analytical approximations of the dispersion relation of the plasmonic modes propagating around a...

Analytical approximations of the dispersion relationof the plasmonic modes propagating around

a curved dielectric-metal interface

Massimiliano Guasoni

Dipartimento di Ingegneria dell’Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy([email protected])

Received November 1, 2010; revised March 25, 2011; accepted April 4, 2011;posted April 13, 2011 (Doc. ID 137493); published May 18, 2011

We find some useful and effective approximations of the dispersion relation of the fundamental surface plasmonwave propagating around a curved dielectric-metal interface. These estimations are valid even with a verysmall curvature radius, which is the case of interest for nano-optics applications, and are validated throughthe numerical solution of the characteristic equation of the curved interface. © 2011 Optical Society of America

OCIS codes: 260.3910, 130.2790, 240.6680, 260.2030.

1. INTRODUCTIONThe field of plasmonics has emerged in the last years as one ofthe most promising ways for the realization of nanoscalestructures [1,2]. The surface plasmon wave (SPW) propagat-ing at a metal-dielectric interface has a spatial confinement ofthe order λ=10 (λ is the vacuum wavelength), thus much smal-ler than the conventional modes of dielectric waveguides: thisis the basic principle that leads to the miniaturization of photo-nic devices. In the last years a lot of effort has been spent inorder to realize plasmonic waveguides by means of straightmetal-dielectic interfaces [3], wedges [4], grooves [5], orchains of ordered and disordered metal nanoparticles [6,7].Moreover, recent studies have led to the demonstration ofplasmonic prisms as well as of lenses [8].

In this context the study of curved metal-dielectric inter-faces has an important role because a small curvature radiusprevents extra space from filling up. In the last years the anal-ysis of radiation and absorption losses of an SPW propagatingaround a curved interface or waveguide, as well as the anal-ysis of the coupling between this SPW and the SPW of astraight metal-dielectric interface, has been the subject of alot of studies, numerical simulations [9–11], and experimentalworks [12]. As far as the single curved interface is concerned,its characteristic equation for TE and TMmodes is well known[10], but its numerical implementation encounters difficultiesdue to the complexity in the evaluation of Bessel functions ofhigh argument and order, which makes this approach timeconsuming and sometimes not accurate. For this fact, differ-ent approaches have been developed; a usual technique con-sists of applying a conformal mapping that transforms thecurved waveguide into a straight and inhomogeneous one[13], whose propagation constant is then approximated bymeans of the Wentzel-Kramers-Brillouin (WKB) method [14],the perturbation theory [15], or the local field analysis [16]. Inparticular, the WKB and the perturbation approach have beenwidely used in the study of curved dielectric waveguides, butthey are not accurate enough when the curvature radius is toosmall. In this last case, the numerical finite element method or

finite-difference time-domain simulations become of primaryimportance in order to obtain good accuracy, but they do notallow a gain in physical insight.

In this paper we get some useful and effective fully analy-tical approximations for the propagation constant of the fun-damental SPW propagating around a curved dielectric-metalinterface. These approximations appear to work well evenwith a small curvature radius so that the propagation constantis put in direct relation to the main system parameters servingas a powerful tool for deeper insight into the SPW properties.

We show that the use of a perturbation approach is effec-tive over a large fraction of the frequency band considered.We also develop two other methods that provide direct formu-las giving accurate results when the perturbation theory fails:the first one is based on the use of asymptotic forms of theBessel functions and the second one on the Poynting theorem.

The paper is organized as follows: In Section 2 the geometryis introduced and the characteristic equation of the curved in-terface is briefly derived. In Sections 3–5 the three techniques(use of asymptotic forms of the Bessel functions, perturbationtheory, and use of the Poynting theorem) are described. InSection 6 some numerical simulations are shown confirmingthe validity of the developed formulas and the complementar-ity of the three techniques. In Section 7 the conclusions arepresented.

