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- Springer Series in Optical Sciences 204
- Verlag: Springer / Springer, Berlin
- Artikelnr. des Verlages: 978-3-319-55437-2
- 1st ed. 2017
- Seitenzahl: 416
- Erscheinungstermin: 10. November 2017
- Abmessung: 241mm x 160mm x 28mm
- Gewicht: 768g
- ISBN-13: 9783319554372
- ISBN-10: 3319554379
- Artikelnr.: 47570705
Editors: Arti Agrawal, Trevor Benson, Richard DeLaRue, Gregory Wurtz
Guided wave interaction in photonic integrated circuits - a hybrid analytical / numerical approach to coupled mode theory
By: Manfred Hammer, University of Paderborn
Computational tools are indispensable in the field of photonic integrated circuits, for specific design tasks as well as for more fundamental investigations. Difficulties arise from the usually very limited range of applicability of purely analytical models, and from the frequently prohibitive effort required for rigorous numerical simulations.Hence we pursue an intermediate strategy. Typically, an optical integrated circuit consists of combinations of elements (waveguide channels, cavities) the simulation and design of which is reasonably well established, usually through more or less mature numerical solvers. What remains is to predict quantitatively the interaction of the waves (modes) supported by these elements. We address this task by a quite general, "Hybrid" variant (HCMT) of a technique known as Coupled Mode Theory. Using methods from the realm of finite-element numerics, the optical properties of a circuit are approximated by superpositions of eigen-solutions for its constituents, leading to good quantitative, reasonably low-dimensional, and easily interpretable models. This chapter describes the theoretical background, explains its limitations, hints at implementational details, and discusses a series of 2-D and 3-D examples that illustrate the versatility of the technique.
Keywords: Photonics, guided wave (integrated) optics, coupled mode theory,
1. Introduction motivation, background
2. Hybrid analytical / numerical coupled mode theory
setting: frequency domain, 3D / 2D ...
2.1 Coupled mode field template specifically for an example, general
2.2 Amplitude discretization specifically for an example, general
2.3 Projection & algebraic procedure
2.4 Remarks on theory and implementation avoiding heuristics, relation to numerics, scaling properties, types of basis fields, boundary conditions, spectral scans, material dispersion
2.5 Eigenfrequencies of composite systems "supermodes"; perturbational analysis
2.6 Variational approach: restriction of a functional
3. Examples, 2D
3.1 Basis elements
modes of straight channels, bend modes, eigenmodes of cavities, WGMs
3.2 Straight parallel waveguides
3.2 Waveguide crossing
3.3 Waveguide Bragg reflector
3.4 Chains of coupled square cavities
3.5 Resonators with Ring and disc cavities, bend- and WGM templates
3.6 Coupled resonator optical waveguide
4. 3D HCMT
implementation, first results, outlook
5. Concluding remarks brief summary of results, additional comments
Finite Element Time Domain Method for Photonics
By: B M A Rahman, R Kabir, A Agrawal, City University London
In this chapter we will discuss the development of a finite element based time domain approach, the use of perforated mesh to increase numerical efficiency and evaluation of numerical dispersions for both 2-D and 3-D photonic devices. Comparison of speed enhancement over the finite difference time domain method will be shown particularly comparing their numerical dispersions. Finally several examples are shown, including waveguide corners, Bragg gratings, and simulation of metamaterials.
1.2 Time domain approaches
1.3 Finite-Element Time-Domain Method
1.4 Results1.4.1 Mesh representation
1.4.2 Numerical Dispersion in 2-D photonics structures
1.4.3 Numerical Dispersion in 3-D photonic structures
1.4.4 Numerical advantage over FDTD method
1.4.5 Numerical Results
Simulation of Second Harmonic generation of photonic nanostructures using the Discontinuous Galerkin Time Domain method
By: Jens Forstner, University of Paderborn
Nonlinear optical properties of photonic nanostructures are of great scientific and technological interest. The problem of numerical simulation of the nonlinear response from such structures is multi-scale that makes it hard for existing numerical methods. Indeed, these are subwavelength metallic objects with sizes ranging from tens to hundreds of nanometers and the mechanisms responsible for the second harmonic generation apparently act on scales of the order of 1 nm. The Discontinuous Galerkin Time Domain (DGTD) method appears to be a good choice here. It is based on the unstructured meshing that simplifies the problem through the local mesh refinement. Second, its nonlinear stability property allows numerical solution of nonlinear PDEs and, correspondingly, construction of a model that describes nonlinear processes in a metal with Maxwell-Vlasov hydrodynamic equation.
