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arXiv:cond-mat/0410334v1 [cond-mat.mtrl-sci] 13 Oct 2004

Resonant transmission of light through finite chains of subwavelength holes

J. Bravo-Abad,1F.J. Garc´ ıa-Vidal,1and L. Mart´ ın-Moreno2

1Departamento de F´ ısica Te´ orica de la Materia Condensada, Universidad Aut´ onoma de Madrid, E-28049 Madrid, Spain

2Departamento de F´ ısica de la Materia Condensada, ICMA-CSIC, Universidad de Zaragoza, E-50009 Zaragoza, Spain

In this paper we show that the extraordinary optical transmission phenomenon found before in 2D

hole arrays is already present in a linear chain of subwavelength holes, which can be considered as the

basic geometrical unit showing this property. In order to study this problem we have developed a new

theoretical framework, able to analyze the optical properties of finite collections of subwavelength

apertures and/or dimples (of any shape and placed in arbitrary positions) drilled in a metallic film.

PACS numbers: 78.66.Bz, 42.25.Bs, 41.20.Jb, 73.20.Mf

After the discovery of extraordinary optical transmis-

sion (EOT) through 2D square arrays of subwavelength

holes in an optically thick metallic film [1], several works

have appeared in order to understand the basics of this

remarkable phenomenon. From the theoretical side, the

studies can be divided in those considering the simpler

1D analog of arrays of subwavelength slits [2, 3, 4] and

the 2D arrays of holes [5, 6, 7, 8]. Several of these works

explained EOT in terms of the existence of surface elec-

tromagnetic (EM) resonances, something pointed out by

the original experiments [1] and definitely corroborated

by recent experiments [9]. However, a question that still

remains open is what is the minimal system that shows

EOT, which is interesting both from the basic point of

view and for possible future applications.

In this letter we move a step forward in this direction

and consider the optical transmission properties of finite

chains of subwavelength holes, a basic structure with less

symmetry than the original 2D array which, up to our

knowledge, had not been considered before.

show that EOT is also present in these 1D finite systems.

As an important byproduct, we develop a formalism ca-

pable of treating the optical properties of even thousands

of indentations (with any shape and placed arbitrarily)

in metal films, something not possible with the present

numerical methods, which are restricted to just a few of

such indentations.

Here we

FIG. 1: A chain of circular holes in a metal film with thickness

h, with a schematic representation of the terms appearing in

the theoretical formalism presented in the text.

Let us first present the formalism, which is a non-trivial

extension of the simpler one developed for sets of 1D in-

dentations and that was successfully applied [10, 11] for

the understanding of enhanced transmission and beam-

ing of light in single apertures flanked by periodic corru-

gations [12, 13]. Here, we analyze the EM transmission

through a planar metal film (with finite thickness h and

infinite in the x-y plane) with a set of indentations at

both input and output interfaces. These indentations

may be either holes or dimples. Furthermore, each one

of them may have any desired shape and may be placed

in any position we wish. The only approximation in the

formalism is that the metal is treated as a perfect con-

ductor (ǫ = −∞). Then, within this approximation, the

method is virtually exact. We have demonstrated in pre-

vious works that this model captures the basic ingredi-

ents of the enhanced transmission phenomena, being even

of semi-quantitative value in the optical regime for good

conductors like silver or gold [14]. Additionally, results

obtained within the perfect conductor approximation are

scalable to different frequency regimes.

In our method, we assume a rectangular supercell, with

lattice parameters Lx and Ly, along the x and y axes,

respectively. This supercell may be real (if we are con-

sidering a bona-fide periodic system) or artificial, if the

system is finite. In this latter case, the limit Lx,Ly→ ∞

must be taken.

For an incident plane wave with parallel wavevector

?k0[15] and polarization σ0, the EM field at z = 0−(at

the metal interface in which radiation is impinging on)

can be written, in terms of the reflection coefficients r?kσ,

as

|?E(z = 0−) > = |?k0σ0> +

?

?kσ

r?kσ|?kσ >

(1)

−? uz× |?H(z = 0−) > = Y?k0σ0|?k0σ0> −

?

?kσ

r?kσY?kσ|?kσ >

where we have used the Dirac notation, and expressed

the bi-vectors?E = (Ex,Ey)Tand ?H = (Hx,Hy)T

(T standing for transposition) in terms of the EM

vacuum eigenmodes, |?kσ

these EM vacuum eigenmodes in real space are:

>.The expressions for

<

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? r|?kp >= (kx,ky)Texp(ı?k? r)/?LxLy|k|2and < ? r|?ks >=

(−ky,kx)Texp(ı?k? r)/?LxLy|k|2. The electric and mag-

netic fields in Eq. (1) are related through the admittances

Y?ks= kz/kωand Y?kp= kω/kz(for s- and p- polarization,

respectively), where kω= ω/c (ω is the frequency and c

the speed of light) and |k|2+ k2

according to Bloch’s theorem,?k =?k0+?G,?G being a

(supercell) reciprocal lattice vector.

