Epsilons Near Zero limits in the Mie scattering theory

Transcription

1 Epsilos Near Zero limits i the Mie scatterig theory M. Tagviashvili * Adroikashvili Istitute of Physics, 6 Tamarashvili st, 77, Tbilsi, Georgia * madoa.tagviashvili@grt.ge Abstract: The classical Mie theory - electromagetic radiatio scatterig by the homogeeous spherical particles - is cosidered i the epsilo ear zero limits separately for the materials of the particles ad the surroudig medium. The maxima of a scattered trasverse electrical (TE field for the surroudig medium materials with the epsilo ear zero limits are revealed. The effective multipole polarizabilities of the correspodig scatterig particles are ivestigated. The possibility to achieve magetic dipole resoace ad accordigly to costruct metamaterials with egative refractive idex for the aggregates spherical particles i surroudig medium with the epsilo ear zero limits is cosidered.

2 The scatterig, absorptio ad extictio of the plae electromagetic wave by spherical homogeeous particles was iitially developed by Gustav Mie at 95 ad has a great umber of applicatios i may fields of sciece ad egieerig. The Mie theory was itesively elaborated durig the last cetury described more geeralize scatterig process with differet kid of the scatterig particles ad the surroudig medium. I this article, the dielectric properties of the scatterig particles ad surroudig medium bulk materials are cosidered i the epsilo ear zero limitsε. The peculiarities of epsilo ear zero materials [] revealed ew features ad sigularity i Mie theory also. As was show by classical Mie theory, the optical behaviour of the system πδ particles ad its surroudig medium ca be described by ratio x = ad by the λ refractive idices of the particles - surroudig medium = ε = ε (idexed by ad of the (idexed by, where δ is the particles radius, λ -the icidet electromagetic radiatio s wavelegth i vacuum, ε, ad ε, are the dielectric permittivity ad magetic permeability of the materials iside particles ad of the surroudig medium cosequetly. The icidet, scattered ad passed waves are expressed by the expasio i series - modes of the orthogoal spherical fuctios. The Mie coefficiets are the weights of the expasio series. They are derived from cotiuity coditios of electrical ad magetic fields tagetial compoets at the spherical surface of the particles. Particularly, the scattered fields are described by Mie scatterig coefficiets a ad b, where coefficiets b a apply to the trasverse magetic (TM scattered fields ad coefficiets apply to the trasverse electric (TE scattered fields. O the other had, it is well kow [], that scattered electromagetic field of a isolated sphere is equivalet to (that of the coheret esemble of ideal poit multipoles with appropriately chose effective multipole polarizabilities. Each mode of the scattered electromagetic field could be represeted as idepedet collective respose drive by the electric ad magetic fields of the correspodig multipole amplitudes i the orthogoal expasio drivig wave. It ca be show that each of effective multipole polarizabilities are proportioal to the correspodig partial wave amplitudes of the scattered wave expressed via the Mie coefficiets a ad b. I particular the effective electric dipole polarizability of a isolated sphere ca be expressed via the TM field s first-dipole modes Mie scatterig coefficiet a : 6π a e = i 3 a ( k The magetic dipole polarizability of a isolated sphere ca be foud from the TE field s first - dipole modes Mie coefficiet b : e

3 a 6π = i b ( k 3 The scatterig coefficiets a ad b deomiator s zeros correspods to the electric multipole ad magetic multipole resoaces correspodigly. Ordiarily [3], [4], [5], [6] the surroudig medium is cosidered to be passive ad o absorptive. The system is described by the relative refractive idex k = = ad by the ratio of particle s radius to the radiatio s wavelegth λ i k the surroudig medium z πδ = kδ = = x. Accordigly the coefficiets λ ad b are commoly expressed as fuctios of the variables z ad z : b a ψ ( z ( z ψ ( z ( z ψ ( z ζ ( z ζ ( z ( z ( z ( z ψ ( z ( z ( z ζ ( z ζ ( z ( z = (3 ψ = (4 ψ πy Where ψ ( y = J + / ( y - is the modified Bessel fuctio ad πy πy ζ ( y = H + / ( y = ( J + / ( y in + / ( y - is the modified Hekell fuctio. The expasio i series of the expressios (3 ad (4 with small values of z (the case for small particles whe λ >> πδ gives well kow Rayleigh approximatio for the first mode s scatterig coefficiets: 3 ε ε 5 a = i ( z + O( z (5 3 ε + ε ( 3 b ( 5 = i z + O z (6 3 + The higher, -th modes scatterig a ad b coefficiets i the Rayleigh approximatio has the smalless of the order equal to + : z a I the case of omagetic particles = ad surroudig medium with =, the coefficiet ad accordigly the magetic dipole polarizability is times less the coefficiet b a ad accordigly the electric dipole polarizability. No magetic dipole πδ resoace exists for small particles z =. λ z 3