2. CHARACTERISTIC EQUATION OF THECURVED INTERFACEThe geometry analyzed in this paper is shown in Fig. 1. Thesystem consists of two homogeneous and isotropic regions S1

and S2 with refractive indexes n1 and n2 and divided by acurved interface of radius R on the y − z plane. Along the xaxis the system is uniform. We can distinguish TE and TMmodes that propagate around the interface, whose compo-nents of the electromagnetic field, respectively, are Ex, Hr ,Hθ and Hx, Er , Eθ where ðx; r; θÞ are the cylindrical coordi-nates (r and θ are used on the y − z plane). If the modes pro-pagate on the y − z plane then their x dependence can be

1396 J. Opt. Soc. Am. B / Vol. 28, No. 6 / June 2011 Massimiliano Guasoni

0740-3224/11/061396-08$15.00/0 © 2011 Optical Society of America

neglected, so we write Exðx; r; θÞ ¼ exðrÞeiθγ (for the TE case)and Hxðx; r; θÞ ¼ hxðrÞeiθγ (for the TM case), where exðrÞ andhxðrÞ are described by the Helmotz equation both in S1 and S2:

δrraðrÞ þ δraðrÞ=r þ ðk20n2i − γ2=r2ÞaðrÞ ¼ 0 ði ¼ 1; 2Þ:

ð1Þ

In Eq. (1) k0 is the vacuum wave vector and aðrÞ is exðrÞ inthe TE case or hxðrÞ in the TM case. The propagation constantγ is the unknown quantity and has to be searched in thecomplex plane; its imaginary part represents the radiationand absorption losses of the mode. In S1 (0 ≤ r ≤ R)aðrÞ ¼ AJγðk0n1rÞ, where Jγ is the Bessel function of orderγ; in S2 (r > R) aðrÞ ¼ BHð1Þ

γ ðk0n2rÞ, where Hð1Þγ is the Hankel

function of the first kind of order γ [9]. A and B are arbitraryconstants. In r ¼ R the continuity of Ex and Hθ is imposed forthe TE case, while of Hx and Eθ for the TM case, from whichthe characteristic equation of the modes is obtained:

δrHð1Þγ ðk0n2RÞ

Hð1Þγ ðk0n2RÞ

¼ ρ · δrJγðk0n1RÞJγðk0n1RÞ

: ð2Þ

In Eq. (2) ρ ¼ 1 for the TE modes, while ρ ¼ n22=n

21 for the

TMmodes. Let us now focus on the TM case, which allows theexistence of SPW, and let us write x1 ¼ k0n1R and x2 ¼ k0n2R.Then Eq. (2) can be rewritten as

δrHð1Þγ ðx2Þ

Hð1Þγ ðx2Þ

¼�x2x1

�2 δrJγðx1ÞJγðx1Þ

: ð3Þ

3. APPROXIMATION OF THE DISPERSIONRELATION BY MEANS OF ASYMPTOTICFORMS OF THE BESSEL FUNCTIONSEquation (3) has to be numerically solved for γ. In this sectionand in the next two we want to find an analytical approxima-tion of γ that highlights its relation to the system parameters.The system is defined d-m interface when in S1 there is a di-electric and in S2 a metal, m-d interface in the opposite case.Both the d-m and them-d interface allow only one TM-SPW topropagate with acceptable losses, which is the fundamentalmode [9,10]. In our analysis we focus only on the d-m inter-face, because as it is shown in Section 6, in this case thefundamental SPW has a sufficiently low loss in a proper fre-quency band even with a very small curvature radius, which isthe situation of interest for nano-optics applications. From

here and for the rest of the paper we define low-frequencyband a the range of frequencies ω for which x1 ≪ 1. Let usnow consider x1 ≪ 1 (we will see in Section 6 that for our pur-poses we need x1 ≤ 1) and let us suppose jγj ≪ 1, then it ispossible to approximate the left-hand side of Eq. (3) in thesubsequent way:

