extraordinary optical reflection from sub-wavelength cylinder arrays

Extraordinary optical reflection fromsub-wavelength cylinder arrays

Raquel Gomez-MedinaDonostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4,

20018 San Sebastian, Spain

Nanophotonics and Metrology Laboratory, Swiss Federal Institute of Technology,1015 Lausanne, Switzerland

[email protected]

Marine Laroche and Juan Jose SaenzDepartamento de Fısica de la Materia Condensada and Instituto “Nicolas Cabrera”,

Universidad Autonoma de Madrid, E-28049 Madrid, Spain.

[email protected]

[email protected]

http://www.uam.es/mole

Abstract: A multiple scattering analysis of the reflectance of a periodicarray of sub-wavelength cylinders is presented. The optical propertiesand their dependence on wavelength, geometrical parameters and cylinderdielectric constant are analytically derived for boths- and p-polarizedwaves. In absence of Mie resonances and surface (plasmon) modes, and forpositive cylinder polarizabilities, the reflectance presents sharp peaks closeto the onset of new diffraction modes (Rayleigh frequencies). At the lowestresonance frequency, and in the absence of absorption, the wave is perfectlyreflected even for vanishingly small cylinder radii.

© 2006 Optical Society of America

OCIS codes:(050.1960) Diffraction theory ; (290.4210) Multiple scattering

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(C) 2006 OSA 1 May 2006 / Vol. 14, No. 9 / OPTICS EXPRESS 3730#68096 - $15.00 USD Received 14 February 2006; revised 7 April 2006; accepted 13 April 2006

mailto:[email protected]



http://www.uam.es/mole

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sion,” Sol. State Comm.84, 457 (1992).30. P.M. Morse and H. Feshbach,Methods of Theoretical Physics(McGraw-Hill, New York, 1953), Chap. 7.31. R.E. Collin and W.H. Eggimann,“Dynamic Interaction Fields in a Two-Dimensional Lattice,” IRE Trans. on

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1. Introduction

The study of light scattering from periodic structures has been a topic of interest during thelast century. Already in 1902, Wood [1, 2] reported remarkable effects (known as Wood’sanomalies) in the reflectance of one-dimensional (1D) metallic gratings. Two different types ofanomalies were definitely identified by Fano [3]. One is associated to the discontinuous change


of intensity along the spectrum at sharply defined frequencies and was already discussed byRayleigh [4]. The other is related to a resonance effect. It occurs when the incoming wavecouples with quasi-stationary waves confined in the grating. The nature of the confined wavesdepends on the details of the periodic structure [5] and is usually associated to surface plasmonpolaritons in shallow metallic gratings, standing waves indeep grating grooves [5] or guidedmodes in dielectric coated metallic gratings [6].

Since the observation of enhanced transmission through a metallic film perforated by a 2Darray of sub-wavelength holes [7], there has been a renewed interest in analyzing and under-standing the underlying physics of both reflection and transmission “anomalies” in both 2Dhole [8, 9, 10, 11, 12, 13, 14] and 1D slit [15, 8, 16, 17, 18] arrays. Although the enhancedtransmission is commonly associated to the excitation of surface plasmons [7, 8, 9], dynamicaldiffraction resonances [10, 19, 20] are also invoked as the origin of the effect. Recent theo-retical and experimental works put forward that the excitation of any surface plasmons wasnot required [11, 20]. Consequently, there remains some controversy surrounding the transmis-sion mechanism [21, 22, 23, 20, 24]. In this work we discuss the physics behind the (Babinet)complementary problem: the extraordinary reflection from aperiodic array of sub-wavelengthscatterers.

In absence of resonant surface modes (or surface plasmons for metallic particles) the scatte-ring cross section of subwavelength-sized particles is very small [25, 26]. However, in a periodicarray of small particles, the coupling of the scattered dipolar field with diffraction modes mayinduce ageometricresonance close to the onset of new propagating modes (i.e. close to theRayleigh frequencies). Following a multiple scattering approach [27, 28, 29], we analyticallyderive the resonance conditions as a function of the geometry and the polarizabilities of eachindividual scatterer. As we will show, in absence of absorption, it is possible to have a perfectreflected wave even for vanishingly small scatterers.

