Classification of electromagnetic resonances in finite inhomogeneous three-dimensional stru

a r X i v :p h y s i c s /0509259v 1 [p h y s i c s .c l a s s -p h ] 30 S e p 2005

Classi?cation of electromagnetic resonances in ?nite

inhomogeneous three-dimensional structures

Neil V.Budko

Laboratory of Electromagnetic Research,Faculty of Electrical Engineering,Mathematics and Computer Science,

Delft University of Technology,Mekelweg 4,2628CD Delft,The Netherlands ?

Alexander B.Samokhin

Department of Applied Mathematics,Moscow Institute of Radio Engineering,Electronics,and Automatics (MIREA),Verndasky av.78,117454,Moscow,Russian Federation ?

(Dated:February 2,2008)We present a simple and uni?ed classi?cation of macroscopic electromagnetic resonances in ?nite arbitrarily inhomogeneous isotropic dielectric 3D structures situated in free space.By observing the complex-plane dynamics of the spatial spectrum of the volume integral operator as a function of angular frequency and constitutive parameters we identify and generalize all the usual resonances,including complex plasmons,real laser resonances in media with gain,and real quasi-static reso-nances in media with negative permittivity and gain.

It is hard to overestimate the role played by macro-scopic electromagnetic resonances in physics.Phenom-ena and technologies such as lasers,photonic band-gap materials,plasma waves and instabilities,microwave de-vices,and a great deal of electronics are all related or even entirely based on some kind of electromagnetic res-onance.The usual way of analysis consists of deriving the so-called dispersion equation,which relates the wave-vector k or the propagation constant |k |of a plane elec-tromagnetic wave to the angular frequency ω.The solu-tions of this equation may be real or complex.In the ?rst case we talk about a real resonance ,i.e.such that can be attained for some real angular frequency and there-fore,in principle,results in unbounded ?elds.In reality,however,ampli?cation of the ?elds is bounded by other physical mechanisms,e.g.nonlinear saturation.If solu-tion is complex,then we have a complex resonance and,depending on the sign of the imaginary part,the asso-ciated ?elds are either decaying or growing with time.This common approach is rather limited and does not include all pertaining phenomena.Indeed,more or less explicit dispersion equations can only be obtained for in-?nite (unbounded)homogeneous media,as often done in plasma and photonic studies.Other approaches im-pose explicit boundary conditions and can handle res-onators and waveguides with perfectly conducting walls,and idealistic piece-wise homogeneous objects (e.g.plane layered medium,circular cylinders,a sphere).On the other hand,very little can be said in the general case of a ?nite inhomogeneous dielectric object situated in free space.Due to the absence of an explicit dispersion equa-tion and explicit boundary conditions,even the existence and classi?cation of resonances in such objects is still an open problem.

We describe here an alternative mathematically rig-orous approach to electromagnetic resonances,based on the volume integral formulation of the electromagnetic scattering,also known as the Green’s function method

and the domain integral equation method.This formu-lation is equivalent to the Maxwell’s equations and is perfectly suited for bounded inhomogeneous objects in free space.Despite its generality,nowadays the volume integral equation is mostly used as a numerical tool,for instance,in near-?eld optics and geophysics.The main limitation seems to be the implicit mathematical struc-ture of this equation resisting any straightforward anal-ysis and interpretation.Recently,however,we have suc-ceeded in deriving useful mathematical bounds on the spatial spectrum of the volume integral operator proving,in particular,that along with the usual discrete eigenval-ues this operator has a dense essential spectrum as well [1].Below we reiterate our results and show how to use them in the analysis of resonances.Then,we proceed with a step by step classi?cation of all known complex and real resonances.In particular,we generalize the no-tion of a complex plasmon,real laser resonance,and a real quasi-static resonance in an exotic material contain-ing a negative permittivity part and a part with gain.Recently,several authors have suggested [2]–[6]that this type of material may be an answer to some urgent tech-nological questions ranging from surface plasmon lasers (SPASER)to loss compensation in media with negative refraction (perfect lens).We believe that our analysis provides a necessary generalization and a handy analyt-ical tool for these and other studies,especially in what concerns the resonant light con?nement.

