EDITOR-IN-CHIEF Peter W Hawkes CEMES-CNRS Toulouse, France VOLUME ONE HUNDRED AND EIGHTY FIVE ADVANCES IN IMAGING AND ELECTRON PHYSICS Edited by PETER W HAWKES CEMES-CNRS, Toulouse, France AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Cover photo credit: David Agard et al Single-Particle Cryo-Electron Microscopy (Cryo-EM): Progress, Challenges, and Perspectives for Further Improvement Advances in Imaging and Electron Physics (2014) 185, pp 113–137 Academic Press is an imprint of Elsevier 32 Jamestown Road, London NW1 7BY, UK 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2014 Copyright Ó 2014 Elsevier Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online by visiting the Elsevier web site at http://www.elsevier com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-800144-8 ISSN: 1076-5670 For information on all Academic Press publications visit our Web site at http://store.elsevier.com/ Printed in the United States of America PREFACE The three chapters in this latest volume span most of the regular themes of the series: an ingenious extension of geometrical optics, electron microscopy and mathematical morphology We begin with a very complete account of complex geometrical optics by P Berczynski and S Marczynski This variant of traditional geometrical optics allows diffraction phenomena to be studied It has two forms, one ray-based, the other eikonal-based, and the authors describe these fully After presenting the underlying theory, the approach is used to study the propagation of Gaussian beams in inhomogeneous media This is followed by a summary of the present state of cryo-electron microscopy for the study of unstained biological macromolecules by D Agard, Y Cheng, R.M Glaeser and S Subramaniam The subject is not new but a large step foward has recently been made with the introduction of a new type of electron-detection camera I leave the authors to set out the advantages of this innovation but we can be sure that many valuable new results can be anticipated To conclude, we have a long and authoritative account by M Welk and M Breuß of the image-adaptive structuring elements known as amoebas It has been shown that for iterated median filtering with a fixed structuringelement, the process is closely related to a partial differential equation associated with the image in question Here, the authors examine the relation between discrete amoeba median filtering and their (continuous) counterparts based on partial differential equations I have no doubt that this clear and very complete account of the subject will be widely appreciated Peter Hawkes vii j FUTURE CONTRIBUTIONS H.-W Ackermann Electron micrograph quality J Andersson and J.-O Str€ omberg Radon transforms and their weighted variants S Ando Gradient operators and edge and corner detection J Angulo Mathematical morphology for complex and quaternion-valued images D Batchelor Soft x-ray microscopy E Bayro Corrochano Quaternion wavelet transforms C Beeli Structure and microscopy of quasicrystals M Berz (Ed.) Femtosecond electron imaging and spectroscopy C Bobisch and R M€ oller Ballistic electron microscopy F Bociort Saddle-point methods in lens design K Bredies Diffusion tensor imaging A Broers A retrospective R.E Burge A scientific autobiography A Carroll Refelective electron beam lithography N Chandra and R Ghosh Quantum entanglement in electron optics A Cornejo Rodriguez and F Granados Agustin Ronchigram quantification N de Jonge and D Peckys Scanning transmission electron microscopy of whole eukaryotic cells in liquid and in-situ studies of functional materials ix j x J Elorza Fuzzy operators A.R Faruqi, G McMullan and R Henderson Direct detectors M Ferroni Transmission microscopy in the scanning electron