2. Scattering theory for s-polarized waves (Electric field parallel to the cylinder axis)

Let us consider an infinite set of parallel cylinders with their axis along thez-axis, relativedielectric constantε and radiusa much smaller than the wavelength. The cylinders are locatedat rn = nDux = xnux (with n an integer number). For simplicity, we will assume incomingplanewaves with wave vectork0⊥uz ( i.e. the fields do not depend on thez-coordinate)

k0 = ksinθ ux +kcosθ uy≡Q0ux +q0uy, (1)

andk = ω/c.Let us first consider an incoming wave with the electric field parallel to the cylinder axis

(s-polarized wave, see Fig. 1),E = E(r)uz = E0eiQ0xeiq0yuz. The scattered field from a givencylindern, can be written as [25, 26]:

Escattn (r) = αzzEin(rn)k

2G0(r , rn) (2)

where

αzz≈ πa2(ε−1)[1− i

π4

(ka)2(ε−1)]−1

, (3)

G0(r , rn) = (i/4)H0(k|r− rn|) is the free-space Green function (H0 is the Hankel function), andEin(rn) is the incident field on the scatterer. Since for a periodic array Ein(rm) = Ein(r0)eiQ0xm,the total scattered field can be written as

Escatt(r) =(αzzEin(r0)

)k2G(r) (4)


Q0D/2

z

yx

D/ c

R

Q0=0

Q0=0.8 /D

E

k

Q0D/2

z

y

D/ c

R

Q0=0

Q0=0.8 /D

Q0=0

Q0=0.8 /D

E

k

cD

x

Fig. 1. (s-polarization) Calculated reflectanceR in a frequencyω versus in-plane wavenumberQ0 = (ω/c)sin(θ). The reflectance along the vertical lines is shown in the inset.

where the total Green functionG(r) is given by:

G(r)≡+∞

∑n=−∞

eiQ0xnG0(r , rn) =1D

∞

∑m=−∞

ei(Q0−Km)x(

i2qm

eiqm|y|)

(5)

with Km = 2πm/D andk2 = q2m+(Q0−Km)2.

Multiple scattering effects manifest themselves in the actual incidentfield on each cylinder[27, 29]: Ein(r0) is given by the incoming plane wave plus the scattered fields from othercylinders, i.e.

Ein(r0) = E0 +αzzk2 ∑

m6=0

Ein(rm)G0(r0, rm) = E0 +αzzk2Ein(r0)Gb

=(1−αzzk

2Gb)−1

E0 (6)

whereGb = limr→r0 [G(r)−G0(r , r0)]. The calculation ofGb involves the sum of a (poorlyconverging) series of Hankel functions. The convergence can be improved by using the wellknown result∑n=1e−byn/n =− ln(1−e−by) [30, 31]. Finally,Gb is found to be given by

Gb = i

{1

2Dq0−

14

}+

12D

∞

∑m=1

(i

qm+

iq−m−

2Km

)+

12π

(ln

{kD4π

}+ γE

). (7)

The total scattered field (eq. 4) can then be rewritten as

Escatt(r) = αzzE0k2G(r) (8)

where all multiple scattering effects are included in the renormalized polarizability,αzz,

k2αzz =

(1

k2αzz−Gb

)−1

. (9)


Fig. 2. (s-polarization) (a) Plot ofℜ{Gb} andℜ{1/(k2αzz} versus frequency along theconstantQ0 line in Fig. 1 (Q0 = 0.8π/D). The crossing points (open circles) correspond todifferent resonant frequenciesω0m. (b) Calculated reflectanceRversusω. The inset showsa zoom-out of theω01 resonance. Dashed lines corresponds to the approximate expressiongiven in eq. 15 with no fitting parameters.

In order to calculate the transmittance/reflectance we consider the far-field limit (|y| → ∞)where only propagating diffraction orders (or channels) contribute to the scattered power. Thetotal field, for bothsandp-polarizations (see below) can be written in the general form

ψ(r) = ψ0eiQ0xeiq0y +ψ0Prop

∑m

i4π f (qm,q0)

2Dqmei(Q0−Km)xeiqm|y| (10)

where 4π f (qm,q0) is the scattering amplitude and the sum runs only over modes having|Q0−Km| < k. The transmittanceT (reflectanceR), defined as the ratio between transmitted(reflected) power and incoming power is shown to be

T = 1−2ℑ{4π f (q0,q0)}

2Dq0+

1D2

Prop

∑n

(|4π f (qn,q0)|

2

4qnq0

)(11)

R =1

D2

Prop

∑n

(|4π f (−qn,q0)|

2

4qnq0

). (12)

For s-polarization (4π f (qm,q0)≡ αzzk2) the scattering amplitude is isotropic and

R =ℑ{G(0)}

2Dq0|αzzk

2|2. (13)

sinceℑ{G(0)}= ∑Propn

12Dqn

.