δrHð1Þγ ðx2Þ

Hð1Þγ ðx2Þ

≈δrHð1Þ

0 ðx2ÞHð1Þ

0 ðx2Þ

¼ k0n2

�ð1=2ÞHð1Þ−1 ðx2Þ − ð1=2ÞHð1Þ

1 ðx2ÞHð1Þ

0 ðx2Þ

�

¼ −k0n2

�Hð1Þ

1 ðx2ÞHð1Þ

0 ðx2Þ

�: ð4Þ

In Eq. (4) we used the relations δxHð1Þγ ðxÞ ¼ ð1=2ÞHð1Þ

γ−1ðxÞ −ð1=2ÞHð1Þ

γþ1ðxÞ andHð1Þ−1 ðxÞ ¼ −Hð1Þ

1 ðxÞ. Being x1 ≪ 1 the Besselfunction Jγðx1Þ can be well approximated with its asymptoticform Jγðx1Þ ≈ ðΓðγ þ 1ÞÞ−1 · ðx1=2Þγ , where Γ is the Euler func-tion. For the term δrJγðx1Þ we exploit the relation δxJγðxÞ ¼ðγ=xÞJγðxÞ − Jγþ1ðxÞ so that

δrJγðx1ÞJγðx1Þ

¼ k0n1

� γx1

−Jγþ1ðx1ÞJγðx1Þ

�

≈ k0n1

� γx1

−ð1=Γðγ þ 2ÞÞðx1=2Þγþ1

ð1=Γðγ þ 1ÞÞðx1=2Þγ�

¼ k0n1

� γx1

−x1

2ðγ þ 1Þ�: ð5Þ

Taking into account Eqs. (4) and (5), we can rewrite Eq. (3)in the subsequent way:

−Hð1Þ

1 ðx2ÞHð1Þ

0 ðx2Þ¼ x2

x1

� γx1

−x1

2ðγ þ 1Þ�: ð6Þ

Equation (6) is now easily solvable for γ with a second de-gree equation. A further simplification can be made by consid-ering that if jγj ≪ 1 then ðγ þ 1Þ ≈ 1, so from Eq. (6) we get

γ ¼ x21

�12−

Hð1Þ1 ðx2Þ

x2Hð1Þ0 ðx2Þ

�: ð7Þ

From the equation above we get two important features ofthe fundamental SPW in the d-m interface. First, for low fre-quencies (x1 ≪ 1) and for typical values of the refractive in-dex n2 of the metal (that can be calculated by means of theDrude model as in Section 6) γ → 0, thus the fundamentalSPW has losses that tend to zero whatever the metal is. Thisis not true in the case of the m-d interface, as we will see inSection 6. Second, the derivative δγ=δω → 0, differently fromthe plasmonic dispersion function of the straight interfacewhose real part strictly follows the light line (see results inSection 6).

Fig. 1. (Color online) The system considered in this paper consists oftwo homogeneous and isotropic regions S1 and S2 divided by a curvedinterface of radius R on the y − z plane. The cylindrical coordinatesare centered in O.

Massimiliano Guasoni Vol. 28, No. 6 / June 2011 / J. Opt. Soc. Am. B 1397

4. APPROXIMATION OF THE DISPERSIONRELATION: PERTURBATION THEORYAPPROACHThe approximations developed in the previous section aregood when x1 ≤ 1 and γ is sufficiently small. In this sectionwe use a perturbation approach to estimate γ when the aboveconditions are not respected. We will see that the more x1 > 1the more the perturbation approach gives correct results, so itis complementary to the use of the asymptotic forms.

