of 5
Quantitative determination of optical transmission through subwavelength slit arrays in Ag films:
Role of surface wave interference and local coupling between adjacent slits
D. Pacifici, H. J. Lezec,
*
and Harry A. Atwater
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
J. Weiner
IRSAMC/LCAR, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
and IFSC/CePOF, Universidade de São Paulo, Avenida Trabalhador São-Carlense, 400-CEP 13566-590 São Carlos, São Paulo, Brazil

Received 11 February 2008; published 7 March 2008

Measurement of the transmitted intensity from a coherent monomode light source through a series of
subwavelength slit arrays in Ag films, with varying array pitch and number of slits, demonstrates enhancement

suppression

by factors of as much as 6

9

when normalized to the transmission efficiency of an isolated slit.
Pronounced minima in the transmitted intensity are observed at array pitches corresponding to

SPP
,2

SPP
, and
3

SPP
, where

SPP
is the wavelength of the surface plasmon polariton

SPP

. The position of these minima
arises from destructive interference between incident propagating waves and pi-phase-shifted SPP waves.
Increasing the number of slits to four or more does not increase appreciably the per-slit transmission intensity.
A simple interference model fits well the measured transmitted intensity profile.
DOI:
10.1103/PhysRevB.77.115411
PACS number

s

: 42.25.Fx, 73.20.Mf, 78.67.

n
I. INTRODUCTION
Since the first experimental report of “extraordinary opti-
cal transmission” through subwavelength hole arrays,
1
con-
siderable theoretical effort has been devoted to interpreting
the essential physics of the process in both hole
2
7
and
slit
8
13
arrays. Roughly speaking, two points of view have
emerged. Proponents of the first school
2
,
3
,
8
,
14
have inter-
preted the transmission spectrum as excitation of the delocal-
ized surface plasmon Bloch modes and identified transmis-
sion
maxima
with resonant excitation of these modes at
wavelengths equal to integer multiples of the array pitch. The
second school
11
13
,
15
,
16
has emphasized interference between
incident and surface waves and localized coupling between
adjacent structures. Adherents of this approach predict trans-
mission
minima
at the same positions where the Bloch mode
excitation predicts maxima.
Experimental studies subsequent to the initial report
1
demonstrated a number of unexpected features. Spectral
transmission measurements
17
revealed that, normalized to
transmission of a single aperture, suppression, as well as en-
hancement, was a characteristic property of hole and slit ar-
rays. Interferometric studies
18
20
showed that the contribu-
tion of transient diffracted surface modes is as important as
the surface plasmon polariton

SPP

guided mode in the im-
mediate vicinity of the subwavelength object. The experi-
mental setups of Refs.
1
and
17
consisted of an incoherent,
broad-band light source dispersed through a scanning spec-
trophotometer and focused on fixed-period subwavelength
hole and slit arrays. Transmitted intensity was detected in the
far field as a function of the scanned wavelength. In that
work, the spectral resolution and coherence length of light
incident on the arrays therefore depended on instrumental
parameters, and these, in turn, can affect the position and
shape of the measured spectral features. Furthermore, the
frequency dependence of the dielectric constant of Ag and
other real metals is non-negligible in the range of typical
wavelength scans from 450 to 900 nm.
II. EXPERIMENT
In order to test the predictions of the two interpretive
schools and to remove measurement ambiguities, we have
undertaken a series of high-resolution measurements of the
transmission through a series of slit arrays in which the spec-
tral source is coherent, monomode, and at fixed frequency.
Rather than scan the light source wavelength, we increment
in 5 nm steps the array pitch of a series of slit arrays. The
transmission measurement setup consists of a

0
=514.5 nm,
5 mW, TEM
00
light beam from an Ar ion laser aligned to the
optical axis of an inverted microscope. The beam is focused
at normal incidence onto the sample surface through the mi-
croscope condenser and polarized TM

magnetic
H
-field
component parallel to the long axis of the slits

. Light inten-
sity transmitted through each slit array is then gathered by a
50

microscope objective with a numerical aperture of 0.45
and detected with a liquid-nitrogen-cooled, charged-coupled
device

CCD

array detector. Light intensity is obtained by
integrating the signal over the entire region of interest in the
CCD image and subtracting the background originating from
electronic noise. Per-slit transmission intensities are obtained
by correcting the transmitted intensity for the calculated col-
lection efficiency of the microscope objective lens and nor-
malizing the transmitted intensity for each series of gratings
to the intensity collected from a single-slit structure. The
series of slit arrays were milled with a focused-ion beam