4 The ovelty of this work cosist i cosideratio of the expressio (3 ad πδ (4 as fuctios of the idepedet variables z = = x ad z λ istead of the variables z ad πδ = = x, λ = That meas to ivestigate the depedece of the scattered field o the dielectric ad magetic properties of the surroudig medium ad the particles separately. Multiplyig the (3 ad (4 expressios umerator ad deomiators by the x we receive the ew expressios for the coefficiets a ad b, were each of the spherical fuctios have argumets as oe of the two idepedet variables z [8]: a z ψ ( z ( z zψ ( z ( z ( z ζ ( z zζ ( z ( z ( z ( z zψ ( z ( z ( z ζ ( z z ζ ( z ( z = (7 zψ b z ψ = (8 zψ The expressios (9 ad ( ca be examied separately relatively to the variables z ad. The first terms i z ζ z fuctios expasio series are: z ψ ( ad ( ( z = A z O( z z ad ψ (9 i ζ ( z = B + O ( z z + where coefficiets A ad B (! ca be expressed as: A = ad ( ++! (! B =! If the epsilo ear zero limits characterizes the bulk materials iside the scatterig particles as ad the surroudig medium is with a arbitrary ad idepedet parameter z, the Mie coefficiets a ad b ted to the expasio procedures usig b a z z as the parameter of smalless: ( z O( z ( z zψ ( z ( + ψ ( z z ζ ( z ( + ζ ( z ψ = ( ζ + = O( z ( + All first terms of coefficiets a ad b expasios are periodical fuctios of variable z with the equal magitudes ad without ay poits of sigularity. The Q sca scatterig efficiecy has expressio [6] 4

5 ( / z ( + { a + b } = Q = (3 sca The calculatio of the sum (5 is roughly limited by four modes. It was show that scatterig efficiecy is a smooth fuctio with the maximum about the Q sca poit z = π (Fig.. As Opposite to above, the scatterig coefficiets of the particles aggregate i the surroudig medium with the epsilo ear zero limit has certai sigular poits at certai z medium greatly exceeds particles radius values. I this limit, the icidet radiatio s wavelegth i the z λ >> πδ. If we Cosider the parameter z as a idepedet certai fiite value, the the limit teds to the followig expasios of the Mie coefficiets a ad b : z A ( + ( ( + a = + i z + O z (4 B A ( + ( ( ψ z zψ z + = ( ( ( ( + b i z + O z B ψ z + zψ z (5 Mie coefficiets a ad b of -th modes are times less the ad - the z a b coefficiets of the ( -th modes correspodigly. The dipole, quadruple, octuple, hexadecapole ad higher multipole electric ad magetic polarizabilities are a b proportioal to k, + k accordigly. Each of them has the k order of + smalless. The coefficiets z appear whe parameter is the solutio of equatio (8 b have their sigular poits. The TE field s resoaces ( z z ψ ( = ψ z +. (6 Fig. presets Mie coefficiet a ad b i a logarithmic coordiates versus to particle s parameter z for give small z =.+ i *.. If the particles ad surroudig medium are omagetic =, ad =, the first mode s Mie first mode s coefficiet ca be expressed as : 3 a = i z (7 3 ( 3 z si( z 3z cos( z 3 b = i z (8 3 z si( z Despite of Rayleigh approximatio i case of the epsilo ear zero limit for the surroudig medium, the effective electric ad magetic polarizabilities i the 3 omagetic medium are commoly characterised with the equal- smalless of order. The effective magetic polarizabilities peaks of the maxima appare are at the poits z satisfyig the equatio si ( z =. z 5

6 Fig. 3 demostrates the scatterig efficiecy Qsca as a fuctio of z ad z. The sum is reasoably limited by four modes. The scatterig efficiecy peaks represet πδ the resoaces of the TE scattered field s first modes at z = = πm, λo m =,,3,.... I other words, we have evidece of the existece of the effective magetic dipole resoaces i the medium, where a dipole approximatio λ >> πδ is valid. The Clausius-Mossotti relatios ( ad ( which describe the effective eff eff permittivity ε ad a effective permeability of the media with implemeted particles have also reasoable validities [7]. Nae + 3ε eff ε = ε (9 Nae 3ε eff + a N / 3 = ( a N / 3 The volume desity of the particles i the material is deoted by N = ad 3 d 3 the fillig fuctio is give by f = 4πNδ / 3 = 4π / 3( δ / d 3 3, where d is a volume of the uit cell. Usig the expressios for the effective magetic dipole ad electric dipole polarizabilities ( ad (, the Clausius-Mossotti relatios ca be expressed as: eff + f ε = ε ( f ( 3 z si( z 3z cos( z + f eff z si( z = ( ( 3 z si( z 3z cos( z f z si( z The effective permittivity ad effective permeability simultaeously achieve eff eff the egative values ε ε,, if the surroudig medium s bulk b dielectric permittivity is ε ear zero ad egative ε < ad a radius δ of idetical particles are equal to m λ, where is particles bulk refractive idex, m =,,3,... ad λ is the wavelegth i a vacuum. That meas that it is reasoable to cosider the egative effective refractive idex for the aggregates of idetical spherical particles implated i the surroudig medium with a epsilo ear zero. 6