Given a relative dielectric function ϵðy; zÞ that varies on they − z plane, then a generalized Helmotz equation can be writ-ten for the field Hxðy; zÞ in the TM case:

∇2ðHxÞ −δyϵϵ δyHx −

δzϵϵ δzHx þ k20ϵHx ¼ 0: ð8Þ

We now rewrite Eq. (8) in cylindrical coordinates, assumingHx ¼ hxðrÞeiγθ and considering the case in which ϵðr; θÞ de-pends only on r, which is our case of study being ϵðrÞ ¼ n2

1

when 0 ≤ r ≤ R and ϵðrÞ ¼ n22 when r > R. We obtain

δrrhþ δrhr

−δrϵϵ δrhþ

�k20ϵ −

γ2r2

�h ¼ 0; ð9Þ

where h ¼ hxðrÞ and ϵ ¼ ϵðrÞ.Let us now apply to Eq. (9) the conformal transformation

r ¼ Reu=R from the r domain to the u domain. We obtain

δuuh −δuϵϵ δuhþ

�k20ϵe2u=R −

γ2R2

�h ¼ 0: ð10Þ

In the new u domain ϵðuÞ ¼ n21 when u ≤ 0, ϵðuÞ ¼ n2

2 whenu > 0. When R → ∞ then e2u=R → 1 and the curved d-m inter-face becomes a straight d-m interface. In this case Eq. (10)turns into Eq. (11):

δuuh −δuϵϵ δuhþ

�k20ϵ −

γ2R2

�h ¼ 0: ð11Þ

Starting from here we use the subscript c when referring tothe curved interface and the subscript s when referring to thestraight one. The plasmonic solution of Eq. (11) is well known:

γ=R ¼ β ¼ k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin21n

22

n21 þ n2

2

s;

hsðuÞ ¼ ek1u�u ≤ 0k1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ2 − k20n

21

q �;

hsðuÞ ¼ e−k2u�u > 0k2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiβ2 − k20n

22

q �:

ð12Þ

In order to solve Eq. (10) in the case of the curved d-m inter-face, we now consider the fundamental SPW of the curvedinterface as a perturbed version of the plasmonic solution re-ported in Eq. (12). Let us first rewrite Eqs. (10) and (11) asSturm-Liouville problems:

δu�1ϵ δuhc

�þ k20e

2u=Rhc ¼λcϵ hc; ð13Þ

δu�1ϵ δuhs

�þ k20hs ¼

λsϵ hs; ð14Þ

where λc ¼ ðγ2c=R2Þ and λs ¼ ðγ2s=R2Þ ¼ β2, with β defined in(12). We define two operators Lc and Ls as:

Lc ¼ ϵ�δu�1ϵ δu

�þ k20e

2u=R

�; ð15Þ

Ls ¼ ϵ�δu�1ϵ δu

�þ k20

�: ð16Þ

In this way Eqs. (13) and (14) can be written as Lc½hc� ¼ λchcand Ls½hs� ¼ λshs. Let us now consider the subsequent scalarproduct between two functions f 1ðuÞ and f 2ðuÞ:

hf 1; f 2i ¼Z

∞

−∞

f 1ðuÞf 2ðuÞϵðuÞ δu: ð17Þ

It is easy to show that Lc and Ls are Hermitian with respectto the above scalar product. The operator Lc can be thought ofas a perturbed version of Ls, so that Lc ¼ Ls þ δL, withδL ¼ k20ϵðe2u=R − 1Þ. We write also hc and λc as perturbed ver-sions of hs and λs, so that hc ¼ hs þ δh and λc ¼ λs þ δλ. Byexploiting the Hermiticity of Ls, it is possible to calculatethe first-order correction δλ as

δλ ¼ hhs; δL½hs�ihhs; hsi

; ð18Þ

In Eq. (18) δλ ¼ ðγ2c − γ2sÞ=R2 ¼ ððγc − γsÞ=RÞððγc þ γsÞ=RÞ≈2βðγc − γsÞ=R, so that ðγc − γsÞ ¼ δγ ¼ R · δλ=ð2βÞ, with δλgiven by Eq. (18). The integrals in Eq. (18) admit a simpleanalytical solution, so that we easily get

δλ ¼ 1R2

1=ð2T1 þ 2Þ − 1=ðT1Þ þ 1=ð2T2 − 2Þ − 1=ðT2Þ1=ð2T1x21Þ þ 1=ð2T2x22Þ

;

T1;2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−x41;2=ðx21 þ x22Þ

q:

ð19Þ

Equation (19) is valid even when the metal absorption istaken into account (that is when x2 is complex) and allowsus to directly get a fully analytical formula of γ ¼ β þ δγ forthe fundamental SPW in the curved d-m interface.

