Phase Imaging: A Compressive Sensing Approach
Since Wolfgang Pauli posed the question in 1933, whether the probability densities |Ψ(r)|² (real-space image) and |Ψ(q)|² (reciprocal space image) uniquely determine the wave function Ψ(r) [1], the so called Pauli Problem sparked numerous methods in all fields of microscopy [2, 3]. Reconstructing the complete wave function Ψ(r) = a(r)e-iφ(r) with the amplitude a(r) and the phase φ(r) from the recorded intensity enables the possibility to directly study the electric and magnetic properties of the sample through the phase. In transmission electron microscopy (TEM), electron holography is by far the most established method for phase reconstruction [4]. Requiring a high stability of the microscope, next to the installation of a biprism in the TEM, holography cannot be applied to any microscope straightforwardly. Recently, a phase retrieval approach was proposed using conventional TEM electron diffractive imaging (EDI). Using the SAD aperture as reciprocal-space constraint, a localized sample structure can be reconstructed from its diffraction pattern and a real-space image using the hybrid input-output algorithm [5]. We present an alternative approach using compressive phase-retrieval [6]. Our approach does not require a real-space image. Instead, random complimentary pairs of checkerboard masks are cut into a 200 nm Pt foil covering a conventional TEM aperture (cf. Figure 1). Used as SAD aperture, subsequently diffraction patterns are recorded from the same sample area. Hereby every mask blocks different parts of gold particles on a carbon support (cf. Figure 2). The compressive sensing problem has the following formulation. First, we note that the complex-valued reciprocal-space wave-function is the Fourier transform of the (also complex-valued) real-space wave-function, Ψ(q) = F[Ψ(r)], and subsequently the diffraction pattern image is given by |Ψ(q)|2 = |F[Ψ(r)]|2. We want to find Ψ(r) given a few differently coded diffraction pattern measurements yn = |F[HnΨ(r)]|2, where the matrices Hn encode the mask structure of the aperture. This is a nonlinear inverse problem, but has been shown to be solvable even in the underdetermined case [6]. Since each diffraction pattern yn contains diffraction information from selected regions of the same sample, the differences in each pattern contain local phase information, which can be combined to form a full estimate of the real-space wave-function[7]. References: [1] W. Pauli in “Die allgemeinen Prinzipien der Wellenmechanik“, ed. H Geiger and W Scheel, (Julius Springer, Berlin). [2] A. Tonomura, Rev. Mod. Phys. 59 (1987), p. 639. [3] J. Miao et al, Nature 400 (1999), p. 342. [4] H. Lichte et al, Annu. Rev. Mater. Res. 37 (2007), p. 539. [5] J. Yamasaki et al, Appl. Phys. Lett. 101 (2012), 234105. [6] P Schniter and S Rangan. Signal Proc., IEEE Trans. on. 64(4), (2015), pp. 1043. [7] Supported by the Chemical Imaging, Signature Discovery, and Analytics in Motion initiatives at PNNL. PNNL is operated by Battelle Memorial Inst. for the US DOE; contract DE-AC05-76RL01830.