Takagi-Taupin description of x-ray dynamical diffraction from diffractive optics with large numerical aperture

We present a formalism of x-ray dynamical diffraction from volume diffractive optics with large numerical aperture and high aspect ratio, in an analogy to the Takagi-Taupin equations [Acta Crystallogr. 15, 1311 (1962); Bull. Soc. Fr. Mineral. Crystallogr. 87, 469 (1964)] for strained single crystals. We derive a set of basic equations for dynamical diffraction from volume diffractive optics, which enable us to study the focusing property of these optics with various grating profiles. We study volume diffractive optics that satisfy the Bragg condition to various degrees, namely, flat, tilted, and wedged geometries, and derive the curved geometries required for ultimate focusing. We show that the curved geometries satisfy both the Bragg condition everywhere and phase requirement for point focusing and effectively focus hard x rays to a scale close to the wavelength. Our calculations were made for an x-ray wavelength of $0.064\mathrm{nm}$ $\left(19.5\mathrm{keV}\right)$.

Figure 1

Types of multilayer Laue lens (MLL) layer arrangements: flat, tilted, wedged, and curved.

Figure 2

A mapping of layers between a Fresnel zone plate and a periodic structure composed by binary bars with thickness ratio 1:1.

Figure 3

The zone plate structure can be constructed by a set of sine waves with varying period that obey zone plate law.

Figure 4

Schematic of a plane wave diffracted by a tilted MLL with flat zones, in which many diffraction orders are excited.

Figure 5

(Color online) (a) The radial efficiency distributions of different diffraction orders on the exit surface of a flat MLL at $\theta =0$ (top), $\theta =1.6$ (middle), and $\theta =3.2\mathrm{mrad}$ (bottom), $w=13.5\mu \mathrm{m}$. (b) The efficiency variation of the negative first-order diffraction inside the zone plate at $\theta =0$ (left) and $\theta =1.6\mathrm{mrad}$ (right). (c) The radial efficiency distribution of the negative first-order diffraction on the exit surface at different thicknesses and tilting angles.

Figure 6

(Color online) (a) Isophotes near the focus at different tilting angles, $\theta =0$, 1.1, 2.1, and $3.2\mathrm{mrad}$, in (i), (ii), (iii), and (iv). The intensity is plotted in a logarithmic scale. (b) The intensity profiles on the best focal plane.

Figure 7

(Color online) (a) The radial intensity distribution of the negative first-order diffraction on the exit surface of a wedged MLL with an outmost zone width of $0.25\mathrm{nm}$. Two-beam approximation is assumed. The dashed curve represents the phase deviation from a perfect converging spherical wave front. (b) The intensity profile on the focal plane, showing a peak width $\mathrm{FWHM}=0.34\mathrm{nm}$. The inset on the top is the isophotes near focus; the intensity is plotted in a logarithmic scale. The one on the bottom is a sketch of the wedged MLL structure.

Figure 8

The domain of dependence in an MLL (left), which has a curved edge due to the change of diffracted direction with position, and the inverse Borrmann fan in a single crystal (right), which is a triangle that is invariant with position.

Figure 9

(a) A schematic of satisfying Bragg’s law everywhere for an incident plane wave. All diffracted wave vectors point to the focal point $F$. (b) A structure consisting of a set of confocal paraboloids obtained from (a), serving as an ideal structure to focus a plane wave into a point.

Figure 10

(Color online) (a) The radial intensity distribution of the negative first-order diffraction on the exit surface of an MLL with ideal parabolic zones and outmost zone width of $0.25\mathrm{nm}$, based on the two-beam simulation. The dashed curve represents the phase deviation from a perfect converging spherical wave front. (b) The intensity profile on the focal plane, showing a peak with $\mathrm{FWHM}=0.21\mathrm{nm}$. The inset on the top is the isophote pattern near focus; the intensity is on a logarithmic scale. The one on the bottom is a sketch of the ideal structure.

Figure 11

(a) A schematic of satisfying Bragg’s law everywhere for an incident spherical wave. All diffracted wave vectors point to the focal point $F$. (b) A structure consisting of a set of confocal ellipsoids, serving as an ideal structure to focus a point in object space into a point in image space.

Figure 12

(Color online) The focused beam profile on the plane of best focus and the isophotes (logarithmic scale) near the image point (the inset on the top) for 1:1 imaging. The inset on the bottom shows the difference of an MLL with flat zones (solid line) and an ideal MLL (dashed line).