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Chapter 1

Introduction

1.1 What are Plasmon Resonances?

The purpose of this introductory chapter is to discuss the physical origin ofplasmon resonances in metallic nanoparticles, to describe their basic proper-ties and to outline the approach to the study of plasmon resonances adoptedin this book. The presentation in this chapter is mostly descriptive andavoids as much as possible mathematical technicalities which are provided inthe subsequent chapters.

To start the discussion, consider a macroscopic piece of metal (gold orsilver, for instance) subject to optical radiation (see Figure 1.1). In this sit-uation, no unique physical phenomena of distinction or long remembranceoccur; this macroscopic piece of metal is “lifeless.” However, if the dimen-sions of this metallic piece are made smaller and smaller and are eventuallyreduced to nanoscale, the resulting metallic nanoparticle may come to lifewhile being subject to optical radiation. It may glow, it may resonate andit may become a very powerful, nanoscale localized source of light. This is,in descriptive terms, the essence of the phenomena of plasmon resonances inmetallic nanoparticles. These powerful localized sources of light are usefulin different areas of science and technology which include scanning near-fieldoptical microscopy [1, 2], nano-lithography [3], biosensor applications [4, 5],surface enhanced Raman scattering (SERS) [6]-[9], nanophotonics [10]-[13],optical and magnetic data storage [14, 15], etc.

The question can be immediately asked: “What is the physical mecha-nism of plasmon resonances?” In other words: “What is there to resonate?”

1

2 Plasmon Resonances in Nanoparticles

Figure 1.1

Experiments show that metallic (gold and silver) nanoparticles may exhibitresonance behavior at certain frequencies at which the following two condi-tions are satisfied: 1) the nanoparticle dielectric permittivity is neg-ative and 2) the free-space wavelength is large in comparison withthe nanoparticle dimensions. The latter condition clearly suggests thatthese resonances are electrostatic in nature. When dielectric permittivityis negative, the uniqueness theorem of electrostatics is not valid. For thisreason, source-free electrostatic fields may appear for certain negative valuesof dielectric permittivity. This is the manifestation of resonances, and thecorresponding source-free electrostatic fields are resonant plasmon modes.

It is important to stress that plasmon resonances in metallic nanoparti-cles are intrinsically nanoscale phenomena. This is because the two resonanceconditions (negative dielectric permittivity and smallness of the particle di-mensions in comparison with free-space wavelength) can be simultaneouslyand naturally realized at the nanoscale.

The question can be asked why it is possible and relevant to speak ofdielectric permittivity of metallic nanoparticles subject to optical radiation.The reason is that conduction electrons in metallic nanoparticles are pinnedby optical radiation and execute tiny back-and-forth oscillations around some“equilibrium” positions. In this sense, these conduction electrons are indistin-guishable from bound charges in dielectrics. That is why metallic nanoparti-cles behave at optical frequencies as dielectric particles with dispersion. Thelatter means that dielectric permittivity depends on frequency. It turns outthat for a certain frequency range its real part may assume negative values.As discussed later in this chapter, this frequency range for metals is near their

Chap. 1: Introduction 3

Figure 1.2

plasma frequencies, where the dispersion relations ε(ω) are fully governed bythe interaction between electromagnetic radiation (light) and the conductionelectrons. For good conductors such as silver and gold, plasma frequencies arein the visible frequency range, and this explains why silver and gold nanopar-ticles are usually employed in plasmon resonance studies and applications.It is worthwhile to remark that plasmon resonances may occur not only inmetallic nanoparticles but in any nanoparticle whose permittivity exhibitsdispersion and whose real part may assume negative values. One example isthe silicon carbide (SiC) material whose negative permittivity is not due tothe interaction of conduction electrons with light but rather due to specificproperties of lattice vibrations in polar crystals.

It has been already mentioned that the second condition of smallnessof particle dimensions in comparison with free-space wavelength reveals thephysical nature of plasmon resonances in nanoparticles as electrostatic reso-nances. Indeed, due to this condition, time-harmonic electromagnetic fieldswithin the nanoparticles and in their vicinity vary almost with the samephase. As a result, at any fixed instant of time these fields look like electro-static fields. The electrostatic nature of plasmon resonances in nanoparticlesand their occurrence for negative values of dielectric permittivity immedi-ately suggest the enhancement of local electric fields inside nanoparticlesand their vicinities. To illustrate this fact, consider an example of a spheri-cal nanoparticle with negative permittivity ε subject to uniform external fieldE0 (see Figure 1.2). Since ε < 0, the polarization vector P has the direc-tion opposite to E0 and this results in surface electric charges which createthe “depolarizing” field E′ with the same direction as E0. This naturally


Figure 1.3

leads to the enhancement of the total electric field E = E0 + E′ insidethe spherical nanoparticle∗. To better appreciate this fact, it is instructiveto consider the case of a spherical nanoparticle with positive permittivity(Figure 1.3) where the depolarizing field E′ results in the attenuation of theexternal field E0.

