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Fano-Like Resonance in Cylinders Including Nonlocal Effects

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2014 Chinese Phys. Lett. 31 087302

(http://iopscience.iop.org/0256-307X/31/8/087302)

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CHIN.PHYS. LETT. Vol. 31, No. 8 (2014) 087302

Fano-Like Resonance in Cylinders Including Nonlocal Effects *

LI Liang-Sheng(李粮生)**, YIN Hong-Cheng(殷红成)Science and Technology on Electromagnetic Scattering Laboratory, Beijing 100854

(Received 17 March 2014)We investigate the optical response of a metallic wire calculated from the classical electromagnetic theory. TheDrude (local) approach is compared with the semi-classical hydrodynamical theory calculations that reveal theFano-like resonances of subsidiary peaks originated from the nonlocality. The bulk plasma resonances containingthe nonlocal effects could be depressed by increasing the dissipation, while the blue shift of the surface localizedplasma resonances could be enhanced by increasing the Fermi velocity.

PACS: 73.20.Mf, 78.67.Pt, 78.68.+m DOI: 10.1088/0256-307X/31/8/087302

Inspired by the progress in chemical synthesis andnanofabrication techniques, researchers are now ableto explore and manipulate the interactions betweenlight and artificial plasmonic media.[1−5] The typi-cal optical characters of plasmonic media are almostdominated by the collective response of the valenceelectrons.[6−8] This collective excitation of localizedplasmons leads to numerous phenomena, such as lo-cal field enhancement and Fano-like resonance, allow-ing, for instance, detection of single-molecule bind-ing events by large spectral shift in the resonancefrequency.[9−11] In the design of plasmonic devices,usually the Drude (simplest) conductivity is adequateto model the complex permittivity of some metals andsemiconductors. However, recent experiments and cal-culations have shown the impact of intrinsic nonlo-cality in the optical resonance of noble metal, wherethe local model fails to predict the peak resonancefrequency shift of surface plasmon modes.[12,13] Thisshift could be explained by the semi-classical hydrody-namical approach, essentially capturing the physicalnature of nonlocality.[12,14] Another feature of the hy-drodynamical model is that a series of regularly spacedsubsidiary resonant peaks appear above the plasmafrequency, since longitudinal bulk plasmon modes areexcited.[15] Nonetheless, a systematic study of sub-sidiary resonance with the hydrodynamical model isstill lacking.

In this Letter, we consider light scattering in de-tail by a nonmagnetic (𝜇r = 1) cylinder with radius(𝑎) embedded in a vacuum with given optics constants𝜀r = 1 and 𝜇r = 1. It is convenient to introduce the di-mensionless backward scattering cross section (𝑄BS),forward scattering cross section (𝑄FS) and scatteringefficiency (𝑄sca) to characterize the scattering proper-ties of the cylinder. These quantities for wires can bedefined by formulas[10]

𝑄BS =2

𝜋𝑞

𝑛=+∞∑𝑛=−∞

(𝑖)𝑛𝑒−𝑖𝜋𝑛/2𝑏𝑛

2, (1)

𝑄FS =2

𝜋𝑞

𝑛=+∞∑𝑛=−∞

(−𝑖)𝑛𝑒−𝑖𝜋𝑛/2𝑏𝑛

2, (2)

and

𝑄sca =2

𝑞

𝑛=+∞∑𝑛=−∞

|𝑏𝑛|2, (3)

in the definition of 𝑞 = 𝑘𝑎 and 𝑘 = 𝜔/𝑐, where 𝑐 is thespeed of light in a vacuum. To accurately simulate thescattering properties of the cylinder by using the Mietheory, we kept 60 multipolar terms in the cylinder-centered expansions of the electromagnetic fields andset the relative residual error tolerance in the numer-ical solution to 10−10.

