PHYSICAL REVIEW A 89, 033805 (2014)

Fundamental correction of Mie’s scattering theory for the analysisof the plasmonic resonance of a metal nanosphere

Masafumi Fujii*

Graduate Research Division of Science and Engineering, University of Toyama, Toyama 930-8555, Japan(Received 9 December 2013; revised manuscript received 6 February 2014; published 6 March 2014)

It is shown that Mie’s solution to Maxwell’s equations no longer holds for the analysis of resonance of aplasmonic metal nanosphere. The conventional Mie’s solution is based on the spherical Bessel and the sphericalHankel functions of an outgoing wave, whereas the permittivity of metals of a negative real part leads to aphase velocity that directs inward to the sphere, which is opposite from the direction of the energy flow asoften discussed for negative-index metamaterials. This is a fundamental problem overlooked for a long time; acorrection can be found from the viewpoint of a time-reversal problem involving negative permittivity media.The continuity of the field solution at the sphere surface is shown to be corrected by replacing the sphericalHankel function of an outgoing wave with that of an incoming wave, i.e., by adopting the complex conjugate ofthe conventional solutions. The corrected theory has been verified by the analyses of various metal nanospheres.In addition, the derivation of the scattering cross sections based on the corrected theory has elucidated that theconservation law of energy holds and that, more importantly, the conventional Mie’s solution gives the sameamplitude of the cross sections when they are obtained for real, not complex, frequency.

DOI: 10.1103/PhysRevA.89.033805 PACS number(s): 42.25.Fx, 42.68.Mj, 78.67.Bf

I. INTRODUCTION

Maxwell’s equations were solved analytically in the spher-ical coordinate system more than 100 years ago [1], in whichan electromagnetic field on relatively large metal spheres ofapproximately 100 μm in radius is analyzed. This theoreticalframework, called Mie’s scattering theory, has been appliedto various problems of metal spheres and dielectric spheres,and is described in a number of textbooks [2–6]. Recently,plasmonic metal nanoparticles have attracted a lot of attentionsfor the application in nanophysics such as surface enhancedRaman scattering [7], optical tweezers [8,9], and opticalmanipulation of nanoparticles [10–12]. Although experimentalinvestigations have been performed enthusiastically, theoreti-cal approaches would still enhance the technological impact,and Mie’s theory would have played such an important role.

However, it has been pointed out that the conventionalMie’s theory gives erroneous results when it is appliedto a metal nanosphere in the optic regime, where typicalmetals have a wavelength-dependent complex permittivityof negative real part [12,13]. It is well known that theelectromagnetic resonance of a sphere is analyzed by solvinga transcendental equation of dispersion based on Mie’s theory.For metal spheres, however, the conventional Mie’s theorygives physically unreasonable results of a positive imaginarypart in the complex resonance frequency, i.e., resulting ina field solution that grows with time. This fact has beenoverlooked for a long time, while the plasmonic nanopar-ticles have been investigated extensively. The error occursbecause the negative real part of the permittivity leads to aphase velocity opposite from the group velocity inside thesphere, which is particularly prominent for metal nanosphereswhere the field penetrates deep into the sphere. In other words,for a larger metal sphere, the field penetrates only into theshallow region of the sphere, which gives only marginal effects

*mfujii@eng.u-toyama.ac.jp

to the field outside the sphere, thus the conventional Mie’stheory has appeared to hold true.

In order to correct the mismatch of the fields inside andoutside of the metal sphere, the function of the external fieldpropagating outward from the sphere should be replaced bythat of a field propagating inward to the sphere. In the conven-tional Mie’s theory, the field outside the sphere is representedby the spherical Hankel function of the first kind when thetime variation is assumed e−iωt . This should be replaced bythe spherical Hankel function of the second kind, which isthe complex conjugate of the first kind function [12,13]. Itshould be noted that the time variation of eiωt is adopted inRefs. [1,3] and e−iωt is adopted in Refs. [2,4–6], for which thedirections of wave propagation are defined oppositely by thesame spherical Hankel functions.

In Refs. [12,13], however, the theoretical derivation ofthe correction has not been given rigorously, or the physicalinterpretation has not been discussed sufficiently. Hence,in this paper, a thorough theoretical proof is given to theabove-mentioned correction by extending the formula, withthe same notation as in Ref. [2], to the general cases ofoutgoing and incoming fields determined by the sphericalHankel functions of the first and the second kinds, respectively.This verifies that the correction is necessary and sufficient.The physical interpretation is then discussed by considering atime-reversal problem, where the backward (namely, inward)propagating wave in the external space is compensated byintroducing a hypothetical negative time −t ; in addition, thesolution to the transcendental equation is shown to be anegative frequency −ω. Consequently, the time dependencye−i(−ω)(−t) = e−iωt is invariant for the external field, which ismatched with the internal field of the metal sphere having anegative permittivity. This may appear confusing, but is shownto be correct theoretically as often discussed for the negativeindex metamaterials [14,15].