1. Introduction and review of the developments of the DGTD method and its applications in plasmonics
2. Parallel implementation of the DGTD method
3. Incorporation of a nonlinear Maxwell-Vlasov hydrodynamic model
4. Simulation of the second harmonic generation in selected plasmonic nanostructures
The Modelling of Fibre Lasers for Mid-Infrared Wavelengths
By: L. Sojka, A. B. Seddon, S. Sujecki, T. M. Benson, University of Nottingham
Mid-infrared light sources are one of the most intensively developing subjects in the photonic area in recent years. The reason for this is that they can find many applications, for example in remote sensing, medicine, and security and military applications including the sensing of toxic gases, the detection of explosives and infrared counter-measurements.
Over the past decades silica fibre has revolutionised optical technology. Using a silica-based erbium doped fibre amplifier (EDFA) the signal in an optical network can be amplified and transmitted over several hundreds of kilometres directly in the optical domain. However, the working wavelength bands of commercially available rare earth (RE) doped fibre lasers are from 0.5-3 µm, with a significant spectral gap beyond 3.5 µm. The reason for this gap is the high multi-phonon energy of commercially available glasses (silica, ZBLAN). In order to construct a mid-infrared fibre laser a proper host material, with low multi-phonon energy, has to be manufactured. An ongoing research aim of the authors is to investigate and develop novel materials for near-infrared and mid-infrared fibre lasers based on chalcogenide glasses. Chalcogenide glasses offer significant advantages in the mid-infrared wavelength region. These advantages include the lowest phonon energy, from 400 cm-1 up to 230 cm-1 depending on the glass composition. Consequently, chalcogenide glasses present low non-radiative decay rates and wide infrared transparency. These glasses also have a high refractive index which results in higher absorption and emission cross-sections in the RE doped glasses.
This chapter describes numerical investigations of some 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 numerical fibre laser models and the experimental techniques for extracting model parameters. We present the results of our studies towards mid-infrared laser action in doped chalcogenide glasses doped with Dy3+, Pr3+ or Tb3+ ions.
1. Introduction to Mid Infrared Fibre Lasers
1.1. Overview of mid-infrared fibre lasers; technology drivers and potential applications
1.2. The physics of fibre laser systems
Interaction between light and atomic systems in gain media
Absorption and emission cross-sections
Gain in active media
Three and four level laser systems
Rare earth doping
1.3. Materials and dopants
Short review of mid-infrared optical fibre materials
Overview of mid-infrared emission from chalcogenide glasses and fibres doped with Pr3+, Dy3+, Tb3+.
2. Laser rate and propagation equations
3. Obtaining spectroscopic parameters as inputs to the numerical model
3.1 Radiative lifetime
3.2 FTIR absorption in bulk and fibre
3.3 Absorption and emission cross-sections
4. Models for fibre lasers
4.1 Numerical calculation procedure.
4.2 Boundary conditions
4.3 Numerical algorithm
5. Numerical and theoretical studies of mid-infrared laser action in chalcogenide glasses doped with Dy3+, Pr3+ or Tb3+
5.1 Fibre lasers based on Pr3+ doping
5.2 Fibre lasers based on Dy3+ doping
5.3 Fibre lasers based on Tb3+ doping
5.4 General discussion
6. Discussion and Conclusions
Hydrodynamic Model for Nonlinear Plasmonics
By: Alexey V. Krasavin, Pavel Ginzburg, Gregory A. Wurtz, and Anatoly V. Zayats, Department of Physics, King's College London, Strand, London WC2R 2LS, United Kingdom
Nonlinear optical interactions give rise to a variety of phenomena and therefore are extensively used in numerous applications such as lasers, classical and quantum optical information processing, bioimaging, and sensing. Consequently, tailoring, enhancing, and controlling nonlinear processes, which are inherently weak and require high light intensities, is a task of prime importance.