Ga
+
ions, 30 keV

in a 200 nm thick layer of silver evapo-
rated onto a flat fused-silica microscope slide. The layout of
slit-array structures consisted of a matrix of 9 rows and 140
columns. Each row was indexed by the slit number
N
in the
array and varied from
N
=1 to 9. Each column was indexed
by the array pitch
p
starting from the first column at
p
=150 nm and incremented by 5 nm with each successive
column. Thus, the pitch varied over a range from
p
=150 to 845 nm, from less than

SPP
to greater than 2

SPP
.
In subsequent measurements, the array pitch was extended to
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/115411

5

©2008 The American Physical Society
115411-1
3

SPP
. Each slit was milled 50 nm wide, 200 nm deep, and
10

m long, as shown in Fig.
1
. The structured silver layer
was covered by a second microscope slide, optically con-
tacted to the silver surface by index-matching fluid

n
=1.46

so that the index change at the dielectric-silver inter-
face was identical at both the input

incident

and output

transmitted

planes. The transmitted intensity of each suc-
cessive array along a given row was recorded in the far field
by the CCD as the sample was stepped using an
X
-
Y
trans-
lation stage. The results are summarized in Fig.
2
. Taking
into account the collection efficiency of the microscope ob-
jective and the far-field angular distribution of the slit grating
diffraction modes, we define

=

H
N

2
/

H
1

2
as the ratio of
the magnetic field intensity at the output aperture of each slit
in an array of
N
slits to the magnetic field intensity at the
output aperture of an isolated slit.
III. EXPERIMENTAL RESULTS
Figure
2
plots

vs array pitch for
N
=1–9

except
N
=5,
omitted due to defective fabrication

. The results show that
the transmission intensity for all arrays exhibits very similar
behavior with transmission dropping to a minimum of

0.1

at an array pitch equal to

SPP
, then rising to a broad
maximum of

6

before repeating similar behavior around
2

SPP
. The position of the minima is in accord with earlier
predictions
11
,
12
and simulations
13
,
21
and at variance with
theory
2
,
3
predicting transmission maxima at

SPP
.
The wavelength

SPP
was calculated from the usual for-
mula for the guided wave on a flat,
n
SPP
=


m

d

m
+

d
,
metal surface,
22

SPP
=

0
n
SPP
,

1

where

0
is the incident wavelength,

m
and

d
are the di-
electric constants of the metal and adjacent dielectric, respec-
tively, and
n
SPP
is the effective surface index of refraction. In
the present experiments, the dielectric constant of the struc-
tured silver sample was measured directly by ellipsometry at

0
=514.5 nm and determined to be

m
=−9.3+0.18
i
. The di-
electric constant of the fused-silica substrate is

d
= +2.13,
and therefore

SPP
=309

1 nm.
Figure
3
plots the maximum and minimum values of

for
each of the
N
grating series. Enhancement above single-slit
transmission up to a factor of

6 is observed as
N
increases
up to
N
=4. Above
N
=3, adding additional grating elements
FIG. 1. Two typical elements in the overall structure layout.
Panel

a

shows
N
=4,
p
=450 nm. Panel

b

shows
N
=4,
p
=700 nm. Each slit is focused-ion beam milled through a 200 nm
thick silver layer. Dimensions of each slit are 50 nm wide and
10

m long.
FIG. 2.

Color online

Normalized transmission intensity

vs
grating pitch

in micrometers on the lower abscissa and normalized
to

SPP
on the upper abscissa

for a series of slit arrays
N
=1–9.
Gratings with
N
=5 were omitted due to defective fabrication. The
wavelengths

0
,

1
, and

SPP
are, respectively, the free-space wave-
length, the wavelength in fused silica

n
=1.46

, and the wavelength
of the surface plasmon polariton.
FIG. 3.