In the region of transmission, the electric field at z =

h+can be expressed as a function of the transmission

amplitudes t?kσas |?E(z = h+) >=?

where the magnetic field can be readily calculated.

The EM fields inside the indentations can be written,

in terms of the expansion coefficients Aα,Bα, as:

z= k2

ω. Notice that,

?kσt?kσ|?kσ >, from

|?E(z) > =

?

α

?

α

|α >?Aαeıqzαz+ Bαe−ıqzαz?(2)

−? uz× |?H(z) > =|α > Yα

?Aαeıqzαz− Bαe−ıqzαz?

In the previous equations, α runs over all ”objects”,

which we define as any EM eigenmode considered in the

expansion. An object is, therefore, characterized by the

indentation it belongs to, by its polarization and by the

indexes related to the mode spatial dependence. All that

is required to be known are the electric field bi-vectors

|α >[16] and the propagation constants qzα associated

to the objects, as the admittance Yα= qzα/kω for TM

modes, while for TE modes Yα = kω/qzα. For inden-

tations with such simple cross sections as rectangular

or circular, the required expressions for |α > and qzα

can be found analytically [17]; otherwise, they can be

numerically computed [18]. By matching the EM fields

appropriately on all interfaces, we end up with a set of

linear equations for the expansion coefficients. We find

it convenient to define the quantities Eα ≡ Aα+ Bα

and E′

amplitudes of the electric field at the input and output

interfaces of the indentations, respectively. Objects cor-

responding to holes are represented by two modal ampli-

tudes, while objects in dimples require only one (as, in

this case, Aαand Bαare related through the boundary

condition at the dimple closed end). The set {Eα,E′

must satisfy:

α≡ Aαeiqzαh+ Bαe−iqzαh, which are the modal

α}

(ǫα+ Gαα)Eα+

?

β?=α

GαβEβ+ GV

αE′

α= Iα

(ǫγ+ Gγγ)E′

γ+

?

ν?=γ

GγνE′

ν+ GV

γEγ = 0 (3)

The different terms in these “tight-binding” like equa-

tions have a simple interpretation: Iα ≡ 2 <?k0σ0|α >

takes into account the direct initial illumination over ob-

ject α. ǫαis related to the bouncing back and forth of the

EM-fields inside object α and is ǫα= Yα(1+Φα)/(1−Φα)

where, for holes, Φα = exp(2iqzαh) and the same ex-

pression applies for dimples, but replacing h by Wα, the

depth of the dimple. The main difference between holes

and dimples is the presence of GV

coupling between the two sides of the indentation. For a

hole GV

The term Gαβ =?

trols the EM-coupling between indentations.

into account that each point in the object β emits EM

radiation, which is ”collected” by the object α. If the

system is periodic, Gαβ can be calculated through the

previous discrete sum, by including enough diffraction

waves. If the considered supercell is fictitious, the limit

Lx,Ly→ ∞ transforms the previous sum into an integral

over diffraction modes. It is then convenient to calculate

Gαβ through Gαβ =< α|ˆG|β >. The integral defining

the dyadicˆG(? r?,? r′

obtaining

α, which reflects the

α= 2YαΦα/(1 − Φα), while for a dimple GV

?kσY?kσ< α|?kσ ><?kσ|β > con-

α= 0.

It takes

?) =< ? r|ˆG|? r′

?> can be evaluated,

ˆGij(? r?,? r′

?) = g(d)δij+ (2δij− 1)∂2g(d)

∂di∂dj

(4)

where i,j can be either x or y, δij is Kronecker’s delta,

d ≡ kω|? r?−? r′

tional to the scalar free-space Green function associated

to the Helmholtz equation in 3D.

ˆG turns out to be the in-plane components of the

EM Green function dyadic associated to an homogeneous

medium in three dimensions [17]. As a technical note,

the calculation of Gαβ for objects in the same indenta-

tion from < α|ˆG|β > suffers from the problems associated

to the divergence ofˆG at |? r?−? r′

we evaluate it directly from its integral over diffraction

modes.

Therefore, our method reduces the calculation of EM

fields into finding the EM field distribution right at the

indentation openings, which is extremely efficient when

the openings cover a small fraction of the metal sur-

face. By projecting these fields into indentation eigen-

modes, convergence (as a function of number of eigen-

modes needed) is reached very quickly, especially in the

subwavelength regime. Notice that, although derived in

order to treat finite systems, Eq.(3) can also be used for

an infinite periodic array of indentations, by imposing

Bloch’s theorem on the set {Eα,E′

make use of when analyzing the case of an infinite linear

chain (see below).