5. LOSSES ESTIMATION BASED ON THEPOYNTING THEOREMAs it is shown in the next section where some examples arediscussed, the first-order correction of Eq. (19) works welleven with a small curvature radius, but at low frequenciesit does not succeed properly in detecting the low losses ofthe fundamental SPW. In order to overcome this problem,we make use of the Poynting theorem that allows us to geta fully analytical estimation for the losses in the range of fre-quencies where Eq. (19) does not work efficiently. We will seethat this approximation is even more effective than the oneobtained by means of the asymptotic forms.

A way to estimate the losses α ¼ ImðγÞ is to make useof the Poynting theorem in the volume V enclosed by thepath ABC as in Fig. 2, so that ReðϕAC þ ϕAB þ ϕBCÞ ¼−RV2

Imðϵ2Þðω=2Þj~Ej2δV , where V2 is the fraction of the


volume V occupied by the metal, ϵ2 ¼ ffiffiffiffiffin2

pand ϕAC , and ϕAB

and ϕBC are the fluxes of the Poynting vector over the surfacesAC, AB, and BC.

We assume that in S1 hxðrÞ ¼ Jγðk0n1rÞ and in S2

hxðrÞ ¼ C0 ·Hð1Þγ ðk0n2rÞ, with C0 ¼ Jγðx1Þ=Hð1Þ

γ ðx2Þ in orderto have the continuity of hx in r ¼ R. Setting δθ → 0 in Fig. 2,then ϕAB þ ϕAC ¼ 2L · δθ · ðα=LÞ · ϕAB and ϕBC ¼ ð1=2Þδθ · L·eθðL; 0Þh�xðL; 0Þ, with L ¼ AB, so that basically in order toobtain an estimation of α we need to calculate ϕAB andRV2

j~Ej2δV . The main problem when dealing with ϕAB ¼RLr¼0ð1=2Þ · erh�xδr is that it requires the calculation ofRLr¼RðjHð1Þ

γ ðk0n2rÞ2j=rÞδr which does not admit an analyticalrepresentation; similar problems arise when consideringRV2

j~Ej2δV because of the presence of the Hankel functions.If we set L ¼ R these difficulties are overcome because noHankel functions appear in the integrals and V2 is null; thenwe get

ϕAB ¼Z

R

r¼0

erh�x2

δr ¼ γc0μ02k0ϵ1

ZR

r¼0

jJγðk0n1rÞj2r

δr; ð20Þ

ZV2

j~Ej2δV ¼ 0: ð21Þ

In Eq. (20) c0 is the vacuum light speed, μ0 is the vacuumpermeability, and ϵ1 ¼ ffiffiffiffiffi

n1p

.When calculating ϕBC two options are available. If we con-

sider L ¼ R−, then the fields eθ and hx of region S1 have to beconsidered, while if we set L ¼ Rþ the fields in S2 need to betaken into account. Let us choose the second option; we willexplain soon why it is better then the first. We indicate withPTXðγ; ϵ1Þ the term 2R · δθ · ReðϕABÞ and with PLossðγ; ϵ2Þ theterm ReðϕBCÞ ¼ ð1=2Þδθ · R · Re½eθðRþ; 0Þh�xðRþ; 0Þ�. BeingReðϕAC þ ϕAB þ ϕBCÞ ¼ 0 we get

αR¼ −PLossðγ; ϵ2Þ=PTXðγ; ϵ1Þ: ð22Þ

Actually γ ¼ ReðγÞ þ i · α, so that the right-hand side ofEq. (22) depends on α. We then make use of the subsequentTaylor expansion around ReðγÞ:

