of 9
Surface-wave interferometry on single
subwavelength slit-groove structures
fabricated on gold films
F. Kalkum [1] , G. Gay, O. Alloschery, J. Weiner
IRSAMC/LCAR
Universit ́
e Paul Sabatier, 118 route de Narbonne,
31062 Toulouse, France
jweiner@irsamc.ups-tlse.fr
H. J. Lezec
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology,
Pasadena, California 91125 USA
Centre National de la Recherche Scientifique, 3, rue Michel-Ange, 75794 Paris cedex 16,
France
Y. Xie, M. Mansuripur
College of Optical Sciences, University of Arizona, Tucson, Arizona 85721 USA
Abstract:
We apply the technique of far-field interferometry to measure
the properties of surface waves generated by two-dimensional (2D) single
subwavelength slit-groove structures on gold films. The effective surface
index of refraction
n
surf
measured for the surface wave propagating over
a distance of more than 12
μ
m is determined to be
n
surf
=
1
.
016
±
0
.
004,
to within experimental uncertainty close to the expected bound surface
plasmon-polariton (SPP) value for a Au/Air interface of
n
spp
=
1
.
018. We
compare these measurements to finite-difference-time-domain (FDTD)
numerical simulations of the optical field transmission through these
devices. We find excellent agreement between the measurements and the
simulations for
n
surf
. The measurements also show that the surface wave
propagation parameter
k
surf
exhibits transient behavior close to the slit,
evolving smoothly from greater values asymptotically toward
k
spp
over the
first 2-3
μ
m of slit-groove distance
x
sg
. This behavior is confirmed by the
FDTD simulations.
© 2007 Optical Society of America
OCIS codes:
(050.1220) Diffraction and gratings ; Apertures (050.2770) Diffraction and grat-
ings ; Gratings (230.7400) Optical devices ; W
aveguides, slab (240.6680) Optics at surfaces ;
Surface plasmons (240.6690) Optics at surfaces ; Surface
waves
(260.3910) Physical optics ;
Metals, optics of
References and links
1. Permanent address: Physikalisches Institut, Universit ̈
at Bonn, Wegelerstrasse 8, 53115 Bonn, Germany.
2. G. Gay, O. Alloschery, B. Viaris de Lesegno, J. Weiner, and H. Lezec, “Surface Wave Generation and Propagation
on Metallic Sub
wavelength Structures Measured by Far-Field Interferometry,” Phys. Rev. Lett.
96
, 213901-1–4
(2006).
3. G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O’Dwyer, J. Weiner and H. J. Lezec, “The optical response of
nanostructured surfaces and the composite diffracted evanescent
wave
model,” Nature Phys.
2
, 262-267 (2006).
#77673 - $15.00 USD
Received 30 November 2006; revised 13 February 2007; accepted 14 February 2007
(C) 2007 OSA
5 March 2007 / Vol. 15, No. 5 / OPTICS EXPRESS 2613
4. H. Raether,
Surface Plasmons on Smooth and Rough Surfaces and on Gratings
, (Springer-Verlag, Berlin, 1988).
5. P. Lalanne and J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nature
Phys.
2
, 551-556 (2006).
6. G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Comment on ‘The
Response of Nanostructured Surfaces in the Near Field’,” Nature Phys.
2
, 792 (2006).
7. G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O’Dwyer, M. Sukharev, and T. Seideman, “Surface quality and
surface
waves on
subwavelength-structured silver films,” Phys. Rev. E
75
, 016612-1–4 (2007).
8. Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, “Transmission of light through slit apertures in metallic
films,” Opt. Express
12
, 6106-6121 (2004).
9. Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in
a thick metallic film,” Opt. Express
13
, 4485-4491 (2005).
10. A. Zakharian, J. Moloney, and M. Mansuripur, “Transmission of light through small elliptical apertures,” Opt.
Express
12
, 2631-2648 (2004).
11. P. Johnson and R. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B
6
, 4370-4379 (1972).