In addition, the energy conservation and the scattering crosssections of the metal nanosphere are derived. As discussed inthe last section, the phenomena itself is the reflection of an

1050-2947/2014/89(3)/033805(10) 033805-1 ©2014 American Physical Society

MASAFUMI FUJII PHYSICAL REVIEW A 89, 033805 (2014)

electromagnetic wave on a spherical surface, in which systemthe energy conservation law is satisfied. More importantly,it is shown that the conventional Mie’s theory gives the sameresults as the corrected theory when it is applied to the analysisof scattering, absorption, and extinction cross sections for real,not complex, frequency.

II. MIE’S SOLUTION TO MAXWELL’S EQUATIONS FORA METAL NANOSPHERE

A. Spherical waves

We consider a general wave equation in terms of a vectorfield C as in Ref. [2],

∇2C − με∂2C∂t2

− μσ∂C∂t

= 0, (1)

where μ, ε and σ are permeability, permittivity, and conduc-tivity, respectively, of the medium to be considered. When Ccan be decomposed into independent field components, (1) isreduced to a scalar wave equation in terms of a scalar field ψ ,

∇2ψ + k2ψ = 0. (2)

A general solution to (2) has a form,

ψ = ξ (R,θ,φ)e−iωt , (3)

where the time dependency e−iωt is assumed in this paper asin Refs. [2,4–6], with ω = 2πν the angular frequency and ν

the frequency; arguments R, θ , and φ are radius, azimuthangle, and polar angle in the spherical coordinate system,respectively.

Scalar Eq. (2) can be reduced to an equation in terms of thespatial function ξ as

1

R2

∂

∂R

(R2 ∂ξ

∂R

)+ 1

R2 sin θ

∂

∂θ

(sin θ

∂ξ

∂θ

)

+ 1

R2 sin2 θ

∂2ξ

∂φ2+ k2ξ = 0,

(4)

and the solution to (4) is given by

ξ (R,θ,φ) =∞∑

n=0

zn(kR)

[an0P

0n (cos θ )

+n∑

m=1

(anm cos mφ + bnm sin mφ)P mn (cos θ )

],

(5)

where P mn (cos θ ) is the associated Legendre polynomial of

degree n and order m, zn(ρ) = jn(ρ) is the spherical Besselfunction chosen for inside sphere, and zn(ρ) = h( )

n (ρ) for =1,2 is the spherical Hankel function of the th kind chosen forthe outside sphere; for = 1, the wave propagates outwardfrom the sphere, and for = 2, the wave propagates inward tothe sphere; anm and bnm are constants.

B. Electromagnetic resonance of a metal sphere in TM mode

We consider in this paper the transverse magnetic (TM)mode field, which is the dominant mode having an electric type

oscillation, of a metal sphere of radius a. The field componentsof the m,nth TM mode of (5) in this case are written as [2]

Ei,eR (R,θ,φ,t) = −n(n + 1) Y i,e

mn

zn(kR)

kRe−iωt , (6a)

Ei,eθ (R,θ,φ,t) = −∂Y i,e

mn

∂θ

1

kR[kR zn(kR)]′e−iωt , (6b)

Ei,eφ (R,θ,φ,t) = − 1

sin θ

∂Y i,emn

∂φ

1

kR[kR zn(kR)]′e−iωt , (6c)

Hi,eR (R,θ,φ,t) = 0, (6d)

Hi,eθ (R,θ,φ,t) = − k

iωμ

1

sin θ

∂Y i,emn

∂φzn(kR)e−iωt , (6e)

Hi,eφ (R,θ,φ,t) = k

iωμ1

∂Y i,emn

∂θzn(kR)e−iωt . (6f)

For the internal region of the sphere for R < a (withsuperscript “i”) in the polar coordinates, zn = jn is chosen,and for the external region of the sphere for R � a (withsuperscript “e”), zn = h( )

n is chosen; wave number is k = k1 =ω

√ε1μ1 for the internal region, and k = k2 = ω

√ε2μ2 for the

external region; the tesseral harmonics composed of even (withsubscript “e”) and odd (with subscript “o”) terms are given by

Y i,emn = (

Ai,eemn cos mφ + Ai,e

omn sin mφ)P m

n (cos θ ), (7)

where Ai,eemn and Ai,e

omn are arbitrary coefficients. Matching thetangential electric and magnetic fields inside and outside thesphere is imposed to derive a transcendental equation thatdetermines the resonance frequency of the sphere.

For the TM mode dispersion of a dielectric sphere of radiusa surrounded by air, the transcendental equation is given by[2, p. 557]

[Nρ jn(Nρ)]′

N2jn(Nρ)= μ0

μ1

[ρ h( )

n (ρ)]′

h( )n (ρ)

, (8)

where the argument ρ to the spherical Bessel function of ordern, or jn(Nρ), as well as to the spherical Hankel function of the th kind ( = 1,2) of order n, or h( )

n (ρ), is defined by

ρ = k a = √ε0 μ0 ω a, (9)

with ε0 and μ0 the permittivity and permeability of vacuum,respectively. Factor N is the ratio of the refractive index of themetal to that of the surrounding medium, which relates in thispaper to the frequency-dependent relative permittivity of themetal εr (ω) by

N =√

εr (ω), (10)

and the relative permeability of the metal sphere μ1 is assumedto be unity.