PLossðγ; ϵ2ÞPTXðγ; ϵ1Þ

≈PLossðReðγÞ; ϵ2Þ þ i · α · δPLoss=δαPTXðReðγÞ; ϵ1Þ þ i · α · δPTX=δα

; ð23Þ

where the derivative at the numerator and at the denominatorare, respectively, calculated in ðReðγÞ; ϵ2Þ and ðReðγÞ; ϵ1Þ. Theinteresting aspect of this formulation, confirmed by means ofthe simulations, is that when x1 is not too great (typicallyx1 < 1:5) then δPLoss=δα and PLossðReðγÞ; ϵ2Þ are of the sameorder of magnitude and being usually α ≪ 1 it follows thatα · δPLoss=δα ≪ PLossðReðγÞ; ϵ2Þ. Similarly, α · δPTX=δα ≪

PTXðReðγÞ; ϵ1Þ, thus the terms α · δPLoss=δα and α · δPTX=δαcan be neglected so that the dependence on α on the right-hand side in Eq. (22) does not occur anymore.

This simplification would not be possible if we had chosenL ¼ R−. In fact, in this case we should use the fields in S1 sothat PLossðγ; ϵ1Þ ¼ ð1=2Þδθ · R · Re½eθðR−; 0Þh�xðR−; 0Þ� and thecondition α · δPLoss=δα ≪ PLossðReðγÞ; ϵ1Þ would not beverified.

Let us note that the subsequent relation holds:

PLossðγ; ϵ2Þ ¼ ð1=2Þδθ · R · ReðeθðRþÞh�xðRþÞÞ

¼ Re�R · δθ · c0μ0jJγðx1Þj2 · δrHð1Þ

γ ðx2Þ2ðHð1Þ

γ ðx2ÞÞik0ϵ2

�: ð24Þ

That said, it is then possible to estimate α with the threesubsequent steps:

1. We neglect Imðϵ2Þ and estimate ReðγÞ with Eq. (6) orEq. (19). These estimations turn out to be very effective, asit is shown in the next section.

2. We use the estimated ReðγÞ in order to calculatePLossðReðγÞ; ϵ2Þ and PTXðReðγÞ; ϵ1Þ using Eqs. (20) and (24)where ReðγÞ takes the place of γ. Being ReðγÞ, purely realan analytical expressions can be found for the integralEq. (20):

ZR

r¼0

erh�x2

δr ¼ ReðγÞc0μ02k0ϵ1

ZR

r¼0

jJReðγÞðk0n1rÞj2r

δr

¼ ReðγÞc0μ02k0ϵ1

ZR

r¼0

ðJReðγÞðk0n1rÞÞ2r

δr

¼ C00 · HPRða; b;−x21Þ;

C00 ¼ Γð2ReðγÞÞ · γc0μ02k0ϵ1

·

�x212

�2ReðγÞ

;

a ¼ ðReðγÞ; 1=2þ ReðγÞ;b ¼ ð1þ ReðγÞ; 1þ ReðγÞ; 1þ 2ReðγÞÞ; ð25Þ

where HPR is the regularized generalized hypergeometricfunction.

3. We estimate ðα=RÞ ¼ −PLossðReðγÞ; ϵ2Þ=PTXðReðγÞ; ϵ1Þso that a fully analytical estimation of the losses is obtained:

αR¼ Re

�−

jJReðγÞðx1Þj2 · δrHð1ÞReðγÞðx2Þ · ϵ1 · 22ReðγÞ−1

ðHð1ÞReðγÞðx2ÞÞ · iϵ2 · ReðγÞ · x4ReðγÞ1 · Γ · HPR

�; ð26Þ

where Γ ¼ Γð2ReðγÞÞ and HPR ¼ HPRða; b;−x21Þ.

6. RESULTSIn this section some examples highlighting the good quality ofthe approximations previously developed are shown.

