Singularities in Physics and Engineering: Properties, Methods, and Applications

249 Pages·2018·15.95 MB·English

by  Paramasivam Senthilkumaran

Singularities in Physics and
Engineering
Properties, methods, and applications
IOP Series in Advances in Optics, Photonics and Optoelectronics
SERIES EDITOR
Professor Rajpal S Sirohi Consultant Scientist
About the Editor
Rajpal S Sirohi is currently working as a faculty member in the Department of
Physics, Alabama A&M University, Huntsville, Alabama (USA). Prior to this, he
was a consultant scientist at the Indian Institute of Science Bangalore, and before
thathewaschairprofessorintheDepartmentofPhysics,TezpurUniversity,Assam.
During 2000–11, he was academic administrator, being vice chancellor to a couple
ofuniversitiesandthedirectoroftheIndianInstituteofTechnologyDelhi.Heisthe
recipient of many international and national awards and the author of more than
400 papers. Dr Sirohi is involved with research concerning optical metrology,
optical instrumentation, holography, and speckle phenomenon.
About the series
Optics, photonics and optoelectronics are enabling technologies in many branches of
science, engineering, medicine and agriculture. These technologies have reshaped our
outlook,ourwayofinteractionwitheachotherandbroughtpeoplecloser.Theyhelpus
tounderstandmanyphenomenabetterandprovideadeeperinsightinthefunctioning
of nature. Further, these technologies themselves are evolving at a rapid rate. Their
applicationsencompassverylargespatialscalesfromnanometerstoastronomicalanda
very large temporal range from picoseconds to billions of years. The series on the
advances on optics, photonics and optoelectronics aims at covering topics that are of
interesttobothacademiaandindustry.Someofthetopicsthatthebooksintheseries
will cover include bio-photonics and medical imaging, devices, electromagnetics, fiber
optics, information storage, instrumentation, light sources, CCD and CMOS imagers,
metamaterials,opticalmetrology,opticalnetworks,photovoltaics,freeformopticsand
its evaluation, singular optics, cryptography and sensors.
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Singularities in Physics and
Engineering
Properties, methods, and applications
Paramasivam Senthilkumaran
Indian Institute of Technology Delhi
IOP Publishing, Bristol, UK
ªIOPPublishingLtd2018
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ISBN 978-0-7503-1698-9(ebook)
ISBN 978-0-7503-1696-5(print)
ISBN 978-0-7503-1697-2(mobi)
DOI 10.1088/978-0-7503-1698-9
Version:20181101
IOPExpandingPhysics
ISSN2053-2563(online)
ISSN2054-7315(print)
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Dedicated to God

Contents
Preface xi
Author biography xiii
1 Introduction 1-1
1.1 Singularity 1-1
1.2 Singularities in science and engineering 1-2
1.3 Acoustic vortex 1-6
1.4 Singularities in optics 1-6
1.5 Amplitude, phase and polarization 1-6
1.6 Brief historical account of optical phase singularities 1-7
References 1-9
2 Topological features 2-1
2.1 Introduction 2-1
2.2 Wavefront shape 2-1
2.3 Amplitude and phase distribution of an optical vortex beam 2-3
2.4 Topological charge 2-6
2.5 Phase contours and zero crossings 2-6
2.6 Phase gradients of an optical vortex beam 2-8
2.7 Critical points 2-10
2.8 Zero crossings and bifurcation lines 2-13
2.9 Charge, order and index 2-15
2.10 Sign rules 2-16
2.11 Disintegrations or explosions 2-16
2.12 Charge conservation 2-18
2.13 Index conservation 2-18
2.14 Limitation on vortex density 2-18
2.15 Threads of darkness 2-19
2.16 Berry’s paradox 2-19
2.17 Manifolds and trajectories 2-20
2.18 Links and knots 2-27
2.19 Different types of phase defects 2-29
References 2-35
vii
SingularitiesinPhysicsandEngineering
3 Generation and detection methods 3-1
3.1 Introduction 3-1
3.2 Generation 3-1
3.2.1 Spiral phase plate 3-1
3.2.2 Fork grating 3-3
3.2.3 Spiral zone plate 3-7
3.2.4 Tilts 3-8
3.2.5 Interference methods 3-11
3.2.6 Speckles 3-16
3.2.7 Spatial light modulators 3-17
3.2.8 Dammann vortex grating 3-18
3.2.9 Mode conversion methods 3-19
3.2.10Intra-cavity methods 3-21
3.2.11Adaptive helical mirror 3-24
3.2.12Vortex generation in optical fibers 3-24
3.2.13q-wave plates for vortex generation 3-27
3.3 Detection 3-27
3.3.1 Interference methods 3-28
3.3.2 Diffraction methods 3-33
3.3.3 Detection using lens aberrations 3-44
3.3.4 Detection of vortices in computational optics 3-46
References 3-49
4 Propagation characteristics 4-1
4.1 Introduction 4-1
4.2 Wave equations and solutions 4-1
4.3 Slowly varying envelope approximation—paraxial Helmholtz equation 4-4
4.4 Gouy phase 4-4
4.5 Divergence of singular beams 4-9
4.6 Near core vortex structure and propagation 4-10
4.7 Propagation dynamics of optical phase singularities 4-13
4.8 Propagation of vortices in non-linear media 4-16
References 4-17
5 Internal energy flows 5-1
5.1 Energy flow 5-1
5.2 Internal energy flows 5-3
viii
SingularitiesinPhysicsandEngineering
5.3 Visualizing internal energy flow 5-4
5.3.1 Bekshaev–Bliokh–Soskin method 5-4
5.3.2 Helmholtz–Hodge decomposition method 5-6
5.4 Focusing of singular beams—effect of aberrations 5-7
5.5 Experimental detection 5-11
5.6 Energy circulations in diffraction patterns 5-16
References 5-17
6 Vortices in computational optics 6-1
6.1 Introduction 6-1
6.2 Diffused illumination in holography 6-2
6.3 Synthesized diffusers 6-3
6.4 Phase synthesis in computer generated holograms 6-4
6.5 Stagnation problem in IFTA 6-5
6.6 Solution to the speckle problem 6-8
6.7 Phase unwrapping in the presence of vortices 6-9
6.7.1 Residue theorem 6-9
6.8 Non-Bryngdahl transforms using branch points 6-10
6.9 Diffraction of singular beams 6-11
6.10 Phase retrieval 6-12
References 6-17
7 Angular momentum of light 7-1
7.1 Introduction 7-1
7.2 Linear momentum 7-1
7.3 Angular momentum 7-2
7.4 Orbital and spin angular momentum of light 7-4
7.4.1 Angular momentum due to circular polarization 7-4
7.4.2 Angular momentum due to azimuthal phase dependence in the 7-5
beam
7.4.3 Angular momentum due to spatially varying 7-6
circular polarization
7.5 Intrinsic and extrinsic angular momenta 7-6
References 7-8
8 Applications 8-1
8.1 Metrology 8-1
ix