Prior to a discussion of the scattering properties ofcylinders within the nonlocal effects, we consider thedielectric function only including local effects. Thisallows us to distinguish between the effects that areinherent to the local model and those which are associ-ated with the nonlocal contributions. In the design ofmetamaterials, the frequency dispersive permittivity,often approximated by using the Drude (local) model,has the form as follows:

𝜀 = 1−𝜔2p

𝜔(𝜔 + 𝑖𝛾), (4)

with a plasma frequency 𝜔p and an electron collisionfrequency 𝛾. The scattering amplitudes 𝑏𝑛, for thecylinders satisfying the local model, are defined bythe TE-mode (𝐸||𝑧) and TM-mode (𝐸⊥𝑧) formulas,respectively,[10] where

𝑏TE𝑛 =

𝐽 ′𝑛(𝑞)𝐽𝑛(𝑚r𝑞)−𝑚r𝐽

′𝑛(𝑚r𝑞)𝐽𝑛(𝑞)

𝐻 ′𝑛(𝑞)𝐽𝑛(𝑚r𝑞)−𝑚r𝐽 ′

𝑛(𝑚r𝑞)𝐻𝑛(𝑞),

(5)

𝑏TM𝑛 =

𝑚r𝐽′𝑛(𝑞)𝐽𝑛(𝑚r𝑞)− 𝐽 ′

𝑛(𝑚r𝑞)𝐽𝑛(𝑞)

𝑚r𝐻 ′𝑛(𝑞)𝐽𝑛(𝑚r𝑞)− 𝐽 ′

𝑛(𝑚r𝑞)𝐻𝑛(𝑞),

(6)

and 𝑚r =√

𝜀/𝜀0 is a relative complex refractive in-dex. Here the optical properties of cylinders obtainedfrom Eqs. (1)–(6), for typically 𝛾/𝜔p = 0.01, couldcover most characters of physical interest, as shown inFig. 1. The identification of plasmonic resonances re-quires surface plasmons that are excited by an incidentwave. As a result, in TE polarization light scatteringby a homogeneous wire, plasmonic resonances do not

*Supported by the National Defense Research Foundation under Grant No 109750014.**Corresponding author. Email: liliangsheng@gmail.com© 2014 Chinese Physical Society and IOP Publishing Ltd

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CHIN.PHYS. LETT. Vol. 31, No. 8 (2014) 087302

occur in 𝑄BS, 𝑄FS and 𝑄sca. However, for TM polar-ization waves, the surface plasma resonances appearas peaks at the Mie frequency 𝜔/𝜔p = 1/

√2 in 𝑄BS,

𝑄FS and 𝑄sca.

0.00

0.02

0.04

0.06

0.08

0.10

-18

-16

-14

-12

-10

-8.0

-6.0

-4.0TE

-17

-15

-13

-11

-9.0

-7.0

-5.0-4.0

0.00

0.02

0.04

0.06

0.08

0.10

0.00

0.02

0.04

0.06

0.08

0.10

-10-9.0-8.0-7.0-6.0-5.0-4.0-3.0-2.0-1.0

0 2 4 6 8 10

-35

-30

-25

-20

-15

-10

-24

-20

-16

-12

-8.0

-4.0-2.0

0 2 4 6 8 10

-15

-12

-9.0

-6.0

-3.0

0.0

γ/ωp=0.01

TEγ/ωp=0.01

TEγ/ωp=0.01

TMγ/ωp=0.01

TMγ/ωp=0.01

TMγ/ωp=0.01

Log(QBS)Log(QBS)

Log(QFS)Log(QFS)

Log(Qsca)Log(Qsca)

ωpa⊳c

ωpa⊳c

ωpa⊳c

ω⊳ωp ω⊳ωp

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 1. Optical responses of a wire within the usual Drudemodel. Backward scattering cross section, forward scatter-ing cross section and scattering efficiency as a function of𝜔/𝜔p and 𝜔p𝑎/𝑐. Panels (a)–(c) and (d)–(f) correspondto TE and TM polarizations, respectively.

We now turn to consider the interaction of lightwith wires in the nonlocal response regime, and use asemi-classical hydrodynamic model for the jellium re-sponse of the free electron gas.[14] Here it is assumedthat the nonlocal responses of the naoncylinders aredominated by the nonlocality induced by free elec-tron gas, while the bound electrons only contributeto local responses.[12] This assumption should be sat-isfied when the interactions between free electrons arestronger than the original dipole-dipole interactionsbetween bound electrons. Thus in the hydrodynamicmodel, the dielectric properties of wires are charac-terized by both the usual Drude transverse dielectricfunction in Eq. (6) and the hydrodynamic longitudinaldielectric function