We consider an air-silver interface, and we use a Drudemodel to describe the relative dielectric constant of

Fig. 2. (Color online) The Poynting theorem discussed in Section 5 isapplied at the volume V enclosed by ABC. The arc BC subtends anangle δθ and AB ¼ L. The points A, B, and C have, respectively, cy-lindrical coordinates (r ¼ 0, θ ¼ 0), (r ¼ L, θ ¼ 0), and (r ¼ L,θ ¼ δθ). The black arrows indicate the outward versors normal tothe surfaces AB, AC, and BC.


silver, which is ϵAgðωÞ ¼ ϵ2 ¼ 1 − ω2p=ðω2 þ iω · νÞ, where

wp ¼ 10:9 · 1015 s−1 is the plasmonic frequency and ν ¼1:6 · 1014 s−1 is the collision frequency used to take intoaccount for absorption. The relative dielectric constant ofair is ϵ1 ¼ 1.

We set R ¼ 100nm, and we numerically calculate by meansof Eq. (3) the dispersion functions of the first four TM modes.The situation depicted in Fig. 3 is representative of all thecurved d-m interfaces, in which only one mode can propagatewith acceptable losses in a certain frequency range. Thismode is the fundamental one and it is referred to with thesubscript fun. All the other modes have ReðγÞ < ReðγfunÞand suffer huge losses.

In Figs. 4 and 5 we focus on the fundamental SPW compar-ing the numerical results with the estimations developed inthe previous sections and with the dispersion function ofthe SPW in the straight interface. For this purpose γ is normal-ized with respect to R, because in the limit R → ∞ the ratio γ=Ris the propagation constant β of the straight interface [seeEqs. (11) and (12)]. First of all, we observe that at low frequen-cies (x1 ≪ 1, which corresponds to the case ω ≪ 3 · 1015 s−1 inthis example) the dispersion function of the fundamental SPWin the curved interface has Reðδðγ=RÞ=δωÞ → 0, as predictedby Eq. (7); then, it is very different with respect to the plas-monic dispersion function of the straight interface, whichhas Reðγ=RÞ proportional to ω. On the contrary, the morethe frequency approaches the surface plasmonic frequency

ωsp ¼ ωp=ffiffiffiffiffiffiffiffiffiffiffiffiffiϵ1 þ 1

p, the more the two dispersion curves are si-

milar, and in fact the correction Eq. (19) decreases and tendsto zero. In Figs. 4 and 5 they become quite indistinguishablewhen ω > 7:2 · 1015.

Let us now indicate with AE, PE, and PoyE the analyticalestimations obtained respectively by means of the asymptoticforms, of the perturbative approach and of the Poynting the-orem. The PE appears to be good over all the spectra consid-ered, especially for high values ofω for which the fundamental

Fig. 3. (Color online) Dispersion functions ωðγÞ of the first four TM-SPW propagating around a curved air-silver interface of radiusR ¼ 100nm. The black bold line represents the fundamental mode(FM). The thin dashed line is the light line ω ¼ k0n1.

Fig. 4. Real part of the dispersion function of the fundamental SPWpropagating around a curved air-silver interface of radius R ¼ 100nm.The three diagrams represent the function in different frequencyranges (respectively, 0 < ω < 3 · 1015, 3 · 1015 < ω < 6 · 1015, and6 · 1015 < ω < 8 · 1015). The gray line is the dispersion function ofthe air-silver straight interface; the black line represents the numericalresults obtained by means of Eq. (3); the dotted line represents theestimation obtained by means of the AE; and the circles representthe estimation obtained by means of the PE. The dashed line is thelight line.