The prime in Eq. (8) represents the differentiation in termsof the argument. The solution ρ to Eq. (8), and consequentlyω, are known to be a complex number, i.e., ρ = ρ

′ + iρ′′,

ω = ω′ + iω

′′, where

′and

′′denote the real and the imaginary

parts, respectively, and i is the imaginary unit.The complex permittivity of metal is then considered by

adopting the fitting models of Lorentz-Drude (LD) type [16]both for the analytical solution of the transcendental equationand for the numerical solution of the finite-difference time-domain (FDTD) analysis [12]. Since we consider metal

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FUNDAMENTAL CORRECTION OF MIE’s SCATTERING . . . PHYSICAL REVIEW A 89, 033805 (2014)

0.01

0.1

1

10

100

1000

0 200 400 600 800 1000 1200

2000 1000 500 400 300 250

Com

plex

rel

ativ

e pe

rmitt

ivity

Frequency [THz]

Wavelength [nm]

Ag -εr’εr’εr’’

0 1 2 3 4 5h-ω [eV]

FIG. 1. Multipole Lorentz-Drude model of complex relativepermittivity for Ag [16].

spheres of radius larger than 10 nm, no quantum size effect istaken into account [5, p. 83]. The LD model of the complexrelative permittivity of metals for e−iωt time dependency isgiven by

εr (ω) = 1 − �2p

ω2 + iω�0+

K∑j=1

fj ω2p(

ω2j − ω2

) − iω�j

, (11)

where �p = √f0 ωp, K = 5, the plasma frequency ωp,

Lorentz pole frequencies ωj for j = 1,2, · · · ,K , the strengthfactor of the polarization fj , and damping factors �j forj = 0,1,2, · · · ,K are found in Ref. [16].

For readers’ convenience, the complex relative permittivityεr = ε

′r + iε

′′r , ε

′′r > 0 is shown in Figs. 1–4 for Ag, Au, Cu,

and Al, respectively, as a function of frequency. The accuracyof the LD model is satisfactory up to 1200 THz (5 eV) for Ag,up to 1400 THz (6 eV) for Au and Cu, and up to 2400 THz(10 eV) for Al, although the discrepancies between the LDmodel and the experimental data become larger beyond thesepoints [16–18]. In Fig. 5 the complex relative permittivity isshown for Ag, of which data is found in Ref. [19] and fitted bythe standard least squares method to the Drude and single-pole

0.01

0.1

1

10

100

1000

0 200 400 600 800 1000 1200

2000 1000 500 400 300 250

Com

plex

rel

ativ

e pe

rmitt

ivity

Frequency [THz]

Wavelength [nm]

Au -εr’εr’εr’’

0 1 2 3 4 5h-ω [eV]

FIG. 2. Multipole Lorentz-Drude model of complex relativepermittivity for Au [16].

0.01

0.1

1

10

100

1000

0 200 400 600 800 1000 1200

2000 1000 500 400 300 250

Com

plex

rel

ativ

e pe

rmitt

ivity

Frequency [THz]

Wavelength [nm]

Cu -εr’εr’’

0 1 2 3 4 5h-ω [eV]

FIG. 3. Multipole Lorentz-Drude model of complex relativepermittivity for Cu [16].

Lorentz model in this paper; these experimental data are oftenused for the analysis of noble metals, and would be worthcomparing to those in Ref. [16]; in general, the permittivityin Ref. [19] have lower loss than that in Ref. [16] due to thedifference in the measurement techniques and the preparationof the specimens. We have obtained the fitting parametersof (11) for the Drude and single-pole Lorentz model (K = 1) asshown in Table I, which is effective for a range ω = 0.1–3.7 eV(20–900 THz) as shown in Fig. 5.

C. Solution of transcendental equation of dispersion relation

The nonlinear Eq. (8) is solved by using a Matlab nonlinearsolver with including the LD model permittivity of Eq. (11)for n = 1,2,3, and 4.

Due to the complicated behavior of the metal permittivity,the resonance conditions are met for a certain frequency rangewhen the real part of the permittivity approaches −2 as writtenin the literature [4, p. 327]. This is clearly seen for the Agsphere; the real part of the permittivity approaches −2 at twofrequency points around 850 THz and 1200 THz as in Fig. 1,and two sets of solutions, low frequency (LF) mode and high

0.01

0.1

1

10

100

1000

0 200 400 600 800 1000 1200

2000 1000 500 400 300 250

Com

plex

rel

ativ

e pe

rmitt

ivity

Frequency [THz]

Wavelength [nm]

Al -εr’εr’’

0 1 2 3 4 5h-ω [eV]

FIG. 4. Multipole Lorentz-Drude model of complex relativepermittivity for Al [16].

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0.01

0.1

1

10

100

1000

0 200 400 600 800 1000 1200

2000 1000 500 400 300 250

Com

plex

rel

ativ

e pe

rmitt

ivity

Frequency [THz]

Wavelength [nm]

Ag -εr’ (JC11 fit)εr’’(JC11 fit)

-εr’ (JC11 data)εr’’(JC11 data)

0 1 2 3 4 5h-ω [eV]

FIG. 5. Single-pole Lorentz-Drude model of complex relativepermittivity for Ag, of which data are reported in Ref. [19]. Thinlines are those of Ref. [16] in Fig. 1 for comparison.

frequency (HF) mode, have been obtained as shown in thefollowing.

