(050.1940) Diffraction and grating : Diffraction; (260.3090) Infrared, far.
References and links
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of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength
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(C) 2009 OSA 16 February 2009 / Vol. 17, No. 4 / OPTICS EXPRESS 2631#102909 - $15.00 USD Received 20 Oct 2008; revised 13 Dec 2008; accepted 14 Dec 2008; published 9 Feb 2009
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1. Introduction
Extraordinary transmission of electromagnetic waves through a thin metal film perforated
with periodic arrays of subwavelength circular holes was investigated in 1998 by Ebbesen et.
al. [1]. This was explained by the coupling between photon and surface plasmon polaritons
(SPPs), the SPPs are the fluctuations in the electron density at the interface between metal and
dielectric materials excited by photon, and then the resonant surface wave reconverts to
photon and enhances the transmission light. The influence of holes shape on extraordinary
transmission including the polarization-dependent transmission intensity [2] and the shift of
the spectral position of the resonances [3] have attracted lots of attention. In particular, a large
enhancement in transmission intensity has been shown to appear when the polarization of the
incident light is perpendicular to long edge of rectangular holes. In subsequent research, the
strong enhanced transmission through rectangular hole arrays were investigated both in
experiments [4] and theory [5, 6]. Klein et. al. [7] proved that the shape resonance originated
from the contribution of individual hole by arranging the rectangular holes in random
distribution. Mary et. al. [8] theoretically demonstrated that SPPs and localized resonance
modes are all present in 2-D array of rectangular holes. Chen et. al. [9] found that the cross
shaped hole arrays gave rise to larger transmission of light than those perforated with square
or rectangular hole. The transmission peaks red-shifted when the aspect ratio of the cross
increased, whereas the transmission dip representing the Wood’s anomalies stayed at the same
wavelength. It is interesting to know how the Wood’s anomaly affects the peak wavelength of
the transmission spectrum of the shape resonance.
In this paper, the rectangular hole arrays were arranged in a rectangular lattice to tailor the
position of Wood’s anomaly away from the peak of localized resonance. The real peak
position of the localized waveguide resonance resλ was observed which is different from
that appears at the fundament shape resonance 2res Lλ ≈ in free standing structure, where L
is the hole length. The simplified transmission-line model was utilized to fit the experimental
values.
2. Experiment
Fig. 1(a) and 1(b) show the top and side views of the square array of rectangular holes,
respectively. The 100 nm silver thin film was deposited on the periodic photoresist rod array
and lifted off to form a silver film perforated with periodic hole array on top of a
doubly-polished n-type silicon wafer. The period of the array along x and y directions, ax and
ay, are fixed at 15 µm . The rectangular holes have different aspect ratios (L/W), i.e., R=2, 4,
6, 7, 9, 11, and ∞ (1-D grating), the L is hole length and W is hole width. All the patterns
were designed by fixed the hole area the same (18 µm2). For 1-D grating, the width of the
hole is 1.2 µm . It is well known that both of the length and width in the
(C) 2009 OSA 16 February 2009 / Vol. 17, No. 4 / OPTICS EXPRESS 2632#102909 - $15.00 USD Received 20 Oct 2008; revised 13 Dec 2008; accepted 14 Dec 2008; published 9 Feb 2009
( ) ( )1 1
2 2 2 2WA da i jλ ε
−= +
1 12 2 2 2( ) ( )
d mspp
d m
a i jε ε
λε ε
−
+= +
rectangle could affect the peak transmission wavelength due to the coupling between surf-
Fig. 1. The schematic diagrams showing the (a) top and (b) side view of the rectangular
hole array in a square lattice. (c) The measurement setup and the sample lies in the xy plane
with light incident in z direction.
ace plasmons on the long edges of hole [10]. Therefore, the coupling mechanism on the
geometrical shape will be first investigated. Fig. 1(c) shows the experimental setup and the
sample lies in the xy plane with light incident in z direction while the polarized light was
along y direction. A Bruker IFS 66 v/s system was used to measure transmission spectra. The
dispersion relations along Ky direction were measured by rotating samples around x axis 1o per
step form 0o to 50
o.
3. Transmission results and discussions
For normal incident light, the free-space wavelength of surface plasmon polaritons (SPPs),
sppλ , in square lattice is given by [11]
(1)
For normal incident light, the free-space wavelength λWA that satisfies Wood- Rayleigh
anomaly condition and results in a transmission minimum is given by
(2)
where i and j are integers corresponding to the specific order of the SPP and WA mode, a (=15
µm ) is the lattice constant, εd (=11.7 at 52 µm ) and εm (= -9.2x104 at 52 µm ) are the
dielectric constants of silicon and silver, respectively. According to Eq. (1) the (1, 0) Ag/Si
SPP mode should exhibit a peak wavelength at 52 µm , whereas the Wood’s anomaly shows a
minimum at 51.3 µm . Fig. 2 shows the zero-order transmission spectra of rectangular holes
in square lattice array with different aspect ratios R = 2, 4, 6, 7, 9, 11, and ∞ (1-D) while the
polarized light is along y direction.
