Norm-Minimized Scattering Data from Intensity Spectra

Abstract

We apply the ℓ 1 minimizing technique of compressive sensing (CS) to non-linear quadratic observations. For the example of coherent X-ray scattering we provide the formulae for a Kalman filter approach to quadratic CS and show how to reconstruct the scattering data from their spatial intensity distribution. PACS numbers: 02.30.Zz, 02.50.-r, 07.85.-m I. INTRODUCTIONThe rapidly growing Si technology in semiconductor electronics [2,5] opens the possibility to grow III-V inorganic nanowires, such as GaAs, InAs or InP, which were supposed to have potential [14,17] to become building blocks in a variety of nanowire-based nanoelectronic devices, for example in nanolaser sources [29] or nanoelectronics [26]. Such epitaxially grown nanowires are repeating the crystal orientation of the substrate and usually grow in Wurtzite (WZ) or Zinc-Blende (ZB) structure differing in the stacking sequence ABABAB and ABCABCABC respectively of the atomic bilayers. Theoretical predictions on the electronic properties [1] of these nanowires show that stacking sequences with WZ and ZB segments considerably differ in the conductivity. However, during the nanowire growth stacking faults, the mixing of ZB and WZ segments takes place, and twin defects [4] appear. As these defects have their own impact on the conductivity and band structure there is great interest in knowing the exact stacking sequence which can be studied by e.g. Transmission Electron Microscopy [38]. But, as this is a destructive method, it is impossible to use the nanowire after the structural studies. Nowadays the 3rd generation synchrotron sources and rapidly developing focusing devices like Fresnel Zone Plates opens new fields of non-destructive X-ray imaging. For example in the Coherent X-ray Diffraction Imaging experiments (CXDI) an isolated nanoobject is illuminated with coherent X-ray radiation and the scattered intensity is measured by a 2D detector [10,34] under the Bragg