𝜀L(𝑘, 𝜔) = 1−𝜔2p

𝜔(𝜔 + 𝑖𝛾)− 𝛽𝑘2, (7)

where 𝛽 = (3/5)𝑣2F, and 𝑣F is the Fermi velocity. Thelongitudinal plasma waves obey the dispersion rela-tion 𝜀L(𝑘L, 𝜔) = 0. For the TE-mode, no bulk plas-mon can be excited, so that nonlocality has no impacton this mode.[12] So we will only deal with the TM-mode for which the excitation of longitudinal modesoccurs. In nonlocal cases, the traditional Maxwell’sboundary conditions are not sufficient to determinethe scattering amplitudes. To avoid this underdeter-mined problem, additional boundary conditions mustbe introduced at the cylindrical interface.[15] Then, thescattering amplitudes of wires with nonlocal effects aregiven by[14]

𝑏NL𝑛 =

𝑚r𝐽′𝑛(𝑞)𝐽𝑛(𝑚r𝑞)−𝐽 ′

𝑛(𝑚r𝑞)𝐽𝑛(𝑞)−𝑎𝑛𝐽𝑛(𝑞)

𝑚r𝐻 ′𝑛(𝑞)𝐽𝑛(𝑚r𝑞)−𝐽 ′

𝑛(𝑚r𝑞)𝐻𝑛(𝑞)−𝑎𝑛𝐻𝑛(𝑞),(8)

where

𝑎𝑛 =𝑛2

𝑘L𝑎

𝐽𝑛(𝑘L𝑎)

𝐽 ′𝑛(𝑘L𝑎)

𝐽𝑛(𝑚r𝑞)(𝑚r

𝑞− 1

𝑚r𝑞

). (9)

When 𝑏NL𝑛 = 0, the scattering amplitudes could be re-

duced to the local TM-mode resulted in Eq. (5). Weexplore the backward scattering properties of wireswith nonlocal effects by using the illuminated hydro-dynamic model. Figure 2(a) shows 𝑄BS, for 𝛾/𝜔p =0.01 and 𝑣F/𝑐 = 1/300, as a function of reduced fre-quency 𝜔/𝜔p and plasma size parameter 𝜔p𝑎/𝑐. InFigs. 2(b)–2(d), the comparisons between Drude andnonlocal theory for the same parameters are shown forthree different plasma size parameters 0.001, 0.01 and0.1. The results of nonlocal scattering theory exhibitthe main (first) peaks shift towards the high frequencyside, where these peaks come from the excitation ofsurface plasmons. When 𝜔p𝑎/𝑐 increasing, the blue-shift distance of the main peak decreases and tendsto zero. Furthermore, the subsidiary peaks are ob-served above the plasma frequency, where these peaksoriginate from the excitation of bulk plasmons. When𝜔p𝑎/𝑐 is continuously increased, the number of sub-sidiary peaks also increases, while the amplitude ofthese peaks related to the main peaks decreases, asshown in Figs. 2(c) and 2(d).

0 2 4 6 8 10-40

-35

-30

-25

-20

-15

0 2 4 6 8 10

-30

-25

-20

-15

0 2 4 6 8 10-20

-15

-10

-5

(a)

Nonlocal

Drude (TM)

(b)

Nonlocal

Drude(TM)

(c) (d) Nonlocal

Drude (TM)

0 2 4 6 8 100.00

0.02

0.04

0.06

0.08

0.10

-35

-30

-25

-20

-15

-10Nonlocal

γ/ωp=0.01

γ/ωp=0.01

Log(QBS)

Log(Q

BS)

Log(Q

BS)

ωpa⊳c/⊲

γ/ωp=0.01

ωpa⊳c/⊲

γ/ωp=0.01

ωpa⊳c/⊲

ω⊳ωp ω⊳ωp

ω⊳ωpω⊳ωp

ωpa⊳c

Fig. 2. (a) Backward scattering cross section obtainedfrom the nonlocal hydrodynamic model as functions of𝜔/𝜔p and 𝜔p𝑎/𝑐. Backward scattering cross section versusincident frequency for plasma size parameter 𝜔p𝑎/𝑐: (a)0.001; (b) 0.01; and (c) 0.1. The red dashed curves showthe result of the Drude model.