Once the self-consistent {Eα,E′

straightforward to find all expansion coefficients, and

from them the EM fields in all space.

tal transmittance through subwavelength holes, we find

T =?

We have tested our formalism against published results

for two extreme systems: a single circular hole [22] and a

square array of square holes [7]. In both cases we recover

the known results, showing that our method is free from

numerical instabilities and that, in finite systems, there

are no spurious effects related to the Lx,Ly→ ∞ limit.

?| and g(d) = −ıkωexp(ıd)/(2πd) is propor-

?| → 0 [19, 20, 21], and

α}, something we will

α} are found, it is

For the to-

αRe[GV

αE∗

αE′

α]/Y?k0σ0.

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0.951 1.051.1

0

0.5

1

1.5

|Gs+ε|

|Gv|

Re(Gs+ε)

Im(Gs+ε)

(b)

0

1

2

3

4

5

Normalized-to-area transmittance

N=1

N=5

N=11

N=21

N=41

N=81

N=161

Infinite chain

0 200 400 600 800

N

2

3

4

Tmax

0.9511.05 1.1

λ/d

0

1

2

3

4

5

T

(a)

λ/d

FIG. 2: (a) Normalized-to-area transmittance (see text) ver-

sus λ/d for a linear array of N holes with a/d = 0.25 and metal

thickness h = a. Top inbox: transmittance peak value, as a

function of N, dashed line showing the value obtained for an

infinite chain. Bottom inbox: Normalized-to-area transmit-

tance versus λ/d calculated for an infinite 2D hole array with

the geometrical parameters defining the 1D arrays. (b) GS+ǫ

and GV (see text for the definition of these magnitudes) as a

function of λ/d.

In the rest of this letter we apply this formalism to

the study of linear chains of subwavelength circular holes

(see Fig.1). For proof of principle purposes, we choose

holes with radius a/d = 0.25 perforated in a metallic

film of thickness h/a = 1, d being the first-neighbor dis-

tance between holes. These are typical geometrical val-

ues in experiments in 2D hole arrays. Fig.2a shows the

evolution of the transmittance versus wavelength as a

function of the number of holes, N [23]. The incident

plane wave impinges normally, and is polarized with the

E-field pointing along the direction of the chain. For the

other polarization, the boost in transmittance is negli-

gible. In Fig.2a, the total transmission is normalized to

the hole area and then divided by the corresponding N.

The most interesting feature of these spectra is that, as

N is increased, a transmittance peak emerges at λ close

to d, showing that enhanced transmission is also present

in linear chains of subwavelength holes. The transmit-

tance peak value, Tmax, grows almost linearly with N

(for small N) eventually reaching saturation (see top in-

box of Fig.2a). In order to gain physical insight into

the origin of this transmission resonance, it is interest-

ing to analyze a simplified model for the infinitely long

chain. In this model, only one mode per hole is consid-

ered: the least evanescent mode with the electric field

pointing mainly along the chain axis. Therefore, for all

wavelengths, each hole behaves as a small dipole. In this

geometry, Bloch’s theorem implies Eα = E exp(ıkxαd),

and E′

equations (3) easily solvable. For the case we are consid-

ering (normal incidence, kx= 0) this procedure yields,

α= E′exp(ıkxαd), which renders the system of

[(ǫ + GS)2− G2

V]E = I(ǫ + GS) (5)

where GS= Gαα+?

Fig. 2b shows that the spectral location of the transmit-

tance peaks for the infinite chain coincide with the cuts

between |ǫ + GS| and |GV|, implying that the origin of

EOT relies on a resonant denominator. Therefore, EOT

is associated, as in the case of 2D hole arrays, to the ex-

citation of coupled surface EM resonances, which radiate

into vacuum as they propagate along the surface. This is

yet another instance of surface EM modes (in this case

a leaky mode) appearing in a perfect conductor due to

the presence of an array of indentations[24]. The results

for finite chains (top inbox of Fig.2a) show that this EM

resonance is characterized by a typical length, LD (for

the case considered LD ≈ 80d) . When the size of the

finite chain is smaller than LD, the resonance is not fully

developed and the transmittance is smaller than the one

obtained for an infinite chain. However, for large enough

N, the system can effectively be considered as infinite and

its associated transmittance peak approaches the asymp-

totic value (with a 1/N contribution coming from holes

located at a distance ≈ LD/2 from the chain ends). We

have found that LD is completely governed by the geo-

metrical parameter a/d, increasing as a/d decreases.