The frequency-domain Maxwell’s equations describing the electromagnetic ?eld in a non-magnetic isotropic in-homogeneous object occupying ?nite spatial domain D lead to the following strongly singular integral equation:

E in

(x ,ω)=

I +

1

2

where I denotes a unit tensor(3×3identity matrix),

whereas the explicit form of the Green’s tensor G is of no importance here,but can be found in[7]and[8].Here,

E in is the incident?eld in vacuum background,where the wavenumber is k0=ω/c,and the total electric?eld in

the con?guration is denoted by E.Constitutive parame-

ters of the object are contained in the so-called contrast functionχ(x,ω)=εr(x,ω)?1,whereεr is the relative dielectric permittivity of the object.In operator notation equation(1)can be written simply as

Au=u in.(2) The spatial spectrum of operator A is de?ned as a set σ(λ)of complex numbersλfor which operator

[A?λI]?1(3) fails to exist in one or another way.We need to distin-guish here two cases.The?rst is when for someλthe homogeneous equation[A?λI]uλ=0has a nontrivial solution uλ=0.In addition,this solution has a?nite norm,i.e. uλ <∞.If the latter condition is satis?ed, thenλis called an eigenvalue and the corresponding uλ–an eigenfunction(eigenmode).It happens that eigen-values constitute,although possibly in?nite,but discrete subset of the complex plane–a set of isolated points,in other words.

The second case is when equation[A?λI]uλ=0is formally satis?ed by some uλ,which either does not have a bounded norm,i.e. uλ →∞,or is localized to a sin-gle point in space.The set ofλ’s corresponding to such cases is often a dense subset of the complex plane,some-times referred to as essential spectrum.An even more rigorous analysis would also require distinction between the continuous and the residual spectra,however,so far we cannot come-up with a simple formal rule to identify and separate them in the electromagnetic case.It is quite easy to?nd the physical interpretation of uλ →∞.For example,in the L2norm suggested by the electromag-netic energy considerations(Pointing’s theorem),such functions are a plane wave and the Dirac’s delta func-tion,which both have in?nite L2norms.The essential spectrum associated with plane waves is common for in-?nite periodic structures,where it surrounds photonic band gaps,and in in?nite plasma models,where it gives rise to certain types of plasma waves.

In[1]we prove that the strongly singular integral oper-ator of equation(1)has both the dense essential spectrum and the discrete eigenvalues.Moreover,for any inhomo-geneous object withχ(x,ω)H¨o lder-continuous in R3(i.e. inside the object as well as across its outer boundary)the essential spectrumλess is given explicitly as

λess=εr(x,ω),x∈R3.(4) In other wordsλess will consist of all values ofεr,which it admits in R3.Thus it will always contain the real unit,since it is the relative permittivity of vacuum,and a curve or even an area of the complex plane emerging from the real unit and running through all other values, which macroscopicεr takes inside the object.This part of the spectrum does not depend on the object’s size or shape,or even the relative volume occupied by di?erent inhomogeneities.

In addition to the essential spectrum operator A has the usual discrete eigenvalues located within the follow-ing wedge-shaped bounds:

Imεr(x,ω)[1?Reλ]+

[Reεr(x,ω)?1]Imλ≤0,x∈D.

(5)

It is also known that|λ|≤ A ,and that A <∞for anyχ,H¨o lder-continuous in R3.Exact distribution of eigenvalues in the complex plane is unknown to us and depends on the object’s shape.The eigenfunctions (modes)associated with these eigenvalues are global(not localized)and,in general,can only be found numerically.