12. Present experimental results. Value of
n
surf
determined from data in the far-zone
13. Present FDTD simulations. Value of
n
surf
determined from FDTD simulations in the far-zone
14. Measurements from Ref. [2] predominantly in the transient near-zone.
15. Measurements from Ref. [3] predominantly in the transient near-zone.
16. Value of
n
surf
determined from FDTD simulations of Refs. [7, 6] in the far-zone.
17. H. J. Lezec and T. Thio, “Diffracted evanescent
wave
model for enhanced and suppressed optical transmission
through sub
wavelength hole arrays,” Opt. Express
12
, 3629-3651 (2004).
18. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of
sub-
wavelength slits in metallic hosts,” Opt. Express
14
, 6400-6413 (2004).
1. Introduction
Recent measurements [2, 3] have characterized surface waves arising from optical excitation of
a series of subwavelength slit-groove structures fabricated on silver films. The amplitude, wave-
length and phase of these surface waves have been measured over a slit-groove distance of a few
microns. After an initial rapid amplitude decrease extending to
3
μ
m from the slit edge, the
interference fringe persists over several microns with a near-constant amplitude and a contrast

0
.
3. The “near-zone” of rapidly changing surface wave character indicates initial transient
phenomena, while the longer-range “far-zone” settling to constant amplitude and contrast is the
signature of a guided mode surface plasmon polariton (SPP). The measured interference fringe
wavelength
λ
surf
=
814
±
8 nm resulted in the determination of an effective index of refraction
n
surf
=
1
.
047, significantly higher than expected from conventional theory[4],
n
spp
=
1
.
015.
The question naturally arose as to whether this result is simply a consequence of interference
fringe sampling over an interval predominantly in a transient near-zone peculiar to silver films,
or was related to the specific surface properties of the silver films deposited on fused silica [5],
or may indicate a more
generic
phenomenon related to the transient properties of surface waves,
generated by subwavelength slits, within 2-3 wavelengths of the edge of origin. Subsequent in-
vestigation of the physical-chemical properties of the silver films confirmed that they were free
of surface contaminants, and FDTD simulations showed excellent agreement with the experi-
mental results in silver films [6, 7].
In order to explore these surface waves further we have carried out a series of experiments
similar to those already reported but using evaporated gold films instead of silver. We per-
formed only “output-side” experiments[2] because they yield a phase modulated interference
fringe less susceptible to noise than the amplitude modulation signal of the “input-side” ex-
periments [3]. In addition we have compared the measurements to detailed field amplitude and
phase maps generated by FDTD numerical solutions [8, 9] to the Maxwell equations in the
vicinity of the slit-groove structure. Both experiments and FDTD simulation show that the
surface wave exhibits transient properties in wavelength and amplitude in the near-zone. This
transient behavior can be interpreted in terms of surface modes all of which dissipate beyond
the near-zone except for
k
spp
, the bound surface mode.
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Fig. 1. Diagram showing interfering
wavefronts, optical path difference between
E
t
and
E
g
,
and far-field detection. Slit dimensions are 100 nm width, 20
μ
m length. Groove dimen-
sions are 100 nm width, 100 nm nominal depth, 20
μ
m length. The evaporated gold layer
deposited o
na1mm
fused silica substrate has a 400 nm nominal thickness.
Fig. 2. Goniometer setup for measuring far-field light intensity and angular distributions.
See text for description. Stabilized laser source is tuned to 852 nm.
