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

Abstract

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 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 the Bragg condition everywhere and phase requirement for point focusing, and effectively focus hard x-rays to a scale close to the wavelength.

Fabrication of wedged multilayer Laue lenses

“…At the same time the radius of curvature of the super-surface must scale by a factor t, equivalent to scaling the deposition rate gradient by 1/t. These conditions are achieved with a deposition rate that varies across the substrate as z 1/2 , which is not surprising given that ideal Laue lenses can be constructed from confocal parabolas [8]. Such a deposition profile could be attained in the mask penumbra to good approximation for small focal separations, as is seen in Fig.…”

“…These were built as analogues of the thin diffractive optics fabricated by lithography for soft X-rays, but with an increased depth to achieve a significant phase difference on transmission through the two different layer materials. Much higher diffraction efficiency could be achieved by tilting the lens relative to the incident beam to ensure that the Bragg condition is satisfied, at least over some region of the lens [2,7,8]. When the numerical aperture (NA) of a lens exceeds the Darwin width of the reflection at any part of the lens then the lens will be severely apodized and the effective NA will be limited by the diffraction efficiency.…”

“…This so-called wedged MLL (Fig. 1) closely approximates the ideal volume zone plate consisting of confocal parabolic layers for an incident plane wave or confocal ellipsoidal layers for spherical wave illumination [8]. Fig.…”

PtyNAMi: Ptychographic Nano-Analytical Microscope at PETRA III: interferometrically tracking positions for 3D x-ray scanning microscopy using a ball-lens retroreflector

“…Recent advances in x-ray optics have pushed these limits to the single-digit nanometer range, 10, 11 and theoretical considerations predict focusing in the (sub-)nanometer range. 12,13 Improving the numerical aperture to these physical limits is very challenging and currently limited by technology.…”

Focusing hard x rays beyond the critical angle of total reflection by adiabatically focusing lenses

“…Indeed, besides for waveguides, 1 this assumption holds for mirror optics based on total external reflection 2 and refractive x-ray lenses, [3][4][5] limiting x-ray nanobeams to tens of nanometers. 1,4 For x-ray optics based on diffraction, such as Fresnel zone plates, multilayer Laue lenses, or multilayer mirrors, it was shown [6][7][8] that their numerical aperture is not bounded from above by the critical angle of total external reflection. Also kinoform lenses can in principle overcome this limitation.…”