Color online

Normalized per-slit transmission intensity
maxima and minima vs the number of slit elements in array
N
. The
labels 1–3 refer to the transmission intensity maxima and minima
from left to right shown in Fig.
2
. The solid lines are guides for the
eyes.
PACIFICI
et al.
PHYSICAL REVIEW B
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2008

115411-2
to the array does not significantly enhance the transmission.
Similar behavior is observed for the transmission minima.
IV. DISCUSSION OF RESULTS
These measurements support the view that transmission
enhancement is dominated by nearest-neighbor slit scattering
with the strength of the local interaction effectively screening
contributions from more distant array elements. If Bloch sur-
face modes, delocalized over the full extent of the array,
played a dominant role, one would expect the per-slit inten-
sity to increase with the number of elements in the array. The
imaginary part of
k
SPP
determines the propagation distance
along the surface limited by absorptive loss in the silver
film.
22
The estimated propagation distance is 8.5

m, and
therefore absorptive losses are negligible over the spatial ex-
tent of the grating structures.
Since the positions of transmission suppression and en-
hancement as a function of period are essentially indepen-
dent of
N
, we can analyze the mechanism responsible for
modulation by concentrating on the simplest case
N
=2. The
normalized per-slit transmission intensity

of an array of slit
pairs with varying pitch
p
is shown in Fig.
4

b

. The intensity

is plotted on a linear scale as a function of
p
for devices
milled into a Ag film of thickness
t
=300 nm

hollow circles

in addition to the Ag film of thickness
t
=200 nm described
earlier

solid circles, replotted from Fig.
3

. Comparison of
the two data sets shows that the


p

modulation varies little
with
t
and is therefore essentially governed by the interaction
between the two slits mediated by surface waves running
along both facets of the structured metal film. Periodic
minima are measured at slit-slit distances corresponding to
integer multiples of

SPP

p
=
n

SPP
,
n
=1,2,...

. This obser-
vation is consistent with recent theoretical predictions for a
two-slit system
15

albeit for a structure with only one SPP-
sustaining surface

.
A. Transmission model
We have developed a simple model for


p

. Figure
4

a

shows the essential idea.
H
0
designates the
H
-field amplitude
of the incident wave and

H
0
the amplitude component dif-
fracted into the surface plasmon polariton mode at slit 1.
This mode with

k
SPP

=2
/

SPP
propagates to the right along
the surface until it reaches slit 2. The phase accumulated by
the surface wave at slit 2 is
e
ik
SPP
p
, where
p
is the distance
between the two slits. At slit 2, the SPP reconverts to a
propagating mode with efficiency


and interferes with the
incident field. An identical process with the roles of slits 1
and 2 interchanged takes place at slit 1. The superposition
fields at the entrance side of slits 1 and 2,
H
0
+


H
0
e
ik
SPP
p
,
are transmitted to the exit side with some overall efficiency
T
. A similar process takes place on the exit side of the film.
Counterpropagating surface waves are again launched by dif-
fractive scattering at the slit exits and interfered with the
directly propagating mode at the opposite slit exit location.
The total
H
-field amplitude at the output aperture of each slit
is then given by the following expression
H
N
=2
=

H
0
+


e
ik
SPP
p
H
0

T

1+


e
ik
SPP
p

=
H
0
T

1
+


e
ik
SPP
p

2
.

2

The
H
field at the output aperture of a
single
slit is given by
H
N
=1
=
H
0
T
.

3

This interference process yields a net transmission intensity

normalized to that of a single slit

given by


1


p

=

1+


0

0


2
+2

0

0

cos

2

SPP
p
+
2
,

4

where

0

0

=




, and the phase
=arg




is the phase
associated with the SPP
propagating wave conversion, ex-
clusive of the phase accumulated along the surface,

2
/

SPP

p
. Refining the model by taking into account mul-
tiple surface wave reflections at the slits results in the fol-
lowing closed-form expression:
FIG. 4.

Color online

Panel

a

shows a schematic of the inter-
ference model used to fit the two-slit transmission intensity. Panel

b

shows the two-slit transmission intensity normalized to single-
slit transmission as a function of slit-slit separation and plotted on a
linear scale. The data

filled circles

show the transmission profile
for a 200 nm thick Ag film; open circles show similar data for a
300 nm thick Ag film. The dashed line shows a fit using the first-
order interference model of Eq.

4

blue curve

and the solid line
shows a fit using the infinite-order model of Eq.

5

red curve

.
QUANTITATIVE DETERMINATION OF OPTICAL
...
PHYSICAL REVIEW B
77
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2008

115411-3




p

=

1+


0

0


2
−2

0

0

cos

2

SPP
p
+
−2
.

