Sparsity-based super-resolved coherent diffraction imaging of one-dimensional objects

Phase-retrieval problems of one-dimensional (1D) signals are known to suffer from ambiguity that hampers their recovery from measurements of their Fourier magnitude, even when their support (a region that confines the signal) is known. Here we demonstrate sparsity-based coherent diffraction imaging of 1D objects using extreme-ultraviolet radiation produced from high harmonic generation. Using sparsity as prior information removes the ambiguity in many cases and enhances the resolution beyond the physical limit of the microscope. Our approach may be used in a variety of problems, such as diagnostics of defects in microelectronic chips. Importantly, this is the first demonstration of sparsity-based 1D phase retrieval from actual experiments, hence it paves the way for greatly improving the performance of Fourier-based measurement systems where 1D signals are inherent, such as diagnostics of ultrashort laser pulses, deciphering the complex time-dependent response functions (for example, time-dependent permittivity and permeability) from spectral measurements and vice versa.


Supplementary Figure 4: Super-resolved CDI of a 1D dense piecewise-constant object. (a)
The "original" 1D object. (b) Power spectrum of the original object with 43dB noise. (c) Truncated power spectrum that corresponds to the part used to simulate the measured data. (d) The blurred reconstruction calculated by inverse Fourier transform of the "measured" power spectrum presented in (c), assuming full knowledge of the spectral phase. (e) Sparsity-based reconstruction (dashed red) compared with the original image (solid blue). The reconstruction uses the "measured" power spectrum (of (c)) and the prior information that the original image is sparse in the frame of shifted rectangular functions with different widths. Extrapolated power spectrum (f) and recovered spectral phase (g) calculated via sparsity-based reconstruction (dashed red) compared with the original image (solid blue).

Demonstration of ambiguity in 1D CDI of example objects in the paper
It is well known that 1D CDI is generally an ill-posed problem: Generic compact support objects correspond to the same far-field intensity pattern 2-3 . Still, there are some uncommon objects for which 1D CDI can yield unique solutions where knowing the support is the only prior information. In this section, we demonstrate that the theoretical ( Fig. 1) and experimental (Fig. 3) examples in the paper do not belong to those unusual cases, but rather belong to the general class of signals that do suffer from the ambiguity problem characteristic to the problem of phase retrieval of 1D objects.
We begin with the object in Fig. 1 which is also plotted in Supplementary Figure 1 Supplementary Figures 1c and 1e, respectively). This shows that it is sparsity that distinguishes between the correct signal and the ambiguous signals. Supplementary x-ray CCD camera (1024×256 pixels). The sought information in our object is practically 1D; hence we integrate the detected intensity pattern along the non-diffraction (horizontal) dimension (256 pixels). Examining the performance of our sparsity-based algorithm (utilizing prior knowledge that the sought information is sparse in the basis of shifted rectangles) by comparing the reconstructed information with the original image (Supplementary Figure 3 (g,f,h)) leads to ~6 times resolution enhancement (while we measured the power spectrum of the object up to spatial frequency 0.073um -1 , we reconstructed its spatial spectral amplitude and phase with good fidelity up to 0.4um -1 ). The experimental object in this example is symmetric; a fact that in principle could assist the phase retrieval, yet symmetry was not used in our reconstruction.

Sparsity-based algorithm for super-resolved 1D CDI of objects that consist of rectangles with only approximately known widths.
Our problem of super-resolved phase retrieval can be written mathematically as

|| ||
Here is the magnitude-squared of a point DFT of a vector with , is the i th -row of a DTF matrix, is a dictionary where and || || Algorithm for basis update.

Input:
Measurements , rectangle width value and uncertainty with step , threshold parameter , maximal number of iterations and .

Output:
Estimate ̂ of

Initialize:
Construct dictionary consisting of shifted bars with constant width . Solve problem (1) with new dictionary and obtain solution ̂ .

End While
Return ̂ ̂ Sparsity-based super-resolved 1D CDI of objects that are sparse in a frame of rectangles with various widths.
The "sparsity basis" is an important component in sparsity-based CDI (in all dimensions). The basis that was used in the paper -rectangle functions with a fixed widthis not an essential  That is, the incomplete power spectrum has led to considerable loss of resolution even if the spectral phase is known. Next, we implement sparsitybased reconstruction on the truncated spatial power spectrum, without assuming any knowledge on the spectral phase. As a model, we assume that the object is constructed from a small Clearly, the reconstructed object, its complete power spectrum and its reconstructed spectral phase match the original object very well despite usage of the noisy truncated spectrum as "measured data" and the lack of any knowledge on the spectral phase.