In the literature [2,4], it is stated for dielectric spheresthat the spherical Hankel function of the first kind representsthe spherical wave propagating outward from the sphere,and the spherical Hankel function of the second kind, whichis the complex conjugate of the first kind function, representsthe spherical wave propagating inward to the sphere. Orig-inally, Eq. (8) is solved for = 1 to obtain the dispersionrelation on a dielectric sphere. In this paper, however, Eq. (8)has been solved for Ag, Au, Cu, and Al spheres with both = 1 and 2, and the solutions are shown in Figs. 6–18 forcomparison. It is found in those results for the metal spheres inthe optic regime, where the real part of the permittivity of metalis negative, if (8) is solved with = 1, the imaginary parts ofρ and ω take positive values for the sphere radius a � 70 nm,while they take negative values for a � 70 nm. This behaviorfor = 1 is physically unreasonable; the positive imaginarypart of the resonance frequency ω results in the optical fieldsthat grow with time. To avoid such unphysical solutions, itis also found that Eq. (8) should be solved by adopting thespherical Hankel function of the second kind, i.e., = 2.

It can be shown that the difference in the field distribution ofthe Hankel functions of the first kind and that of the second kinddoes correspond to the difference in the field distribution of thesurface wave on a planar dielectric surface (represented by a

TABLE I. Lorentz-Drude parameters of (11) for Ag data byJohnson and Christy [19]. Effective for ω = 0.1–3.7 eV (20–900 THz).

Unit Value for Ag

ωp (eV) 9.01f0 (1) 1.0370�0 (eV) 0.01649f1 (1) 1.1348�1 (eV) 0.4272ω1 (eV) 5.980

0

200

400

600

800

1000

1200

0 100 200 300 400

1

2

3

4

Re

[Fre

q.]

[TH

z]

Sphere radius [nm]

Ag

n=1,LFn=1,HF

FDTD,LF

-250

-200

-150

-100

-50

0

50

0 100 200 300 400

-1

0

Im [F

req.

] [T

Hz]

Sphere radius [nm]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 6. Complex resonance frequency of the Ag sphere for n = 1mode. Thick lines are for the corrected Mie’s theory with h(2)

n , andthin lines are for the conventional Mie’s theory with h(1)

n . Note thatω+ of (14b) is plotted, while the direct solution of (8) is −ω+ forthe metal nanosphere as discussed in the text; this applies also toFigs. 6–18. For the LF mode, the imaginary part for h(1)

n (thin solidline) approaches that for h(2)

n (thick solid line) at radius a ≈ 70 nm(arrow).

sinusoidal function) and that of the surface plasmon on a planarmetal surface (represented by a hyperbolic function) [20].Similarity exists for both fields on the surface of metal sphereand on the planar metal surface; due to the negative real part ofthe permittivity for metals, the phase velocity inside the metalsphere is in the opposite direction from the group velocity.The fields for the inside and the outside regions of the metalsphere match by opposing the directions of the phase velocityas well. To the contrary, the group velocities remain the samedirections for the inside and the outside of the sphere, thusretain the energy conservation. The energy conservation in thescattering process is clearly demonstrated later in the contextof the scattering cross sections.

1. Physical interpretation of complex resonance frequency ofdamped oscillation in negative permittivity medium

To show the consistency of the analysis, one can considerthe general complex resonance frequency of damped oscil-lation. Suppose we have a particle oscillating in a dampedmedium with characteristic angular frequency ω0 and dampingfactor γ > 0, of which equation of motion is represented by

d2x

dt2+ γ

dx

dt+ ω2

0x = 0. (12)

The general solution to (12) is readily given by the superpo-sition of two oscillation terms, in analogy with the e−iωt timedependency in this paper,

x(t) = A exp[−iω+t] + B exp[−iω−t], (13)

where

ω+ = ω − iγ

2, (14a)

ω− = −ω − iγ

2, (14b)

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FUNDAMENTAL CORRECTION OF MIE’s SCATTERING . . . PHYSICAL REVIEW A 89, 033805 (2014)

and ω is the angular frequency of oscillation under damping,i.e.,

ω =√

ω20 − γ 2

4. (15)

For the analysis of the resonance on the metal nanospheres,we expect two complex resonance frequencies of ω+ and ω−as in (13), both having a negative imaginary part. The dampedoscillation (13) is an impulse response of the system of whichcomplex resonance frequency ω = ω

′ + iω′′

leads to the realand the imaginary parts ω

′ = ±ω and ω′′ = − γ

2 . These can beobtained with numerical analyses such as the FDTD methodwith the Fourier transform of an impulse response; ω is thecenter frequency and γ is the full width at half maximum(FWHM) of the Fourier spectral peak.