To use these results in the analysis of electromagnetic resonances we note that both the essential spectrum and the eigenvalues are parametric functions of the angular frequencyω.In general,a perfect(real)resonance would occur,if for someωthe spatial spectrum of A would ac-quire a zero eigenvalue.If,on the other hand,for someωthe spatial spectrum does not contain zero,but gets close to it,while moves away for otherω’s,then we have a com-plex resonance.With this in mind,one should try to vi-sualize the dynamics of the spatial spectrum as it‘moves’in the complex plane,paying attention to the eigenvalues and portions of essential spectrum,which?rst approach zero and then move away from it.Expression(4)is very important in this respect as it tells us that the motion of essential spectrum is explicitly related to the temporal dispersion of the relative permittivity.We also know(see below)that the eigenfunctions related to this spectrum are highly localized.Thus from(4)and the known spa-tial distribution ofεr(x,ω)we can immediately tell where exactly in D would a local resonance occur.The motion of discrete eigenvalues,on the other hand,is quite un-predictable,with the general tendency to spread out at higher frequencies.While doing so,some of these eigen-values may pass through or close to zero,which will be an indication of a global resonance.We propose here a useful rule of thumb for visualizing the eigenvalue bound (5).Imagine a line drawn through the real unit and any value ofεr inside the object.If you now stand in the com-plex plane and look from the real unit towards that value ofεr,then the eigenvalues can only be to your right. Finally,we have also been able to prove that in the static limitω→0or D→0all discrete eigenvalues are located within the convex envelope of essential spectrum

x ∈D

|??λ(x )|2d x

,(6)

where ?λis a scalar static mode.Formally,our essential spectrum (4)can be derived from this expression as well,by taking |??λ(x )|2～δ(x ?x ′).This also proves that the eigenfunctions associated with the essential spectrum are highly localized in space.Another important observa-tion is about the discrete eigenvalues outside the convex envelope of essential spectrum.Since those do not ex-ist in the quasi-static regime and appear only at higher frequencies and object sizes,we may conclude that the corresponding eigenfunctions are not of static type,but more of the wave-like type,i.e.oscillating in space.

Now,we have everything we need for a uni?ed descrip-tion of resonances.We shall illustrate our conclusions by numerically computed spectra for an inhomogeneous cube consisting of two equal halves with di?erent permit-tivity values.The side of the cube is half of the vacuum wavelength.

In objects consisting of lossy dielectric materials only complex resonances can be observed.For example,in Fig.1(left)we show the spectrum for the case of a lossy dielectric with both Re εr >0and Im εr >0.The ac-tual values of relative permittivity and the real unit are given as circles.Numerical equivalent of essential spec-trum (there is no such thing as dense or continuous spec-trum with matrices)always looks like a set of line seg-ments emerging from the real unit [1].One should simply keep in mind that in a continuously inhomogeneous ob-ject this spectrum may be a rather arbitrary curve or an area.Other,o?-line eigenvalues are within the bounds prescribed by (5).As the angular frequency varies,some of these latter o?-line eigenvalues may get close,but not equal to zero.These are the complex resonances,corre-sponding to complicated global wave-like spatial modes.

In Fig.1(middle)we illustrate the case where due to strong anomalous dispersion one of the object’s parts has Re εr <0and Im εr >0at a certain angular frequency.The line of essential spectrum proceeds close to the zero of the complex plane.For other angular frequencies this line will move away from zero.It is well-known that this combination of materials supports complex plasmon reso-nances.Hence,we may safely conclude that we deal here with one of them.As an extra con?rmation we see that this resonance is related to the highly localized modes of essential spectrum.Further we conclude that,in gen-eral,complex plasmons may exist not only at an inter-face between two homogeneous objects,but along rather arbitrary surfaces inside a continuously inhomogeneous object with strong anomalous dispersion.The precise lo-cation of this surface is determined by that value of εr inside D ,which appears to be the closest to zero.Recalling the rule of thumb about the location of eigen-values we realize that a discrete eigenvalue can be equal to zero,only if the relative permittivity at some point inside the object happens to have a negative imaginary part,i.e.Im εr <0.This corresponds to the so-called negative losses or gain as in pumped laser media.In Fig.1(right)the numerical spectrum for a cube with one lossy half and another half with gain is shown.Two of the discrete eigenvalues are very close to zero,mean-ing that the whole con?guration is in the vicinity of a real laser resonance.It is,however,very hard to come-up with an exact real resonance in this way.For a given temporal dispersion of the medium,one has to optimize the geometrical parameters of the object until the reso-nance is achieved,which is a very challenging numerical problem.One thing we can be sure about,though:for such con?gurations the zero eigenvalue will always be outside the convex envelope of the essential spectrum.Therefore,real laser resonances correspond to wave-like spatial modes and,thus,can only be achieved in struc-tures whose size is comparable to or greater than the

FIG.2:Real quasi-static resonance in an object with negative permittivity and gain.

medium wavelength.This is con?rmed by the standard theory of lasers.

