Two-dimensional electron beam size measurements with x-ray heterodyne near field speckles

In this paper we report on recent two-dimensional (2D) electron beam size measurements with a nonconventional synchrotron radiation interferometric technique based on x-ray heterodyne near field speckles (HNFS). The method relies on Fourier analysis of the random speckle patterns generated by a water suspension of nanospheres to assess the full 2D transverse coherence of the incoming x rays. The horizontal and vertical electron beam sizes are then retrieved by means of statistical optics approaches. The manuscript thoroughly describes the HNFS technique, and shows experimental results obtained at the ALBA Synchrotron Light Source. By changing the machine coupling, beam sizes as small as $5\mu \mathrm{m}$ are measured, thus improving on past measurements reported in the literature and proving the HNFS diagnostics suitable for low-emittance particle beams.

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Figure 1

Geometry for the self-referencing interference between the intense transilluminating x-ray beam and the weak scattered spherical wave. The case of interest in this paper of a horizontally elongated electron beam is depicted. From left to right: 2D electron beam distribution at the source plane, time-resolved field distribution of the emitted x rays impinging onto the scatterer, and finally time-integrated single-scatterer interference pattern.

Figure 2

Simulated single-scatterer interference pattern in the case of a filament electron beam (a), a round electron beam (b) and a horizontally elongated electron beam (c). Plots refer to Eqs. (3) and (6), namely the interference pattern generated by a single scatterer on top of the (uniform) x-ray intensity distribution. The side of each square is $140\mu \mathrm{m}$. Simulation parameters: $\lambda =0.1\mathrm{nm}$, $Z_0=30\mathrm{m}$, $z=2\mathrm{m}$, $23\mu \mathrm{m}$ (H) × $23\mu \mathrm{m}$ (V) electron beam in (b), $45\mu \mathrm{m}$ (H) × $11\mu \mathrm{m}$ (V) electron beam in (c).

Figure 3

Simulated single-scatterer interference pattern (a) and corresponding 2D power spectrum (b) for a horizontally elongated electron beam. Simulated heterodyne speckles (c) and corresponding 2D power spectrum (d) for the same electron beam configuration as in (a) and (b). Notice that the power spectrum of heterodyne speckles closely resembles the power spectrum of the single-scatterer interferogram.

Figure 4

Summary of the HNFS technique showing the relation between the 2D profile of the electron beam [(a), (g), and (m)], the modulus of the 2D CCF $\mu (\mathrm{\Delta }\overrightarrow{x})$ of the emitted synchrotron radiation [(b), (h), and (n)], the single-scatterer interference fringes $i_{\mathrm{ref}}(\overrightarrow{x},z)$ [(c), (i), and (o)] and the corresponding 2D power spectrum $I_{\mathrm{ref}}(\overrightarrow{q},z)$ [(d), (j), and (p)], the heterodyne speckle pattern $i_{\mathrm{het}}(\overrightarrow{x},z)$ generated by a large number of scatterers [(e), (k), and (q)] and finally the 2D power spectrum of heterodyne speckles $I(\overrightarrow{q},z)$ [(f), (l), and (r)]. The case of a filament electron beam [(a), (b), (c), (d), (e), and (f)], a round electron beam [(g), (h), (i), (j), (k), and (l)] and a horizontally elongated electron beam [(m), (n), (o), (p), (q), and (r)] are depicted. Notice the relation between the 2D transverse profile of the electron beam, the 2D CCF of the emitted synchrotron light and the envelope of the Talbot oscillations in the 2D power spectrum.

Figure 5

Effect of 3D scattering [(a), (c), and (e)] and walk-off [(b), (d), and (f)] on Talbot oscillations for three progressively increasing sample-detector distances $z_1=5\mathrm{cm}$ [(a) and (b)], $z_2=25\mathrm{cm}$ [(c) and (d)] and $z_3=50\mathrm{cm}$ [(e) and (f)] and for the actual experimental parameters described in Sec. 3.

Figure 6

Simulated horizontal profiles of 2D power spectra for three different distances $z_1=10\mathrm{cm}$, $z_2=30\mathrm{cm}$ and $z_3=50\mathrm{cm}$ as a function of Fourier wave vectors (a) and transverse displacements (b). Upon the spatial scaling, Talbot maxima and minima clearly fit onto the two unique master curves of the upper and lower envelope $u(\mathrm{\Delta }x)$ and $w(\mathrm{\Delta }x)$, respectively, as described by Eq. (17).

Figure 7

HNFS experimental setup at the NCD-SWEET beamline at ALBA.

Figure 8

Acquired raw data (a) and corresponding differential heterodyne speckles resulting from the DFA (b). Notice that the speckle pattern is remarkably uniform, despite the intensity modulations and stray-light contributions clearly visible in the raw image. The side of each square image is roughly $260\mu \mathrm{m}$ in real space. Data refer to $\kappa =0.50\%$ and $z=1.1\mathrm{m}$.

Figure 9

2D power spectra at $\kappa =0.50\%$ (a) and $\kappa =2.80\%$ (b). Data refer to $z=1.1\mathrm{m}$.

Figure 10

Master curves describing the upper and lower envelopes of Talbot oscillations along the horizontal (a) and vertical (b) direction for $\kappa =2.80\%$. Different colors represent different sample-detector distances $z$.

Figure 11

Measured horizontal (a) and vertical (b) beam sizes as a function of the beam coupling measured at FE34. Expected values based on independent pinhole and LOCO measurements are also reported for comparison.