It should be mentioned that the solution to the transcen-dental Eq. (8) obtained directly by our nonlinear solver isnegative frequency, i.e., corresponding to −ω+ = −(ω − i

γ

2 )in (13). The solution of negative frequency −ω is clearlyexplained from the viewpoint of a time-reversal problem,where the hypothetical backward wave is considered. Thenegative time −t of the time-reversal system and the solutionof the transcendental equation in negative frequency −ω

retain the time-dependency e−i(−ω)(−t) = e−iωt invariant forthe external field. The external field of h(2)

n , propagating inwardto the sphere, is that of the time-reversal system, which ismatched with the internal field of jn at the surface of thesphere having a negative permittivity. Note the spherical Besselfunction jn represents a standing wave, not a propagatingwave. If the argument ρ to jn(ρ) has a negative real partdue to the negative permittivity, jn matches only with thecomplex conjugate of h(1)

n , i.e., with h(2)n . If the time is restored

forwardly, it is equivalent that the external field propagatesoutward with time.

When substituting (11) into (8) and (10), the sign of ω

must also be chosen negative, i.e., εr (−ω), so that the complexpermittivity is consistent with the definition for e−iωt . Notealso that (11) is a complex function that is expandable to thecomplex argument of ω. For the sake of intuitive comparison,however, ω+ = ω − i

γ

2 is plotted in all Figs. 6–18.

2. Numerical results of the complex resonancefrequency of metal nanosphere

In Figs. 6–18, thick lines (solid lines and dotted lines)show the results for h(2)

n , and thin lines (solid lines anddotted lines) show those for h(1)

n . The solid lines show thelow frequency (LF) modes, and the dotted lines show thehigh frequency (HF) modes. For the Ag sphere as shownin Figs. 6–9, if analyzed with h(1)

n , the imaginary part ofthe complex resonance frequency exists strangely in boththe positive and the negative regions, whereas the solutionsfor h(2)

n exist consistently in the negative region. The realpart of the resonance frequency even differs for h(1)

n andh(2)

n by 1%–2%. The solution of the positive imaginary partis physically unreasonable, leading to a field growing withtime. In order to check the results, an FDTD analysis wasperformed [12] and the results are compared in Fig. 6 forn = 1; they agree well with the complex resonance frequencyfor h(2)

n . In these results, it is suggested that the conventional

0

200

400

600

800

1000

1200

0 100 200 300 400

1

2

3

4

Re

[Fre

q.]

[TH

z]

Re

[h-ω

] [e

V]

Sphere radius [nm]

Ag

n=2,LFn=2,HF

-250

-200

-150

-100

-50

0

50

0 100 200 300 400

-1

0

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Sphere radius [nm]

FIG. 7. Complex resonance frequency of the Ag sphere for n = 2.Thick lines for h(2)

n , and thin lines for h(1)n apply also to Figs. 8–18. The

imaginary part for h(1)n approaches that for h(2)

n at radius a ≈ 200 nm(arrow).

Mie’s theory with h(1)n may give wrong results; instead, h(2)

n

should be used for the analysis of a metal nanosphere. Inall the results of Figs. 6–18, it should be interesting to notethat the imaginary parts for h(1)

n and h(2)n are symmetric with

respect to the zero frequency at the infinitesimal sphere radius.Moreover, as the radius becomes larger, the imaginary part forthe LF mode of h(1)

n approaches asymptotically that of h(2)n (thin

and thick solid lines), which is clearly seen in Figs. 6–9. Thisfact would have also confused the interpretation of the results,and misled to the wrong solution by h(1)

n . In other words, forthe metal spheres of radius larger than approximately severalhundred nanometers, the conventional solutions by h(1)

n maystill be effective.

The tendency for the results of Au sphere inFigs. 10–13 is similar to that of the Ag sphere; the imaginarypart of the complex resonance frequency exhibits partlypositive and partly negative values for h(1)

n , while it exhibitsalways negative values for h(2)

n for all the higher-order modes.The real part of the complex resonance frequency is close

0

200

400

600

800

1000

1200

0 100 200 300 400

1

2

3

4

Sphere radius [nm]

Ag

n=3,LFn=3,HF

-250

-200

-150

-100

-50

0

50

0 100 200 300 400

-1

0

Sphere radius [nm]

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 8. Complex resonance frequency of the Ag sphere for n = 3.The imaginary part for h(1)

n approaches that for h(2)n at radius a ≈

300 nm (arrow).

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0

200

400

600

800

1000

1200

0 100 200 300 400

1

2

3

4

Sphere radius [nm]

Ag

n=4,LFn=4,HF

-250

-200

-150

-100

-50

0

50

0 100 200 300 400

-1

0

Sphere radius [nm]

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 9. Complex resonance frequency of the Ag sphere for n = 4.The imaginary part for h(1)

n approaches that for h(2)n at radius a ≈ 400

nm (arrow).

but behaves differently for h(1)n and h(2)

n ; the low frequency(LF) mode varies smoothly, but the high frequency (HF) modeseems rather distorted due to the complicated behavior of thepermittivity of Fig. 2 at 800 to 1000 THz.

In Figs. 14–17, the results for a Cu sphere are shown.Because the complex permittivity of Fig. 3 varies onlygradually, the variation of the resonance frequency is alsomoderate; nevertheless the tendency is similar to those of Agand Au; moreover, the imaginary part exists always in thepositive region. In those cases of Cu, the real parts approachbetween the LF and the HF modes, which is due to the gradualvariation of ε

′r for Cu in Fig. 3.

