X-Ray Optics on a Chip: Guiding X Rays in Curved Channels

We study the propagation of hard x rays in single curved x-ray waveguide channels and observe waveguide effects down to surprisingly small radii of curvature $R\simeq 10\mathrm{mm}$ and a large contour length $s\simeq 5\mathrm{mm}$, deflecting beams up to 30°. At these high angles, about 2 orders of magnitude above the critical angle of total reflection ${\theta }_c$, most radiation modes are lost by “leaking” into the cladding, while certain “survivor” modes persist. This may open up a new form of integrated x-ray optics “on a chip,” requiring curvatures mostly well below the extreme values studied here, e.g., to split and to delay x-ray pulses.

Figure 1

(a) Schematic of the channels defined in the XWGchip and the experimental geometry. The individual channels are selected by focusing the beam into the entrance. Guiding of a beam in a curved channel is evidenced by measurement of the far-field pattern. (b) Finite difference simulation of beam propagation in the curved WG channels. At $z=0$ a plane wave of unit amplitude and 7.9 keV photon energy impinges on a channel of $w=100\mathrm{nm}$ and $R=40\mathrm{mm}$ in Ta (initial data). The intensity distribution is shown in logarithmic scale within a rectangular region of length $\mathrm{\Delta }z=0.5\mathrm{mm}$ and width $\mathrm{\Delta }x=5\mu \mathrm{m}$, along with a zoom of the central region with $\mathrm{\Delta }z=100\mu \mathrm{m}$ and $\mathrm{\Delta }x=1.5\mu \mathrm{m}$. The transmission is $T=0.842$, after a propagation length of 0.5 mm. The spikes show that evanescent waves depart into the cladding at certain locations corresponding to reflections in the geometric optical description. (c) Fundamental mode of a channel in tantalum (Ta) with $w=100\mathrm{nm}$ and $R=10\mathrm{mm}$ as simulated by solving the wave equation with a tilted potential well.

Figure 2

Measured far-field patterns of the curved WG series. (a) Experimental geometry. Intensity distribution on the pixel detector moved out of the plane of incidence to record the exit WG beam at a deflection angle $\theta$ as given by the exit tangent of the curved channel, for channels with (b) $R=10\mathrm{mm}$, (c) $R=20\mathrm{mm}$, (d) $R=40\mathrm{mm}$, and (e) $R=80\mathrm{mm}$. Intensity due to radiation losses of leaky modes is observed along the entire horizon, but the propagating modes of the waveguide couple out only at the end of the 5-mm-long WG chip, leading to a pronounced peak in the WG pointing direction. (f) The intensity distribution after integration over the exit angle ${\alpha }_f$, as a 1D plot for the example of the $R=40\mathrm{mm}$ guide shows that the exit peak of the transported modes is significantly higher than the radiation loss at smaller $\theta$.

Figure 3

Radiation loss and waveguide exit flux. (a) High-resolution far-field intensity distribution (logarithmically scaled color code) around the horizon of the $R=40\mathrm{mm}$ channel exit at ${\theta }_{\mathrm{WG}}=7.12°$ (red colored intensity maximum in right detector panel) with a total waveguide exit intensity $I_{\mathrm{WG}}=0.97\times {10}^8\mathrm{ph}/\mathrm{s}$, as computed from the corresponding region of interest in the right detector panel. A fine structure in the leakage radiation (horizontal stripe) is observed, originating from interference of radiation tunneling out of the leaky modes. Scale bar: 100 pixel, corresponding to 5.5 mm on the detector, or $\mathrm{\Delta }\theta =0.42°$. (b) Integrated leakage intensity along the contour as a function of $\theta$ for different channels (not including the WG exit radiation). (c) Waveguide exit intensities as a function of $1/R$, normalized to the flux impinging onto the channel entrance, along two exponential curves $\exp (-R_0/R)$ as a guide to the eye, with $R_0=99\mathrm{mm}$ (solid line) and $R_0=23\mathrm{mm}$, respectively. Note that in order to obtain the correct transmission of the channels, these values have to be corrected for coupling efficiency and leakage due to the open top (see Ref. [20]).