Spatial resolution improvement for an optical transition radiation monitor by asymmetric light collection

: The applicability of optical transition radiation (OTR) for measurements of micron sized transverse electron beam profiles is limited not only by the optical system resolution which has a fundamental limit imposed by the uncertainty principle. In the case of OTR generation, a single electron crossing the boundary between vacuum and screen cannot be considered as a single emitting point with isotropic angular distribution. On the contrary, the radiation is emitted from an area with a transverse range that is defined by the radial extension of the electron’s Lorentz contracted Coulomb field and is typically estimated as γλ (with γ the Lorentz factor and λ the wavelength of observation). The OTR angular distribution has a characteristic “funnel” shape. As a result the one-dimensional image of a single electron measured with an ideal thin lens has a double lobe shape, and the resolution of any OTR based imaging system is determined by this double lobe function which is also known as OTR Point Spread Function (PSF). As a consequence, the reconstruction of micron sized electron beam profiles is hampered not only due to the fundamental diffraction limit, but also due to the PSF lobe shape. In this paper we present two approaches to improve the spatial resolution of an OTR monitor based on asymmetric light collection using a traditional optical system which allows blocking of one of the lobes. With such a scheme, an OTR PSF can be achieved that is comparable to the one of an ideal point source (Airy distribution).


Introduction
Transverse beam profile diagnostics based on optical transition radiation (OTR) is a routine technique at most modern electron linear accelerators (linacs) [1][2][3]. The mechanism of OTR generation possesses a practically prompt response ( -15 Δt~/ ΔE, ΔE~1 eV, Δt~10 s h ) and a linear dependency on the initial beam intensity, neglecting coherent radiation emission effects driven by the microwave instability which is sometimes observed at modern linac based Free-Electron-Lasers (FELs) [4,5]. For electron accelerators with energies higher than 50 MeV for instance, the intensity of the OTR light collected by an optical system with a numerical aperture of ~0.1 rad can achieve ~3 10 − photons/electron, assuming an optical filter with relative bandwidth of /~5 λ λ ∆ %. The spatial resolution of a transverse beam size monitor based on OTR is determined by the so-called Point Spread Function (PSF) or, in other words, by the response of the monitor optical system to a point charge crossing the target. With knowledge of the PSF, in principle it is possible to reconstruct beam size and beam shape from an electron bunch passing through the target applying a deconvolution algorithm to the measured OTR image.
In the first approximation for E > 100 MeV the PSF can be obtained based on approximations of classical optics. In this approach the field of a relativistic point charge is considered as a set of the electromagnetic waves (the pseudo photon approximation) see, for example [6][7][8][9]. A typical sketch illustrating the observation geometry of backward OTR is presented in Fig. 1. The generation of backward OTR is considered from a target with ideal conductivity according to [9] ( )  (1) where ω is the frequency of the emitted photons, γ is the Lorentz-factor, x y θ ,θ are the projection angles with respect to the specular reflection direction,  is the Planck constant, and α is the fine structure constant. It should be mentioned that formula (1) is valid for the ultra-relativistic case and for the far-field zone only.
As it was shown in [10,11], for distances 2 a γ λ < away from the target (λ is the wavelength of the emitted OTR photon), the angular distribution of OTR is affected by the transverse size of the particle Coulomb field which is in the order of ~γλ and which induces polarization currents at the target surface in a region with radius γλ (so-called the prewave zone effect). As consequence the OTR angular distribution is broadened and depends on the ratio 2 / R a γ λ = . However, in the case of OTR imaging where the detector is placed in the image plane, the pre-wave zone effect can be neglected for arbitrary ratios R [6][7][8]. While simulations of OTR diagnostic techniques were recently performed using the Zemax code [12], the approach which is presented in the following is based on classical diffraction theory where the electromagnetic field of the point charge is considered as a set of transverse electromagnetic waves.
Such a simple approach allows to obtain general features of the OTR images and compare them for different wavelengths, apertures, electron energies, etc. Zemax's simulations can be used effectively for taking into account characteristics of the optical system, defocusing effects and so on after choosing of the experimental scheme which can be done on the classical diffraction model.