In our study of plasmon resonances in nanoparticles, we shall follow thetraditional approach when all losses are first neglected and resonance modesand resonance frequencies are first found for lossless systems. A similar ap-proach is used, for instance, in the study of resonance modes in metalliccavities. For such cavities, the resonance mode problem is mathematicallyformulated as an eigenvalue problem for specific differential equations derivedfrom the Maxwell equations. It will be demonstrated in this book that theproblem of plasmon resonance modes can be also mathematically formulatedas an eigenvalue problem which is posed, however, not for differential equa-tions but rather for specific boundary integral equations. There is anotherimportant difference between plasmon resonances in metallic nanoparticlesand resonances in metallic cavities. In the latter case, the resonance fre-quencies depend on the shape and dimensions of the metallic cavities. Forinstance, in the case of rectangular resonant cavities (see Figure 1.4), the

∗It is interesting to mention that a similar phenomenon of enhancement of interiormagnetic fields occurs in type-I superconductors, which are ideal diamagnets. For thisreason, the magnetic fields inside type-I superconductors may exceed a critical field, whileapplied exterior magnetic fields remain below this field. This leads to the formation of theLandau “intermediate state” of type-I superconductors when normal and superconductingregions (“domains”) coexist inside type-I superconductors.


Figure 1.4

resonance frequencies are given by the formula

ωmnk =π√µ0ε0

√

m2

a2+n2

b2+k2

c2, (1.1)

which clearly reveals their dependence on cavity dimensions a, b and c. Itwill be demonstrated in this book that in the case of plasmon resonances inmetallic nanoparticles, resonance frequencies are scale-invariant. This meansthat these frequencies depend only on particle shapes but not on their dimen-sions, provided that these dimensions are appreciably smaller than resonancefree-space wavelengths. This scale invariance implies, among other things,that in the case of ensembles of almost self-similar (of the same shape butdifferent dimensions) metallic nanoparticles, they may resonate at practicallythe same wavelength. Consequently, plasmon resonances can be simultane-ously excited in all these nanoparticles.

It turns out that the eigenvalue formulation of the plasmon resonanceproblem has another important feature. This eigenvalue formulation leads tothe direct calculation of the negative values of dielectric permittivity at whichplasmon resonances may occur. These negative values of dielectric permittiv-ity can then be used for any dispersion relation of nanoparticle to determinethe corresponding resonance frequencies. In this way, the properties of plas-mon resonances which depend on nanoparticle shapes can be fully separatedfrom those which depend on the material properties of nanoparticles whichdefine their dispersion relations. In other words, the solution of the plasmonresonance eigenvalue problem for a nanoparticle of specific shape can be usedfor different materials of this nanoparticle to find resonance frequencies.

Having found plasmon resonance modes and their resonance frequenciesfor lossless metallic nanoparticles through the solutions of the appropriateeigenvalue problem, the next step is to study the excitation of these plas-mon modes by the incident radiation and the effect of ohmic and radiationlosses. This approach is adopted in the book and it reveals that, by and


Figure 1.5

Figure 1.6


large, dipole plasmon modes are excited by incident optical radiation cre-ated by remote sources. This is because the electric field of this radiationis practically uniform over the nanoparticle region due to the smallness ofnanoparticle dimensions in comparison with the wavelength of the incidentradiation. It also turns out that the incident radiation is most efficientlycoupled to a plasmon mode if the dipole moment of this mode is parallel tothe direction of electric field of the incident radiation. The plasmon modeexcitation analysis also reveals that the quality of plasmon resonances, i.e.,the local enhancement of the incident optical radiation, is controlled by theratio of the real part of dielectric permittivity to its imaginary part at theresonance frequency. For gold and silver, this ratio is most appreciable whenthe free-space wavelength is within the ranges of 650-1000 nm and 600-1400nm (see Figures 1.5 and 1.6, respectively, which are based on experimentaldata of P. B. Johnson and R. W. Christy [16]). Therefore, plasmon reso-nances in gold and silver nanoparticles can be most efficiently excited in thecorresponding frequency ranges. It is also worthwhile to observe that the ra-tio of real and imaginary parts of dielectric permittivity is appreciably higherfor silver than for gold. Thus, as far as the quality of plasmon resonances isconcerned, silver is “gold” and gold is “silver.” This fact has long been ap-preciated in the area of surface enhanced Raman scattering (SERS) researchwhere silver nanoparticles have been predominantly used in experiments. Ofcourse, silver oxidation presents some experimental difficulties that must bedealt with.

1.2 Dispersion Relations

It is clear from the presented discussion that the dispersion relation ε(ω) of ametallic nanoparticle is instrumental for the analysis of plasmon resonances.For this reason, it is worthwhile to briefly review the simplest analytical mod-els for the dispersion relations. We start with the case of the free-electronplasma dispersion relation, when all electron collisions are neglected. In ad-dition, since the force on electrons arising from interaction with the magneticfield of optical radiation is typically much smaller (several orders of magni-tude) than the electric force, the former force is neglected as well. Then, theequation of motion for electrons can be written as follows:

md2ridt2

= −eE(t), (1.2)

where ri is the position vector of the i-th electron.


