization.

 

Fig. 3 Experimental set-up for ODRI angular distribution measurement. The DR from two aluminum-coated silicon screens, one at 90° and the other at 45° with respect to the beam axis, with their centers 25 mm far, are recombined in a focal plane of a custom apochromatic convex lens (f=527 mm). Here, a 16-bit CCD camera (Hamamatsu ORCA II-BT-512G model type C4742-98-26LAG2) is placed. An interferential filter is used to choose the wavelength to be analyzed, while a polarizer selects only the horizontal polarization.

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Fig. 4 Comparison of central line profiles of bunch simulation (blue dashed line), single electron simulation (orange dashed line) and experimental data (red continuous line). The error bars are computed following a Poissonian distribution of pixel intensities, since the CCD camera worked in photon counting mode.

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In Fig. 4 the comparison between simulations, experimental data and fit is shown. The wavelength has been chosen equal to 500 nm. Both single electron and bunch simulations well reproduce the experimental asymmetry. Indeed, the two slits are not perfectly centered: the first slit (1 mm aperture) has a +50 µm vertical offset with respect to the beam trajectory, while the second one is displaced by −6 µm. Moreover, by taking into accounto beam transverse size and divergence, the overlap between bunch simulation and experimental fit is excellent (detail on the fit tool can be found in [6]). At the same time, a single electron analysis (orange dashed curve in Fig. 4) does not allow to properly retrieve the radiation properties. Indeed, the central minimum and the side lobes, sensitive to beam transverse properties, are quantitative different with respect to the full bunch simulation. Therefore, it is not meaningful to characterize the electron source.

3.2. Transition radiation: THz imaging system

The experimental data have been taken at the SPARC_LAB Test Facility in Frascati (Italy) [23]. Here, an experiment concerning the characterization of the THz source [16] has been performed. In detail, coherent TR (CTR) in the THz spectral range is generated through ultra-short (~100 fs), relativistic electron bunches, accelerated up to 115 MeV in a high brightness photo-injector. The total charge was 400 pC. The transverse size, measured on a YAG:Ce screen placed on the same holder of the TR source, was σ = 600µm.

The longitudinal form factor for the analyzed wavelength, corresponding to a frequency equals to 1 THz, is F_‖(λ) ∼ 0.8 therefore the radiation has been considered totally coherent and our algorithm has been run accordingly. It is important to emphasize that, conversely to the incoherent case, CTR in the THz regime is more sensitive to diffraction effects due to the target size, because of the greater wavelength [24]. Consequently, it is very sensitive to the experimental constraints. For this reason, the comparison of simulations with real data is more meaningful with respect to an analytical description that does not take into account of optical diffraction effects as well as any possible source of misalignment.

The experimental set-up is shown in Fig. 5: the source on the aluminum foil at 45° degrees in ultra high vacuum environment (10⁻⁹mbar) is collected and re-collimated by a first 90° off axis parabolic (OAP) mirror with focal length f₁ = 152mm; after a flat mirror, a second 90° OAP mirror with f₂ = 50 mm focused the THz radiation onto a Pyrocam III camera, with a (12.4 × 12.4) mm² active area and 100 µm pixel size. Since the camera is placed in the focal plane of the second OAP mirror, the initial source is a replica with a magnification factor equals to 0.33. Moreover, in order to select the desired wavelength, a band-pass filter, centered at 300 µm and with 15% bandwidth FWHM, has been used.

 

Fig. 5 Experimental set-up for THz measurement from coherent TR. The radiation from an aluminum-coated silicon screen is collected by a OAP mirror (f₁ = 152 mm) and sent to a second OAP (f₂ = 50 mm) focusing it on a Pyrocam III camera placed in its focal plane. A band-pass filter, centered at 300µm and with 15% FWHM bandwidth, has been installed.

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In Fig. 6 the simulated line profile is compared with the experimental data. The agreement, in terms of overlap, is excellent. In particular, the source of the peak asymmetry has been investigated. Indeed, it has been possible to identify the experimental issue related to it. In detail, it has been found a 8mm diagonal shift between the centers of the two mirrors and an angular tilt equals to (−9°, +9°) with respect to the horizontal and the vertical axis, respectively.

 

Fig. 6 Comparison of central line profiles of simulated (dashed line) and experimental (solid line) coherent transition radiation. The beam parameters are γ = 230, λ = 300 µm, σ = 600 µm, where σ is relative to the gaussian transverse profile of the electron beam.

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4. Conclusions and perspectives

In this work, simulations made by an innovative code written in Zemax Programming Language have been presented. The effects on the TR and DR due to the whole electron bunch have been properly retrieved for the first time by considering a real optical set-up both for incoherent and coherent radiation. In particular, we have analyzed the results of the optical diffraction radiation interference (ODRI) experiment [6], performed at FLASH (DESY), and a THz imaging experiment [16] installed at the SPARC_LAB Test Facility [23] exploiting coherent TR. In both cases, our simulations are in excellent agreement with the experimental outcomes.

Concerning the DR experiment, the simulation well reproduces the asymmetry due to a non perfect alignment of the two slits with respect to the electron beam. Moreover, the difference with a single electron analysis have been addressed. On the other hand, analyzing THz imaging experiment data, effects due to the beam size on CTR have been studied. The agreement between experiment and simulation, in terms of overlap, is excellent and the analysis allowed to discover the source of the peak asymmetry. Furthermore, some analysis on exotic optical elements, not reported in this work, such as microlens array, have been performed in order to study the feasibility of new diagnostics [8, 9, 25, 26]. Therefore, our algorithm allows to test the feasibility of new diagnostics solutions.

In conclusion, our algorithm based on Zemax allows to simulate TR and DR in any optical system, taking into account the electron bunch properties (spatial extension, energy, angular divergence) generating the radiation. In addiction, it allows to take into account diffraction effects, due to experimental constraints (e.g. limited apertures of optics), not evaluated in an anaytical description. Moreover, in the next future, more efforts will be done in order to implement other beam parameters, e.g. energy spread, temporal profile and emittance.