2. Experimental Setup
The structures were fabricated with a focused ion beam (FIB) station as described previously
in [3]. The experiments were performed on the same home-built goniometer setup used in the
silver studies. For convenience, Figs. 1,2 of [2] are reproduced here to show the principle of
the measurement and the schematic arrangement of the apparatus. Output from a temperature
stabilized diode laser source tuned to 852 nm, is modulated at 850 Hz by a mechanical chop-
per, injected into a monomode optical fibre, focused and linearly polarized (TM polarization,
H-field parallel to the slit long axis) before impinging perpendicularly on the matrix of struc-
tures mounted in a sample holder. The beam waist diameter and confocal parameter of the
illuminating source are 300
μ
m and 33 cm, respectively. The sample holder itself is fixed to a
precision x-y translator, and individual slit-groove structures of the 2-D matrix are successively
positioned at the laser beam waist. A photodiode detector is mounted at the end of a 200 mm
rigid arm that rotates about the center of the sample holder. A stepper motor drives the arm at
calibrated angular increments of 1.95 mrad per step, and the overall angular resolution of the
goniometer is

4 mrad. The photodetector output current passes to a lock-in amplifier refer-
enced to the optical chopper wheel. Data are collected on a personal computer that also controls
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Fig. 3. Left panel: Points are the measured fringe phase
φ
(
x
sg
)
as a function of slit-groove
distance
x
sg
. The straight-line fit is
φ
0
=
k
surf
x
sg
+
φ
int
with constant slope
k
surf
and in-
tercept
φ
int
. Gap in the data in left and right panels of this figure and in the left and right
panels of Fig. 4 is due to defective structures in this interval. Right panel: Fringe amplitude
η
0

1
/
2
C
as a function of
x
sg
where
C
is the interference fringe contrast.
the goniometer drive.
3. Results and Analysis
3.1. Measurements
With the detector rotated perpendicular to the structure plane (
θ
=
0 in Figs. 1, 2) the expression
for the normalized detected intensity
I
/
I
0
as a function of slit-groove distance
x
sg
is given by
I
I
0
1
+
η
2
o
+
2
η
o
cos
(
k
x
x
sg
+
φ
int
)
(1)
where
η
0
is related to the fringe contrast
C
through
C
=
2
η
o
1
+
η
2
o
(2)
In the argument of the cosine term
k
x
=
2
π
/
λ
eff
relates the propagation parameter of the surface
wave to the effective surface wavelength, and the “intrinsic phase”
φ
int
is any phase contribution
not directly due to the propagation path length
x
sg
. It may be associated with phase shifts at
the slit or groove structures. The interferometry measurements were carried out on 4 separate
substrates, each substrate containing about 50 structures in which the the slit-groove separation
was systematically varied from 50 nm to more than 12
μ
m in increments of 50 nm. The left
panel of Fig. 3 plots the measured interference fringe phase against the slit-groove distance
x
sg
.
The fitted value for
k
surf
=
2
π
/
λ
surf
determines the effective surface index of refraction
n
surf
,
and extrapolation to zero slit-groove distance
x
sg
determines the intrinsic phase
φ
int
. The fit
from the left panel of Fig. 3 yields
n
surf
=
λ
0
λ
surf
=
1
.
016
±
0
.
004 and
φ
int
=
0
.
35
π
±
0
.
01
π
(3)
The right panel of Fig. 3 plots
η
0
, the amplitude factor of the interference term in Eq. 1, as a
function of slit-groove distance. From Eq. 2 this amplitude factor can be expressed in terms of
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Fig. 4. Left panel: Fringe phase difference
φ
(
x
sg
)
φ
0
as a function of slit groove distance
x
sg
. Deviation in the near-zone from
φ
0
indicates that early, transient fringe oscillation
is slightly greater and approaches
φ
0
asymptotically in the far-zone beyond
2
μ
m slit-
groove distance. Right panel: Same data as shown in left panel but on an expanded scale of
slit-groove distances to emphasize the curvature in
φ
(
x
sg
)
φ
0
in the near-zone.
the fringe visibility or contrast
C
as
η
0
=
1
1
C
2
C

1
2
C
,
C

1
(4)
Although the fringe contrast shown in the right panel of Fig. 3 is about a factor of 5 below
that measured for silver structures[2], the same rapid fall-off in the near-zone,
x
sg

0
3
μ
m,
followed by a near-constant contrast beyond is observed. This contrast behavior is evidence of
surface-wave transient phenomena in the near-zone. More evidence of this transient behavior
is shown in left and right panels of Fig. 4 that plot
φ
(
x
)
φ
0
vs.
x
sg
, where
φ
0
=
k
surf
x
sg
+
φ
int
is the best-fit linear trace in the left panel of Fig. 3 over the range of slit-groove distances out
to 12
μ
m. A pronounced departure from the asymptotic value of
φ
0
is evident in the near-zone
of slit groove distances, indicating that the fringe oscillation frequency is initially somewhat
greater than the SPP value and smoothly decreases to it beyond the near-zone.