microscope R.G Forbes Liquid metal ion sources A G€ olzh€auser Recent advances in electron holography with point sources J Grotemeyer and T Muskat Time-of-flight mass spectrometry M Haschke Micro-XRF excitation in the scanning electron microscope M.I Herrera The development of electron microscopy in Spain R Herring and B McMorran Electron vortex beams M.S Isaacson Early STEM development K Ishizuka Contrast transfer and crystal images C.T Koch In-line electron holography T Kohashi Spin-polarized scanning electron microscopy O.L Krivanek Aberration-corrected STEM M Kroupa The Timepix detector and its applications B Lencova Modern developments in electron optical calculations H Lichte New developments in electron holography M Matsuya Calculation of aberration coefficients using Lie algebra J.A Monsoriu Fractal zone plates Future Contributions Future Contributions L Muray Miniature electron optics and applications M.A O’Keefe Electron image simulation V Ortalan Ultrafast electron microscopy D Paganin, T Gureyev and K Pavlov Intensity-linear methods in inverse imaging M Pap Hyperbolic wavelets N Papamarkos and A Kesidis The inverse Hough transform S.-C Pei Linear canonical transforms P Rocca and M Donelli Imaging of dielectric objects J Rodenburg Lensless imaging J Rouse, H.-n Liu and E Munro The role of differential algebra in electron optics J Sanchez Fisher vector encoding for the classification of natural images P Santi Light sheet fluorescence microscopy C.J.R Sheppard The Rayleigh–Sommerfeld diffraction theory R Shimizu, T Ikuta and Y Takai Defocus image modulation processing in real time T Soma Focus-deflection systems and their applications P Sussner and M.E Valle Fuzzy morphological associative memories J Valdés Recent developments concerning the Systeme International (SI) G Wielgoszewski Scanning thermal microscopy and related techniques xi CONTRIBUTORS David Agard HHMI and the Department of Biochemistry and Biophysics, University of California, San Francisco, CA 94158, USA Pawel Berczynski Institute of Physics, West Pomeranian University of Technology, Szczecin 70-310, Poland Michael Breuß Brandenburg University of Technology, Cottbus, Germany Yifan Cheng Department of Biochemistry and Biophysics, University of California, San Francisco, CA 94158, USA Robert M Glaeser Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, USA Slawomir Marczynski Faculty of Mechanical Engineering and Mechatronics, West Pomeranian University of Technology, Szczecin 70-310, Poland Sriram Subramaniam Laboratory for Cell Biology, Center for Cancer Research, National Cancer Institute, National Institutes of Health (NIH), Bethesda, MD 20892, USA Martin Welk UMIT, Biomedical Image Analysis Division, Eduard-Wallnoefer-Zentrum 1, 6060 HALL (Tyrol), Austria xiii j CHAPTER ONE Gaussian Beam Propagation in Inhomogeneous Nonlinear Media Description in Ordinary Differential Equations by Complex Geometrical Optics Pawel Berczynski1, Slawomir Marczynski2 Institute of Physics, West Pomeranian University of Technology, Szczecin 70-310, Poland Faculty of Mechanical Engineering and Mechatronics, West Pomeranian University of Technology, Szczecin 70-310, Poland Contents Introduction CGO: Fundamental Equations, Main Assumptions, and Boundary of Applicability Gaussian Beam Diffraction in Free Space CGO Method and Classical Diffraction Theory On-Axis Propagation of an Axially Symmetric Gaussian Beam in Smoothly Inhomogeneous Media 4.1 First-Order Ordinary Differential Equation for Complex Parameter B 4.2 The Second-Order Ordinary Differential Equation for GB Width Evolution in an Inhomogeneous Medium 4.3 The First-Order Ordinary Differential Equation for the GB Complex Amplitude 4.4 The Energy Flux Conservation