To further explore the nonlocal responses of cylin-ders, we render 𝑄FS and 𝑄sca versus 𝜔/𝜔p and theplasma size parameter in Figs. 3(a) and 4(a), respec-tively. When 𝜔p𝑎/𝑐 = 0.001, the main peaks of 𝑄FS

and 𝑄sca are also blue-shifted while subsidiary peaksdo not show up. In Figs. 3(c) and 4(c), subsidiarypeaks appear and the resonances show the clearlyasymmetry profiles. In contrast to 𝑄BS, the curvesof 𝑄FS and 𝑄sca corresponding to the nonlocal theoryat 𝜔p𝑎/𝑐 = 0.1 almost overlap the numerical resultsof the Drude theory, as shown in Figs. 3(d) and 4(d).

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CHIN.PHYS. LETT. Vol. 31, No. 8 (2014) 087302

0 2 4 6 8 10-30

-25

-20

-15

-10

-5

0 2 4 6 8 10-20

-15

-10

-5

0 2 4 6 8 10-15

-10

-5

0

0 2 4 6 8 10

0.00

0.02

0.04

0.06

0.08

0.10 (a)

-24

-19

-14

-9.0

-4.0-2.0Nonlocal

(b) Nonlocal

Drude(TM)

(c)

Nonlocal

(d) Nonlocal

Drude(TM)

Drude(TM)

γ/ωp=0.01

γ/ωp=0.01

Log(QFS)

Log(Q

FS)

Log(Q

FS)

Log(Q

FS)

ωpa⊳c/⊲

γ/ωp=0.01

ωpa⊳c/⊲

γ/ωp=0.01

ωpa⊳c/⊲

ω⊳ωp ω⊳ωp

ω⊳ωpω⊳ωp

ωpa⊳c

Fig. 3. (a) Forward scattering cross section obtained fromthe nonlocal hydrodynamic model as functions of 𝜔/𝜔p

and 𝜔p𝑎/𝑐. Forward scattering cross section versus inci-dent frequency for different choice of 𝜔p𝑎/𝑐: (a) 0.001; (b)0.01; and (c) 0.1. The red dashed curves show the resultof the Drude model.

0 2 4 6 8 10-20

-15

-10

-5

0 2 4 6 8 10

-10

-5

0

0 2 4 6 8 10-9

-6

-3

0

0 2 4 6 8 100.00

0.02

0.04

0.06

0.08

0.10 (a)

-15

-13

-10

-7.5

-5.0

-2.5

0.0Nonlocal

(b)

Nonlocal

Drude(TM)

(c) Nonlocal

Drude(TM)

(d)

Nonlocal

Drude(TM)

γ/ωp=0.01

γ/ωp=0.01

Log(Qsca)

Log(Q

sca)

Log(Q

sca)

Log(Q

sca)

ωpa⊳c/⊲

γ/ωp=0.01

ωpa⊳c/⊲

γ/ωp=0.01

ωpa⊳c/⊲

ω⊳ωp ω⊳ωp

ω⊳ωpω⊳ωp

ωpa⊳c

Fig. 4. (a) Scattering efficiency obtained from the nonlo-cal hydrodynamic model as functions of 𝜔/𝜔p and 𝜔p𝑎/𝑐.The red dashed curves show the result of the Drude model.

-24

-21

-18

-15

-24

-21

-18

-15

-24

-21

-18

-15

0 2 4 6 8 10

-24

-21

-18

-15

-12

(a)

(b)

(c)

(d)

γ/ωp=0.01

Log(Q

BS)

ωpa⊳c/⊲

ω⊳ωp

vF/c=1/600

vF/c=1/300

vF/c=1/150

vF/c=1/100

Fig. 5. The value of 𝑄BS as a function of 𝜔/𝜔p at𝜔p𝑎/𝑐 = 0.01 and 𝛾/𝜔p = 0.01 for different ratios 𝑣F/𝑐:(a) 1/600; (b) 1/300; (c) 1/150; and (d) =1/100. The ma-genta, royal, and yellow solid disks highlight the shift ofmain, second, and third peaks, respectively.