When the Cartesian components of vector E(t) are time-harmonic func-tions of angular frequency ω, the last equation can be written in the phasorform

−ω2mri = −eE, (1.3)

where ri and E are the notations for the phasors of ri(t) and E(t), respec-tively. From the last equation, we find

ri =e

mω2E. (1.4)

Now, by using the definition of the polarization vector P, we obtain

P = −N∑

i=1

eri = − e2N

mω2E, (1.5)

where N is the electron density, i.e., the number of electrons per unit volume.By using formula (1.5), we derive

D = ε0E+ P = ε0

(

1− e2N

ε0mω2

)

E, (1.6)

which implies thatD = ε(ω)E(ω), (1.7)

with

ε(ω) = ε0

(

1−ω2p

ω2

)

. (1.8)

In the last formula, ωp is the plasma frequency defined by the formula

ω2p =

e2N

ε0m. (1.9)

It is clear from the dispersion relation (1.8) that

ε(ω) < 0 if ω < ωp. (1.10)

The derivation of the dispersion relation (1.8) has been based on the greatlysimplified equation of motion (1.2). For this reason, it is natural to questionits validity and accuracy. To test the latter, we shall compare this disper-sion relation with the available experimental data for gold and silver. The


Figure 1.7

remarkable fact is that the electron density N for gold is practically the sameas for silver, namely,

NAu = 5.90× 1022 cm−3, NAg = 5.86× 1022 cm−3. (1.11)

This implies that according to formulas (1.8) and (1.9) the dispersion re-lation ε(ω) must be practically identical for gold and silver in the opticalfrequency range. By using the experimental data from P. B. Johnson andR. W. Christy [16], the dispersion relations for gold and silver are plotted inFigure 1.7, which reveals that the real parts of dielectric permittivities ofgold and silver are indeed approximately the same for the free-space wave-length range between 500 nm and 2000 nm. In addition, the comparisonbetween ε(ω) computed by using formulas (1.8) and (1.9) and the experi-mentally measured [16] real parts of dielectric permittivity of gold and silverare shown in Figures 1.8 and 1.9, respectively. These figures suggest that thedispersion relation (1.8)-(1.9) is indeed fairly accurate in the 500 nm-2000 nmwavelength range, and it is in this wavelength range that plasmon resonancesin metallic nanoparticles are usually studied.

Formulas (1.8) and (1.9) suggest that the dispersion relation ε(ω) canbe controlled by manipulating the electron density N . This may be espe-cially attractive in the case of semiconductors where the conduction electron


Figure 1.8

density can be manipulated optically. This may open opportunities for op-tical controllability (optical gating) of plasmon resonances in semiconductornanoparticles.

The constitutive relation (1.7) is local in the frequency domain. However,due to the dispersion of dielectric permittivity, the constitutive relation be-tween D(t) and E(t) is non-local in the time domain. It is instructive andsomewhat interesting to find the non-local-in-time constitutive relation forD(t) and E(t) which corresponds to the dispersion relation (1.8)-(1.9). Thesimplest way to do this is to integrate twice the equation of motion (1.2) andto perform the integration by parts in the resulting double integral. Thisleads to the following expression:

ri(t) = − e

m

∫ t

0

(t− τ)E(τ)dτ. (1.12)

Next, by using the same definitions of P and D as in formulas (1.5) and(1.6), we arrive at the following constitutive relation:

D(t) = ε0

[

E(t) + ω2p

∫ t

0

(t− τ)E(τ)dτ

]

. (1.13)


Figure 1.9

In general, non-local time-domain relations between D(t) and E(t) can bequite complicated. Mathematically, these non-local-in-time constitutive re-lations can be treated as pseudo-differential operators [17] and the experi-mentally measured dispersion relations ε(ω) as their symbols. This pseudo-differential operator interpretation of dispersion may be especially promisingfor the study of spatially dispersive media, but this is beyond the scope ofthis book.

The greatly simplified equation of motion (1.2) leads to the real-valueddielectric permittivity ε(ω) given by formulas (1.8) and (1.9). Experimentsreveal that the actual dielectric permittivity of metals in the optical frequencyrange has an imaginary part as well which accounts for energy losses. It turnsout that the equation of motion (1.2) can be modified to obtain the complex-valued dispersion relation ε(ω). Indeed, if the effect of electron collisions ismodeled by introducing a “friction” term representing some average loss ofelectron momentum, then this results in the following equation of motion:

md2ridt2

− γmdridt

= −eE(t). (1.14)


Now, by using the same line of reasoning as in the derivation of formulas(1.8) and (1.9), we arrive at the following dispersion relation

ε(ω) = ε0

[

1−ω2p

ω(ω + jγ)

]

,(

j =√−1

)

, (1.15)

which is quite often referred to as the Drude model for dielectric permittivity.From the last formula, the following expressions for the real ε′ and imag-

inary ε′′ parts of dielectric permittivity can be easily derived,

ε′ = ε0

[

1−ω2p

ω2 + γ2

]

, (1.16)

ε′′ = ε0ω2pγ

ω(ω2 + γ2). (1.17)

In the typical (for plasmon resonances) case when 1 ≪ ω2p/(ω

2 + γ2), fromthe last two formulas we find

ε′

ε′′≃ −ω

γ. (1.18)

The Drude model will be extensively used in this book for the analytical studyof time-dynamics of excitation and dephasing (decay) of plasmon modes. Inparticular, it will be shown that 1/γ can be identified as the decay time forthe light intensity of plasmon modes in the absence of optical excitation.