7 The metals could be regarded as the most preferred alteratives of a epsilo ear zero material. The dielectric property of the metals are described via Drude- Sommerfeld model: ω p ε ( ω = ε (3 ω ω + iγ ( where γ is the electro relaxatio rate, ε -the bad electros cotributio, ω p - bulk plasma frequecy could be accepted from the refereces [8]. For example, the egative epsilo ear zero medium for gold takes place, whe the icidet light wavelegth is about 43 m ( ω p = 9. ev, ε = 9. 84, γ =. 35c. For silver materials, the egative epsilo ear zero ca be achieved, whe the icidet light wavelegth is about 6 m ( ω p = 9. ev, ε = 3. 7, γ =. 5c. Correspodig magetic dipole resoaces occurs for the particles with radius δ = 5m, if the particles are implated i gold ad filled with vacuum =. We have magetic dipole resoaces for the particles filled with o dispersive materials with = implated i the gold whe the radius of the particles is about 7 m. For the particles implated i silver with a epsilo ear zero permittivity magetic dipole resoace occurs for the particles with radiuses 3 m ad 65 m if the particles itself cosist of materials with refractive idexes =ad = cosequetly. Fig. 4 represets the effective refractive idex profile for silver medium with implated spherical idetical particles as a fuctio of the particles radius δ. The bulk refractive idex of particles is =, the icidet lights wavelegth is equal λ = 6 m ad the system is characterized with fillig fuctio f =.. Fig. 5 represets the effective egative idex areas of the wavelegth i vacuum λ ad of the particle radius δ for the silver surroudig medium ad implated idetical dielectric spherical particles ( =, the Fig 5.a ad ( =, Fig 5b. Fig. 6 represets the effective egative idex areas of the wavelegth i vacuum λ ad the particle radius δ for gold surroudig medium ad implated aggregate of particles with bulk refractive idex = (the Fig 6.a ad = ( the Fig 6.b. The ivestigatio of the Mie scatterig for homogeeous (ot layered spherical ao-sized particles i the surroudig medium with epsilo ear zero limits reveal the magetic multipole resoaces for s i the visible rage of icidet radiatio. Similar to the other Mie resoaces this pheomea ca have applicatios i differet fields of physics. Particularly, the ivestigatio demostrates possibility to achieve egative refractive idex metamaterials costructed of the aggregates of spherical idetical dielectric particles implated i the metal surroudig medium i the visible rage of the radiatio. The author is immesely grateful to Prof. V. Berezhiai for his commets ad discussio. The work was supported by ISTC uder Grat No.G366. 7

8 . Adrea Al`u, Nader Egheta, Richard W. Ziolkowski, Sigle-Negative, Double-Negative, ad Low-idex Metamaterials ad their Electromagetic Applicatios, IEEE Trasactios o Ateas ad Propagatio, Magazie, Volume 49, Issue, February 7, pages W. T. Doyle, Optical properties of a suspesio of metal spheres, Phys. Rev. B 39, 985 ( A. Stratto, Electromagetic Theory, New York: McGraw-Hill, H. C. va de Hulst, Light scatterig by small particles, New York, Dover, M. Kerker, The scatterig of light ad other electromagetic radiatio. New York, Academic, C. F. Bohre, D. R. Huffma, Absorptio ad scatterig of light by small particles. New York, Wiley-Itersciece, Mark S. Wheeler, J. Stewart Aitchiso, ad Mohammad Mojahedi, Threedimesioal array of dielectric spheres with a isotropic egative permeability at ifrared frequecie,s,phys. Rev.B 7, 933 (5. 8. "Optical properties of pure metals ad biary alloys." I K. Hellwege adj. Olse, editors, Metals: Electroic Trasport Pheomea, volume 5b of Ladolt-B orstei.spriger, Berli (985. 8

9 Fig. The scatterig efficiecy Q sca limit z =.+ i *. versus parameter z. for the particles with epsilo ear zero 9

10 Fig. Mie coefficiets a ad b for the surroudig medium with epsilo ear zero limit z =.+ i *.as fuctio of the parameter z.

11 Fig.3. The scatterig efficiecy Q ( z z sca,.

12 Fig.4. The effective refractive idex of silver medium with implated spherical idetical particles versus the particles radiusδ. The particles refractive idex is equal to =, the icidet lights wavelegth - λ = 6 m ad system s fillig fuctio - f =..

13 Fig. 5.The effective egative idex areas of the wavelegth i vacuum λ ad of the particle radius δ for the silver surroudig medium ad implated idetical dielectric spherical particles. The particles refractive idices are (a. = ad (b. =. 3

14 Fig. 6. The effective egative idex areas of the wavelegth i vacuum λ ad of the particle radius δ for the gold surroudig medium ad implated idetical dielectric spherical particles (a. = ad (b. =. 4