In Fig. 5, we compare the backward scattering re-

sults for different Fermi velocities. For the small ratiobetween the Fermi velocity and the speed of light in avacuum (𝑣F/𝑐), the amplitude of the main peak (ma-genta disk) is higher than the second (royal disk) andthird peaks (yellow disk). As 𝑣F/𝑐 increases, the sec-ond and third peaks shift to higher frequencies dueto the fact that the effective longitudinal wave vectorbecomes smaller. For the ratio 𝑣F/𝑐 ≥ 1/150, the am-plitude of the second peak becomes higher than themain peaks, and we emphasize that this trend, theo-retically reported here, might be confirmed in futureexperiments. Additionally, when 𝑣F/𝑐 increases, thesubsidiary peaks of 𝑄FSand 𝑄sca also blue shift andamplitudes are increased, where these results are sim-ilar to 𝑄BS.

1.9 2.0 2.1-20

-19

-18

-17

-16

1.5 1.6 1.7 1.8-12

-11

-10

-9

-8

1.5 1.6 1.7 1.8-8

-7

-6

-5

(a)

Log(

BS)

p

a

(b)

p(c)

Log(

sca)

p

/ p

/ p

/ p

/ p

c

Fig. 6. Backward scattering cross section (a), forwardscattering cross section (b) and scattering efficiency (c) asa function of 𝜔/𝜔p at 𝜔p𝑎/𝑐 = 0.01 for different frequencyratios 𝛾/𝜔p: (black solid line) 0.01; (red dashed line) 0.02;(blue dotted line) 0.05; and (dark cyan dash-dotted line)0.1.

1.5 1.6 1.7 1.8 1.9 2.00

1

2

3

sca (

10

-5)

p

/ p=0.01 Fano-fit =1

sca

Fig. 7. Scattering efficiency in the vicinity of the secondpeak described by the Fano lineshape, Eq. (10) (red solidline).

The effect of dissipation is investigated in Fig. 6,where 𝑄BS, 𝑄FS and 𝑄sca are shown as a functionof 𝜔/𝜔p, for different values of frequency ratio 𝛾/𝜔p,in the vicinity of the second peak. Figure 6 clearlydemonstrates that those resonances are quite robust

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CHIN.PHYS. LETT. Vol. 31, No. 8 (2014) 087302

against dissipation, so that they are not completelydepressed by the presence of energy loss. Specif-ically, 𝑄FS and 𝑄sca display Fano-like resonanceswith an asymmetric line shape. The Fano-like res-onance, originated from the interaction between de-structive and constructive interferences, is character-ized as follows:[16−18]

𝐹 = 𝜅(𝑥+ 𝜍)2

(𝑥2 + 1), (10)

where 𝜍 and 𝜅 are the asymmetry and amplitude pa-rameters, respectively. The scaling parameter is de-fined by 𝑥 = 𝛼(𝜔 − 𝜔F), where 𝜔F is the Fano reso-nance frequency and 𝛼 is the scaling parameter. InFig. 6, the Fano fitting line well describes the behav-ior of the numerical scattering efficiency and confirmsthe existence of Fano-like resonance. In contrast tothe pioneer works, this result demonstrates that theFano-like resonance can be contributed by not onlythe excited surface plasmons, but also the bulk excitedplasmons. However, here the Fano-like resonances areunconventional, which arises from the interference ofdifferent electromagnetic modes with the same multi-pole moment.[11] The identification of unconventionalFano resonances requires examining the Fano profilein the dipolar moments as shown in Fig. 7.

In conclusion, we have presented a classical elec-tromagnetic study of the optical response of a plas-monic cylinder with the Drude and hydrodynamicalmodel. For thin cylinders, the main peaks of 𝑄BS,𝑄FS and 𝑄sca, due to the surface plasmon excitation,are blue shifted from its classical position, and sub-sidiary peaks, due to the bulk plasmon excitation, oc-cur above the plasma frequency. These results withnonlocal effects show that the Fano-like resonance canbe identified in subsidiary peaks of 𝑄FS and 𝑄sca. Thestrength of nonlocal effects is dependent on the phys-

ical properties of the wire and on the polarization ofthe incident field. These effects increase with decreas-ing the dissipation of the cylinder, and hence are mostprominent for a perfect dielectric wire (𝛾 → 0). Ourresults are not only valid for the cylinder consideredhere but also can be extended to general geometricbodies within nonlocal effects.

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