In absence of absorption (T +R= 1), ℑ{1/(k2αzz) =−ℑ{G0(0)}, we obtain

R =ℑ{G(0)}

2Dq0

(ℜ2{

1k2αzz

−Gb

}+ℑ2{G(0)}

)−1

(14)

(whereℜ2{x} ≡ (Real{x})2 andℑ2{x} ≡ (Imag{x})2). Figure 1 presents the calculated re-flectance in aω vs. Q0 map (frequency versus in-plane wave numberQ0 = ksinθ ). For sim-plicity, we have considered a real dielectric constant (ε > 1) independent of the frequency (theresults correspond to(2πa/D)2(ε − 1) ≈ 4/9). The physics of the reflectance/transmittancecan be understood from a simple argument (see Fig. 2): For small cylinders andε > 1,ℜ{1/(k2αzz)}> 0 is large and dominates the renormalized polarizability. However, approach-

ing the threshold of a new propagating channel (i.e.ω → ω(−)m = c|Q0−Km|) qm goes to zero.

Then, the contribution of the lowest evanescent mode toℜ{Gb} outweighs all the others andℜ{Gb} ≈ (2Dqm)−1 diverging at the threshold. The precise compensation of these two largeterms atω = ω0m (see Fig. 2(a) ) gives rise to a geometric resonance. Very close to eachresonance, the reflectance along the vertical lines in Fig. 1(i.e. for a givenQ0) can then beapproximated as

R(ω . ωm)≈ Rmax

1

γ2

1−

√ω2

m−ω20m

ω2m−ω2

2

+1

−1

(15)

whereRmax= (2Dq0ℑ{G(0)})−1 andγ = ℑ{G(0)}/ℜ{

1/(k2αzz

)}. As shown in Fig. 2(b),

the reflection resonances present typical asymmetric Fano line shapes: The reflectance presentssharp maximaR= Rmax at frequenciesω = ω0m / ωm. Just at the onset of a new diffractionchannel, i.e. at the Rayleigh frequenciesω = ωm, the reflectance goes to zero. Forω = ω01,i.e. at the lowest resonance frequency, there is a perfect reflection (R= 1) even for vanishinglysmall cylinders (although, in these extreme cases, the resonance widthΓ ≈ γ(ω2

m−ω20m)/ωm

goes to zero). Notice that for metallic cylinders or strips (with α < 0) there will be no sharpresonances. This is consistent with the Babinet complementary system of a periodic array ofslits [15, 8, 16, 17, 18] where, forp-polarized waves, sharp transmittance peaks only appear fordeep enough gratings (i.e. when the phase shift inside the slit changes the sign of the effectivepolarizability).

3. Scattering theory for p-polarized waves (Magnetic field parallel to the cylinder axis)

Let us now consider an incoming wave with the magnetic field parallel to the cylinder axis(p-polarized wave),H = H(r)uz = H0eiQ0xeiq0yuz. The scattered (magnetic) field from a givencylinder [25, 26] can be cast in the form:

Hscattn (r) = −

{αyy∂xHin(r)

}r=rn

∂xG0(r , rn)−{

αxx∂yHin(r)}

r=rn∂yG0(r , rn) (16)

where

αxx = αyy≈ 2πa2 ε−1ε +1

[1− i

π4

(ka)2 ε−1ε +1

]−1

(17)

αxy = αyx = 0 (18)

and Hin(rn) = Hin(r0)eiQ0xn is the incident field on the scatterer. Forp-polarization, multiplescattering effects manifest themselves in the actual incident magnetic fieldgradienton each


Q0D/2

H

k

R

D/ c

Q0=0.6 /D

Q0=0.2 /D

z

yxc

D

Fig. 3. (p-polarization) Calculated reflectanceR in a frequencyω versus in-plane wavenumberQ0 = (ω/c)sin(θ). The reflectance along the vertical lines is shown in the inset.

