Table Of ContentSpringer Series in Optical Sciences 204
Arti Agrawal
Trevor Benson
Richard M. De La Rue
Gregory A. Wurtz E ditors
Recent
Trends in
Computational
Photonics
Springer Series in Optical Sciences
Volume 204
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William T. Rhodes, Georgia Institute of Technology, Atlanta, USA
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Ali Adibi, Georgia Institute of Technology, Atlanta, USA
Toshimitsu Asakura, Hokkai-Gakuen University, Sapporo, Japan
Theodor W. Hänsch, Max-Planck-Institut für Quantenoptik, Garching, Germany
Ferenc Krausz, Ludwig-Maximilians-Universität München, Garching, Germany
Barry R. Masters, Cambridge, USA
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AmCoulombwall1
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Arti Agrawal Trevor Benson
(cid:129)
Richard M. De La Rue Gregory A. Wurtz
(cid:129)
Editors
Recent Trends
in Computational Photonics
123
Editors
Arti Agrawal Gregory A.Wurtz
Department ofElectrical andElectronic Department ofPhysics
Engineering University of NorthFlorida
City,University ofLondon Jacksonville
London USA
UK
and
Trevor Benson
Schoolof Electrical andElectronic Department ofPhysics
Engineering King’sCollege
University of Nottingham London
Nottingham UK
UK
Richard M.De LaRue
Schoolof Engineering
University of Glasgow
Glasgow
UK
ISSN 0342-4111 ISSN 1556-1534 (electronic)
SpringerSeries inOptical Sciences
ISBN978-3-319-55437-2 ISBN978-3-319-55438-9 (eBook)
DOI 10.1007/978-3-319-55438-9
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Preface
Electromagnetic simulation is an established tool across many scientific and
technological disciplines, buoyed by the development of high-power computer
platforms. This holds true in the field of photonics, where recent years have seen
major advances in theoretical, numerical, and computational techniques. These
advances have been driven by the emergence of sophisticated integrated photonic
systems,ontheonehand,andsub-wavelengthdevicestructuringtoenhancelight–
matter interactions on the other, all aimed towards the generation, transmission,
localisation, manipulation, detection, and use of light.
The annual Optical Wave and Waveguide Theory and Numerical Modelling
(OWTNM) Workshop has, since 1992, provided a forum for lively debates,
intended to bring forward new ideas in the field of theoretical and computational
photonics. This book brings together some of the cutting-edge work on computa-
tional methods in photonics presented at recent OWTNM meetings and addresses
the physical understanding, mathematical description, and the computational
treatment of guided optical waves, and related optical effects in micro- and
nanostructures, including multi-physics effects.
The first section of this book describes some numerical methodologies for
computational photonics. Chapter 1 discusses the development of a finite-element-
based time-domain approach; the use of a perforated mesh increases numerical
efficiency, which is compared against the well-established finite-difference
time-domain (FDTD) method. Mid-infrared light sources have been intensively
investigatedinrecentyearssincetheycanenablemanyapplications,forexamplein
remotesensingandmedicine.Chapter2describesnumericalinvestigationsofsome
of the possibilities for obtaining mid-infrared laser action in rare earth-doped
chalcogenide glass fibres, starting from some basic laser physics and progressing
through the development of powerful numerical fibre laser models and the exper-
imentaltechniquesusedtoextractthemodelparameters.Chapter3thendescribesa
hybrid analytical–numerical approach to coupled mode theory which leads to
quantitative,computationallyefficientandreadilyinterpretablemodelsforphotonic
integrated circuits. The application of these models is illustrated for components
suchassingleandparallelwaveguides,waveguidecrossings,micro-resonatorswith
v
vi Preface
circular elements, and photonic molecules excited by straight waveguides.
Chapter 4 provides a rigorous analysis of acousto-optic interactions in optical
waveguides using a vector finite-element approach, in particular of nonlinear
Stimulated Brillouin Scattering (SBS). The numerical approach presented can be
usedforthestudyofawiderangeofpracticalopticalwaveguideswitheitherco-or
anti-guiding acoustic modes
The twenty-first century has seen a remarkable growth within the field of pho-
tonicsintheunderstanding,realisation,anduseofnovelmaterials,suchasphotonic
crystals (PCs) and metamaterials (MMs), the electromagnetic properties of which
are engineered to obtain properties that are unattainable in naturally existing
materials. This is often achieved by using structuring on a sub-wavelength scale.
Chapter5providesatheoreticalexaminationofphotoniccrystalsandmetamaterials
coupled with gain and explains how the incorporation of a gain material in such
systems can be treated numerically with the FDTD technique. The following two
chapters (Chaps. 6 and 7) describe the theory and numerical modelling of
Parity-Time (PT)-symmetric structures in photonics, starting with the study of a
PT-symmetric Bragg gratingstructure in which unique scattering phenomena have
been discovered. A dispersive and saturable gain model, implemented within the
time-domain Transmission-Line Modelling (TLM) method, is then introduced
which enables the impact of realistic (non-ideal) material properties on the beha-
viourofthesedevicestobeinvestigated.AfurtherextendedTLMmodel,including
material nonlinearity, is then used to study the behaviour of nonlinear PT Bragg
gratingsasaninnovativeall-opticalmemorydevice.Thesecondofthetwochapters
describing PT-symmetric structures in photonics investigates the spectral beha-
viour, and real-time operation, of PT-symmetric coupled resonators. A Boundary
Integral Equation (BIE) model is developed to study these structures in the fre-
quency domain. Then, the TLM method is used to study the impact of realistic
gain/lossmaterial propertiesontheoperationofPT-symmetric coupledresonators.