(c)
X
Y
E
Incident Light
(a) hole
Silver
ax
ay
L
W
(b) Silver
n-Si wafer
(C) 2009 OSA 16 February 2009 / Vol. 17, No. 4 / OPTICS EXPRESS 2633#102909 - $15.00 USD Received 20 Oct 2008; revised 13 Dec 2008; accepted 14 Dec 2008; published 9 Feb 2009
Fig. 2. Zero-order transmission spectra of rectangular hole array in a square lattice with aspect
ratios R = 2, 4, 6, 7, 9, 11, and ∞ (1-D grating). All the lattice constants are 15 µm .
It is clear that the transmission peaks shift from 52 µm to longer wavelength (76 µm )
as the aspect ratio of the rectangular holes increase from 2 to 11, and to much longer
wavelength (27360 µm ) as R→∞ (1-D grating). This phenomenon had been observed
before [3, 7]. One of the reasons for red shift is due to the shift of the cut-off wavelength of
rectangular waveguide to longer wavelength, but because the metal thickness is thin, the
Fabry-Perot resonant mode may not be able to exist in this short waveguide [12-15]. It is also
noticed that the wavelength of transition minima are almost the same for all samples which
are attributed to the Wood’s anomalies [14].
The dispersion relations of rectangular hole arrays along Ky direction with different aspect
ratios, i.e., R = 4, 7, 9, and ∞ (1-D) are shown in Figs. 3(a)-(d), respectively. The top views
of the samples were shown in the inset of the figures. In Fig. 3(a), aspect ratio R = 4, the
transmission maxima and minima are represented by the yellow and blue curves, respectively.
It can be seen that the yellow curve is accompanied by the slightly higher energy blue curve
representing the (0, -1) WA mode. A slightly flat dispersion curve was observed near low Ky
values, and then the curve follows and mixes with the (0, -1) Ag/Si SPP mode at high incident
angles, showing that a localized mode coexists with the SPP mode [4]. When the aspect ratio
is increased to 7, the yellow curve becomes flat indicating a localized resonance mode. This
confirms the previous reports [5, 16] that the transparency can only be realized through the
shape resonance. To see the fine details, the transmission spectra of samples with aspect ratio
of 6 and 7 with different incident angle (from 0o to 50
o) are shown in Fig. 4(a) and 4(b),
respectively. It can be seen definitely in Fig. 4(b) for R = 7 that the long wavelength tail of the
transmission peak keeps the same profile when measured at different incident angles as
compared to that shown in Fig. 4(a) for R=6. This is because the localized resonance
dominates the transmission through the structure for R = 7 sample, deriving from the
competition between propagating SPPs and the cut-off wavelength of the rectangular
waveguide [8]. A Wood’s anomaly cannot shift resonances. It can make a transmission peak
seem to be at a different position than the resonant position. For the sample with R = ∞
(1-D grating), only transmission minimum (Wood’s anomalies) was observed, the
transmission maximum which is related to the cut-off wavelength of the waveguide and
determined by the length of the long edge of the rectangle moves all the way to long
wavelength outside the detection range due to the infinite long edge of the 1-D grating.
(C) 2009 OSA 16 February 2009 / Vol. 17, No. 4 / OPTICS EXPRESS 2634#102909 - $15.00 USD Received 20 Oct 2008; revised 13 Dec 2008; accepted 14 Dec 2008; published 9 Feb 2009
Fig. 3. The dispersion relations and transmission intensity of rectangular hole array as a
function of photon energy and yk��
. The aspect ratios of rectangular holes are (a) 4, (b) 7, (c) 9
and (d) ∞ (1-D grating). All the lattice constants are 15 µm .
Fig. 4 Zero-order transmission spectra of rectangular hole array in a square lattice with
different incident angles. The lattice constants are 15 µm , and the aspect ratios R of
rectangular holes are (a) 6 and (b) 7, respectively.
To see the effect of Wood’s anomalies and cut-off condition on the spectral shape of the
transmission spectra, the rectangular hole arrays in a rectangular lattice were designed and
fabricated. The purpose was to tailor the position of Wood’s anomalies [17] away from the
peak position of shape resonance. In Fig. 5, the transmission spectra for the specific
rectangular holes with aspect ratio of 9, i.e., the length is 12.7 µm and the width is 1.4 µm ,
were arranged in a rectangular lattice with a fixed period of 15µm along x axis and different
periods from 3 to 18 µm along y axis. As the period along y axis ay is decreased from 18 µm ,
the transmission peak gradually shifts to shorter wavelength until the period reaches 6 µm , the
(a) (b)
(b) x 10-3
Ph
oto
n e
ner
gy
(eV
)
R=7
Localized mode
Ky (1/nm) (a)
Ph
oto
n e
ner
gy
(eV
)
(0, -1) Ag/Si
Ky (1/nm) x 10-3
R=4
(0, -1) WA
(d)
(0, -1) WA
(0, 1) WA
(0, -2) WA
(0, 2) WA
Ph
oto
n e
ner
gy
Ky (1/nm) x 10-3
R=∞ (1-D)
(c)
Ph
oto
n e
ner
gy
(eV
)
Ky (1/nm) x 10-3
Localized mode
R=9
(C) 2009 OSA 16 February 2009 / Vol. 17, No. 4 / OPTICS EXPRESS 2635#102909 - $15.00 USD Received 20 Oct 2008; revised 13 Dec 2008; accepted 14 Dec 2008; published 9 Feb 2009
peak is then fixed at 52.5 µm . This is because Wood’s anomaly moves further to shorter
wavelength, away from the shape resonance, its influence on the shape of the transmission
peaks diminishs. The transmission peaks appear at the fundamental shape resonance
2res Lλ ≈ as observed in the free-standing structure [16], but in our sample this value should
be multiplied by the effective refractive index effn since the metallic periodic structure is
sandwiched between the Si substrate and air.