Principle in GB Cross Section Generalization of the CGO Method for Nonlinear Inhomogeneous Media Self-Focusing of an Axially Symmetric Gaussian Beam in a Nonlinear Medium of the Kerr Type The CGO Method and Solutions of the Nonlinear Parabolic Equation Self-Focusing of Elliptical GB Propagating in a Nonlinear Medium of the Kerr Type Rotating Elliptical Gaussian Beams in Nonlinear Media Orthogonal Ray-Centered Coordinate System for Rotating Elliptical Gaussian Beams Propagating Along a Curvilinear Trajectory in a Nonlinear Inhomogeneous Medium 10 Complex Ordinary Differential Riccati Equations for Elliptical Rotating GB Propagating Along a Curvilinear Trajectory in a Nonlinear Inhomogeneous Medium 11 Ordinary Differential Equation for the Complex Amplitude and Flux Conservation Principle for a Single Rotating Elliptical GB Propagating in a Nonlinear Medium 12 Generalization of the CGO Method for N-Rotating GBS Propagating Along a Helical Ray in Nonlinear Graded-Index Fiber 13 Single-Rotating GB Evolution of Beam Cross Section and Wave-Front Cross Section 14 Pair of Rotating GBS Advances in Imaging and Electron Physics, Volume 185 ISSN 1076-5670 http://dx.doi.org/10.1016/B978-0-12-800144-8.00001-X Ó 2014 Elsevier Inc All rights reserved 11 15 15 16 17 18 18 20 21 23 26 28 32 33 36 48 j Pawel Berczynski and Slawomir Marczynski 15 Three- and Four-Rotating GBS 16 Conclusion References 75 106 109 INTRODUCTION In the traditional understanding, geometrical optics is a method assigned to describe trajectories of rays, along which the phase and amplitude of a wave field can be calculated via diffractionless approximation (Kravtsov & Orlov 1990; Kravtsov, Kravtsov, & Zhu, 2010) Complex generalization of the classical geometrical optics theory allows one to include diffraction processes into the scope of consideration, which characterize wave rather than geometrical features of wave beams (by diffraction, we mean diffraction spreading of the wave beam, which results in GB having inhomogeneous waves) Although the first attempts to introduce complex rays and complex incident angles started before World War II, the real understanding of the potential of complex geometrical optics (CGO) began with the work of Keller (1958), which contains the consistent definition of a complex ray Actually, the CGO method took two equivalent forms: the ray-based form, which deals with complex raysdi.e., trajectories in complex space (Kravtsov et al., 2010; Kravtsov, Forbes, & Asatryan 1999; Chapman et al 1999; Kravtsov 1967)dand the eikonal-based form, which uses complex eikonal instead of complex rays (Keller & Streifer 1971; Kravtsov et al., 2010; Kravtsov, Forbes, & Asatryan 1999; Kravtsov 1967) The ability of the CGO method to describe the diffraction of GB on the basis of complex Hamiltonian ray equations was demonstrated many years ago in the framework of the ray-based approach Development of numerical methods in the framework of the ray-based CGO in the recent years allowed for the description of GB diffraction in inhomogeneous media, including GB focusing by localized inhomogeneities (Deschamps 1971; Egorchenkov & Kravtsov 2000) and reflection from a linear-profile layer (Egorchenkov & Kravtsov 2001) The evolution of paraxial rays through optical structures also was studied by Kogelnik and Li (1966), who introduced the concept of a very convenient ray-transfer matrix (also see Arnaud 1976) This method of transformation is known as the ABCD matrix method (Akhmediev 1998; Stegeman & Segev 1999; Chen, Segev, & Christodoulides 2012; Agrawal 1989) The eikonal-based CGO, which deals with complex eikonal and complex amplitude was essentially influenced by quasi-optics (Fox 1964), 2.6 × 10 2GB/RefLin 25: beam spots −10 