For an Al sphere, the complex resonance frequency exhibitsonly the low frequency mode as shown in Fig. 18 for n = 1,2,3,and 4, due to the almost monotonously decreasing permittivityin Fig. 4. The complex resonance frequency decreases alsomonotonously for the real part, and varies smoothly for boththe real and the imaginary parts. For h(1)

n the imaginary partobviously exists both in the positive and the negative regions,

0

200

400

600

800

1000

1200

1400

0 100 200 300 400

1

2

3

4

5

Sphere radius [nm]

Au

-400

-300

-200

-100

0

100

200

300

0 100 200 300 400

-1

0

1

Sphere radius [nm]

n=1,LFn=1,HF

FDTD,LF

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 10. Complex resonance frequency of the Au spherefor n = 1.

0

200

400

600

800

1000

1200

1400

0 100 200 300 400

1

2

3

4

5

Sphere radius [nm]

Au

-400

-300

-200

-100

0

100

200

300

0 100 200 300 400

-1

0

1

Sphere radius [nm]

n=2,LFn=2,HF

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 11. Complex resonance frequency of the Au spherefor n = 2.

0

200

400

600

800

1000

1200

1400

0 100 200 300 400

1

2

3

4

5

Sphere radius [nm]

Au

-400

-300

-200

-100

0

100

200

300

0 100 200 300 400

-1

0

1

Sphere radius [nm]

n=3,LFn=3,HF

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 12. Complex resonance frequency of the Au spherefor n = 3.

0

200

400

600

800

1000

1200

1400

0 100 200 300 400

1

2

3

4

5

Sphere radius [nm]

Au

-400

-300

-200

-100

0

100

200

300

0 100 200 300 400

-1

0

1

Sphere radius [nm]

n=4,LFn=4,HF

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 13. Complex resonance frequency of the Au spherefor n = 4.

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FUNDAMENTAL CORRECTION OF MIE’s SCATTERING . . . PHYSICAL REVIEW A 89, 033805 (2014)

0

200

400

600

800

1000

0 100 200 300 400

1

2

3

4

Sphere radius [nm]

Cu

n=1,LFn=1,HF

-200

-100

0

100

200

0 100 200 300 400-1

0

1

Sphere radius [nm]

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 14. Complex resonance frequency of the Cu spherefor n = 1.

0

200

400

600

800

1000

0 100 200 300 400

1

2

3

4

Sphere radius [nm]

Cu

n=2,LFn=2,HF

-200

-100

0

100

200

0 100 200 300 400-1

0

1

Sphere radius [nm]

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 15. Complex resonance frequency of the Cu spherefor n = 2.

0

200

400

600

800

1000

0 100 200 300 400

1

2

3

4

Sphere radius [nm]

Cu

n=3,LFn=3,HF

-200

-100

0

100

200

0 100 200 300 400-1

0

1

Sphere radius [nm]

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 16. Complex resonance frequency of the Cu spherefor n = 3.

0

200

400

600

800

1000

0 100 200 300 400

1

2

3

4

Sphere radius [nm]

Cu

n=4,LFn=4,HF

-200

-100

0

100

200

0 100 200 300 400-1

0

1

Sphere radius [nm]

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 17. Complex resonance frequency of the Cu spherefor n = 4.

while for h(2)n it exists always in the negative region. The

real part of the complex resonance frequency agrees for bothh(1)

n and h(2)n . The plots for n = 2 in Fig. 18 stopped halfway

because the nonlinear solver did not work well for this branch.As shown in Fig. 19, since the permittivity data by Johnson

and Christy [19] have lower loss than those in Ref. [16], theimaginary part of the complex frequency is smaller than thatin Figs. 6–9 at the infinitesimal sphere radius. The overalltendency of the complex frequency is similar to those inFigs. 6–9; the conventional Mie’s theory with h(1)

n resultedin an unphysical solution. The real parts for both h(1)

n andh(2)

n agree well, and the imaginary parts for both approachasymptotically as the sphere radius increases.

D. Scattering cross sections of a metal sphere

Scattering, extinction, and absorption cross sections of asphere are discussed in this section. The formulation followsalso that of Ref. [2], comparing the expression for dielectricspheres with the spherical Hankel function of the first kindof order n, or h(1)

n , and the expression for metal spheres with

0

500

1000

1500

2000

2500

0 100 200 300 400

1

2

3

4

5

6

7

8

9

10

Sphere radius [nm]

Al

n=1,LFn=2,LFn=3,LFn=4,LF

-350

-300

-250

-200

-150

-100

-50

0

50

100

0 100 200 300 400

-1

0

Sphere radius [nm]

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 18. Complex resonance frequency of the Al sphere for n =1,2,3, and 4. Only the LF modes are shown.

033805-7

MASAFUMI FUJII PHYSICAL REVIEW A 89, 033805 (2014)

0

200

400

600

800

1000

0 100 200 300 400

1

2

3

4

Sphere radius [nm]

Ag JC11

-250

-200

-150

-100

-50

0

50

100

0 100 200 300 400

-1

-0.75

-0.5

0

0.25

Sphere radius [nm]

n=1,LFn=2,LFn=3,LFn=4,LF

Re

[Fre

q.]

[TH

z]

Im [F

req.