The imaginary part ε′′ of the dielectric permittivity can be used for com-putation of power losses in metallic nanoparticles. As discussed before, plas-mon resonances result in appreciable enhancement of electric fields insidenanoparticles. This, in turn, results in substantial increase of power losses,which leads to the peaks in extinction (total or forward-scattering) cross sec-tion. These peaks of extinction cross sections are used in experiments toidentify plasmon resonances and the corresponding resonance frequencies.

As discussed, the real part of dielectric permittivity for dispersive mediamay assume negative values (ε′(ω) < 0). This makes the classical formula

we =1

4ε′∣

∣

∣E∣

∣

∣

2

(1.19)

for the time-average stored electric energy density meaningless. It turns outthat the last formula can be properly modified to be valid for slightly lossy(transparent) dispersive media [18, 19]. The modified formula is

we =1

4

d (ωε′(ω))

dω

∣

∣

∣E∣

∣

∣

2

. (1.20)


It is apparent that in the case of non-dispersive media the last formula isreduced to formula (1.19). It is also clear that in accordance with the lastformula we > 0 in the case of the dispersion relation described by formula(1.8) as well as by the Drude model (1.16) under the transparency conditionω > γ.

1.3 Overview of Book Contents

It is apparent from the previous discussion that the approach to the studyof plasmon resonances in metallic nanoparticles adopted in this book is theeigenmode approach. This means that the problem of resonant plasmonmodes (and the corresponding resonance frequencies) in lossless metallicnanoparticles is first framed as the eigenvalue problem for specific bound-ary integral equations. After this problem is solved and plasmon modes areidentified, the excitation of these plasmon modes by incident optical radia-tion is studied along with the radiation corrections. This approach is quitedistinct and different from techniques frequently used in scientific literature.Indeed, resonances in metallic nanoparticles are often found experimentallyand numerically by using a “trial-and-error” approach, i.e., by probing metal-lic nanoparticles of complex shapes with optical radiation of various frequen-cies and polarizations [20]-[23]. The numerical analysis is typically performedby using the finite-difference time-domain (FDTD) technique. It seems to usthat the eigenmode technique adopted in this book has important advantagesover FDTD and other techniques.

First, the eigenmode technique leads to the direct calculation of the reso-nance values of dielectric permittivity which can then be used for any disper-sion relation of a metallic nanoparticle to immediately find the correspondingresonance frequency. In this way, the properties of plasmon resonances whichdepend on nanoparticle shapes are clearly separated from those propertieswhich depend on dispersion relations. In contrast, separate FDTD compu-tations have to be performed for different dispersion relations.

Second, the eigenmode technique has analytical capabilities for the time-domain analysis of plasmon resonance modes. As discussed in this book,the eigenmode technique explicitly reveals the coupling conditions betweenspecific plasmon modes and the incident radiation. It leads to analytical for-mulas for time dynamics of plasmon modes (their excitation and dephasing).It results in analytical expressions for steady-state amplitudes of plasmon


resonance modes and for steady-state amplitudes in the case of off-resonanceexcitations; the latter reveal the sharpness of plasmon resonances. In con-trast, FDTD is a purely numerical technique without analytical capabilities.

Furthermore, FDTD requires discretization of three-dimensional space,while the technique advocated in this book is a surface integral equationtechnique, which requires discretization of two-dimensional boundaries ofthe nanoparticles. Since only a finite region of three-dimensional space canbe discretized and used in computations, FDTD requires the introduction ofartificial external boundaries. Special absorbing boundary conditions mustbe posed on these boundaries to minimize distortions and errors caused bythe introduction of artificial boundaries. In the eigenmode technique, no suchartificial boundaries are required.

Finally, plasmon resonances occur in dispersive media with non-local-in-time (convolution-type) constitutive relations between electric displacementand electric field. These non-local constitutive relations lead to finite differ-ence schemes with the electric field coupling at all previous time steps. Thispast history-coupling of the electric field may diminish the effectiveness ofFDTD for plasmon resonance computations.

The book consists of four chapters. The detailed review of the bookcontent is given below, chapter by chapter. The review is presented withoutinvoking complicated mathematical formulas, but rather emphasizing thephysical aspects of the matter.