Model for OTR PSF calculation
Under ultra-relativistic approximation the particle Coulomb field can be sufficiently described by its transverse components and it is possible to write down the OTR field in the lens plane in analogy with wave scattering at a finite size conducting screen: the coordinates of target surface and lens plane, T S is the target surface area, 2 2 cos . For a transverse target size which is much larger than the field radius γλ it is possible to extend the integration region over the target surface to infinity.
The OTR fields in the image plane using thin lens approximation can be written in the following way [7,8]: Here the integration is performed over the lens aperture L S . For the sake of simplicity dimensionless variables are introduced in the following: Inserting Eq. (2) in Eq. (3) and taking into account the variable substitution, the following 4fold integration has to be solved: M b a = is the optical system magnification (in the following all simulations will be carried out for the case M = 1, i.e. a = b). The condition of ideal focusing was used in order to obtain Eq. (5): For geometries with azimuthal symmetry the radial component of the OTR field in the image plane can be reduced to a more simple expression, see [7,8]: r dr K r G r r r i R G r r r r dr J r r J r r r r J r r J r r r J r r J r r r r A spatial resolving detector located in the image plane will measure a two-dimensional distribution which is proportional to the intensity of OTR. Therefore, the OTR PSF under assumption of target ideal reflectivity can be can be finally written as

OTR imaging
Measurements of OTR images using a conventional optical system as depicted in Fig. 1 allow avoiding the influence of the pre-wave zone effect [7,8]. The spatial resolution of such a scheme is determined by the lens aperture and by the wavelength of observation and can be characterized by the PSF. Results of radial PSF distribution calculations according to Eqs. (7) and (8) are shown in Fig. 2(a). The calculation parameters are as follows: for the blue curve and max 100 for the red curve. The lens diameter for the latter case, for instance, is defined as Equations (7) and (8) are written in the universal form with dimensionless variables. Therefore, the same plots presented in Fig. 2 Figure 3 shows the one-dimensional horizontal PSF distribution. In order to calculate the OTR image from a beam with Gaussian transverse profile, this PSF can be approximated by a simple analytical function with 0 1 , q q being free fit parameters [13]. While q0 is a simple amplitude factor, the parameter q1 characterizes the PSF and can be used to give a resolution estimate as will be shown below. The result of the convolution of a Gaussian beam profile with rms size σ and the PSF according to Eq. (9) can be expressed by the following analytical function [13]: The calculated function was fitted with the fit function Eq. (9), resulting in 1 2.0 um q = . As can be seen, there is a good agreement especially in the central part between the fit and the PSF calculation based on Eq. (5). In order to illustrate the sensitivity of an OTR based beam profile measurement on the PSF influence, the PSF distribution from Fig. 4a was convoluted with Gaussian beams with sizes σ = 0.5; 1.0; 2; 4 um according to Eq. (10). The results are plotted in Fig. 4(b). As can be seen, for beam sizes σ < 1 q the profiles exhibit a two-lobe structure. For this case (so called PSF dominated regime) the problem of reconstructing the initial beam profile is rather complicated, see [12,13]. Only in the case 1 10q σ > the PSF influence is so small that fitting the convoluted data with a Gaussian function results in a rms beam size which practically coincides with the input size for the beam [13].

