DOCSLIB.ORG

The Multipole Description of Complex Plasmonic Nanostructures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Multipole Description of Complex Plasmonic Nanostructures The Multipole Description of Complex Plasmonic Nanostructures Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat der Physikalisch-Astronomischen Fakultät der Friedrich-Schiller-Universität Jena von Dipl.-Phys. Jörg Petschulat geboren am 12.02.1981 in Zwickau. Gutachter / Referees: 1. Prof. Thomas Pertsch (Friedrich-Schiller-Universität Jena) 2. Prof. Kurt Busch (Universität Karlsruhe) 3. Prof. Olivier J. F. Martin (Ecole Polytechnique Federale De Lausanne) Tag der Disputation / Day of defense: 12.04.2011 For Nadja and Mathilde. Contents 1 Introduction 3 2 Theoretical and experimental basics 8 2.1 Theory . .8 2.1.1 Averaging microscopic Maxwell’s equations - the role of multipole moments8 2.1.2 Multipole radiation . 18 2.1.3 Localized plasmon polaritons . 21 2.1.4 Propagating plasmon polaritons . 26 2.1.5 Rigorous numerical methods . 31 2.2 Experimental . 37 2.2.1 Far-field spectroscopy . 38 3 Electric dipole excitations in metal nanostructures 40 3.1 Optical properties of dipole nanoantennas . 40 3.1.1 Near-field enhancement . 40 3.1.2 Polarization dependence . 45 3.2 Combining localized dipole and propagating surface plasmon modes . 49 3.2.1 Doubly resonant nanoantenna arrays . 49 3.2.2 Surface-Enhanced Raman Scattering . 56 3.2.3 Application of doubly resonant nanoantenna arrays to Surface-Enhanced Raman Scattering . 59 4 Higher-order multipole properties of optical metamaterials 63 4.1 Linear optical properties . 63 4.1.1 Cut-Wire metamolecule . 63 4.1.2 Split-Ring Resonator metamolecule and planar modifications enabling asymmetric transmission . 70 4.2 Nonlinear optical properties . 76 4.2.1 Multipole nonlinearity - Second-harmonic generation . 77 4.2.2 Comparison with other nonlinear approaches . 85 4.3 Scattering patterns of isolated metamolecules . 88 4.3.1 Multipole decomposition of the scattered field . 88 4.3.2 Origin dependence for the multipole expansion . 94 5 Summary and perspective 97 1 6 Zusammenfassung II 7 Curriculum Vitae VI 8 Bibliography XI 9 List of Figures XXVI 2 1 Introduction The peculiarities of the interaction of light with confined metal inclusions have been of interest for a long time. The most prominent examples are the coloring effects achieved by certain noble metals included in windows in churches1. The theoretical description of such effects has been performed by G. Mie [Mie08]. Furthermore, the theoretical prediction of bound elec- tromagnetic modes at metal dielectric interfaces by R. H. Ritchie [Rit57] comprises another important milestone. In the following decades many interesting applications have been inves- tigated, e.g., extra-ordinary transmission of light through metal films perforated with holes of sub-wavelength dimensions [ELG+98, BDE03]. During the last years, the subject of light-matter interaction with metal objects of sub- wavelength size has been experienced further interest [BFL+07, Ram05, Sha07]. The reasons for this development are manyfold. On one side the fabrication processes have been contin- uously improved due to the growing demands for smaller circuit sizes in the semiconductor industry. Among other fabrication schemes, mostly top-down approaches2, such as electron beam lithography, have successfully been applied to realize highly reproducible structures covering large areas. Thus, versatile fabrication methods have been established to provide structures with feature sizes far below the wavelength of optical radiation. On the other side, the classical electrodynamic description of light-matter interaction with arbitrarily-shaped ob- jects got enriched by the exploration and application of powerful numerical techniques. These methods allow the verification of experimental findings in the same manner as new effects can be predicted and finally observed in application-oriented realizations of specific structures. Moreover, the numerical methods are valuable tools to bridge the gap between the physics, taking place at the scale of nanometers, and the macroscopic observation. The dramatical developments in these two fields provided the optimal breeding ground for the emerging field of metamaterials. Eventually, everything that was required to launch this topic was a physical