beam spot beam spot 2.4 2.2 S (m ) 1.8 1.6 1.4 1.2 0.005 0.01 0.015 τ (m) 0.02 PLATE 17 (Figure 57 on page 67 of this Volume) 2GB/RefLin 25: curvature 1,2 2GB/RefLin 25: max curvature 1,2 900 κ1 κ1 κ2 κ2 max −100 max 800 −200 700 (1/m) 600 500 max −500 κ1 κ1 , κ2 max −400 , κ2 (1/m) −300 400 −600 300 −700 200 −800 −900 0.005 0.01 0.015 τ (m) 0.02 100 0.005 0.01 0.015 τ (m) PLATE 18 (Figure 58 on page 67 of this Volume) 0.02 −6 11.5 −5 2GB/RefLin 27: widths 1,2 × 10 2.2 × 10 2GB/RefLin 27: max widths 1,2 w1min w1max w2 w2 11 max 2.1 10.5 w1max, w2max (m) 9.5 1.9 1.8 1.7 8.5 1.6 1.5 7.5 0.005 0.01 0.015 τ (m) 1.4 0.02 0.005 0.01 0.015 τ (m) PLATE 19 (Figure 61 on page 69 of this Volume) 2.4 × 10 2GB/RefLin 27: beam spots −10 beam spot beam spot 2.2 1.8 S (m ) w1min, w2min (m) 10 1.6 1.4 1.2 0.005 0.01 τ (m) 0.015 0.02 PLATE 20 (Figure 62 on page 70 of this Volume) 0.02 2GB/RefLin 27: curvature 1,2 2GB/RefLin 27: max curvature 1,2 −100 1100 κ1min −400 800 κ1max, κ2max (1/m) (1/m) 900 , κ2 κ1 −700 −600 700 600 500 −800 400 −900 300 −1000 200 −1100 0.005 0.01 0.015 τ (m) κ2max 1000 −300 −500 κ1max κ2min −200 100 0.02 0.005 0.01 0.015 τ (m) 0.02 PLATE 21 (Figure 63 on page 70 of this Volume) −5 1.4 × 10 −5 2GB/RefLin 30: widths 1,2 2.2 × 10 2GB/RefLin 30: max widths 1,2 w1min w1 w2 w2 max max 1.3 1.2 1.8 w1 , w2 (m) w1max, w2max (m) 1.1 1.6 0.9 1.4 0.8 1.2 0.7 0.6 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 22 (Figure 66 on page 72 of this Volume) 2GB/RefLin 30: beam spots −10 3.5 × 10 beam spot beam spot S (m ) 2.5 1.5 0.5 0.005 0.01 0.015 τ (m) 0.02 PLATE 23 (Figure 67 on page 73 of this Volume) 2GB/RefLin 30: curvature 1,2 2GB/RefLin 30: max curvature 1,2 400 1600 κ1min 1200 −200 1000 −400 −600 −800 600 400 200 −1200 −1400 −200 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) max 800 −1000 −1600 κ2 1400 κ1max, κ2max (1/m) κ1min, κ2min (1/m) κ1max κ2 200 −400 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 24 (Figure 68 on page 73 of this Volume) 13 × 10 3GB/RefLin 2: widths 1,2,3 × 10 3GB/RefLin 2: max widths 1,2,3 w1min w1max w2 w2 12 max w3 w3max 3.5 10 w1max, w2max, w3max (m) w1min, w2min, w3min (m) 11 2.5 1.5 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 25 (Figure 71 on page 76 of this Volume) 4.5 3GB/RefLin 2: beam spots × 10 beam spot beam spot beam spot 3.5 S (m ) 2.5 1.5 0.5 0.005 0.01 τ (m) 0.015 0.02 PLATE 26 (Figure 72 on page 76 of this Volume) 3GB/RefLin 2: curvature 1,2,3 3GB/RefLin 2: max curvature 1,2,3 1000 5000 κ1 κ1 κ2min κ2max κ3 −2000 −3000 max 3000 2000 1000 −4000 −5000 κ3 4000 κ1max, κ2max, κ3max (1/m) −1000 0 −1000 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 27 (Figure 73 on page 77 of this Volume) 12 × 10 3GB/RefLin 6: widths 1,2,3 3.5 × 10 3GB/RefLin 6: max widths 1,2,3 w1min w1max w2 w2 11 max w3min w3max 10 w1max, w2max, w3max (m) w1min, w2min, w3min (m) κ1min, κ2min, κ3min (1/m) max 2.5 1.5 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) 0.5 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 28 (Figure 76 on page 79 of this Volume) 3GB/RefLin 6: beam spots × 10−10 beam spot beam spot beam spot 3.5 S (m2) 2.5 1.5 0.5 0.005 0.01 0.015 0.02 τ (m) PLATE 29 (Figure 77 on page 80 of this Volume) 16 × 10 3GB/RefLin 14: widths 1,2,3 5.5 × 10 3GB/RefLin 14: max widths 1,2,3 w1 w1 max w2 w2 max w3min 14 w3max 4.5 12 10 w1 , w2 , w3 (m) w1max, w2max, w3max (m) 3.5 2.5 1.5 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) 0.5 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 30 (Figure 80 on page 82 of this Volume) −10 3GB/RefLin 14: beam spots × 10 beam spot beam spot beam spot S (m ) 0.005 0.01 0.015 0.02 τ (m) PLATE 31 (Figure 81 on page 83 of this Volume) × 10 3GB/RefLin 22: widths 1,2,3 × 10 3GB/RefLin 