PHYSICS OF NONLINEAR OPTICS

188 Pages·2021·3.546 MB·English

by  YVGS. VIJAYAN C. MURTI

Y. V. G. S. Murti
C. Vijayan

Physics of Nonlinear
Optics
Second Edition
Physics of Nonlinear Optics
Y. V. G. S. Murti C. Vijayan
(cid:129)
Physics of Nonlinear Optics
Second Edition
123
Y.V.G.S.Murti C. Vijayan
Formerly, atDepartment of Physics Department ofPhysics
Indian Institute of Technology Madras Indian Institute of Technology Madras
Chennai, Tamil Nadu,India Chennai, Tamil Nadu,India
ISBN978-3-030-73978-2 ISBN978-3-030-73979-9 (eBook)
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“...Jyotirgamaya...”
(Lead, kindly, to light !)
BrhadAraNyaka Upanishad
Preface to Second Edition
Thefirsteditionofthisbook,publishedin2014,waswelcomedenthusiasticallyand
appreciatedbystudents,researchers,andteachersofnonlinearoptics.Asthereisa
demand for thenextedition, it isquite naturalto take note of theseveral advances
that have occurred in the subject over the past five years and to decide which
of these are appropriate for inclusion at the textbook level, in view of the funda-
mentalnatureandthesignificanceofthesubjectarea.Thisistheprimemotivation
of bringing out a revised second edition.
Among the newer mechanisms and materials, we introduce supercontinuum
generation, which arises from an excellent interplay of the various mechanisms of
opticalnonlinearity,mostofwhichwereincludedinthefirstedition.So,itisquite
naturaltoincludethistopicintherequireddetailinthesecondedition.Anotherarea
of optical nonlinearity, owing origin to basic nonlinear light–matter interaction, is
that of multi-photon microscopy, which has demonstrated the potential to revolu-
tionize applications in medical imaging and materials characterization. So, a full
chapter has been added on this important emerging area of nonlinear optics. Our
sincerehope isthat themodifications and new additions will render thebook even
more useful to the beginner in nonlinear optics.
Several colleagues and students have helped in shaping up the second edition
of the book, and feedback on the first edition from many readers was a valuable
input indeed. In addition, I am particularly grateful to my Ph.D. student
Ms. Gayathri Radhakrishnan for contributing the newly added part on supercon-
tinuum generation and for designing the cover page of this book.
Chennai, India Y. V. G. S. Murti
C. Vijayan
vii
Preface to First Edition
Nonlinearopticalinteractionofmatterwithintenseopticalfieldsformsthebasisof
Photonics and holds the key to a deep and comprehensive understanding of
light-matter interaction as well as to the development of several significant facets
of the technology of tomorrow. Training of students equipped with a clear
understandingofthebasic Physicsinvolvedandenthusiasticawarenessofthevast
scopeofthefieldappeartobemoreessentialthaneverinviewoftherequirements
for the delineation of the basic physical mechanisms as well as the optimal
exploitation of the application potential in areas such as control, communication,
and computing.
AninterestingobservationabouttheavailablebooksonNonlinearOpticstoday
isthewidevarietyincontent,style,coverageofspecifictopics,relativeemphasisof
areas and the depth of treatment. Excellent classics on this topic by Bloembergen,
Boyd, Shen, Butcher and Cotter, Yariv are resources providing adequate coverage
and for researchers. insight. The handbooks by Sutherland and Paras Prasad,
Fischer,andZyssandChemlaandthebookonNonlinearFiberopticsbyAgrawal
arealsoworthspecialmention.Worthyofspecialmentionisaveryusefulresource
letter published recently, listing the details of all text books, reviews and other
documents related to nonlinear optics with brief descriptions on each. Details
of these and the recent books are given in the bibliography.
However,whileteachingcoursesonnonlinearopticstostudentsofbothscience
andengineeringduringthepasttwodecades,theauthorshavefeltastrongneedfor
an introductory level book that caters to the requirements of students at the level
of the bachelors and masters courses. This book is a humble attempt in this
direction and is presented as a textbook designed at college/University level for
undergraduate students of Science and Engineering and a one semester course can
easilybedesignedbasedonthis.Carehasbeentakentoincluderigorouslyfeatures
such as:
(cid:129) elucidation of relevant basic principles of Physics.
(cid:129) clear exposition of the ideas involved at the beginners’ level.
(cid:129) coverage of the physical mechanisms of nonlinearity.
ix
x PrefacetoFirstEdition
(cid:129) overview of physical processes in emerging photonic materials.
(cid:129) exploratory questions and problems at the end of the chapters.
The topics covered include a detailed formalism and phenomenology of non-
linear wave mixing processes, quantum mechanics of nonlinear interaction of
matter and radiation and an elucidation of specific processes as harmonic genera-
tion, optical phase conjugation, selffocusing, self phase modulation and nonlinear
absorption,leadingtoanappreciationoftheirapplicationpotentialinareassuchas
tunable laser action, pulse modification, photonic switching and bistability. Major
highlights of the book are detailed chapters on the symmetry aspects in nonlinear
opticsandonthevariousspecificphysicalmechanismsofopticalnonlinearitywith
examplesofmaterialsystemsinwhichtheseareoperative,takingcaretointroduce
emerging trends as well. However, an exhaustive coverage of all nonlinear optical