Chapter 2 deals with the modal analysis of plasmon resonances in metal-lic nanoparticles. This analysis is first framed as an eigenvalue problemfor boundary integral equations with respect to surface electric charges dis-tributed over nanoparticle boundaries. These virtual (fictitious) charges areintroduced on S in free space in order to mimic the boundary conditionswhich occur for electric fields of plasmon modes on the nanoparticle sur-face. As a result, these surface charges create in free space the same electricfields E which exist for actual plasmon modes in the presence of metallicnanoparticles. This replication of the boundary conditions for E leads tothe eigenvalue problem for a specific boundary integral equation with re-spect to surface electric charges which has nonzero solutions only for specificnegative values of dielectric permittivity. In practice, metallic nanoparticlesare placed on dielectric substrates. These substrates can be accounted forin the mathematical formalism by using the appropriate Green functions inthe kernel of the boundary integral equations. In the case of flat substrates,this Green function can be found by using the method of images. Some


analytical results for plasmon modes in spherical and ellipsoidal nanoparti-cles are presented here as well. In particular, it is demonstrated that forany negative value of dielectric permittivity an ellipsoidal nanoparticle withthe appropriate aspect ratio can be found that will resonate for this valueof permittivity and, consequently, for the corresponding value of frequency(wavelength) of optical radiation. This suggests that ellipsoidal nanoparticlescan be used (at least in principle) for the solution of the tunability problemfor plasmon resonances.

It turns out that there exists a dual eigenvalue approach to the study ofplasmon modes in metallic nanoparticles. In this approach, virtual (fictitious)double layers of electric charges are introduced on S in free space to reproduceon S the same boundary conditions for electric displacement D which existfor actual plasmon modes in the presence of metallic nanoparticles. Thiseventually leads to the eigenvalue problem for boundary integral equationsfor double layer densities. These integral equations are adjoint to the integralequations for simple (single) layers of electric charges.

The detailed study of general properties of the spectrum of the derivedboundary integral equations (which is the plasmon spectrum) is then fol-lowed. It is demonstrated that for any shape S of nanoparticles the spec-trum is discrete and real, and that, as expected, the corresponding valuesof dielectric permittivity are negative. It is also shown that the spectrumis scale-invariant. Since the kernels of the corresponding integral equationsare not symmetric, the eigenfunctions are not orthogonal on S. However,the corresponding electric fields of plasmon modes are orthogonal. More-over, strong orthogonality conditions hold for plasmon mode electric fields.The term “strong” means that these fields are orthogonal not only in theentire space but they are also separately orthogonal inside and outside thenanoparticles.

Unique spectral properties of plasmon resonances occur for metallic nano-wires, i.e., in the two-dimensional case. In this case, for any cross-sectionalshape of nanowires, the spectrum of the corresponding integral equationsconsists of pairs (couples) of positive and negative eigenvalues of the samemagnitude. In other words, each positive eigenvalue has its twin (counter-part) of negative value and the same magnitude. This phenomenon of twinspectrum is due to unique symmetry properties of the mathematical for-mulation which appear in the two-dimensional case due to the existence ofthe stream function which is conjugate to the electric potential. The twinspectrum phenomenon results in two distinct bands of plasmon resonances


with relative dielectric permittivities which are reciprocal to one another fordifferent bands.

It is apparent that the tunability of plasmon resonances may be of valuein many applications. It turns out that a wide range of tunability can beachieved by using metallic nanoshells [24, 25] and controlling the plasmonresonance frequencies via adjustment of the shell thickness. It is demon-strated in the chapter that the plasmon mode analysis in metallic nanoshellscan be reduced to a generalized eigenvalue problem for specific boundary (in-tegral) equations and the detailed analysis of plasmon spectrum in metallicnanoshells is carried out.

The chapter is concluded with the discussion of interesting relation ofeigenvalue treatment of plasmon resonances to the Riemann hypothesis.Namely, it is pointed out that for any sufficiently regular shapes of nanowirecross sections Fredholm determinants D(λ) of integral equations used in thecalculation of plasmon spectrum form the class of entire functions with prop-erties that have been conjectured or proved for the Riemann xi-function ξ(λ).For this reason, the problem can be posed to find such curved boundary Lof nanowire cross section that

ξ(λ) = D(λ), (1.21)

which will prove the Riemann hypothesis. The last formula is consistent witha spectral interpretation of the Riemann hypothesis asserted in the Hilbert-Polya conjecture. Finally, a promising approach to the proof of the Riemannhypothesis based on the Grommer theorem (and suggested by formula (1.21))is discussed as well.

Chapter 3 deals with analytical and numerical analysis of plasmon res-onances in metallic nanoparticles. Before proceeding with the discussionof numerical issues, analytical solutions for plasmon modes in certain nanos-tructures are presented. There are two reasons why these analytical solutionsare of importance. First, these analytical solutions are used in the book fortesting the accuracy of numerical computations. Second, these analyticalsolutions for plasmon modes are of interest in their own right because theyare derived for nanostructures that have appeared (or will appear) in variousapplications of plasmon resonances. These analytical solutions are obtainedby using the method of separation of variables in various coordinate systems.It is known that in this method possible solutions are expressed as productsof functions, each of which depends only on one of the variables of the coor-dinate system used. These product solutions are plasmon modes, and they


are realized for specific negative values of dielectric permittivity of metallicnanoparticles or nanowires. These specific negative (resonance) values ofpermittivity are found from the interface boundary conditions.