idea. Exactly this idea was presented in the work of J. B. Pendry in 1999 [PHRS99]. He proposed the concept of artificial materials, i.e., metamaterials3, that are composed of sub-wavelength structures, acting similarly as atoms and molecules do in ordinary matter. These structures were termed metamolecules and have in common that their dimensions are much smaller than the wavelength of the impinging radiation. Thus, in a simplified picture, the illumination cannot resolve the structural details and behaves like in an effective medium with propagation 1The Lycurcus Cup, exposed in the British museum in London, is a frequently referred example for its different color appearance depending on the illumination conditions. 2The term top-down accounts for fabrication schemes for which the entire structure is created starting from a predefined arrangement, e.g., a stack of layers, that is finally processed, typically by electron or ion beam lithography. Differently, bottom-up approaches denote schemes that are starting with entities at a smaller scale than the final structure. Examples are chemical synthesis of gold spheres and other self-organized fabrication schemes. 3The term metamaterial originates from the greek word "meta" which can be translated by "beyond". Hence, the expression metamaterials should account for materials beyond the capabilities of natural matter. 3 properties dictated by the elementary metamolecules. These metamolecules, similar to ordi- nary matter, predominantly interact with electromagnetic waves by their carriers. In contrast to ordinary matter, for which the carrier dynamics is determined by the atomic assembly of the respective element, the carrier dynamics can be tailored by the geometry of the meta- molecules in metamaterials. Due to their high electron density and mobility, noble metals such as gold or silver are typically selected as composite materials for metamolecules. The associated resonant carrier excitations are so-called localized surface plasmon-polaritons that can be controlled by adjusting the shape and the composition of the metamolecule. As it may be anticipated from the terminology, such excitations are located at the surface of the respective metamolecule, due to the limited penetration depth in the metal on the one side and an evanescent decay in the ambient dielectric medium on the other side. Furthermore, they are hybrid excitations, since the electron dynamics (plasmon) induces an electric field (polariton) and vice versa. With these considerations it becomes clear that a resonance in the carrier dynamics is accompanied with a resonant and thus enhanced local electromagnetic field in the vicinity of the structure. In contrast to the research associated with metamaterials, localized surface plasmon-polari- tons have been known for a long time. A famous and fundamental theoretical description related to such resonances is the work published by G. Mie [Mie08] which has been already mentioned above. Although Mie did not use the terminology of localized surface plasmon- polaritons, he developed a rigorous electrodynamic formalism to solve the scattering problem of electromagnetic waves at isolated spheres with arbitrary diameter and composition. With this approach he was able to predict the resonances occurring in the scattering spectra of colloidal gold solutions. These resonances have been observed in the form of coloring effects mentioned in the beginning. Hence, the physics of metal structures with sub-wavelength dimensions has been known for decades and was subject of intense investigations. The dramatic new feature of structures that are typically investigated in the field of metamaterials is the possibility to additionally create a magnetic response besides the electrical one. Exactly this intriguing difference can be considered as the major achievement during the last years, since metamaterials can exhibit optical magnetism, i.e., magnetic light-matter interaction at optical frequencies [PHRS99]. Such an effective magnetic response has never been observed in natural matter before and opens the gate to alter both, the magnetic and the electric properties for light propagation by structured materials with an unprecedented diversity. Among other famous works which investigated a potential magnetic response there is one publication that is of particular importance in the field