Figure 6 shows the transmission spectra of the rectangular hole arrays with the aspect ratio
of 9 on doubly-polished Si and Ge substrates. There are two different periods along y axis ay,
i.e., 4 and 6µm , and a fixed period of ax = 15 µm along x axis. Regardless of the ay, the
spectra show that the transmission peaks only correlates with the wafer type, the peak shifts
from 52.5 µm to the longer wavelength of 61 µm when the Si substrate was replaced by Ge
substrate. To estimate the peak wavelength, a transmission-line model that describes the
effective index variation with different substrates was applied [18-19]. For a capactive strip
grating, the electric field direction is perpendicular to the long edge of rectangular hole, the
equivalent circuit is a capacitor with capacitance ( )2 2
1 2 / 2n n+ times of the same grating in free
standing structure, and its reactance cX is given by
Fig. 5. The transmission spectra of the rectangular hole array with aspect ratio of 9, i.e., the
length is 12.7 µm and the width is 1.4 µm , in a rectangular lattice with the fixed period of
15 µm along x axis and the different periods from 3 to 18 µm m along y axis.
Fig. 6. The transmission spectra of the rectangular hole arrays on Si or Ge substrate with aspect
ratio of 9 in a rectangular lattice. The periods are 15 µm along x axis for all four samples, the
periods along y axis ay are either 4 or 6 µm .
0
0
0
1
2 2
1 2
24 ln csc
c
s
X a
Z n n g
π ω ωω
ω ω
−′ ′= − ′+
(3)
Where n1 and n2 are the refraction indexes of the two adjacent media, i.e. air and Si (Ge)
(C) 2009 OSA 16 February 2009 / Vol. 17, No. 4 / OPTICS EXPRESS 2636#102909 - $15.00 USD Received 20 Oct 2008; revised 13 Dec 2008; accepted 14 Dec 2008; published 9 Feb 2009
substrate, a is one half of the space between two adjacent holes, and g is the period of the
periodic structure. The 0ω′ is the resonant frequency of the periodic metal arrays
sandwiched between two dielectric medium with n1 and n2 refraction indexes, respectively.
0ω′ is related to the resonant frequency of the same metallic arrays in free standing
structure 0ω by
0 02 2
1 2
2
n nω ω′ =
+ (4)
When the periodic metal structure deposited on Si or Ge substrate, the refraction index n1 of
air is unity, the refraction index of substrates n2 (Si) and n2 (Ge) are 3.35 and 4 at the peak
wavelengths, respectively. From Eq. (4), the cut-off resonance wavelength
( ( )2 2
1 22 2
2res eff
n nL Lnλ
+≈ = ) will be modified by effective refraction index of 2.47 (Si) and
2.92 (Ge). The real aperture of the rectangular hole with aspect ratio of 9 has a length of 11.5
µm and width of 1.3 µm . The theoretical resonance peaks were estimated to be 56.8 µm
and 67.2 µm for Si and Ge substrates, respectively. The measured values, i.e., 52.5 µm for
Si substrate and 61 µm for Ge substrate, are slightly smaller than the theoretical value.
4. Conclusions
In conclusion, the extraordinary transmission through the periodic metal arrays perforated
with subwavelength rectangular holes was investigated. When the aspect ratio of rectangular
holes exceeds 7, the propagating SPP modes transform to the localized resonant modes that
was attributed to the cut-off wavelength of the rectangular waveguide beyond the wavelength
of SPP mode. It was found that the Wood’s anomalies play an important role in shaping the
spectral profile of the transmission peak. When the Wood’s anomalies are moved far apart
form the shape resonance by using rectangular lattice, the real peak position of the localized
resonance mode appears and stays at the same wavelength irrespective of the period of the
lattice. The peak position of the localized or shape resonance was demonstrated to be relevant
to the substrate type and can be explained by a transmission-line model.
Acknowledgment
This research was carried out with the financial support of the National Science Council of the
Republic of China under the Contract No. NSC 96-2221-E-002-242
(C) 2009 OSA 16 February 2009 / Vol. 17, No. 4 / OPTICS EXPRESS 2637#102909 - $15.00 USD Received 20 Oct 2008; revised 13 Dec 2008; accepted 14 Dec 2008; published 9 Feb 2009
