Ptychography is a scanning Coherent Diffraction Imaging (CDI) method for reconstructing a complex valued object function from intensity measurements recorded in the Fraunhofer or Fresnel diffraction region. The technique provides a solution to the so-called ‘phase problem’ and is found to be very suitable for Extreme Ultraviolet (EUV) and X-ray imaging applications due to its high fidelity and its minimum requirement on optical imaging elements. Moreover, abundant studies show that ptychography is able to provide a wide field-of-view and retrieve the illumination probe also. During the last two decades, ptychography has been successfully demonstrated with X-ray radiation sources, electron beams and visible light sources.

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Objective and impact

For performing phase retrieval in the EUV regime more efficiently, developing polychromatic ptychography is desirable. As an alternative to the existing ptychographic information multiplexing (PIM) method, we develop an another scheme where all monochromatic exit waves are expressed in terms of the amplitude of the transmission function and the thickness function of the object. Our proposed algorithm is a gradient based method and its validity is studied numerically. In addition, the sampling issue which appears in the polychromatic ptychography scheme and its influence to the reconstruction quality are discussed. The results show that the proposed method performs nicely for a known probe, and that a reasonable level of noise can be handled by the algorithm. Although in principle performing ptychography with a polychromatic light source can give higher SNR and shorter acquisition time, the reconstruction quality is not necessarily better than the monochromatic ptychography results due to the missing data issue. Compared to the PIM method, is less sensitive to the missing data issue.

We investigate the performance of ptychography with noisy data by analyzing the Cram\'{e}r Rao Lower Bound (CRLB). The lower bound of ptychography is derived and numerically computed for both top-hat plane wave and structured illumination. The influence of Poisson noise on the ptychography reconstruction is discussed. The computation result shows that, if the estimator is unbiased, the minimum variance for Poisson noise is mostly determined by the illumination power and the transmission function of the object. Monte Carlo analysis is conducted to validate our calculation results for different photon flux numbers. Furthermore, the performance of the maximum likelihood method and the approach of amplitude-based cost function minimization is studied in the Monte Carlo analysis. Our result can help to understand the defects that occur in the ptychograghy reconstruction from noisy data.

We develop a parameter retrieval method which combines ptychography and additional prior knowledge about the object. The proposed method is applied to two applications: (1) Parameter retrieval of sub-wavelength particles using Fourier ptychography with dark field measurements only. For this example the configuration is in line with the ‘RapidNano’ particle scanner developed by TNO. The particle scanner is supposed to detect nano-particles on an EUV reticle. Since only dark field images are recorded in the scanner, the part of the spatial spectrum of the object in the neighborhood of origin is lost. The missing data can in principle be filled in by analytic continuation using the fact that the object has bounded support, however, this method is unstable with noisy measurement and leads in practice to incorrect reconstructions. The proposed parameter retrieval algorithm is able to extract information of sub-wavelength particles from the incomplete data.; (2) Parameter retrieval of rectangular objects using real-space ptychography. This example can also be applied to the metrology of EUV reticles. The influence of Poisson noise is discussed in the second part of this work. The CRLB in both applications is computed and Monte Carlo analysis is used to verify the calculated lower bound. With the computation results we report the lower bound for various noise levels and the correlation of particles in application 1. For application 2 the correlation of parameters of the rectangular structure is discussed.

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Publications

• Cramér Rao lower bound and maximum likelihood estimation in ptychography with Poisson noise

  X. Wei, H.P. Urbach and W.M.J. Coene, Phys. Rev. A 102, 043516 (2020)

• Parameter retrieval methods in ptychography

  X. Wei, H.P. Urbach, P. Van der Walle and W.M.J. Coene, arXiv:2009.01652

• Ptychography with multiple wavelength illumination

  X. Wei and H.P. Urbach, Optics Express 27, 25 (2019)