Methods to improve the OTR spatial resolution for beam imaging
The author of [8] proposed to use the parameter 0 1.12 i m R λ θ ≈ for the characterization of One possible ways to improve the spatial resolution was proposed in [6,14]. The authors proposed to shield the central part of the lens aperture by a mask which is placed just in front of the lens. The shielding effect results in a suppression of the PSF side lobes at larger distances such that only a double lobe structure remains. In [8] a criterion was proposed for the optimal choice of the mask aperture: Figure 5 presents calculated PSFs taking into account the influence of the shielding mask. The optimal mask effect is shown by the green curve (see Fig. 5). Despite a decrease of the OTR yield, the oscillatory behavior in the distribution tails (i.e. the region where the PSF is in the range ) is suppressed. Evidently, the volcano-like part of the OTR distribution in the lens plane is transformed into a two-lobe shape as before. As was noticed in [8] "… the effect of the mask can be interpreted as increasing the uncertainty in the momentum for to reduce the uncertainty in the position …". In the course of the preceding work two methods are proposed by which the OTR PSF contribution can further be reduced close to the level of the Airy function which represents the ultimate resolution level of a point source.
Basis of the first method is the installation of a mask in front of the lens with appropriate size at a position which is asymmetric with respect to the optical axis. If mask size and position are selected in a proper way it is possible to block one of the lobes of the horizontal angular OTR distribution. In this case information is extracted only from the second lobe in the lens plane or in the x y k k − phase space. Due to the properties of the Fourier transformation the output in the D D X Y − plane (i.e. the detector plane) will also have a single maximum distribution. Figures 6 and 7 represent the evolution of the OTR PSF as function of the mask position using the horizontally polarized radiation component. As expected, in the case of full blocking of one of the lobes in the lens plane ( Fig. 6(f)), the resulting PSF possesses a single maximum only (Fig. 7(f)). In all cases under consideration, the PSF distributions were calculated based on Eq. (13) which is the far-field zone approximation for the radiation field, using a lower limit for the integration over L X as indicated in Fig. 6  The single maximum PSF in this case can be approximated by a Gaussian distribution as shown in Fig. 8 Fig. 8. Gaussian fit using the PSF for the case shown in Fig. 6(f) (c.f. also Fig. 7(f)). Fig. 9. Comparison of PSF distributions for masking with different asymmetric masks (blue -50% screening, red -75% screening).
In Fig. 9 a PSF comparison is shown for two cases, using masking of 75% and of 50% (i.e. the case shown in Figs. 6 and 7(f)) of the lens aperture. As can be seen, with increasing screening the PSF is broadened. Again this effect is connected with the uncertainty principle because the narrowing of the distribution in the x y k k − phase space results in a broadening of x-y distribution. The second method to improve the spatial resolution of an OTR monitor has the same purpose to create a PSF with a single maximum. Again, the OTR horizontal polarization component will be considered in the following. However, in this case the idea is to displace the imaging lens asymmetrically as depicted in Figs. 10 and 11. If the lens optical axis is disoriented with respect to the specular reflection direction from the OTR target at an angle of 0 Fig. 10, the OTR light will be collected from the outside region of the OTR lobe-shape distribution, c.f. Figure 11. Evidently, for such imaging geometry the part of the horizontal OTR distribution which is accepted by the lens possesses only a single maximum. The PSF calculation for this asymmetric light collection geometry is based on Eq.
Results of such PSF calculations using asymmetric light collection geometries are presented in Fig. 12 for 100 / 0.1rad m θ γ = = (green points) and 50 / 0.05 rad θ γ = = (red points). The remaining parameters for the calculation are the same as before. um for the second one (see Fig. 12).