Nonlinear optics is another topic that has triggered the discovery of important
phenomenaanddevelopeddeepunderstandingsoffundamentalopticaleffects,and
which has enabled a large variety of applications. As described in Chap. 8, non-
linear optical interactions can be significantly enhanced using the local electro-
magnetic field enhancement achievable with plasmonic effects and metallic
nanostructures. The chapter reviews coherent nonlinear plasmonic effects due to
intrinsic electronic metallic nonlinearities, which are rigorously described by a
hydrodynamicmodel,focusingonharmonicgenerationandrelatedphenomena.An
illustrativeanalyticalsolutionofthehydrodynamicequationsrevealstheimportant
role of resonance symmetries and establishes a criterion for the generation of
nanoscale localised second harmonic generation (the analogue of macroscale
phase-matching at the nanoscale). A further example predicts the formation of
cascaded surface plasmon solitons. Then, a comprehensive non-perturbative
numerical implementation of the hydrodynamic model is presented that allows
theinvestigationofnonlinearopticalinteractionsinmetallicnanostructureswithout
any approximations. Nonlinear harmonic and broadband white light supercontin-
uum generationinmetallic nanorods andmetallic nanospirals isconsidered; thisis
significantly enhanced by the interplay between the topology of the nanostructure
Preface vii
and the non-local response of the metal. Chapter 9 continues the nonlinear optics
theme andappliesaDiscontinuous Galerkin Time-Domain(DGTD)methodtothe
numerical simulation of the second harmonic generation from various metallic
nanostructures. A Maxwell–Vlasov hydrodynamic model is used to describe the
nonlinear effects in the motion of the excited free electrons in a metal. Results are
compared with experimental measurements for split-ring resonators and plasmonic
gap antennas. Chapter 10 extends the nanophotonic theme to the study of high
refractive index dielectric nanoparticles and presents several basic approaches for
the numerical study of their collective optical response. It is shown that these
dielectric nanoparticles offer new phenomena for the manipulation of directional
lightscatteringandnano-antennaapplications,andadvantagesovertheirplasmonic
counterparts in terms of reduced losses and the resonant enhancement of both
electric and magnetic fields which can bring novel functionalities to simple
nanoparticle geometries.
The following two chapters give further detail on the field of computational
plasmonics. The first of these, Chap. 11, covers essential theoretical background
materialandprovidessomesimplemodelsoflight–matterinteractions.Thefocusis
on the physical properties of bulk plasmons, surface plasmon polaritons, and
localised plasmons. Analytical and numerical examples are given, and
plasmon-enhanced solar cells and other exciting new research directions are
introduced. These applications provide motivation for the more detailed numerical
studies described in Chap. 12, which include modern ab initio methods, and the
standard frequency-domain and time-domain methods of computational electro-
magnetics.Someapplicationsinthefieldsofphotovoltaicsandplasmonic–photonic
crystalsarestudied;thechaptercloseswithadiscussionofsomeopenproblemsthat
showthatthereisstillroomfornewandexcitingmethodsandfurtherdiscoveriesin
the future. The final chapter of the book, Chap. 13, investigates the engineering of
hybrid photonic–plasmonic devices for enhanced light–matter interactions. The
focus is on the design and characteristics of a hybrid photonic–plasmonic
nanoresonator using 3D finite-difference time-domain simulations. The structure is
uniquelycapableofthelocalisationofhigh-intensitylight inasub-wavelengthhot
spot, whilst maintaining a high-quality factor. This makes it suitable for applica-
tions such as near-field optical trapping and manipulation, sensing and spec-
troscopy.Clearpathwaystothepractical realisationandoptimisationofthedevice
arealsoestablished.Thedeviceisdesignedforoperationinthetelecommunication
wavelength range, making it compatible with the existing telecommunication
technology and photonic crystal fabrication processes.
Theeditorsthankalltheauthorsfortheirvaluablecontributionsandtheirtimely
responses to our requests. We also wish to thank members of the Technical
ProgrammeCommitteeoftheOpticalWaveandWaveguideTheoryandNumerical
Modelling (OWTNM) Workshop for their continued support.
