Hanbury Brown–Twiss Interferometry at a Free-Electron Laser

We present measurements of second- and higher-order intensity correlation functions (so-called Hanbury Brown—Twiss experiment) performed at the free-electron laser (FEL) FLASH in the non-linear regime of its operation. We demonstrate the high transverse coherence properties of the FEL beam with a degree of transverse coherence of about 80% and degeneracy parameter of the order ${10}^9$ that makes it similar to laser sources. Intensity correlation measurements in spatial and frequency domain gave an estimate of the FEL average pulse duration of 50 fs. Our measurements of the higher-order correlation functions indicate that FEL radiation obeys Gaussian statistics, which is characteristic to chaotic sources.

Figure 1

(a) Scheme of the experiment. FEL radiation is generated in the undulator and is transmitted through the beam line including one collimating (M1) and two focusing (M2, M3) mirrors, a plane grating (PG), and an exit slit. Intensity profiles of individual femtosecond pulses are measured at the detector. (b),(c) Typical single pulse intensity profiles and (d) an average over $2\times {10}^4$ pulses for a bandwidth of $\Delta E/E=0.8\times {10}^{-4}$. (e)–(g) Projections of the pulse intensities along the vertical direction. The scale bar is 0.5 mm long.

Figure 2

Intensity correlation analysis. (a),(b) Intensity correlation function $g^{(2)}(x_1,x_2)$ for a bandwidth of $0.8\times {10}^{-4}$ (a) and $1.4\times {10}^{-3}$ (b). (c),(d) Intensity correlation function $g^{(2)}(\Delta x)$ taken along the white line in (a),(b) around the center of the beam. Insets in (c),(d) show intensity fluctuations $g^{(2)}(x,x)=⟨I^2(x)⟩/⟨I(x){⟩}^2$ taken along the blue line in (a),(b). The error bars in (c),(d) are obtained by statistical analysis of 20 individual subensembles (${10}^3$ shots each) from the whole set of $2\times {10}^4$ shots. The number of points displayed in this figure with error bars is reduced for better visibility.

Figure 3

Statistical properties of FEL light as a function of the bandwidth. (a) Contrast (points) ${\zeta }_2(D_{\omega })=g^{(2)}(0)-1$ as a function the coherence time ${\tau }_c$. The error bars are same as in Figs. 2c, 2d. Theoretical fit (solid line) corresponds to a pulse duration of $T=47\pm 21\mathrm{fs}$. (b) Measured single pulse (blue solid line) and average (black dotted line) spectra. Gaussian fit (red dashed line) gives an FEL bandwidth of $\Delta E/E=6.7\times {10}^{-3}$ FWHM. The width of the largest exit slit used in the experiment and corresponding to the bandwidth $\Delta E/E=1.4\times {10}^{-3}$ is shown by the gray region. (c) Intensity correlation function $g^{(2)}({\omega }_1,{\omega }_2)$ in the spectral domain. (d) Intensity correlation function $g^{(2)}(\Delta \omega )$ (red circles) taken along the white line in (c). Gaussian fit (blue line) gives an average pulse duration of 27 fs.

Figure 4

Higher order correlation functions for a bandwidth of $0.8\times {10}^{-4}$ (a),(b) and $1.4\times {10}^{-3}$ (c),(d). (a),(c) Third-order correlation function $g^{(3)}(x_1,x_2)$ (shown in the insets) and $g^{(3)}(\Delta x)$ (black solid line) taken along the white line in the insets. (b,d) Fourth-order correlation function $g^{(4)}(x_1,x_2)$ (shown in the insets) and $g^{(4)}(\Delta x)$ (black solid line) taken along the white line in the insets. The same functions obtained under assumption of the Gaussian statistics are shown as blue dashed lines. The number of points displayed in this figure with error bars is reduced for better visibility. Error bars are evaluated in the same way as in Fig. 2.