22: max widths 1,2,3 w1 w1 w2 w2 max 4.5 max w3 w3 max 3.5 w1max, w2max, w3max (m) w1min, w2min, w3min (m) 2.5 1.5 1 0.5 0.005 0.01 0.015 τ (m) 0.02 0 0.005 0.01 0.015 τ (m) PLATE 32 (Figure 84 on page 85 of this Volume) 0.02 2.5 3GB/RefLin 22: beam spots × 10 beam spot beam spot beam spot 2 S (m ) 1.5 0.5 0 0.005 0.01 0.015 τ (m) 0.02 PLATE 33 (Figure 85 on page 86 of this Volume) 3GB/RefLin 22: curvature 1,2,3 3GB/RefLin 22: max curvature 1,2,3 500 1800 κ1 κ1 max κ2 κ3min max κ3max κ1max, κ2max, κ3max (1/m) 1400 −500 , κ2 , κ3 (1/m) −1000 κ1 κ2 1600 1200 1000 800 600 400 −1500 200 −2000 0.005 0.01 0.015 τ (m) 0.02 0 0.005 0.01 0.015 τ (m) PLATE 34 (Figure 86 on page 86 of this Volume) 0.02 3GB/RefLin 23: widths 1,2,3 × 10 × 10 3GB/RefLin 23: max widths 1,2,3 w1min w1max w2 w2 w3 w3max 1.8 max 2.5 w1max, w2max, w3max (m) (m) 1.6 1.2 w1 , w2 , w3 1.4 1.5 0.8 0.6 0.4 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) 0.5 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 35 (Figure 89 on page 89 of this Volume) 3GB/RefLin 23: beam spots −10 4.5 × 10 beam spot beam spot beam spot 3.5 S (m ) 2.5 1.5 0.5 0 0.002 0.004 0.006 0.008 0.01 τ (m) PLATE 36 (Figure 90 on page 89 of this Volume) 0.012 3GB/RefLin 23: curvature 1,2,3 3GB/RefLin 23: max curvature 1,2,3 1500 3000 κ1min κ1max κ2min 1000 κ2max 2500 κ3 κ3 max 500 2000 1500 κ1max, κ2max, κ3max (1/m) κ1min, κ2min, κ3min (1/m) −500 −1000 −1500 1000 500 −2000 −500 −2500 −1000 −3000 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) −1500 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 37 (Figure 91 on page 90 of this Volume) 16 × 10 4GB/RefLin 1: widths 1,2,3,4 × 10 4GB/RefLin 1: max widths 1,2,3,4 w1 w1 w2 w2 w3 w3 max 14 max max w4 w4 10 w1 , w2 , w3 , w4 (m) w1max, w2max, w3max, w4max (m) 12 max 6 2 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 38 (Figure 94 on page 92 of this Volume) −10 4GB/RefLin 1: beam spots × 10 beam spot beam spot beam spot beam spot 0.002 0.004 0.006 τ (m) 0.008 0.01 PLATE 39 (Figure 95 on page 93 of this Volume) −9 × 10 4GB/RefLin 6: beam spots 1.8 beam spot beam spot beam spot beam spot 1.6 1.4 1.2 S (m ) S (m2) 0.8 0.6 0.4 0.2 0 0.002 0.004 0.006 τ (m) 0.008 0.01 PLATE 40 (Figure 100 on page 99 of this Volume) 0.012 0.012 × 10 1.2 −9 4GB/RefLin 10: beam spots beam spot beam spot beam spot beam spot S (m ) 0.8 0.6 0.4 0.2 0 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 41 (Figure 103 on page 102 of this Volume) × 10 4GB/RefLin 14: widths 1,2,3,4 1.2 × 10 4GB/RefLin 14: max widths 1,2,3,4 w1min w1max w2 w2 3.5 max w3min w3max w4 w4 max 2.5 1.5 w1 , w2 , w3 , w4 (m) w1max, w2max, w3max, w4max (m) 0.8 0.6 0.4 0.2 0.5 0 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) 0 0.002 0.004 0.006 0.008 0.01 0.012 τ (m) PLATE 42 (Figure 106 on page 105 of this Volume) 3.5 4GB/RefLin 14: beam spots × 10−9 beam spot beam spot beam spot beam spot 2.5 S (m2) 1.5 0.5 0 0.002 0.004 0.006 τ (m) 0.008 PLATE 43 (Figure 107 on page 106 of this Volume) PLATE 44 (Figure on page 125 of this Volume) 0.01 0.012 PLATE 45 (Figure on page 130 of this Volume) PLATE 46 (Figure 12 on page 197 of this Volume) ... and Wave-Front Cross Section 14 Pair of Rotating GBS Advances in Imaging and Electron Physics, Volume 185 ISSN 1076-5670 http://dx.doi.org/10.1016/B978-0-12-800144-8.00001-X Ó 2014 Elsevier Inc... propagating and diffracting in free space, we obtain in a simple and illustrative way the same result as can be obtained in the standard way within a Fresnel approximation to the Kirchoff integral,... electron optics and applications M.A O’Keefe Electron image simulation V Ortalan Ultrafast electron microscopy D Paganin, T Gureyev and K Pavlov Intensity-linear methods in inverse imaging M Pap Hyperbolic