] [T

Hz]

Im [h

- ω] [

eV]

Re

[h-ω

] [e

V]

FIG. 19. Complex resonance frequency of the Ag sphere for n =1,2,3, and 4 for the permittivity data from Johnson and Christy [19],which have been fitted to a Drude and single-pole Lorentz model.Only the LF modes are shown.

the spherical Hankel function of the second kind of order n,or h(2)

n . The derivation of the cross sections is based on theenergy scattered or absorbed by a sphere; it is thus proved thatthe energy is strictly conserved in the scattering process, nomatter if it is a dielectric sphere or a metal sphere.

The diffraction of a plane wave by a sphere is consideredunder the spherical coordinate system. The vector sphericalfunctions are introduced as in Ref. [2],

M = meomn e−iωt , (16a)

N = neomn e−iωt , (16b)

with

meomn = ∓ m

sin θzn(kR)P m

n (cos θ ) sincosmφ i2

− zn(kR)∂P m

n (cos θ )

∂θ

cossin mφ i3, (17a)

and

neomn = n(n + 1)

kRzn(kR)P m

n (cos θ ) cossin mφ i1

+ 1

kR

∂

∂R[R zn(kR)]

∂

∂θP m

n (cos θ ) cossin mφ i2

∓ m

kR sin θ

∂

∂R[R zn(kR)]P m

n (cos θ ) sincosmφ i3,

(17b)

where the index m is restricted to m = 1 due to the dependenceof sin mφ and cos mφ on φ, and i1, i2, and i3 are the unit vectorsof the spherical coordinates in the R, θ , and φ directions,respectively. Up to this point, the formulas hold in general forany radial functions of zn, which can be replaced either by jn

or h( )n for = 1,2 depending on the region inside or outside

of the sphere.First, the expansion of the incident plane wave

into the vector spherical functions (17a) and (17b)is considered for the electric and the magnetic

fields,

Ei = axE0eik2z−iωt

= E0e−iωt

∞∑n=1

in2n + 1

n(n + 1)

(m(1)

o1n − in(1)e1n

), (18a)

H i = ay

k2

μ2ωE0e

ik2z−iωt

= −k2E0

μ2ωe−iωt

∞∑n=1

in2n + 1

n(n + 1)

(m(1)

e1n + in(1)o1n

), (18b)

where m(1)eo1n and n(1)

eo1n represent the internal vector field given

by substituting zn(kR) = jn(k2R) in (17a) and (17b), ax anday are unit vectors in the x and y directions of the rectangularcoordinate system, respectively.

The incident wave is partly scattered and partly transmittedinto the sphere, and thus the induced secondary field for theexternal region (R > a) is expanded into the vector sphericalfunctions with the expansion coefficients ar

n and brn as

Er = E0e−iωt

∞∑n=1

in2n + 1

n(n + 1)

(ar

nm(3)o1n − ibr

nn(3)e1n

), (19a)

H r = −k2E0

μ2ωe−iωt

∞∑n=1

in2n + 1

n(n + 1)

(br

nm(3)e1n + iar

nn(3)o1n

),

(19b)

where m(3)eo1n and n(3)

eo1n are given for the external vector field by

substituting zn(kR) = h( )n (k2R) for = 1,2 in (17a) and (17b).

Lastly, the transmitted field inside the sphere for R < a isexpanded in a similar manner as

Et = E0e−iωt

∞∑n=1

in2n + 1

n(n + 1)

(at

nm(1)o1n − ibt

nn(1)e1n

), (20a)

H t = −k1E0

μ1ωe−iωt

∞∑n=1

in2n + 1

n(n + 1)

(bt

nm(1)e1n + iat

nn(1)o1n

),

(20b)

with m(1)eo1n and n(1)

eo1n given for the internal field by substituting

zn(kR) = jn(k1R) in (17a) and (17b). The difference from theconventional formulation using h(1)

n is for only the externalfields (19a) and (19b); the expansion coefficients ar

n and brn are

to be used for the calculation of the scattering cross sectionsand are obtained by imposing the boundary conditions at thesphere surface. The expression of them is retained as theconventional one, i.e.,

arn = − μ1jn(Nρ)[ρjn(ρ)]′ − μ2jn(ρ)[Nρjn(Nρ)]′

μ1jn(Nρ)[ρh( )n (ρ)]′ − μ2h

( )n (ρ)[Nρjn(Nρ)]′

, (21a)

for the magnetic type or TE mode, and

brn = − μ1jn(ρ)[Nρjn(Nρ)]′ − μ2N

2jn(Nρ)[ρjn(ρ)]′

μ1h( )n (ρ)[Nρjn(Nρ)]′ − μ2N2jn(Nρ)[ρh

( )n (ρ)]′

,

(21b)

for the electric type or TM mode.

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FUNDAMENTAL CORRECTION OF MIE’s SCATTERING . . . PHYSICAL REVIEW A 89, 033805 (2014)

The scattering cross section is then given by the totalscattered energy divided by the mean energy flow of theincident wave,

Qs = 1

E20

√ε2

μ2Re

∫ π

0

∫ 2π

0(Erθ Hrφ − ErφHrθ )

R2 sin θdθdφ. (22)

By substituting the corresponding field values of (19a)and (19b) into (22), we obtain after lengthy algebraic ma-nipulation,

Q(1)s = 2π

k22

∞∑n=1

(2n + 1)(∣∣ar

n

∣∣2 + ∣∣brn

∣∣2), (23a)

for h(1)n and

Q(2)s = −Q(1)

s , (23b)

for h(2)n .