Discussion
In the present report it was shown that an asymmetric mask in front of the imaging lens or an asymmetric light collection geometry caused by a displacement of the imaging lens both result in light collection from only one lobe of a linearly polarized OTR intensity distribution. As consequence, the PSF possesses also a single maximum in the image plane. In contrast to that, the previously proposed scheme for resolution improvement based on a mask which is oriented coaxially with the lens results in a two-lobe PSF structure with suppression of the outer maxima [8].
In the following PSF distributions calculated for asymmetric OTR light collection (see Fig. 8 and Fig. 12) will be compared with the well-known Airy distribution, describing the PSF of an isotropic emitting point source [15]. It is expressed as  (19) and this parameter is considered as the ultimate resolution of an imaging optics based on an ideal lens using isotropic light emission from a point source. For the following it is more convenient to compare the different PSF characteristics based on the FWHM (full width at half maximum). For the Airy distributions presented in Fig. 13  , red dots (see Fig. 12).
For a Gaussian distribution the relation between σ and FWHM is FWHM = 2.36 σ. Applying this relation for the PSF for off-axis light collection (see Fig. 12   In order to reconstruct the shape of a micron or even sub-micron beam profile from a measurement based on OTR imaging, the usage of a PSF having a single maximum is a task which is much simpler than the approach which was developed in [16,17]. The authors of the work [17] used a conventional OTR monitor scheme with a lens diameter of 30 mm and a focal distance of 120 mm together with a polarizer and an optical bandpass filter (λ = 550 nm). The double-lobe distribution of the vertical polarization component was measured, possessing an inter-peak distance of about 10 μm. In the so-called "PSF dominated regime" an rms beam size could be extracted using a complicated algorithm including a selfcalibration procedure. As a result they have obtained a minimum vertical beam size of 0.75 ± 0.03 μm [17]. In the view of the authors of the present work, the approach presented here will provide the same accuracy but an easier and more direct reconstruction procedure simply by determining rms sizes from the measured image and taking into account the calculated OTR PSF.
In the discussion so far it was assumed that the OTR PSF is the same for all beam electrons. Therefore, the beam image can be constructed from the convolution of the PSF with the beam distribution. In reality however, electrons emitting backward OTR have a certain transverse offset of about , x y σ σ with respect to the beam axis (which coincides with the optical axis in the presented scheme), and their field has to be calculated according to Eq.
(2) while performing the integration and taking into account this offset. Especially in the case of asymmetric light collection schemes this might result in OTR PSFs which depend on the transverse particle position in the beam cross section such that the convolution argument for beam image construction is no longer valid. For the conditions under consideration however ( , x y σ σ ≤ 5 um, a = 500 mm) the argument of the MacDonald function in Eq. (2) is changed within a range of 2 10 − , and the phase change there is even much smaller (in the order of 4 10 − ). As consequence it is expected that the PSF distortion due to a transverse offset will be smaller than a few percent and therefore can be neglected.
It was shown before that asymmetric optical schemes for OTR light collection can provide a significant narrowing of the OTR PSF distribution. Drawback of asymmetric OTR observation geometries however is a decrease in the overall intensity. In the ultra-relativistic approximation (which is valid for γ ≥ 1000 with accuracy better than 1%) it is possible to estimate the OTR photon yield for the cases under consideration using the formula ( ) In the following the yield of the OTR horizontal polarization component is estimated (which is described by the first term in the nominator in Eq. (13). For the shielding geometry presented in Fig. 6  The reduction of the OTR photon yield due to asymmetric light collection schemes will impose only minor restrictions: with the usage of a standard industrial camera for beam imaging based e.g. on the Sony IMX264 CMOS sensor [20] which has a dark noise of two electrons and a quantum efficiency of 68% at λ = 545nm, a single bunch current of 100 pC is sufficient in order to measure a single shot beam image, assuming (i) a Gaussian beam profile is sampled with 10 pixels per 1σ-width, (ii) in the central region of ± 2σx,y the number of photons is well above the camera noise level, and (iii) the conversion between electrons and photons is determined according to Eq. (23). Using a scientific grade camera, single shot beam imaging with even smaller bunch currents will be possible.

Conclusion
It was shown that asymmetric optical schemes for OTR light collection can provide a significant narrowing of the OTR PSF distribution. The achieved spatial distribution with such a scheme can be close to the ultimate resolution of a point source described by the Airy disk, and the FWHM value is even smaller than / m λ θ .
The calculations were carried out under the assumption of an ideal imaging lens, i.e. not taking into account optical imperfections as aberrations which might disturb the achieved resolution especially in the case of off-axis light collection. However, the consideration of higher order optical imperfections was out of focus of the present work. To the authors opinion these effects could be compensated in an experimental setup with a dedicated image lens design, e.g. by using aspheric lenses.
Drawback of asymmetric OTR observation geometries however is a decrease in the overall intensity. As it was shown before, the OTR intensity in this case is sufficient to allow single shot beam profile diagnostics with bunch charges down to 100 pC using standard cameras for industrial machine vision applications, in case of scientific grade cameras the sensitivity can even be increased. As consequence, the detectable OTR intensity will be well above the sensitivity threshold of a standard OTR monitor.
In order to further increase the sensitivity to a level of sub-micron spatial resolution [18], the techniques proposed in the present work could in principle be combined with detection schemes at smaller wavelengths using OTR in the UV and EUV spectral region [19].

Funding
Russian Ministry of Education and Science "Nauka" (3.1903.2017).