of metamaterials. This work has been presented by V. G. Veselago [Ves68, Boa11] who theoretically investigated what impact a potential magnetic response would have early in 1968. As the most important consequence, Veselago considered the case of a negative refractive index [Ves68]. However, only in combi- nation with the structures and the concepts proposed by Pendry, realizations of such effects became feasible [BdWPA10, DWSL07, SCC+05, VZZ+08]. Although the ability to create a magnetic response in addition to the electrical seems to 4 be only an incremental development, the consequences of such an apparently slight change are enormous. For instance, the resolution limit of E. Abbe [Abb73], which is approximately one half of the irradiating wavelength, can be circumvented by perfect lenses [Pen00, SE06, JAN06]. Recently, much smaller details have been resolved with the use of metamaterials [LLX+07, SHD07]. Finally, the fascinating concept of cloaking
Load more

Recommended publications
  • Ph501 Electrodynamics Problem Set 8 Kirk T
    Princeton University Ph501 Electrodynamics Problem Set 8 Kirk T. McDonald (2001) [email protected] http://physics.princeton.edu/~mcdonald/examples/ Princeton University 2001 Ph501 Set 8, Problem 1 1 1. Wire with a Linearly Rising Current A neutral wire along the z-axis carries current I that varies with time t according to ⎧ ⎪ ⎨ 0(t ≤ 0), I t ( )=⎪ (1) ⎩ αt (t>0),αis a constant. Deduce the time-dependence of the electric and magnetic fields, E and B,observedat apoint(r, θ =0,z = 0) in a cylindrical coordinate system about the wire. Use your expressions to discuss the fields in the two limiting cases that ct r and ct = r + , where c is the speed of light and r. The related, but more intricate case of a solenoid with a linearly rising current is considered in http://physics.princeton.edu/~mcdonald/examples/solenoid.pdf Princeton University 2001 Ph501 Set 8, Problem 2 2 2. Harmonic Multipole Expansion A common alternative to the multipole expansion of electromagnetic radiation given in the Notes assumes from the beginning that the motion of the charges is oscillatory with angular frequency ω. However, we still use the essence of the Hertz method wherein the current density is related to the time derivative of a polarization:1 J = p˙ . (2) The radiation fields will be deduced from the retarded vector potential, 1 [J] d 1 [p˙ ] d , A = c r Vol = c r Vol (3) which is a solution of the (Lorenz gauge) wave equation 1 ∂2A 4π ∇2A − = − J. (4) c2 ∂t2 c Suppose that the Hertz polarization vector p has oscillatory time dependence, −iωt p(x,t)=pω(x)e .
    [Show full text]
  • PHYS 352 Electromagnetic Waves
    Part 1: Fundamentals These are notes for the first part of PHYS 352 Electromagnetic Waves. This course follows on from PHYS 350. At the end of that course, you will have seen the full set of Maxwell's equations, which in vacuum are ρ @B~ r~ · E~ = r~ × E~ = − 0 @t @E~ r~ · B~ = 0 r~ × B~ = µ J~ + µ (1.1) 0 0 0 @t with @ρ r~ · J~ = − : (1.2) @t In this course, we will investigate the implications and applications of these results. We will cover • electromagnetic waves • energy and momentum of electromagnetic fields • electromagnetism and relativity • electromagnetic waves in materials and plasmas • waveguides and transmission lines • electromagnetic radiation from accelerated charges • numerical methods for solving problems in electromagnetism By the end of the course, you will be able to calculate the properties of electromagnetic waves in a range of materials, calculate the radiation from arrangements of accelerating charges, and have a greater appreciation of the theory of electromagnetism and its relation to special relativity. The spirit of the course is well-summed up by the \intermission" in Griffith’s book. After working from statics to dynamics in the first seven chapters of the book, developing the full set of Maxwell's equations, Griffiths comments (I paraphrase) that the full power of electromagnetism now lies at your fingertips, and the fun is only just beginning. It is a disappointing ending to PHYS 350, but an exciting place to start PHYS 352! { 2 { Why study electromagnetism? One reason is that it is a fundamental part of physics (one of the four forces), but it is also ubiquitous in everyday life, technology, and in natural phenomena in geophysics, astrophysics or biophysics.