3.2. Numerical simulations
The time-dependent Maxwell equations are solved numerically using an FDTD non-conformal
grid refinement method in Cartesian space coordinates. The methodologyis described in greater
detail in Refs [8, 9, 10].
Figure 5 shows a field map of the z-component of the electric field amplitude, and Fig. 6
shows a field map of the y-component of the magnetic field amplitude. The
|
E
z
|
map clearly
shows the dipolar charge distribution concentrated at the corners of the slit on the input and
output planes of the structure. These corner charge concentrations result from currents induced
on the input side of the gold film by the magnetic field components
|
H
y
|
shown in Fig. 6. The
incident light propagates from below through the fused silica substrate onto the gold film and
through the 100 nm wide, 400 nm thick slit. The incident light is TM polarized and the guided
mode propagating along the
±
z
direction within the slit sets up a standing wave resulting in a
high
|
E
z
|
,
|
H
y
|
amplitudes at the output plane. The groove is at the output side of the gold film;
and, in the simulations depicted, the distance between the center of the slit and the center of
the groove is 3.18
μ
m. The absolute value of the z-component of the electric field amplitude
|
E
z
|
just above (or just below) the gold film is proportional to the surface charge density at the
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Fig. 5. FDTD simulations for slit-groove center-to-center distance of 3.18
μ
m, slit and
groove widths 100 nm, groove depth 100 nm and gold film thickness 400 nm. Map shows
|
E
z
|
, z-components (perpendicular to input and output facets) of the electric field amplitude
in the vicinity of the input and output surfaces.
Fig. 6. FDTD simulations for slit-groove center-to-center distance of 3.18
μ
m, slit and
groove widths 100 nm, groove depth 100 nm and gold film thickness 400 nm. Map shows
|
H
y
|
, y-components (parallel to the slit and groove long axis) of the magnetic field ampli-
tude in the vicinity of the input and output surfaces.
film surface. Note that on the output side surface, in the region between the groove and the slit,
the surface wave excited at the left edge of the slit travels to the groove, is reflected from the
groove’s right edge, then interferes with itself. The standing wave is clearly visible on and near
the output side surface. Within the slit, on the vertical walls, E
z
is fairly strong as well. Here,
however, E
z
is parallel to the metallic surface, and its presence within the skin depth of the slit
walls does not signify the existence of surface charges; instead, the E
z
-field in this region is
responsible for the surface currents that carry the charges back and forth between the entrance
and exit facets of the slit. Figure 6 shows the magnitude of the magnetic field
|
H
y
|
over the same
region as Fig. 5. Interference fringes between the incident and reflected beams on fused silica
substrate are clearly visible. Note also the interference fringes between the excited evanescent
waves (mainly SPP) and the incident beam on the entrance facet of the gold film adjacent the
substrate. Inside the slit,
|
H
y
|
shows a dark band; this is caused by interference between the
upward-moving guided mode within the slit and the reflected, downward-travelling mode.
In addition to the electric and magnetic field components E
z
,
H
y
at the surface, the light
transmission efficiency in the
z
direction through the slit as a function of slit-groove distance
x
sg
was calculated and is shown in Fig. 7. The transmission efficiency
T
is defined as the ratio
of the
z
-component of the Poynting vector
S
out
z
on the output side, integrated over
x
, to the
total energy flux incident on the slit
S
in
z
. The red trace plots the transmission efficiency
T
=
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Fig. 7. FDTD calculations of the transmission efficiency
T
=
S
out
z
/
S
in
z
as a function of
x
sg
.