One of the most promising approaches in this area makes use of metallic nanostructures, supporting highly localised electromagnetic resonances and drastically enhancing the local fields. Although the traditional view here is enhancing nonlinear phenomena in the surrounding media, metals themselves are strongly nonlinear materials. The electromagnetic response of a free-electron gas leads to the inherent nonlinear optical behaviour of nanostructured plasmonic materials enabled through both strong local field enhancements and complex collective electron dynamics. In this Chapter, we will overview our recent results on implementation of the hydrodynamic model for conduction electrons to describe a broad range of nonlinear effects in metallic nanostructures. It will be started with an illustrative hydrodynamic analytical approach, defining the nonlinear coupling of the plasmonic modes in metallic nanoparticles, particularly revealing an important role of the resonance symmetries. In the further development, we will report our very recent numerical time-domain implementation of the hydrodynamic model, which enables non-perturbative studies of nonlinear coherent interactions between light and plasmonic nanostructures without any simplifications. The effects originating from the convective acceleration, the magnetic contribution of the Lorenz force, the quantum electron pressure, as well as the presence of the nanostructure's boundaries are taken into account leading to the appearance of second, third and higher harmonics in the scattering field of a metallic nanorod. We will also report supercontinuum generation in resonantly-tuned metallic nanospirals originating from the nonlocal effects. The proposed time-domain method enables one to obtain a universal, self-consistent numerical solution free from any approximations, enabling investigations of nonlinear optical interactions with arbitrary shaped optical pulses, in contrast to the generally employed plane-wave CW pump assumption, opening unique opportunities to approach realistic experimental scenarios.
2. Hydrodynamic model for free electron gas
3. Analytical description of nanoscale plasmonic nonlinear phenomena
3.1 Nonlinear coupling of plasmonic resonances in metallic nanoparticles.
3.2 Cascaded surface plasmon-solitons.
4. Full non-perturbative numerical model for nonlinear dynamics in metallic nanostructures
4.1 Second and third harmonic generation by metallic nanorods.
4.2 Supercontinuum generation by metallic nanospirals.
Photonic Crystals and Metamaterials with gain
By: Sotiris Droulias, Thomas Koschny and Costas M. Soukoulis, IESL, Forth, Greece
In this chapter we examine Photonic Crystals (PCs) and Metamaterials (MMs) coupled with gain. We explain how incorporation of a gain material in such systems can be treated numerically with the Finite Difference Time Domain (FDTD) technique. We highlight several aspects of the numerical implementation and illustrate, for educational purposes, examples of simple systems. There we establish some fundamental concepts of how the transition from loss to loss compensation and lasing happens. Finally, we demonstrate the route to the design of realistic systems with gain. We show how loss compensation is achieved in MMs and we highlight several aspects of the transition to lasing in certain PCs.
Choosing PCs and MMs over other systems to operate as lasers offers a vast flexibility in design, because their physical properties can be tailored at will and, hence, laser operation can be tuned as desired. This is a direct consequence of the wide variety of the photonic behavior, essentially bandstructures in PCs and resonances in MMs. With so much power over the physical behavior of these systems, important parameters that affect the laser performance can be adjusted and improved, such as the lasing threshold, laser volume, output power, polarization, etc. The FDTD method, which can simulate stationary, as well as strongly transient behavior, very easily gives a route to incorporate nonlinearity, such as the interaction of gain materials with passive systems. The algorithm is implemented in a straightforward manner, offering simplicity in modifications, so that the lasing system can be introduced simply by coupling it to the existing set of Maxwell's equations.
1.1. Photonic Crystals (PCs) and Metamaterials (MMs): importance and applications
1.2 Incorporating gain in PCs and MMS
1.3 Modeling gain
2. Theory and numerical implementation
2.1 Theory of four-level gain systems coupled with Maxwell's equations
2.2 Numerical implementation with the FDTD
2.3 Initiating lasing and measuring the lasing threshold
2.4 Examples of simple lasing systems.
3. Realistic systems
3.1 PCs with gain: lasing threshold control
3.2 MMs with gain: mechanism of the gain material coupled with the MM
Dirichlet-to-Neumann map method for modelling photonic crystals
By: Ya Yan Lu, City University Hong Kong
2. Photonic crystals and modelling problems
3. Dirichlet-to-Neumann maps
4. Eigenvalue problems
5. Boundary value problems
6. Nonlinear problems
By: Faris Mohammed, Robert Warmbier and Alexander Quandt, School of Physics, Centre of Excellence in Strong Materials (CoE-SM), Materials for Energy Research Group (MERG); University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
Over the last decade, affordable massive computing technologies in combination with new developments in numerical simulation methods have reached a level of sophistication, which should allow for the modelling of plasmonic devices down to the atomic level of the constituent materials. We will report about past and ongoing research in this area, describing the progress made in recent years, as well as the key algorithms and numerical methods. On top of that we will sketch some of the missing numerical tools, which would be necessary to achieve a complete multi-scale modelling of plasmonic devices.