    [Show full text]
  • Electrodynamics II: Lecture 9 Multipole Radiation
    Electrodynamics II: Lecture 9 Multipole radiation Amol Dighe Sep 14, 2011 Outline 1 Multipole expansion 2 Electric dipole radiation 3 Magnetic dipole and electric quadrupole radiation Outline 1 Multipole expansion 2 Electric dipole radiation 3 Magnetic dipole and electric quadrupole radiation ~ rad Potential A! for monochromatic sources We are interested in calculating the radiative components of EM fields and related quantities (like radiated power) for a charge / current distribution that is oscillating with a frequency !. The results for a general time dependence can be obtained by integrating over all frequencies (inverse Fourier transform), of course. We have already seen that it is enough to know about the current distribution (we are interested only in radiative parts), since the charge distribution is related to it by continuity. In such a case, we know that ~ ~0 rad µ Z eikjx−x j A~ (~x) = 0 ~J (~x0) d 3x 0 (1) ! 4π ! j~x − ~x0j ~ Given A!, the rest of the quantities can be easily calculated in terms of it. We shall omit the “rad” label in this lecture, it is assumed to be everywhere except when specified. ~ rad ~ rad B! and E! for monochromatic sources The radiative part of the magnetic field is then ~ ~ ~ B! = r × A! = ik^r × A! (2) Note that here ~r = ~x, to be consistent with standard convention. The radiative part of the electric field can be obtained in this ~ ~ monochromatic case by using r × B! = µ00(−i!)E! (note that there is no current at large r): ic2 E~ = r × B~ = cB~ × ^r (3) ! ! ! ! ~ ~ Thus, E! and B! fields are orthogonal to ^r, orthogonal to each other, and their magnitudes differ simply by a factor of c.
    [Show full text]
  • Classical Electromagnetism
    Classical Electromagnetism Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Maxwell’s Equations 7 1.1 Introduction . .................................. 7 1.2 Maxwell’sEquations................................ 7 1.3 ScalarandVectorPotentials............................. 8 1.4 DiracDeltaFunction................................ 9 1.5 Three-DimensionalDiracDeltaFunction...................... 9 1.6 Solution of Inhomogeneous Wave Equation . .................... 10 1.7 RetardedPotentials................................. 16 1.8 RetardedFields................................... 17 1.9 ElectromagneticEnergyConservation....................... 19 1.10 ElectromagneticMomentumConservation..................... 20 1.11 Exercises....................................... 22 2 Electrostatic Fields 25 2.1 Introduction . .................................. 25 2.2 Laplace’s Equation . ........................... 25 2.3 Poisson’sEquation.................................. 26 2.4 Coulomb’sLaw................................... 27 2.5 ElectricScalarPotential............................... 28 2.6 ElectrostaticEnergy................................. 29 2.7 ElectricDipoles................................... 33 2.8 ChargeSheetsandDipoleSheets.......................... 34 2.9 Green’sTheorem.................................. 37 2.10 Boundary Value Problems . ........................... 40 2.11 DirichletGreen’sFunctionforSphericalSurface.................. 43 2.12 Exercises....................................... 46 3 Potential Theory
    [Show full text]
  • Arxiv:1007.1996V1
    On the exact electric and magnetic fields of an electric dipole W.J.M. Kort-Kamp∗ and C. Farina† Universidade Federal do Rio de Janeiro, Instituto de Fisica, Rio de Janeiro, RJ 21945-970 Abstract We derive from Jefimenko’s equations1 a multipole expansion in order to obtain the exact expres- sions for the electric and magnetic fields of an electric dipole with an arbitrary time dependence. A few comments are also made about the usual expositions found in most common undergraduate and graduate textbooks as well as in the literature on this topic. arXiv:1007.1996v1 [physics.class-ph] 12 Jul 2010 1 Even in the context of electrostatic, the problem of finding analytically the electric field associated with a localized but arbitrary static charge distribution may become quite in- volved. Of course, simplifications occur when symmetries are present. Due to the difficulty of getting exact solutions, numerical methods and approximation theoretical methods have been developed. One of the most important examples of the latter is the so called multipole expansion method relative to a base point (for convenience, let us take the origin at the base point). Choosing the origin inside the distribution, the multipole expansion method states, essentially, that the field outside the distribution is given by a superposition of fields, each of them being interpreted as the electrostatic field of a multipole located at the origin (see, for instance, Griffiths’ textbook2). The first terms of the multipole expansion correspond to the fields of a monopole, a dipole and a quadrupole, respectively. Hence, for an arbi- trary distribution with a vanishing total charge the first term in the multipole expansion is given, in principle, by the field of the electric dipole of the distribution.