5

The analysis and resulting expression is analogous to that
obtained for a Fabry–Perot cavity containing propagating
modes. Here, the cavity “mirrors” are the slits with an effec-
tive reflection coefficient


and the modes contained in the
cavity are surface plasmon polariton modes. Equation

5

is
fit to the experimental results
Fig.
4

b


by fitting param-
eters,

SPP
,

0

0

, and
.
B. Comparison of model and measurements
Afitof


p

to the combined set of experimental trans-
mission data for both
t
=300 nm and
t
=200 nm is shown in
Fig.
4

b


solid red curve

. Very good agreement is obtained
using fitting parameters

SPP
=307

2 nm,

0

0

=0.23

0.01, and
=

0.03. The best-fit value for

SPP
agrees with

SPP
=309 nm, calculated from Eq.

1

, within
experimental uncertainty. The shape of


p

is reminiscent
of the transmission characteristics of a “lossy” two-mirror
Fabry–Perot resonator of free spectral range,

=

SPP
, and
with full width at half maximum,


=0.29

SPP
. The “fi-
nesse” of the surface wave cavity is then defined by
F
=

/


=3.4. The Fabry–Perot profile suggests the influence
of multiple surface wave reflections at the slit sites but with
rather low reflectivity. It is important to emphasize that po-
sitions of minima and maxima between a conventional
Fabry–Perot cavity and the surface wave cavity are reversed,
and this reversal is due to the
phase shift between the
surface waves and the incident wave. A plot of the first-order
model


1


p

is also included in Fig.
4

b


dashed blue
curve

using the fitting parameters above. The essential pro-
file of the normalized transmission as a function of
p
is al-
ready well reproduced by


1


p

. The first-order fit suggests
that the formation of transmission minima is predominantly
controlled by interference at the slit openings rather than by
the presence of higher order multiple reflections. Assuming
that the amplitudes

0
,

0

conversion efficiencies are equal,
the best-fit value of their product implies that slits convert
incident light to surface waves with an efficiency of almost
50%. The positioning of the minima at
p
=
n

SPP

n
=1,2,...

is due to
=
and is in agreement with the find-
ings of Ref.
15
. It is also consistent with recent calculations
showing that diffracted propagating and evanescent surface
modes are
out of phase.
23
A simple physical explanation
for this phase shift involving surface currents, induced by the
standing wave
H
field at the surface, charging the slits, has
recently been proposed.
24
V. SUMMARY AND CONCLUSIONS
In summary, we have measured the transmitted far-field
intensity through a series of subwavelength slit arrays as a
function of array pitch and have determined that the
mini-
mum
per-slit transmission at the array output facet occurs for
an array pitch equal to an integer number of wavelengths of
the surface plasmon polariton. We have also determined that
the per-slit transmitted intensity does not increase apprecia-
bly above an array size greater than
N
=3. These findings
support the view that the transmission profile is controlled by
two sequential processes: interference between the incident
propagating mode and the principal evanescent surface mode

SPP

on the input side of the film, followed by interference
between the emerging light and the SPP on the exit side of
the film. Furthermore, at least in the case of slit arrays, the
surface wave interaction is essentially confined to adjacent
structures rather than characterized by excitation of collec-
tive Bloch modes delocalized over the entire array. Finally,
in contrast to the minima at pitches equal to integer multiples
of

SPP
, we find no evidence of regular recurring features

maxima or minima

at array pitches equal to integer mul-
tiples of the propagating wavelengths at

0
or

1
.
ACKNOWLEDGMENTS
Support from the Caltech Kavli Nanoscience Institute, the
National Science Foundation under Grant No. DMR
0606472 and the use of facilities at the Center for Science
and Engineering of Materials, an NSF Materials Research
Science and Engineering Center at Caltech, are gratefully
acknowledged. Support from the Ministère délégué à
l’Enseignement Supérieur et à la Recherche under the pro-
gramme ACI-“Nanosciences-Nanotechnologies,” the Région
Midi-Pyrénées

SFC/CR 02/22

, and FASTNet

HPRN-CT-
2002-00304

EU Research Training Network, as well as
from the research foundation FAPESP of the State of São
Paulo, Brazil is also gratefully acknowledged.
*
Present address: Center for Nanoscale Science and Technology,
National Institute of Standards and Technology, 100 Bureau
Drive, Stop 6203, Gaithersburg, MD 20899-8412.
jweiner@irsamc.ups-tlse.fr
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