The Chapter will contain a sound introduction into the field of plasmonics. We also describe the materials properties, which determine the interaction of light of a certain frequency interacting with combinations of metals and insulators. The focus of the book chapter will be on details of modern ab initio methods used to numerically determine these important materials properties like the dielectric function and the loss function. We also present the standard frequency-domain and time-domain methods used in computational electromagnetics. Both fields will be combined into a proper multi-scale approach, which also takes into account larger scale effects of composite materials interacting with light. Finally we show some applications in the fields photovoltaics and for plasmonic photonic quasicrystals, the latter providing interesting paradigms for optical localization phenomena. And we discuss some open problems an possible new directions of research in computational plasmonics.
1.1 Introduction 1.2 Theoretical background 1.2.1 Optical properties of solids 22.214.171.124 Metals and insulators. 126.96.36.199 Frequency dependent dielectric functions. 188.8.131.52 Derived properties: absorption, reflectivity, loss functions, conductivity. 184.108.40.206 Excitons. 220.127.116.11 Example: 1.2.2 Plasmons and Plasmonics 18.104.22.168 Bulk plasmons. 22.214.171.124 Surface plasmon polaritons 126.96.36.199 Localized plasmons 188.8.131.52 Example: 1.3 Basic numerical techniques 1.3.1 Frequency domain methods. 184.108.40.206 Photonic band structures. 220.127.116.11 Perturbation theory. 18.104.22.168 Numerical details. 22.214.171.124 Some recent improvements. 126.96.36.199 Example: 1.3.2 Time domain methods. 188.8.131.52 Outline of the FDTD method. 184.108.40.206 Numerical details. 220.127.116.11 Some improvements. 18.104.22.168 Example: 1.3.3 Ab initio dielectric functions. 1.3.4 Density functional theory. 22.214.171.124 Linear response. 126.96.36.199 Time dependent density functional theory. 188.8.131.52 GW method. 184.108.40.206 Bethe-Salpeter equation. 220.127.116.11 Example: 1.3.5 Multi-scale approaches. 18.104.22.168 Implementation of ab initio dielectric data. 22.214.171.124 Larger scale effects. 126.96.36.199 Example: 1.4 Some applications 1.4.1 Plasmonic photonic crystals. 188.8.131.52 Photonic crystals and surface plasmons. 184.108.40.206 Photonic quasicrystals, surface plasmons and optical localization. 220.127.116.11 Example: 1.4.2 Photovoltaics. 18.104.22.168 Solar cell design. 22.214.171.124 Light trapping strategies to improve the efficiency of solar cells. 126.96.36.199 Glass layers and plasmonic nanospheres. 188.8.131.52 Example: 1.4.3 Open questions. 184.108.40.206 Missing numerical tools and new types of algorithms. 220.127.116.11 New research directions. 18.104.22.168 Example:
All-dielectric nanophotonic structures: Exploring the magnetic component of light
By: Yuri Kivshar, Alex Krasnok, Andrey Miroshnichenko, Pavel Belov, Australian National University
Modern technologies largely depend on the rapidly growing demands for powerful computational capacities and efficient information processing, so the development of conceptually new approaches and methods are extremely valuable. One of the most important advantages of optical technology over electronics is its high operating frequency around 500 THz. However, photons as alternative information carriers have relatively large ``sizes'' determined by their wavelength. That is why they interact weakly with nanoscale (subwavelength) objects such as quantum emitters, subwavelength waveguides, and others. This problem call for challenges in creating and developing novel light control tools at the nanoscale.