Next, the numerical issues of the implementation of the analysis of plas-mon resonances in metallic nanoparticles presented in Chapter 2 are dis-cussed. The discussion starts with the description of the special discretiza-tion technique for the eigenvalue problem for boundary integral equationswith respect to single layers of electric charges. This discretization techniquecompletely circumvents the difficulties associated with (weak) singularitiesof the kernels of the integral equations as well as with singularities of surfaceelectric charge densities on edges and corners of nanoparticle boundaries.

Discretizations of boundary integral equations used in the analysis ofplasmon resonances result in fully populated matrices. For this reason, thesediscretized equations are computationally expensive to solve. This is espe-cially true when many nanoparticles are involved in the design of metallicnanostructures. Fortunately, since the fully populated matrices are gener-ated through discretizations of integral operators with 1/r-type kernels, thiscomputational problem can be considerably alleviated by using the fast mul-tipole method [26] introduced by V. Rokhlin and L. Greengard. This methodgreatly speeds up the matrix-vector multiplications, resulting at most inO(N) computational cost for N × N fully populated matrices. The centralidea of the fast multipole method is to split the computations for near-fieldand far-field regions and then utilize factorized representations for kernels inthese regions in terms of spherical harmonics which follow from “additiontheorem” expansions.

The theoretical discussion of discretization techniques is illustrated in thechapter by numerous computational examples. Many of these computationsare carried out for metallic nanoparticle arrangements that have been alreadystudied experimentally due to their physical interest or possible technologicalapplications. The comparison of computational results with available exper-imental data reveals the coincidence which is mostly within five to sevenpercent. This agreement with experimental data is quite good and, by andlarge, within the accuracy of measuring techniques. This agreement can alsobe construed as some justification for using the macroscopically measureddispersion relation ε(ω) at the nanoscale, i.e., in the constitutive relation formetallic nanoparticles.

The chapter also contains the discussion of a numerical technique forthe solution of inhomogeneous boundary integral equations with singular


kernels. Such integral equations arise in the analysis of extinction crosssections. The unique features of this numerical technique are 1) unique solv-ability of discretized equations for any mesh and any surface geometry ofmetallic nanoparticles and 2) guaranteed convergence of the numerical tech-nique which is valid for any singularity in the kernels of the integral equa-tions. The unique solvability of discretized equations, convergence of theapproximate solutions to the exact solutions and the rate of this convergenceare established under only one natural condition of unique solvability of theinhomogeneous boundary integral equations for any right-hand sides (any“forcing” terms).

The chapter is concluded with the discussion of exact absorbing boundaryconditions for finite-difference time-domain (FDTD) analysis of scatteringproblems. It may seem that this discussion is somewhat out of place in thisbook which advocates alternatives to the FDTD technique. Nevertheless,it is included in the book for two reasons: its distinctness and the exten-sive current use of the FDTD technique in the study of plasmon resonances.The central point of the presented discussion is to use the time-domain ver-sions of Kirchhoff or Stratton-Chu formulas as Dirichlet-type “absorbing”boundary conditions that can be posed on any artificial boundary that en-closes a scattering nanoparticle. These boundary conditions are exact andthey are updated as computations proceed. The physical foundation for thistype of absorbing boundary conditions is the retardation phenomena. TheStratton-Chu-type boundary conditions can be naturally coupled with theYee finite-difference scheme, while the Kirchhoff-type boundary conditionscan be naturally coupled with the standard explicit finite-difference schemefor scalar wave equations written for each Cartesian component of electric(or magnetic) field.

Chapter 4 deals with several topics. The first one is the radiation cor-rections to the electrostatic plasmon mode analysis presented in Chapter 2.These radiation corrections are mathematically treated as perturbations withrespect to small parameters β, which are the ratios of particle dimensions(their diameters) to the free-space wavelengths. To compute these correc-tions, homogeneous Maxwell equations are written in a scaled (perturbative)form which explicitly includes the small parameter β. Then, source-free so-lutions of these Maxwell equations and the values of dielectric permittivitiesfor which they occur are expanded in power series with respect to the smallparameter β. In this way, the appropriate boundary value problems are de-rived for zero-, first- and second-order terms of these power expansions. It


turns out, as expected, that the zero-order approximation coincides with theelectrostatic formulation of plasmon modes studied in detail in Chapter 2.Then, by using the principle of normal solvability of integral equations, thefirst- and second-order radiation corrections with respect to β are studied.Namely, it is demonstrated that the first-order radiation corrections to theresonance value of dielectric permittivity (hence, to resonance frequencies)and to plasmon mode electric fields are always equal to zero for any shapeof metallic nanoparticle. Meanwhile, the first-order radiation corrections forthe magnetic fields of plasmon modes are not equal to zero, and the explicitanalytical expressions for these corrections are found. Finally, the explicitsecond-order radiation corrections are derived for resonance values of dielec-tric permittivity of plasmon modes in terms of the zero-order electrostaticsolutions for these modes. It is demonstrated that in the particular caseof spherical nanoparticles these second-order radiation corrections coincidewith the radiation corrections obtained from the classical Mie theory.