    [Show full text]
  • Arxiv:0812.4679V1 [Physics.Class-Ph] 26 Dec 2008 Fteeetoantcpotentials
    Multipole radiation fields from the Jefimenko equation for the magnetic field and the Panofsky-Phillips equation for the electric field R. de Melo e Souza,∗ M.V. Cougo-Pinto, and C. Farina Universidade Federal do Rio de Janeiro, Instituto de Fisica, Rio de Janeiro, RJ 21945-970 M. Moriconi Universidade Federal Fluminense, Instituto de Fisica, Niteroi, RJ 24210-340 Abstract We show how to obtain the first multipole contributions to the electromagnetic radiation emited by an arbitrary localized source directly from the Jefimenko equation for the magnetic field and the Panofsky-Phillips equation for the electric field. This procedure avoids the unnecessary calculation of the electromagnetic potentials. arXiv:0812.4679v1 [physics.class-ph] 26 Dec 2008 1 I. INTRODUCTION The derivation found in most textbooks of the electromagnetic fields generated by arbi- trary sources in vacuum starts by calculating the corresponding electromagnetic potentials (see for instance, Refs. 1, 2, or 3). After the retarded potentials are obtained (assuming the Lorenz gauge) the electromagnetic fields are calculated with the aid of the relations E = −∇Φ − (1/c)(∂A/∂t) and B = ∇ × A (we use Gaussian units). The resulting ex- pressions for the fields are usually called Jefimenko’s equations because they appeared for the first time in the textbook by Jefimenko.4 Jefimenko’s equations are obtained in Ref. 1 from the retarded potentials and are obtained directly from Maxwell’s equations in Ref. 5. (Heras6 had already derived Jefimenko’s equations directly from Maxwell’s equations.) Ref- erences 8 and 7 obtain a less common form of Jefimenko’s equations for the electric field, but this form is more convenient for studying radiation.
    [Show full text]
  • Classical Electrodynamics for the First Few Values of I the Explicit Forms Are
    1 6 Multipole Fields In Chapters 3 and 4 on electrostatics the spherical harmonic expansion of the scalar potential was used extensively for problems possessing some symmetry property with respect to an origin of coordinates . Not only was it useful in handling boundary-value problems in spherical coordinates, but with a source present it provided a systematic way of expanding the potential in terms of multipole moments of the charge density. For time-varying electromagnetic fields the scalar spherical harmonic expansion can be generalized to an expansion in vector spherical waves. These vector spherical waves are convenient for electromagnetic boundary-value problems possessing spherical symmetry properties and for the discussion of multipole radiation from a localized source distribution . In Chapter 9 we have already considered the simplest radiating multipole systems . In the present chapter we present a systematic development . 16.1 Basic Spherical Wave Solutions of the Scalar Wave Equatio n As a prelude to the vector spherical wave problem, we consider the scalar wave equation. A scalar field V(x, t' satisfying the source-free wave equation, 2 (16. 1 ) °2 V - C 2 at2 = o can be Fourier-analyzed in time a s V(X' t) = . V(X, c ))e-iwt do) (1 6.2) E with each Fourier component satisfying the Helmholtz wave equation , (V2 + kl)V(X, co) = 0 (16.3) 538 [Sect. 16.1] Multipole Fields 539 with V = C021c2. For problems possessing symmetry properties about some origin it is convenient to have fundamental solutions appropriate to spherical coordinates. The representation of the Laplacian operator in spherical coordinates is given in equation (3 .1) .
    [Show full text]
  • Radiation-Notes.Pdf
    In this chapter we discuss the production of electromagnetic waves. The real trick here is to develop techniques that allow us to compute closed form solutions for the electric and magnetic field as opposed to series expansions similar to approach we used towards the end of Physics 435. For example, in one of the problems in the last homework, you had an oscillating solenoid, that would create a Faraday electrical field, that would create a correction to the magnetic field due to the displacement current that would modify the Faraday electric field etc. There is a lot of fascinating physics here. We begin with production of plane polarized waves from an oscillating, infinite current sheet. Here we are able to solve for the fields and radiated energy by matching the B field from the plane polarized waves to the B-field generated by the Ampere term of current sheet since we can show that displacement contribution is negligible. The usual way that radiation problems are handled in closed form is through the potential formulation where we use “retarded” versions of the same scalar and vector potential integrals developed in Phys435. These retarded potentials imply that if one turns on a current, a far away observer doesn’t immediately see an Ampere’s law B-field but must wait for the “information” to get there. Interestingly enough the information travels at the speed of light. We are able to get a simple, closed form solution for this problem, We then switch to radiation from oscillating antennae and work out the electromagnetic fields, the Poynting vector and the total radiated power.