An efficient light manipulation means simultaneous control of its electric and magnetic components. However, the magnetic response of natural materials is very weak because of small electron's spin contribution at high frequency. This is the reason why photonic devices operate mainly with the electric part of light wave. At the same time, magnetic dipoles are very common sources of magnetic
field in nature. The field of the magnetic dipole is usually calculated as the limit of a current loop shrinking to a point. The fields configuration is equivalent to the one of an electric dipole considering that the electric and magnetic fields are exchanged.
The most common example of a magnetic dipole radiation is an electromagnetic wave produced by an excited metal split-ring resonator (SRR), which is a basic constituting element of metamaterials. The real currents excited by external electromagnetic radiation and running inside the SRR produce a transverse oscillating up and down magnetic field in the centre of the ring, which simulates an oscillating magnetic dipole. However, for shorter wavelengths and in particular for visible spectral range this concept fails due to increasing losses and technological difficulties in fabrication of smaller and smaller constituting split-ring elements. Several other designs based on metal nanostructures have been proposed to shift the magnetic resonance wavelength to the visible spectral range. However, all of them are suffering from losses inherent to metals at visible frequencies.
This chapter will review a new, rapidly developing field of all-dielectric nanophotonics which allows to control both magnetic and electric
response of structured matter by engineering the Mie resonances in high-index dielectric nanoparticles. We will discuss opticalproperties of high index dielectric nanoparticles, methods of their fabrication, and also recent advances in all-dielectric metadevices including couple-resonator dielectric waveguides, nanoantennas, and metasurfaces.
2. Electric vs. magnetic response
3. Dielectric nanoparticles as antennas for light
4. Oligomers of nanoparticles and Fano resonance
5. Experimental verifications
6. Conclusion and perspectives
Engineering of hybrid plasmonic-photonic devices for enhanced light-matter interactions
By: Mina Mossayebi, Gaetano Bellanca, Alberto Parini, Mike G. Somekh, Amanda J. Wright, Eric C. Larkins , University of Nottingham
In this chapter, we focus on the design and applications of hybrid plasmonic-photonic nanoresonators using finite-difference time-domain simulations. These hybrid nanodevices are carefully designed to optimise the coupling between a photonic crystal cavitiy and a plasmonic nanoantenna in k-space. This allows us to take advantage of the high spatial confinement of plasmonic nanoantennas and the high quality factor of photonic crystal resonators. The resulting devices have the ability to confine light to extremely small volumes ( 10-16 cm3 or 100 zl) in the gap of the plasmonic nanoantenna, whilst maintaining a high quality factor (~ 5-15k). Finally, these confined fields are externally accessible and the devices are suitable for integration with waveguides and photonic integrated circuits. These characteristics, together with the intensity gradient and field profile inside and near this high optical field region, make them ideal for the exploitation of greatly enhanced and highly efficient light-matter interactions such as optical trapping, nonlinear optical processes, direct optical-radio frequency coupling, coupled photonic-phononic systems and coupling to quantum systems. Practical applications include the sensing of nano and micron-sized particles including molecules, viruses and bacteria etc. This will allow spectroscopy, manipulation and behavioral investigation of these nano-sized samples in much more detail than achieved to date as well as other enhanced light-matter interactions such as quantum interfacing
The chapter will begin with an introduction and explanation of the technology, discussing the potential applications of the hybrid devices, to put the work in context. We will then describe the traditional nanophotonic resonators that form the basis for the hybrid resonator, as well as their capabilities and limitations. This will be followed by the design of a hybrid photonic-plasmonic nanocavity, introducing and demonstrating the use of k-space engineering to optimise the coupling between the photonic crystal resonator and the plasmonic nanoantenna - allowing us to effectively combine their best features. Next, we will investigate the integration of these hybrid devices with photonic integrated circuits. The chapter will conclude with a discussion of relevant figures of merit for the devices and a summary assessment of their expected performance and potential applications.