It is also interesting to compare the peak value calcu-

lated for an infinite linear array (T1D

with the one obtained for an infinite 2D hole array with

the same geometrical parameters (T2D

inbox of Fig.2a). This is, when going from a linear chain

to a 2D hole array, the transmittance per hole is only

increased by 25%. This means that the basic unit of the

extraordinary transmission phenomenon observed in 2D

hole arrays is the linear chain of holes, and that the 2D

hole array can be considered as an array of weakly cou-

pled 1D arrays. Although in this paper we only show

results for normal incident radiation, we have checked

that this conclusion remains valid provided the in-plane

component of the incident E-field points along the direc-

tion of the chain.

In order to study how the EOT in 2D hole arrays

develops from the EOT in linear chains, we have cal-

culated the transmittance of a collection of finite linear

chains separated by a distance d. Figure 3 renders the

normalized-to-area transmittance versus wavelength, for

different stripes formed by several (ranging from 1 to ∞)

β?=αGαβ, GV = GV

α, and ǫ = ǫα.

max≈ 4, see Fig.2a)

max≈ 5, see bottom

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0.951 1.051.1

0

1

2

3

4

Normalized-to-area transmittance

41x1

41x2

41x3

41x4

41x5

41x∞

λ/d

012

dc / d

345

2.4

2.8

3.2

3.6

Tmax

FIG. 3:

several stripes formed by M linear chains of 41 holes with

a/d = 0.25 and h/a = 1. M varies from 1 to 5. Also the

limiting case of M = ∞ is presented. Inset: Transmittance

peak value for the case of two linear chains (each one of them

with the same parameters as before), as a function of their

distance, dc.

Normalized-to-area transmittance versus λ/d for

chains of 41 holes. As shown in the figure, the stronger

effect appears when going from 41×1 to 41×3, suggesting

that the EM coupling between chains of subwavelength

holes is very short-ranged. The short-range nature of the

inter-chain interaction is more clearly seen in the inset to

Fig. 3, which renders the transmission peak value, Tmax,

through two finite linear chains as a function of the dis-

tance between them, dc. As this inset shows, for this set

of geometrical parameters, the chains are already prac-

tically uncoupled when dc= 3d, the maximum coupling

being for dc≈ d.

When linear chains are added up to the structure, the

transmittance peak shifts to longer wavelengths and its

maximum value increases. The increase in transmittance

is due to the fact that each chain takes advantage from

the re-illumination coming from other chains. The peak

redshift is related to the corresponding decrease of fre-

quency of the stripe surface EM mode, due to a reduc-

tion of its lateral confinement. More precisely, in this

case the stripe surface mode that couples to the inci-

dent plane wave with?k = 0 is essentially the sum, with

equal phases, of the single chain leaky modes.

finite 2D hole arrays, two EOT peaks originate from

the resonant coupling (via the holes) of the surface EM

modes, with a narrow transmission peak appearing close

to λ ≈ d (see bottom inbox of Fig.2). Notice that, in

stripes of chains of finite length only one peak is clearly

resolved, the second expected peak appearing as a shoul-

der in the transmission curve (see Fig. 3). This suggests

that finite size effects may prevent the development of

the narrowest peaks. Interestingly, the addition of linear

chains also provokes the birth of a minimum in the trans-

mittance spectrum, appearing at a wavelength slightly

smaller than d. Eventually, in 2D hole arrays, this mini-

In in-

mum leads to a “Wood’s anomaly”, appearing just when

a propagating Bragg diffracted wave becomes evanescent.

In this case the reciprocal lattice vectors involved are

(±1,0)2π/d, and Wood’s anomaly appears due to a di-

vergence in the EM density of states corresponding to

those wavevectors. Notice that, in a linear chain, as the

diffracted field contains a continuum of ky components,

the divergence in the density of states is smeared out; cor-

respondingly no Wood’s anomalies appear in this case.

To summarize, we have found that extraordinary opti-

cal transmission phenomenon is already present in a sin-

gle finite chain of subwavelength holes in metallic films.

For a chain, the transmittance per hole is comparable

to that found in 2D arrays; therefore, the single chain

can be considered as the basic entity of EOT and then

2D arrays can be seen as a collection of weakly EM cou-

pled chains. As a byproduct we have developed a new

theoretical framework which is able to treat the optical

properties of even thousands of indentations (holes or

dimples) placed arbitrarily in a metallic film. This nu-

merical tool will surely help in the design of new arrange-

ments of subwavelength objects pursuing specific optical

functionalities.

Financial support by the Spanish MCyT under

grant BES-2003-0374 and contracts MAT2002-01534 and

MAT2002-00139, and the EC under project FP6-NMP4-

CT-2003-505699 is gratefully acknowledged.

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