London, UK Arti Agrawal
Nottingham, UK Trevor Benson
Glasgow, UK Richard M. De La Rue
London, UK Gregory A. Wurtz
Contents
1 Finite Element Time Domain Method for Photonics.... ..... .... 1
S.M. Raiyan Kabir, B.M.A. Rahman and A. Agrawal
1.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 1
1.2 Derivation from Maxwell’s Equations ... .... .... ..... .... 3
1.2.1 Two Dimensional Formulation .. .... .... ..... .... 4
1.2.2 Three Dimensional Formulation . .... .... ..... .... 5
1.2.3 Discretisation.... .... .... .... .... .... ..... .... 5
1.2.4 Discretised Governing Equations. .... .... ..... .... 7
1.3 The Mesh.... .... ..... .... .... .... .... .... ..... .... 8
1.3.1 The Space Mesh System... .... .... .... ..... .... 9
1.3.2 The Time Mesh System.... .... .... .... ..... .... 12
1.4 Simulation Results . ..... .... .... .... .... .... ..... .... 13
1.4.1 Simulation in Two Dimensions.. .... .... ..... .... 13
1.4.2 Simulation in Three Dimensions. .... .... ..... .... 14
1.5 Numerical Dispersion.... .... .... .... .... .... ..... .... 17
1.5.1 Numerical Dispersion Relation for Two Dimensions.... 17
1.5.2 Numerical Dispersion Relation for Three Dimensions ... 19
1.5.3 Numerical Dispersion for Two Dimensions. ..... .... 20
1.5.4 Numerical Dispersion for Three Dimensions..... .... 23
1.5.5 Comparison with the FDTD Method.. .... ..... .... 24
1.6 Calculating Resolution Reduction Factor . .... .... ..... .... 26
1.6.1 Resolution Reduction Factor for Two Dimensions .... 27
1.6.2 Resolution Reduction Factor for Three Dimensions.... 28
1.7 Theoretical CPU Performance.. .... .... .... .... ..... .... 29
1.7.1 CPU Optimised Form . .... .... .... .... ..... .... 29
1.7.2 CPU Time/Latency of the Governing Equations .. .... 32
1.7.3 Performance Comparison of Proposed FETD and
FDTD for Two Dimensions. .... .... .... ..... .... 33
1.7.4 Performance Comparison of Proposed FETD and
FDTD for Three Dimensions.... .... .... ..... .... 34
ix
x Contents
1.8 Concluding Remarks..... .... .... .... .... .... ..... .... 35
References. .... .... .... ..... .... .... .... .... .... ..... .... 36
2 The Modelling of Fibre Lasers for Mid-Infrared Wavelengths ........ 39
L. Sojka, T.M. Benson, D. Furniss, Z. Tang, H. Sakr, A.B. Seddon
and S. Sujecki
2.1 Introduction to Mid-Infrared Fibre Lasers. .... .... ..... .... 39
2.1.1 Mid-Infrared Lasers; Technology Drivers and
Potential Applications . .... .... .... .... ..... .... 39
2.1.2 Overview of Mid-Infrared Fibre Lasers.... ..... .... 40
2.2 The Physics of Fibre Laser Systems. .... .... .... ..... .... 41
2.2.1 Light-Matter Interactions in Gain Medium . ..... .... 41
2.2.2 Cross(cid:1)Section... .... .... .... .... .... ..... .... 43
2.2.3 Lifetime... ..... .... .... .... .... .... ..... .... 43
2.2.4 Nonradiative Lifetimes .... .... .... .... ..... .... 44
2.2.5 Branching Ratio.. .... .... .... .... .... ..... .... 45
2.2.6 Gain in Active Media . .... .... .... .... ..... .... 45
2.2.7 Laser Systems... .... .... .... .... .... ..... .... 47
2.2.8 Laser Theory.... .... .... .... .... .... ..... .... 47
2.2.9 Rare Earth Doping.... .... .... .... .... ..... .... 48
2.2.10 Fibre Lasers..... .... .... .... .... .... ..... .... 49
2.3 Materials and Dopants ... .... .... .... .... .... ..... .... 49
2.3.1 Short Review of Mid-Infrared Optical
Fibre Materials ................................ 49
2.3.2 Overview of Mid-Infrared Emission from
Chalcogenide Glasses and Fibres Doped
with Pr3þ, Dy3þ, Tb3þ ... .... .... .... ..... .... 51
2.4 Obtaining Spectroscopic Parameters as Inputs
to the Numerical Model .. .... .... .... .... .... ..... .... 54
2.4.1 Absorption Cross-Section Measurements... ..... .... 54
2.4.2 Emission Cross-Section Measurements .... ..... .... 55
2.4.3 Lifetime Measurements.... .... .... .... ..... .... 55
2.5 Models for Fibre Lasers . .... .... .... .... .... ..... .... 57
2.5.1 Laser Rate and Propagation Equations .... ..... .... 59
2.5.2 Boundary Conditions.. .... .... .... .... ..... .... 61
2.5.3 Numerical Calculation Procedure. .... .... ..... .... 62
2.5.4 Numerical Algorithm.. .... .... .... .... ..... .... 64
2.6 Numerical and Theoretical Studies of Mid-Infrared Laser
Action in Chalcogenide Glasses Doped with Dy3þ,
Pr3þ or Tb3þ. .... ..... .... .... .... .... .... ..... .... 65
2.6.1 Parameters Used in Modelling... .... .... ..... .... 66
2.6.2 Modelling of Mid-Infrared Chalcogenide
Fibre Lasers..... .... .... .... .... .... ..... .... 66