Red curve traces
T
, and the blue curve traces a cos
(
2
k
fdtd
surf
·
x
sg
+
φ
fdtd
int
)
fit to the oscillation
in the asymptotic region. Note the decreasing transmission amplitude in the near-zone close
to the slit edge and the higher oscillation frequency compared to the asymptotic harmonic
wave.
Best-fit values for
λ
fdtd
surf
=
2
π
/
k
fdtd
surf
=
839 nm and intrinsic phase
φ
fdtd
int
=
0
.
55
π
rad.
Fig. 8. Phase difference
φ
(
x
sg
)
φ
0
as a function of
x
sg
, analogous to the right panel of
Fig. 4 but derived from the FDTD simulation data. Residual “high frequency” oscillations
in the phase difference, believed to be due to numeric artifacts in the FDTD results, have
been smoothed.
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S
out
z
/
S
in
z
as a function of
x
sg
at the output plane. As expected the transmission efficiency exhibits
pronounced oscillations with a rapid decrease in amplitude in the near-zone followed by a
constant amplitude oscillation out to 6
μ
m, the calculation limit. In the far-zone the blue trace
fits the oscillations to a single cosine function, cos
[
2
(
k
fdtd
surf
·
x
sg
+
φ
fdtd
int
)]
. These oscillations, as
can be seen in Figs. 5 and 6, arise from the superposition of waves launched from the slit and
back reflected at the groove. Because these waves are counterpropagating along
x
, rather than
copropagating along
z
, the intensity fringe frequency is twice the fringe frequency of the far-
field interferometry results expressed by Eq.1. This standing wave at the output plane results
in
T
=
S
out
z
/
S
in
z
2
[
1
+
cos2
(
k
fdtd
surf
·
x
sg
+
φ
fdtd
int
)]
=
2
{
1
+
cos2
[
φ
(
x
sg
)]}
(5)
Taking into account this factor of two in the argument of the cos term, the fringe oscillation
in the asymptotic far-zone is in good agreement with measurement and the expected SPP. In the
near-zone the oscillation
φ
(
x
sg
)
exhibits a definite “chirp,” such that close to the slit edge the
oscillation frequency is higher compared to the asymptotic harmonic wave. Best-fit values for
λ
fdtd
surf
=
2
π
/
k
fdtd
surf
=
839 nm and intrinsic
φ
fdtd
int
=
0
.
55
π
rad. Fig. 8 plots the deviation from the
asymptotic value
φ
0
as function of
x
sg
. Comparing Fig. 8 to Fig. 4 we see that the FDTD results
accord well with deviations in the interferometric fringes measured in the far- field.
4. Discussion
4.1. Surface wave in the far-zone
The index of refraction for the bound surface-plasmon-polariton
n
spp
is given by the Raether
formula[4]
n
spp
=
ε
m
ε
d
ε
m
+
ε
d
(6)
where
ε
m
and
ε
d
are the real parts of the dielectric constants of metal and dielectric at the in-
terface on which the surface wave propagates. Interpolation of reflectivity data for gold[11]
at 852 nm yields
ε
Au
=
28
.
82 and the present results [12, 13]
ε
Au
=
28
.
32. Using these
data and Eq.6, the surface index of refraction for the surface plasmon polariton at the gold-
air interface is
ε
spp
=
1
.
018. The measured surface index of refraction reported here, to within
experimental uncertainty and in the far-zone, is in accord with the SPP prediction. The results
from the FDTD calculations are also in agreement with the experimental results and the SPP
predictions. It appears therefore that in the far-zone, for both silver/air and gold/air surfaces,
far-field interferometryand FDTD calculations show that the surviving long-range surface wave
is indeed the expected bound surface plasmon polariton. Table 1 summarizes the relevant para-
meters, far-field interferometric measurements, and finite-difference-time-dependent (FDTD)
numerical simulations for gold and silver.
Table 1. Summary of
λ
surf
,
n
surf
and
n
spp
determined from far-field interferometric studies
and FDTD simulations in gold and silver
ε
m
λ
surf
n
surf
n
spp
Au, Ref. [12]
28
.