cylinder, ∇Hin(r)∣∣r=rn

:

limr→r0

∂xHin(r) = iQ0H0−αyy ∂xHin(r)∣∣r=r0

∂ 2x Gb = iQ0H0(1+αyy∂ 2

x Gb)−1

(19)

where∂ 2x Gb≡ limr→r0 ∂ 2

x [G(r)−G0(r , r0)]. The resulting series can be written as [31]:

∂ 2x Gb = −

12D

∞

∑m=1

{i(Km−Q0)

2

qm+

i(Km+Q0)2

q−m−2Km−

k2

Km

}

−k2

4π

(ln

{kD4π

}+ γE−

12

)+

16

πD2 + i

(k2

8−

Q20

2Dq0

)(20)

∂ 2y Gb = −

12D

∞

∑m=1

{iqm+ iq−m+2Km−

k2

Km

}

−k2

4π

(ln

{kD4π

}+ γE +

12

)−

16

πD2 + i

(k2

8−

q20

2Dq0

)(21)

(notice that∇2Gb +k2Gb = 0). The total scattered field can now be written as

Hscattn (r) = −iH0αyyQ0∂xG(r)− iH0αxxq0∂yG(r) (22)

where all the multiple scattering effects have been included in a renormalized polarizability

tensor←→α :

αxx =(1+αxx∂ 2

y Gb)−1 αxx (23)

αyy =(1+αyy∂ 2

x Gb)−1 αyy. (24)


The transmittance/reflectance are given by the general equations 11 and 12 with a non-isotropicscattering amplitude

4π f (qm,q0) = αyyQ0 (Q0−Km)+ αxxq0qm. (25)

Figure 3 presents the calculated reflectance in aω −Q0 map. (The results correspond tonon-absorbing cylinders with(2πa/D)2(ε − 1)/(ε + 1) ≈ 8/9). The physics behind the re-flectance presents significant differences with respect tos-polarization. Sharp peaks in the re-flectance (which now appear forε > 1 or ε < −1) are associated to the resonant couplingof electric dipoles pointing along they-axis which lead to the divergence ofℜ{∂ 2

x Gb} ≈−(Q0−Km)2(2Dqm)−1 at the Rayleigh frequencies (in contrastℜ{∂ 2

y Gb} remains finite). Inabsence of absorption, the reflectance at the lowest resonant frequency can be very large but, incontrast withs-waves, strictly less than 1.

4. Conclusion

Similar resonances to the ones discussed above appear in very different contexts under the la-bel of Fano or Feshbach geometric resonances [3, 32, 33]: Geometric resonances had beendiscussed in the context of electronic transport in waveguides [34], ultracold atomic collisions[37] and light-atom interactions in confined geometries [35, 36, 38]. In general, Fano-Feshbachresonances occur when the energy (frequency) of the incoming wave is tuned to the energy (fre-quency) of a quasi-bound-state (QBS). These QBS may have a (multiple scattering -dynamicaldiffraction-) geometricalorigin or can be an internal property of each scatterer. Fromthe dis-cussion above, reflectance resonances, for bothsandp polarized waves, have a geometrical ori-gin for dielectric cylinders. The existence of particle surface modes or plasmons would reflectitself in a resonant behavior of the bare polarizabilities (for p-polarized fields) andℜ{1/αii}would present sharp maxima and minima around each internal resonant frequency. Surfacemodes would then induce new peaks in the reflectance or, when the surface resonance fre-quency is close to a Rayleigh frequency, they would mix with geometrical resonances leadingto more complex reflectance patterns.

To summarize, in this work we have discussed the optical properties of an array of sub-wavelength Rayleigh cylinders. In absence of particle resonant modes, the cross-section ofa single non-resonant cylinder can be extremely small and resonant effects are associated togeometrical resonances. We have shown that, for non-absorbing scatterers, it is possible tohave a perfect reflected wave even for vanishingly small cylindrical radii. We believe that ourstudy of reflectance resonances provide a new physical insight into the general mechanisms oflight interactions with periodic structures of sub-wavelength objects.

Acknowledgements

We thank S. Albaladejo, R. Carminati, J. Garcıa de Abajo, J.J. Greffet, L. Froufe-Perez andM. Thomas for interesting discussions. This work has been supported by the Spanish MCyT(Ref. No. BFM2003-01167) and the EU Integrated Project “Molecular Imaging” (EU contractLSHG-CT-2003-503259).


extraordinary optical reflection from sub-wavelength cylinder arrays

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