As we already mentioned in the beginning,combina-tion of a negative permittivity material and a material with gain is an attractive candidate for several applica-tions.In the quest for a perfect lens [6]the gain is sup-posed to compensate for the inevitable losses in the fre-quency band where the negative permittivity is achieved.Plasmons,which are considered to be ideal candidates for the sub-wavelength manipulation of light,su?er from losses as well.Here too,combination with a gain medium is supposed to compensate for the losses.Some authors argue that in this way the surface plasmon ampli?cation by stimulated emission of radiation (SPASER)can be achieved,similar to the usual laser [2]–[5].While all this is true,and our bounds show that real resonances may ex-ist in such media,we can explicitly show that these reso-nances are not necessarily the localized lossless plasmons,but may as well be associated with global modes.Con-sider the spatial spectrum corresponding to this case –see Fig.2.The upper branch of the essential spectrum is indeed approaching zero as with the usual complex plas-mon.In a continuously inhomogeneous object there may be essential spectrum going right through zero in this case.Hence,perfect real plasmons are possible in classi-cal electromagnetics (at least mathematically).However,in Fig.2it is the discrete eigenvalue,which is now the closest to zero,and it has a global eigenfunction associ-ated with it,not a localized one.Our numerical calcu-lations con?rm that the complex plasmon mode and the

mode of this real resonance indeed look di?erent.Note also that the angular frequency of this resonance may in practice coincide with the one of plasmon.There is an important di?erence,though,between the real laser resonances described above and the present resonance.If the medium parameters are such that the zero of the complex plane is situated inside the convex envelope of the essential spectrum,then a real quasi-static resonance can be achieved.Hence,the mode may be con?ned to a very small volume,if the object’s volume is small.It may even be enough to reduce the volume of the part with gain only to achieve con?nement.

In summary,we have presented a uni?ed approach to macroscopic electromagnetic resonances in ?nite inhomo-geneous three-dimensional objects.We have analyzed the dynamics of the spatial spectrum of the pertaining vol-ume integral operator as a function of the angular fre-quency and constitutive parameters,and were able to recover and generalize all known resonances in this way.In addition,we have con?rmed the possibility and estab-lished conditions for the existence of a real quasi-static resonance in media with negative permittivity and gain leading to the volume-dependent light con?nement.

?Electronic address:n.budko@ewi.tudelft.nl

?

This research is supported by NWO (The Netherlands)and RFBR (Russian Federation).

[1]N.V.Budko and A.B.Samokhin,(under review)SIAM https://www.sodocs.net/doc/7b19105273.html,p.,also see Arxiv version:math-ph/0505013(2005).

[2]M.I.Stockman,S.V.Faleev,and D.J.Bergman,Phys.Rev.Lett.87,167401(2001).

[3]D.J.Bergman and M.I.Stockman,Phys.Rev.Lett.90,027402(2003).

[4]M.P.Nezhad,K.Tetz,Y.Fainman,Optics Expres,.12,4072–4079(2004)[5]J.Seidel,S.Grafst¨o m,L.Eng,Phys.Rev.Lett.94,177401(2005).

[6]S.A.Ramakrishna and J.B.Pendry,Phys.Rev.B 67,201101(R)(2003).

[7]J.Rahola,SIAM https://www.sodocs.net/doc/7b19105273.html,p.,Vol.21,No.5,pp.1740–1754(2000).

[8]A.B.Samokhin,Integral Equations and Iteration Methods in Electromagnetic Scattering ,VSP,Utrecht,2001.

[9]

N.V.Budko and A.B.Samokhin,(accepted)Di?erential Equations ,(2005).