2. Introduction (what we are doing?)
3. Technology and application context (why are we doing it?)
a. Optical trapping
b. Nonlinear optical spectroscopy
c. Other enhanced light-matter interactions
4. Tools and figures of merit (how are we going to evaluate the devices?)
a. Trapping forces
b. K-space analysis
5. Traditional nanophotonic resonators and their limitations and capabilities
a. Photonic crystal (PhC) resonators
b. Plasmonic nanoantennas
6. Design of hybrid nanophotonic-plasmonic devices (Design details of the hybrid devices)
a. Evaluation and selection of photonic crystal resonator structure (H0, L3, L5, L7, DH, etc.)
b. Evaluation and choice of substrate/superstrate (e.g. to control the plasmonic nanoantenna-PhC resonator separation, and for application specific demands such as mechanical robustness)
c. Discussion about the choice of ambient (Vacuum, air, water, biological fluids, etc.)
d. Investigation of various plasmonic nanoantenna designs and studying the effect of the BNA angle, gap size, arrays and distance from the photonic crystal on the quality factor and spatial confinement.
e. Coupling mechanisms between the PhC resonator and the plasmonic nanoantenna.
i. K-space analysis
ii. Spectra analysis
f. Device integration with photonic integrated circuits
i. Addition of waveguides and other resonators
7. Summary and discussions
a. Figures of merit (e.g. Purcell factor, hot-spot volume, Q-factor, optical power requirement/energy efficiency, etc.)
b. Performance potential for envisaged applications. These might include: the trapping and manipulation of nanoparticles and molecules; MIR/THz sensors; Raman spectroscopy of nanoparticles and molecules; the study of the dependence of chemical interactions of molecules upon local vibrational excitation; and even quantum interfacing.
Theory and Numerical Modelling of Parity-Time-Symmetric Structures in Photonics
By: Sendy Phang, Trevor M. Benson, Stephen Creagh, Gabrielle Gradoni, Phillip Sewell, and Ana Vukovic, University of Nottingham
Optical structures with balanced gain and loss, mimicking parity and time (PT) symmetry in quantum field theory, have been the subject of intense investigation in the last few years. PT symmetric structures based on Bragg gratings, couplers, and lattices have been reported and demonstrated functionalities including optical switching, unidirectional invisibility, and memory. Unidirectional invisibility and power oscillation have also been experimentally demonstrated.
In this chapter we explore the effect of material dispersion, nonlinearity, and gain/loss saturation on the behaviour of several PT-symmetric structures from the field of photonics. These include PT-symmetric Bragg gratings, systems of coupled waveguides, and coupled resonators both in isolation and in the presence of coupling waveguides. Coupled mode theory, a Green Integral Equation approach and a time-stepping numerical technique based on the transmission line modelling (TLM) method are developed for this purpose.
1. Introduction to Parity and Time-Reversal (PT) Symmetry
2. The photonics analogue of Quantum Mechanical PT-Symmetric problems
PT-symmetry in electromagnetics
3. Time-domain modelling of saturable, dispersive and non-linear materials in 1D
4. PT-symmetric scatterers in one dimension
Overview of properties
Generalised conservation relations
Studies of PT-symmetric Bragg gratings as information photonics devices, - the effect of saturation, dispersion, and non-linearity
5. * PT-symmetric systems of coupled waveguides
Coupled mode theory for coupled waveguides
Development of 2D TLM model
Application of 2D tools:
- Balancing gain and loss, and the impact of saturation
6. PT-symmetric coupled resonators
Green Integral equation approach to coupled resonator systems.
Whispering Gallery modes in the presence of PT-symmetry
The impact of dispersion and frequency de-tuning
Finite and infinite chains of coupled resonators
Resonators coupled with waveguide structures
7. Conclusions and further perspectives
* this chapter is largely self-contained and could be omitted if necessary.
Modelling of Light-sound interactions in optical waveguides
By: B M A Rahman, N Kejalakshmy, S Srivatanavaree, M M Rahman
In this chapter we discuss the guidance of acoustic waves and light through waveguides. The development of finite element codes for both the propagation models is discussed. The transformation of complex matrix to real symmetrical one for loss-less waveguides and use of symmetry conditions to avoid degenerate modes and extrapolation techniques to increase the numerical accuracy are discussed. Results are presented for the hybrid quasi-transverse and quasi-longitudinal acoustic modes in both low-index and high-index contrast acoustic waveguides. Spatial variations of the dominant and non-dominant displacement vectors for the fundamental and higher order acoustic modes are presented. Finally the interactions between the acoustic and optical modes, the SBS frequency shift and their overlaps are also presented for a range of optical waveguides.
2. Propagation of Acoustic waves
3. Optical waveguides
4. Light-sound interactions
5. Finite Element Method