Simulation of transition radiation based beam imaging from tilted targets

Transverse beam profile diagnostics in linear electron accelerators is usually based on direct imaging of a beam spot via visible transition radiation. In this case the fundamental resolution limit is determined by radiation diffraction in the optical system. A method to measure beam sizes beyond the diffraction limit is to perform imaging dominated by a single-particle function (SPF), i.e. when the recorded image is dominated not by the transverse beam profile but by the image function of a point source (single electron). Knowledge of the SPF for an experimental setup allows one to extract the transverse beam size from an SPF dominated image. This paper presents an approach that allows one to calculate two-dimensional SPF dominated beam images, taking into account the target inclination angle and the depth-of-field effect. In conclusion, a simple fit function for beam size determination in the case under consideration is proposed and its applicability is tested under various conditions.

Figure 1

Sketch of the calculation geometry.

Figure 2

Calculated OTR SPFs according to Eqs. (7)–(9). The parameter set for the calculation is indicated in the text.

Figure 3

Calculation scheme for the inclined target geometry together with geometrical definitions.

Figure 4

SPFs calculated for horizontally displaced radiation emitters from a tilted target surface, emulating OTR emission from individual electrons out of a ${\sigma }_x=30\mu \mathrm{m}$ thick beam profile. The numerical aperture amounts to ${\theta }_m=0.10$.

Figure 5

Calculated two-dimensional beam images for a beam size of ${\sigma }_x=30\mu \mathrm{m}$, ${\sigma }_y=0.4\mu \mathrm{m}$ and for different numerical apertures ${\theta }_m$, in top-to-bottom order ${\theta }_m=0.05$, ${\theta }_m=0.075$, ${\theta }_m=0.10$, and ${\theta }_m=0.19$.

Figure 6

Examples of vertical beam image cross sections, taken from Fig. 5 at two different horizontal positions (a) $x_i=0$ and (b) $x_i/M=40\mu \mathrm{m}$ with the numerical aperture ${\theta }_m$ as parameter: ${\theta }_m=0.19$ (red solid line), ${\theta }_m=0.10$ (green dashed line), ${\theta }_m=0.075$ (blue dash-dotted line), and ${\theta }_m=0.05$ (black dotted line).

Figure 7

Calculated vertical SPF profile (points) and fit (solid line) according to Eq. (11). The SPF calculation parameters are those of Fig. 2 for ideal focusing ($\mathrm{\Delta }a=0$).

Figure 8

Calculated vertical beam profiles at $x_i=0$ (green triangles and yellow points) together with the fitted functions according to Eq. (12) (red solid line) and to Eq. (13) (black dashed line). The vertical beam profile is indicated by the blue dash-dotted line. The beam image calculation parameters are the same as in Fig. 5 for ${\theta }_m=0.1$.

Figure 9

SPF distribution of Fig. 7 plotted on a logarithmic scale (red circles) together with fit based on $f_1^{\text{mod}}(y_i)$ Eq. (13) (red line). In addition, the individual contributions from the fit of the central lobe (blue solid line) and the side lobes (blue dashed line) are shown.

Figure 10

Upper plot: Beam image calculated for ${\theta }_m=0.19$, $\lambda =400\mathrm{nm}$, ${\sigma }_y=1\mu \mathrm{m}$. Lower plot: Comparison of vertical beam profiles taken at different horizontal positions with the fit of the central region according to Eq. (13). The points represent the calculated profiles, the solid lines the fitted curves. The horizontal positions are $x_i=0$ (red squares/line, ${\sigma }_y^{\text{fit}}=1.16\pm 0.06\mu \mathrm{m}$), $x_i/M=10\mu \mathrm{m}$ (green triangles/line, ${\sigma }_y^{fit}=1.08\pm 0.02\mu \mathrm{m}$), and $x_i/M=20\mu \mathrm{m}$ (blue dots/line ${\sigma }_y^{fit}=0.92\pm 0.01\mu \mathrm{m}$).

Figure 11

(a) Comparison of two different fit functions applied to extract vertical beam sizes from calculated beam images. Parameter set for calculation: ${\theta }_m=0.1$, $\lambda =400\mathrm{nm}$, and OTR $\sigma$-polarization component. Red points: proposed fit function Eq. (13); blue points: Gaussian fit function. The black solid line indicates ideal coincidence between input and fitted beam sizes, the gray dashed line shows the beam size at which the visibility becomes equal to zero. (b) Relative difference between input and fitted beam sizes according to Eq. (13).

Figure 12

Comparison of beam profile data (blue points) with fit function Eq. (13) (red solid line) under real experimental conditions. For the profile the data from Fig. 8 were taken for a beam size of ${\sigma }_y=0.9\mu \mathrm{m}$ and a random noise was added.