    [Show full text]
  • Module I: Electromagnetic Waves Lectures 10-11: Multipole Radiation
    Module I: Electromagnetic waves Lectures 10-11: Multipole radiation Amol Dighe TIFR, Mumbai Outline 1 Multipole expansion 2 Electric dipole radiation 3 Magnetic dipole and electric quadrupole radiation Coming up... 1 Multipole expansion 2 Electric dipole radiation 3 Magnetic dipole and electric quadrupole radiation ~ Vector potential A0 for monochromatic sources We are interested in calculating the radiative components of EM fields and related quantities (like radiated power) for a charge / current distribution that is oscillating with a frequency !. The results for a general time dependence can be obtained by integrating over all frequencies (inverse Fourier transform). We have already seen that it is enough to know about the current distribution (we are interested only in radiative parts), since the charge distribution is related to it by continuity. In such ~ 0 ~ 0 −i!t a case, with J(~x ; t) = J0(~x )e , we get 0 µ Z eikj~x−~x j A~ (~x; t) = 0 e−i!t ~J (~x0) d 3x 0 (1) 4π 0 j~x − ~x0j ~ ~ −i!t Given A(~x; t) = A0(~x)e , the rest of the quantities can be easily calculated in terms of it. We shall omit the “rad” label in this lecture, it is assumed to be everywhere except when specified. B~ rad and E~ rad for monochromatic sources At large distances, 0 µ Z e−i~k·~x A~ (~x; t) = 0 ei(kjx|−!t) ~J (~x0) d 3x 0 (2) 4π 0 j~x − ~x0j Taking ~x = r ^r and neglecting terms that go as (1=r 2), the radiative part of the magnetic field is ~ ~ ~ −i!t B(~x; t)= r × A(~x; t) = ik^r × A0(~x)e (3) The radiative part of the electric field can be obtained in this monochromatic case by using (1=c2)@E~ (~x; t)=@t = r × B~ (~x; t) (there is no current at large j~xj): c2 E~ (~x; t)= r × B~ (~x; t) = cB~ (~x) × ^re−i!t (4) (−i!) 0 Thus, E~ (~x; t) and B~ (~x; t) fields are orthogonal to ^r, orthogonal to each other, and their magnitudes differ simply by a factor of c.
    [Show full text]
  • Multipole Radiation Fields from the Jefimenko Equation For
    Multipole radiation fields from the Jefimenko equation for the magnetic field and the Panofsky-Phillips equation for the electric field ͒ R. de Melo e Souza,a M. V. Cougo-Pinto, and C. Farina Universidade Federal do Rio de Janeiro, Instituto de Fisica, Rio de Janeiro, RJ 21945-970, Brazil M. Moriconi Universidade Federal Fluminense, Instituto de Fisica, Niteroi, RJ 24210-340, Brazil ͑Received 12 February 2008; accepted 8 September 2008͒ We show how to obtain the first multipole contributions to the electromagnetic radiation emitted by an arbitrary localized source directly from the Jefimenko equation for the magnetic field and the Panofsky-Phillips equation for the electric field. This procedure avoids the unnecessary calculation of the electromagnetic potentials. © 2009 American Association of Physics Teachers. ͓DOI: 10.1119/1.2990666͔ I. INTRODUCTION tion only in the temporal coordinate. We shall see that this method, although longer than the previous one, avoids any The derivation found in most textbooks of the electromag- possibility of misleading manipulations with retarded quan- netic fields generated by arbitrary sources in vacuum starts tities. The subtleties in calculations involving retarded quan- by calculating the corresponding electromagnetic potentials tities have been discussed in Refs. 13–15. We then present an ͑ ͒ see, for instance, Ref. 1 and 2,or3 . After the retarded alternative method that is a variation of the first one; the ͑ ͒ potentials are obtained assuming the Lorenz gauge the main difference is the order of integration. electromagnetic fields are calculated with the aid of the rela- -١ϫA ͑we use Gauss=t͒ and Bץ/Aץ١⌽−͑1/c͒͑−=tions E ian units͒.