The extinction cross section, which is the sum of scatteredand absorbed energy divided by the mean energy flow of theincident wave,

Qt = − 1

E20

√ε2

μ2Re

∫ π

0

∫ 2π

0

(Eiθ Hrφ + Erθ Hiφ − EiφHrθ − ErφHiθ )

R2 sin θdθdφ. (24)

Similarly, substituting the corresponding field values outof (18b) to (19b) into (24) leads to

Q(1)t = −2π

k22

Re∞∑

n=1

(2n + 1)(ar

n + brn

), (25a)

for h(1)n and

Q(2)t = −Q

(1)t , (25b)

for h(2)n .

The absorption cross section is given for both h(1)n and h(2)

n

by

Q(1,2)a = Q

(1,2)t − Q(1,2)

s . (26)

As seen in these formulas, the absolute values of the crosssections remain the same for both h(1)

n and h(2)n . This is because

the frequency used for the calculation of the cross sections isnot complex but real values. If the cross sections are calculatedwith h(2)

n for negative real frequency, energy flows outwardfrom inside the sphere to the external space as discussed inSec. II C 1, which is consistent with the definition of thenegative value of the cross sections. To illustrate this point,the scattering, extinction, and absorption cross sections arecalculated for an Ag nanosphere of radius a = 100 nm havinga hypothetical constant complex permittivity of, for simplicity,εr = −5 + i1.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

-300 -400-500 -1000 -2000 2000 1000 500 400 300

Cro

ss S

ectio

n [1

0-13

m2 ]

Frequency [THz]

Wavelength [nm]

hn(1)

hn(2)

Qt(1)

Qs(1)

Qa(1)

Qt(2)

Qs(2)

Qa(2)

FIG. 20. Cross sections for a hypothetical Ag sphere of radiusa =100 nm with a constant complex permittivity εr = −5 + i1. Theresults in the positive frequency region are for h(1)

n , and those inthe negative frequency region are for h(2)

n . The curves are exactlysymmetric with respect to the origin.

It is interesting to note in Fig. 20 that the scattering crosssections calculated with h(1)

n for positive real frequency areexactly symmetric with respect to origin to those calculatedwith h(2)

n for negative real frequency. This also shows theconsistency of the solution to the transcendental Eq. (8) havinga negative real part when solved with h(2)

n .From these results, it is concluded that the conventional

Mie’s theory with h(1)n gives correct magnitudes of the cross

sections for metal nanospheres, as well as the corrected Mie’stheory with h(2)

n .

III. CONCLUSIONS

It has been shown that Mie’s scattering theory needscorrection for the analysis of metal nanospheres in the opticregime where the complex permittivity has a negative realpart. For the conventional Mie’s theory, the complex resonancefrequency obtained by solving the transcendental equation hasa positive imaginary part, in particular for sphere radius lessthan about 100 nm, which leads to an unphysical solution ofgrowing field as time progresses.

It has been also shown that the error can be corrected byadopting the radial variation of complex conjugate function;i.e., if e−iωt time variation is assumed as in this paper, thespherical Hankel function of the first kind must be replacedby that of the second kind. This gives a thorough correctionto Mie’s theory for the analysis of a metal nanosphere.Additional consideration of the scattering, extinction, andabsorption cross sections indicates that the corrected Mie’stheory gives the cross sections exactly the same magnitudesas the conventional Mie’s theory, with opposite signs ofthe cross sections. The energy is shown to be conservedin the scattering process represented by the corrected Mie’stheory.

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MASAFUMI FUJII PHYSICAL REVIEW A 89, 033805 (2014)

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York, 1941).[3] H. C. van de Hulst, Light Scattering by Small Particles (Dover

Publications, New York, 1981).[4] C. G. Bohren and D. R. Huffman, Absorption and Scattering

of Light by Small Particles (John Wiley and Sons, New York,1983).

[5] U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters(Springer-Verlag, Berlin, 1995).

[6] M. Born and E. Wolf, Principle of Optics, 7th ed. (CambridgeUniversity Press, Cambridge, 1999).

[7] S. Nie and S. R. Emory, Science 275, 1102 (1997).[8] A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).[9] K. Svoboda and S. M. Block, Opt. Lett. 19, 930 (1994).

[10] S. Foteinopoulou, J. P. Vigneron, and C. Vandenbem, Opt. Exp.15, 4253 (2007).

[11] M. Fujii, Opt. Exp. 18, 27731 (2010).[12] M. Fujii, J. Lightwave Technol. 30, 1284 (2012).[13] M. Fujii, in Frontiers in Optics 2013, Orlando, Florida, edited

by I. Kang, D. Reitze, N. Alic, and D. Hagan, OSA’s 97th AnnualMeeting, Technical Digest (online) (Optical Society of America,Washington, D.C., 2013), paper FTh3A.3.

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4654 (1983).[18] M.-L. Theye, Phys. Rev. B 2, 3060 (1970).[19] P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).[20] H. Raether, Surface Plasmons on Smooth and Rough Surfaces

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033805-10