The chapter also contains the analysis of extinction cross section of metal-lic nanoparticles subject to optical radiation. This analysis is performed byusing the same perturbation technique that has been employed for the cal-culation of radiation corrections. Namely, it is demonstrated that the main(zero-order) term in the power expansion of the solution of the scatteringproblem is electrostatic in nature, and it can be efficiently computed by us-ing inhomogeneous boundary integral equations that have the same kernelsas the integral equations used in Chapter 2 for the analysis of plasmon modes.Then the algorithm for the calculation of first- and second-order terms in βfor scattered electric and magnetic field is presented. Once the scatteredfields are found, the extinction cross section of metallic nanoparticles can becomputed by using the optical theorem. This theorem relates the extinctioncross sections to the far electric field scattered in the forward direction. It isworthwhile to remark that calculations of extinction cross section are help-ful in the analysis of experimental data because the plasmon resonances inmetallic nanoparticles are usually studied through the measurements of thiscross section.

The analysis of extinction cross section is concluded with the discus-sion of nanoparticle-structured plasmon waveguides of light. These wave-guides consist of arrays of metallic nanoparticles with their plasmon reso-nance frequency in the range of optical waveguiding. These nanoparticle-structured waveguides are quite promising for light guiding and bending atthe nanoscale. In the chapter, a technique for calculations of extinction cross


sections as well as guiding (resonance) frequencies of these waveguides ispresented. This technique is based on the boundary integral equation methodand, in this sense, it is the further extension of the techniques already devel-oped in the book.

The next discussion in the chapter deals with temporal analysis of plas-mon resonances in metallic nanoparticles, which is the least studied area ofplasmonics. The purpose of this analysis is to develop techniques for thequantitative characterization of time dynamics of excitation and dephasing(decay) of plasmon resonance modes. The central mathematical elementof this analysis is the biorthogonal expansion of bound electric charges in-duced on nanoparticle boundaries by incident radiation. This expansion isgiven in terms of eigenfunctions of boundary integral equations which aresurface electric charges corresponding to specific plasmon modes. Concep-tually, this analysis is quite similar to the traditional approach to the studyof time dynamics of excitation of resonant cavities. In the above biorthogo-nal expansion, the expansion coefficients depend on time. It turns out thatsimple analytical expressions can be derived for Fourier transforms of thesecoefficients in terms of dispersion relation ε(ω) and the Fourier transformof the electric field of incident radiation. These analytical expressions forthe expansion coefficients reveal that the incident radiation is efficiently cou-pled only to dipole plasmon modes when the incident electric fields have thesame directions as the plasmon mode dipoles. It is also observed that theFourier transforms of the plasmon mode expansion coefficients exhibit reso-nance behavior at plasmon resonance frequencies. This fact is used to derivethe analytical formulas for the steady-state amplitudes of plasmon modesin terms of the real and imaginary parts of dielectric permittivity, ampli-tude of incident field and its spatial orientation with respect to the dipolemoments of the plasmon modes. These formulas reveal that the quality ofplasmon resonances (i.e., the local enhancement of incident electric fields andlight intensity) is fully controlled by the ratio of the real to imaginary partsof dielectric permittivity at resonance frequencies of plasmon modes. Theanalytical expressions for the steady-state amplitudes of plasmon modes foroff-resonance excitations are derived as well. These expressions are presentedin terms of real and imaginary parts of dielectric permittivity evaluated atoff-resonance excitation frequencies. The obtained formulas are instrumentalfor the assessment of the width (sharpness) of plasmon resonances.

The chapter further contains the extensive analytical study of time dy-namics of excitation and dephasing of specific plasmon modes. This study


has been carried out by using the Drude model for the permittivity dispersionrelation. The analytical calculations reveal the nearly linear growth of theamplitude of resonance plasmon modes at the beginning of the excitation pro-cess. This growth eventually saturates due to ohmic losses accounted for bythe imaginary part of dielectric permittivity. This type of excitation dynam-ics is generic for slightly lossy resonance systems. The performed analysis alsosuggests that the reciprocal of the Drude damping factor can be identifiedwith the dephasing time for the light intensity of plasmon modes. In partic-ular, for gold and silver nanoparticles the dephasing time is in the range of5-12 femtoseconds, which is consistent with the available experimental data.Other comparisons of obtained results with the published experimental dataare presented as well.

