Investigations and improvements in ptychographic imaging Peng Li A thesis submitted for the degree of Doctor of Philosophy The University of Sheffield Department of Electronic and Electrical Engineering August 2016 Contents Abstract ····························································································· v Declaration ························································································ vii Acknowledgements ············································································· viii Publications ························································································ x 1 Introduction ···················································································· 1 1.1 Conventional imaging ···································································· 1 1.1.1 Abbé’s theory ······································································· 2 1.1.2 Resolution limit ····································································· 3 1.2 Diffractive imaging ········································································ 4 1.2.1 Crystallography ····································································· 5 1.2.2 Crystalline ptychography ·························································· 6 1.2.3 Direct ptychography ······························································· 8 1.2.4 Fourier holography ································································ 10 1.2.5 Iterative coherent diffractive imaging ·········································· 13 1.2.6 Iterative ptychography ···························································· 18 1.3 Outline of thesis ·········································································· 20 2 Background ···················································································· 25 2.1 The 2D Fourier transform ······························································· 25 2.1.1 Definition ··········································································· 25 2.1.2 Properties ··········································································· 26 2.1.3 Discrete Fourier transform ······················································· 28 2.1.4 The Shannon sampling theorem ················································· 28 2.2 Wave propagation ········································································ 30 2.2.1 Scalar diffraction theory ·························································· 31 2.2.2 Fresnel propagation ······························································· 32 2.2.3 Fraunhofer propagation ··························································· 34 2.2.4 Angular spectrum propagation ·················································· 35 2.3 Iterative ptychography ··································································· 37 2.3.1 The phase problem ································································ 37 i 2.3.2 Coherence··········································································· 38 2.3.3 Geometries of the image and detector ·········································· 42 2.3.4 Sampling requirement in ptychography ········································ 43 2.3.5 Experimental geometries for the illumination ································· 46 2.3.6 The Ptychographic Iterative Engine (PIE) ····································· 48 2.3.7 The extended PIE and its relation with the gradient descent method ······ 50 2.3.8 Reconstruction ambiguities ······················································ 52 2.3.9 Error metrics for the reconstructions ··········································· 55 2.3.10 Some key experimental parameters ··········································· 58 3 Direct Ptychography ········································································· 61 3.1 Direct ptychography of aperiodic objects ············································· 62 3.1.1 The 4D intensity dataset ·························································· 62 3.1.2 The derivation of WDDM ························································ 65 3.1.3 The frequency cut-off ····························································· 69 3.1.4 The projection strategy ··························································· 72 3.2 Noise effects on WDDM ································································ 75 3.2.1 Noise suppression via the probe design ········································ 75 3.2.2 Noise suppression via an iterative method ····································· 78 3.2.3 Model calculations for different noise levels ·································· 80 3.3 Solving for spatial partial coherence via WDDM ···································· 83 3.4 Conclusions ················································································ 87 4 Spatially Mixed State Ptychography ····················································· 89 4.1 Spatially mixed state ptychography ···················································· 89 4.1.1 The reconstruction algorithm ···················································· 90 4.1.2 The reconstruction structure ····················································· 92 4.1.3 Breaking the reconstruction ambiguities ······································· 96 4.1.4 Optical experiments ····························································· 100 4.2 The effects of a diffuser in the presence of spatial partial coherence ··········· 109 4.2.1 The benefits of a diffuser ······················································· 110 4.2.2 X-ray experiments ······························································· 111 4.3 Conclusions ·············································································· 118 5 Temporally Mixed State Ptychography ················································ 120 ii 5.1 Temporally mixed state ptychography ·············································· 120 5.1.1 The reconstruction algorithm ·················································· 120 5.1.2 The reconstruction structure ··················································· 122 5.1.3 Optical experiments ····························································· 125 5.2 An iterative method to remove background noise ································· 128 5.2.1 The background noise update function ······································· 128 5.2.2 Optical Experiments ···························································· 130 5.3 Conclusions·············································································· 133 6 3D Fourier Ptychography ································································· 135 6.1 2D Fourier ptychography ······························································ 136 6.1.1 The reconstruction algorithm ·················································· 138 6.1.2 Optical experiment ······························································ 141 6.2 3D Fourier ptychography ······························································ 144 6.2.1 The multislice method in real-space ptychography ························· 146 6.2.2 The reconstruction algorithm of 3D Fourier ptychography ··············· 148 6.2.3 Optical experiments ····························································· 152 6.2.4 Reconstruction resolution ······················································ 156 6.3 Conclusion and discussion ···························································· 161 7 Ptychographic Tomography ····························································· 163 7.1 Theoretical basics of tomography ···················································· 164 7.1.1 The calculation coordinates ···················································· 164 7.1.2 The Radon transform ··························································· 165 7.1.3 Fourier slice theorem ··························································· 167 7.1.4 Image reconstruction via filtered back projection ·························· 168 7.1.5 Sampling requirement for ptychographic tomography····················· 171 7.2 X-ray ptychographic tomography experiment of glass beads ···················· 172 7.2.1 Data acquisition ·································································· 173 7.2.2 Ptychographic reconstruction ·················································· 174 7.2.3 Tomographic reconstruction ··················································· 176 7.2.4 The possibility of electron ptychographic tomography ···················· 184 7.3 Conclusions·············································································· 187 8 Summary and Future Work ····························································· 191 iii 8.1 Summary ················································································· 191 8.2 Future work ·············································································· 197 Bibliography ····················································································· 202 iv Abstract This thesis has been devoted to investigate and improve ptychography, which is a newly developed coherent diffractive imaging technique that can achieve quantitative imaging (both modulus and phase) at diffraction-limited resolution without imaging lenses. In particular, this thesis has first looked into two solutions of partial coherence in ptychography: the Wigner distribution deconvolution method (WDDM) and mixed state decomposition. WDDM