SPW of the curved interface is just a little perturbed version ofthe plasmonic mode in the straight interface. The main mis-match with respect to the numerical results is at low frequen-cies (ω < 2:5 · 1015) and, in particular, in the estimation ofImðγÞ. Anyway, in this part of the spectrum the AE works verywell because values of x1 are sufficiently small. In particular,from this and other simulations it seems that a critical valueunder which the AE works well is x1 ¼ 1. An even better es-timation of the losses in the low part of the spectrum is givenby the PoyE, which matches very well the numerical resultsfor ω < 4 · 1015. Therefore, Figs. 4 and 5 show that, despite thesmall curvature radius, the proposed analytical estimationsoptimally describe the dispersion curve of the fundamentalSPW. The AE and the PE are complementary to each other,because the first works very well for ω < 2:5 · 1015 and thesecond in the remaining part of the spectrum. Moreover

the PoyE detects quite perfectly the small losses of valuesin the band of interest, which is the band in which lossesare sufficiently small. In this simulation the value of ReðγÞneeded by the PoyE is provided by the AE for ω <2:5 · 1015 and by the PE for the other frequencies, so that itis as accurate as possible.

Similar considerations can be made for the caseR ¼ 600 nm, whose results are shown in Figs. 6 and 7. In thisinstance, the PE succeeds in estimating ReðγÞ well in all thespectra considered except for ω < 0:8 · 1015 where a good es-timation is given by the AE, because x1 < 1 in this frequencyrange. Also in this case, the AE and the PE appear to be com-plementary to each other, at least as far as the estimation ofReðγÞ is concerned. The PoyE and the PE appear instead to becomplementary in the estimation of losses: The first matchesvery well the numerical results for ω < 4:0 · 1015, where itworks even better than the AE, while the second works well

Fig. 5. Same as Fig. 4 but the imaginary part is represented. Also,estimation obtained by means of the PoyE is shown (triangles).

Fig. 6. Same as Fig. 4 but with R ¼ 600nm.


in the remaining part of the spectrum. In this case, the value ofReðγÞ needed by the PoyE is provided by the AE for ω < 0:8 ·1015 and by the PE for the other frequencies.

For the sake of completeness we numerically calculate bymeans of Eq. (3) also the dispersion curves of the TMmodes inthe m-d interface (silver air) when R ¼ 100 nm. In Fig. 8 thefirst four modes are shown, and it is evident that they all haveunacceptable losses. This is a typical scenario for the m-d in-terface. Even the fundamental mode, the one with the lowestlosses, has ImðγfunÞ ≠ 0 when ω → 0. When considering them-d interface, the estimations previously discussed canwork well only with a great curvature radius (typicallyR > 1000 nm). That is not the case examined in this examplewhere the main assumption of the AE (jγfunj ≪ 1) is not ver-ified even when ω → 0, and the PE does not work properlybecause of the great losses that occur and that make the

curved m-d interface to be not simply a little perturbed ver-sion of the straight interface. Consequently, neither doesthe PoyE succeed in estimating losses, because no proper va-lues of ReðγÞ are provided by the AE or the PE.

7. CONCLUSIONSIn this paper we have derived some simple analytical formulasthat accurately describe the dispersion curve of the funda-mental SPW propagating around a dielectric-metal interface.We used three different approaches: the asymptotic forms ofthe Bessel functions, a perturbation model based on the exactknowledge of the plasmonic mode propagating along astraight interface, and a revised version of the Poyntingtheorem.

According to the numerical simulations, the first two esti-mations appear to be very effective even with a small curva-ture radius but in different frequency ranges, so that they arecomplementary. Some problems arise in the estimation of lowlosses of values, but they are overcome by means of the thirdtechnique that gives very good results in the band of interest,which is the useful one in which losses are acceptable.

We also simulated the case of a metal-dielectric interface,highlighting that with a small curvature radius even the funda-mental SPW has unacceptable losses that make it useless fornano-optics applications.

Fig. 7. Same as Fig. 5 but with R ¼ 600nm.

Fig. 8. (Color online) The dispersion functions of the first fourTM-SPW propagating around a curved silver-air interface ofradius R ¼ 100nm. The black bold line represents the FM. The thindashed line is the light line ω ¼ k0n1. Imðγ=RÞ ≠ 0 when ω ¼ 0even in the case of the fundamental mode (in this exampleγ=R ¼ i · 2:50 · 106).


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analytical approximations of the dispersion relation of the plasmonic modes propagating around a...

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