    [Show full text]
  • Chapter Nine Radiation
    Chapter Nine Radiation Heinrich Rudolf Hertz (1857 - 1894) October 12, 2001 Contents 1 Introduction 1 2 Radiation by a localized source 3 2.1 The Near Zone . 6 2.2 The Radiation or Far Zone . 6 3 Multipole Expansion of the Radiation Field 9 3.1 Electric Dipole . 10 3.1.1 Example: Linear Center-Fed Antenna . 13 3.2 Magnetic Dipole . 14 3.3 Comparison of Dipoles . 16 3.4 Electric Quadrupole . 18 3.4.1 Example: Oscillating Charged Spheroid . 21 3.5 Large Radiating Systems . 23 3.5.1 Example: Linear Array of Dipoles . 24 1 4 Multipole expansion of sources in waveguides 26 4.1 Electric Dipole . 26 5 Scattering of Radiation 31 5.1 Scattering of Polarized Light from an Electron . 31 5.2 Scattering of Unpolarized Light from an Electron . 34 5.3 Elastic Scattering From a Molecule . 35 5.3.1 Example: Scattering O® a Hard Sphere . 38 5.3.2 Example: A Collection of Molecules . 40 6 Di®raction 43 6.1 Scalar Di®raction Theory: Kircho® Approximation . 45 6.2 Babinet's Principle . 51 6.3 Fresnel and Fraunhofer Limits . 53 7 Example Problems 55 7.1 Example: Di®raction from a Rectangular Aperture . 55 7.2 Example: Di®raction from a Circular Aperture . 57 7.3 Di®raction from a Cross . 60 7.4 Radiation from a Reciprocating Disk . 61 1 Introduction An electromagnetic wave, or electromagnetic radiation, has as its sources electric accelerated charges in motion. We have learned a great deal about waves but have not given much thought to the connection between the waves and the sources that produce them.
    [Show full text]
  • Radiation of the Electromagnetic Field Beyond the Dipole Approximation
    Radiation of the electromagnetic field beyond the dipole approximation Andrij Rovenchak and Yuri Krynytskyi Department for Theoretical Physics, Ivan Franko National University of Lviv, Ukraine July 9, 2018 Abstract An expression for the intensity of the electromagnetic field radia- tion is derived up to the order next to the dipole approximation. Our approach is based on the fundamental equations from the introduc- tory course of classical electrodynamics and the derivation is carried out using straightforward mathematical transformations. Key words: multipole expansion; radiation; higher multipole mo- ments; anapole; toroidicity. 1 Introduction Multipole expansion is a well-known technique applied in calculations of electromagnetic [1, 2] and gravitational fields [3, 4] at large distances from sources. In stationary cases, such as electrostatic problems, this approach is mainly linked to simple series expansions and certain sym- metrization procedures [5, 6]. Treatment of magnetostatic problems requires more elaborate techniques beyond the lowest order [7]. How- ever, it becomes more tricky in the case of the electromagnetic field radiation. Moreover, in textbooks on classical electrodynamics one arXiv:1803.07889v2 [physics.class-ph] 20 Sep 2018 of the terms beyond the dipole approximation is typically omitted in expressions for the radiated power even if the detailed derivations are given (see, e.g., [8, 9, 10]; for an exception see [11]). The goal of this article is to derive an expression for the radiated power in the approximation next to the dipole one using the basic 1 equations of electrodynamics and the techniques from vector and ten- sor calculus. From the methodological point of view, our approach is advantageous comparing to the derivations based on gauge symmetries [12] or solutions to the scattering problem [13].
    [Show full text]