Finally, the chapter deals with several selective applications of plasmonresonances and some of these applications may be novel. First, plasmon reso-nance enhancement of Faraday rotation in thin garnet films is discussed. It isknown that on the macroscopic level magnetic garnets act as gyrotropic me-dia which discriminate between right-handed and left-handed polarizationsof light. This results in the Faraday rotation. However, on the microscopiclevel, magneto-optic effects are controlled by spin-orbit interaction (coupling)whose Hamiltonian depends on local electric fields. These fields can be opti-cally induced by exciting plasmon resonances in metallic nanoparticles em-bedded in garnets, and these induced fields may eventually lead to the en-hancement of magneto-optic effects. In this way, the plasmon resonances ingarnet-embedded nanoparticles can be utilized for the enhancement of theFaraday effect as well as for the probing of the origin of this effect on thefundamental microscopic (quantum mechanical) level. Some encouraging ex-perimental results in this direction are presented in the chapter. Namely, itis reported that garnet films have been grown by liquid phase epitaxy over(100)-oriented gadolinium gallium substrates populated with gold nanopar-ticles and the noticeable enhancement of Faraday rotation in such films hasbeen observed.

The applications of plasmon resonances to heat-assisted and all-opticalmagnetic recording are next discussed in the chapter. Heat-assisted magneticrecording (HAMR) is currently the focus of considerable research and techno-logical interest. A central issue of high density HAMR is the development ofoptical sources of high intensity and nanometer resolution. The proper pro-filing (shaping) of optical spots is also very important in order to reduce thecollateral heating of adjacent recorded bits. It is demonstrated in the chapter


that plasmon resonances in metallic nanoparticles and perforated nanofilmscan be efficiently used for nano-localization of high-intensity optical radiationand proper profiling of optical spots.

It has been recently demonstrated [27] that magnetization reversal can beconsistently achieved by using only circularly polarized laser pulses, that is,without applying any external magnetic fields. In this all-optical switching,the direction of reversed magnetization is controlled by the helicity of circu-larly polarized light, which acts as an “effective” dc magnetic field alignedwith the light propagation direction. The performed experiments [27] havedemonstrated femtosecond magnetization reversals of 100 µm spots on mag-netic media. This all-optical magnetization switching will be technologicallyfeasible for magnetic recording only if the techniques for delivery of nanoscale-focused circularly polarized light are developed. It is demonstrated in thebook that the focusing of light at the nanoscale and, at the same time, thepreservation of its circular polarization can be simultaneously achieved byexciting circularly polarized plasmon modes in metallic nanoparticles withuniaxial (rotational) symmetry. This rotational symmetry leads to the exis-tence of such circularly polarized plasmon modes and their effective couplingto circularly polarized optical radiation.

Next, the discussion of the electromagnetic mechanism of surface en-hanced Raman scattering (SERS) is presented in the chapter. The essenceof the electromagnetic mechanism of SERS can be briefly outlined as fol-lows. The incoming optical radiation excites a desired plasmon resonancemode in a metallic (usually silver) nanoparticle or a cluster of nanoparticles.This results in strong electric field on the particle boundaries. This strongelectric field causes molecules adsorbed on metal surfaces to radiate at theRaman-shifted frequency. This molecule radiation excites in turn a resonanceplasmon mode in the nanoparticles at the (slightly shifted from resonance)Raman frequency, which may significantly enhance the overall Raman scat-tering. It is apparent from the presented description of the electromagneticmechanism of SERS that the fine-tuning of the following conditions must beperformed to achieve very strong SERS enhancement: a) the incident opticalradiation matches the resonance frequency and the polarization of the desiredplasmon resonance mode, b) adsorbed molecules are in the region where theplasmon mode electric field is the strongest, and c) molecule radiation atthe Raman-shifted frequency closely matches the resonance frequency of theplasmon mode and is efficiently coupled to it. Various ways to achieve thefine-tuning of the above three conditions are discussed in the chapter alongwith supporting computational results.


The chapter is concluded with the discussion of optical controllabilityof plasmon resonances in semiconductor and metallic nanoparticles and theplausible plasmon resonance mechanism for ball lightning formation. Theseapplications of plasmon resonances are speculative in nature at the currentstate of affairs; nevertheless, they are interesting and quite promising. Theoptical controllability of plasmon resonances is attractive because it mayeventually lead to the development of controllable nanoscale light switchesand all-optical nanotransistors. In such devices, one light beam can be usedto generate conduction electrons in semiconductor nanoparticles and, in thisway, to properly manipulate the dispersion relation ε(ω) and drive semicon-ductor nanoparticles into conditions where the desired plasmon mode canbe resonantly excited by another light beam. Another possibility of opticalcontrollability of plasmon resonances in metallic nanoparticles is to controlin time the polarization of incident optical radiation and, consequently, itscoupling to a desired plasmon mode. Finally, it is discussed that the theory ofplasmon resonances and, particularly, their scale invariance, may shed lighton the nucleation and growth of ball lightning, the physical mechanism ofaccumulation of electromagnetic energy in this lightning, as well as providean explanation for its “ball” shape.

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