32 839
±
6nm
1
.
016
±
0
.
004
1.018
Au, Ref. [13]
31
.
62 839 nm
1
.
016
1
.
016
Ag, Ref. [14]
33
.
27 819
±
8nm 1
.
04
±
0
.
01 (near zone) 1
.
015 (far zone)
Ag, Ref. [15]
33
.
27 814
±
8nm 1
.
05
±
0
.
01 (near zone) 1.015 (far zone)
Ag, Ref. [16]
33
.
98 837 nm
1
.
017
1
.
015
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4.2. Surface wave in the near-zone
In the near-zone both experiment and numerical simulation show that the surface wave devi-
ates from pure SPP behavior. The effective propagation parameter
k
x
, originating near the slit
edge, appears greater than
k
spp
and evolves smoothly to the bound mode over the near-zone
interval of a few microns. This behavior may be interpreted either in terms of initial excitation
of a composite evanescent surface “wave packet” in
k
-space at the slit edge [17], followed by
subsequent decay of all surface modes except the bound
k
spp
mode or in terms of detailed field
matching at the boundaries within the slit and near the slit edges [18]. These two points of view
both invoke evanescent modes
k
x
k
spp
in order to satisfy boundary conditions in the vicin-
ity of the slit edge, but standard wave-guide theory dictates that only the SPP mode is stable
against phonon coupling to the bulk metal or to radiative decay. We can estimate the the surface
distance over which the dissipation occurs by appealing to the standard Drude model of a metal
that expresses the frequency dependence of the dielectric constant
ε
(
ω
)
in terms of the bulk
plasmon resonance
ω
p
and a damping constant
Γ
.
ε
(
ω
)=
ε
0
(
ε
ω
2
p
ω
2
+
i
Γ
ω
)
(7)
In Eq. 7
ε
0
is the permittivity of free space and
ε
=
ε
(
ω
)
is the dimensionless infinite
frequency limit of the dielectric constant. The Drude model is based on a damped harmonic
oscillator model of an electron plasma in which the electrons oscillate about positive ion centers
with characteristic frequency
ω
p
, subject to a phenomenological damping rate
Γ
, normally
assumed to be due to electron-phonon coupling. Values for
Γ
are typically
10
14
s
1
, and in
fact for Au the value is 1
.
02
×
10
14
s
1
. For a wave propagating on the surface with group
velocity

3
×
10
8
ms
1
the expected decay length

3
μ
m, consistent with the measurements
and FDTD simulations.
In summary the picture that emerges from far-field interferometry and FDTD simulation
studies of these simple slit-groove structures on silver and gold films is that in the near-zone
of slit-groove distances, on the order of a few wavelengths, the surface wave consists of a
composite of several evanescent modes all of which dissipate within this near-zone. Only the
bound, stable SPP mode survives into the far-zone, and in the studies reported here we have
observed essentially constant SPP amplitude out to 12
μ
m, a distance limited only by our
fabricated structures. Earlier measurements[3] indicate that absorption and surface roughness
scattering should permit propagation lengths as far as
80
100
μ
m.
Support from the Minist`
ere d ́
el ́
egu ́
e`
a l’Enseignement sup ́
erieur et `
a la Recherche under
the programme ACI-“Nanosciences-Nanotechnologies,” the R ́
egion Midi-Pyr ́
en ́
ees [SFC/CR
02/22], and FASTNet [HPRN-CT-2002-00304]EU Research Training Network, is gratefully
acknowledged. F.K. gratefullly acknowledges support from the Deutsche Telekom Stiftung.
Technical assistance and the fabrication facilities of the Caltech Kavli Nanoscience Institute
are also gratefully acknowledged.
#77673 - $15.00 USD
Received 30 November 2006; revised 13 February 2007; accepted 14 February 2007
(C) 2007 OSA
5 March 2007 / Vol. 15, No. 5 / OPTICS EXPRESS 2621