phenomenaisnotattemptedandtopicssuchasRamanandBaudouinscatteringare
not included.
The authors are pleased to place on record the beneficial interactions with stu-
dents and fellow teachers in the evolution of this book. Specifically, inputs from
Professors R. Vijaya, IIT, Kanpur, C. V. Krishnamurthy and Edamana Prasad, IIT
Madras,RejiPhilip,IISc,Bangalore,MajlesAra,TTU,TeheranandP.Nandakumar,
BITSGoaandaregratefullyacknowledged.Sincerethanksarealsoduetostudents
Dileep,RadhikaV.Nair,JaisTomandRadhuandpostdoctoralfellowDr.AnitaR.
Warrierforacarefulproofreadingofthemanuscript.
The authors have been rather slow in putting the book together and the book
could not have taken the shape but for the kind indulgence of the publishers, Ane
Books Pvt Ltd, and particularly Mr. A. Rathinam.
Chennai, India Y. V. G. S. Murti
C. Vijayan
Contents
1 From Optics to Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Charm and Challenge of Photonics . . . . . . . . . . . . . . . . . . . 1
1.2 The Nature of Optical Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Overcoming the Materials Bottleneck. . . . . . . . . . . . . . . . . . . . . 4
1.4 The Expanding Frontiers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 A Phenomenological View of Nonlinear Optics. . . . . . . . . . . . . . . . . 9
2.1 Optics in the Nonlinear World. . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 First-Order Susceptibility . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Second-Order Susceptibility . . . . . . . . . . . . . . . . . . . . . 11
2.1.4 Third-Order Susceptibility. . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Time Domain Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 First-Order Polarization—Time Domain Response . . . . . 13
2.2.2 Higher-Order Polarizations—Time Domain Response. . . 14
2.3 Frequency Domain Response. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 First-Order Susceptibility . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Second-Order Susceptibility . . . . . . . . . . . . . . . . . . . . . 17
2.3.3 General Order (n) Susceptibility . . . . . . . . . . . . . . . . . . 18
2.4 The nth-Order Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Monochromatic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Calculation of the Factor K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6.1 Optical Rectification. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6.2 Second-Harmonic Generation . . . . . . . . . . . . . . . . . . . . 22
2.6.3 Pockels Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.4 Sum and Difference Frequency Generation . . . . . . . . . . 22
2.6.5 Third-Harmonic Generation. . . . . . . . . . . . . . . . . . . . . . 23
2.6.6 Non-degenerate Four-Wave Mixing. . . . . . . . . . . . . . . . 23
2.7 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
xi
xii Contents
3 Symmetry and Susceptibility Tensors . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Crystal Symmetry and Susceptibility Tensors . . . . . . . . . . . . . . . 28
3.2.1 Neumann Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Symmetry of Second-Order Susceptibility . . . . . . . . . . . 34
3.2.3 Second-Harmonic Generation . . . . . . . . . . . . . . . . . . . . 36
3.2.4 Kleinman Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.5 Symmetry of Third-Order Susceptibility . . . . . . . . . . . . 41
3.3 The Dielectric Permittivity Tensor . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 The Refractive Index Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Calculation of Nonlinear Susceptibilities . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1 Physical Quantities in Quantum Physics . . . . . . . . . . . . 50
4.1.2 The Projection Operator . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 The Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Ensembles of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Time-Dependent Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Dipolar Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.6 First-Order Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.7 Second-Order Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.8 Third-Order Density Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.9 Double Integrals in the Expressions for Density Matrix. . . . . . . . 61
4.10 Second-Harmonic Susceptibility. . . . . . . . . . . . . . . . . . . . . . . . . 62
4.11 Relaxation Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.12 Applications to Color Centers . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.12.1 Third-Order Susceptibility. . . . . . . . . . . . . . . . . . . . . . . 64
4.12.2 Second-Order Susceptibility . . . . . . . . . . . . . . . . . . . . . 66
4.13 Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Nonlinear Wave Mixing Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Elements of Electromagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Traveling Electromagnetic Waves in Free Space. . . . . . . . . . . . . 72
5.3.1 Energy Density in the Traveling Wave . . . . . . . . . . . . . 74
5.4 Propagation of Electromagnetic Waves in Linear Materials . . . . . 75
5.5 Propagation of Electromagnetic Waves in Nonlinear
Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5.2 Energy Transfer Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Three-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.6.1 An Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81