scispace - formally typeset

Journal ArticleDOI

Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate

17 Oct 2005-Physical Review B (American Physical Society)-Vol. 72, Iss: 15, pp 153406

Abstract: We demonstrate through both experiment and theory that in the microwave regime one can have high transmittance through narrow fractal-patterned slits in thick metallic plates, whereby the aperture is subwavelength in all cross sectional dimensions. In contrast to the recently discovered extraordinary transmissions via surface plasmon excitations and Fabry-Perot-like resonances, the transmission in the present case is independent of the incident angle, plate thickness, or array periodicity. We show the physics to be governed by the transversal shape resonance localized in the metallic fractal slots. In particular, for the lowest transmission mode the EM field experiences no phase change when transmitting through the metallic plate. Simulation demonstrates the viability of the observed phenomenon as a subwavelength $k=0$ waveguide mode, where $k$ is the axial wave number.

Content maybe subject to copyright    Report

Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate
Weijia Wen, Lei Zhou, Bo Hou, C. T. Chan, and Ping Sheng
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
Received 5 May 2005; revised manuscript received 12 August 2005; published 17 October 2005
We demonstrate through both experiment and theory that in the microwave regime one can have high
transmittance through narrow fractal-patterned slits in thick metallic plates, whereby the aperture is subwave-
length in all cross sectional dimensions. In contrast to the recently discovered extraordinary transmissions via
surface plasmon excitations and Fabry-Perot-like resonances, the transmission in the present case is indepen-
dent of the incident angle, plate thickness, or array periodicity. We show the physics to be governed by the
transversal shape resonance localized in the metallic fractal slots. In particular, for the lowest transmission
mode the EM field experiences no phase change when transmitting through the metallic plate. Simulation
demonstrates the viability of the observed phenomenon as a subwavelength k =0 waveguide mode, where k is
the axial wave number.
DOI: 10.1103/PhysRevB.72.153406 PACS numbers: 73.20.Mf, 42.79.Dj, 41.20.Jb
Electromagnetic EM wave transmission through small
metallic openings has been a topic of considerable fascina-
tion in recent years. Two mechanisms have been reported
involving different types of resonances. In 1998, Ebbesen et
al. showed that EM wave transmission through a silver film
with a periodic array of subwavelength holes can be signifi-
cantly higher than the conventional predictions,
1–5
due to the
excitation of surface plasmon SP. Subsequently, Porto et
al.
6
identified another waveguide mode resonance inside me-
tallic slits due to the Febry-Perot FP interferences.
7–9
In
both cases the length scales relevant to the transmission
mechanism must be comparable to the wavelength: in the SP
case the periodicity must be comparable to the wavelength;
the FP resonance requires at least one dimension of the slit
cross section be comparable to the relevant wavelength, so
that a fundamental TEM propagating wave mode
10
may
exist.
In this work, we demonstrate through both experiment
and theory that in the microwave regime where SP is un-
likely, one can have high transmittance through narrow slits
in thick metallic plates arranged in a fractal geometry, in
which the aperture cross section is subwavelength in all cross
sectional dimensions. In contrast to the evanescent wave
coupling mechanism,
1
here the maximum transmission mag-
nitude can range from 9% for a single fractal aperture to
100% for a 5 5 array of apertures. The high transmission is
independent of periodicity of the array and incidence angle.
In addition, the transmission frequency is also nearly inde-
pendent of the sample thickness, in sharp contrast with the
FP mechanism which depends strongly on the film
thickness.
6
We show the underlying physics to be governed
by the localized resonances of the fractal slit patterns.
The sample was prepared with stainless steel plates with
different thicknesses on which a periodic array of fractal slits
was generated by a diamond wire cutter. The unit cell of the
array consists of a five-level structure, wherein the width of
each slit is 0.8 mm, with the longest slit being 1 cm, as
shown in Fig. 1. A total of 25 fractal slit units were made on
a12 12 cm
2
steel plate. This sample was mounted in the
central window with the same size as the 12 12 cm
2
of a
100 100 cm steel plate so as to prevent microwave trans-
mission through channels other than the fractal slit array. The
microwave spectra were measured by a network analyzer
Agilent 8720ES. Two identical microwave horns
HP11966E were used to generate and receive the signals
separated by a distance of 100 cm. The sample was placed
on a stage, 15 cm from the receiving horn. The microwave
polarization was such that the electric field is perpendicular
to the shortest slits of the fractal pattern defined as E
,
while E
is 90° rotated. All measured spectra are normalized
to the transmission when no sample is mounted. Transmis-
sion measurement for a single fractal slit aperture was car-
ried out by covering all 24 apertures of the array with me-
tallic sheets except the center one.
Figures 2a and 2b show the microwave transmission
spectra through the slit array for two polarizations, where a
and b are for E
and E
, respectively. We note that for the
thin sample, 100% transmittance can be identified at
5.1 GHz, 18 GHz for the case of E
, and 3.0 GHz and
10.0 GHz for the case of E
. The peak transmission fre-
quency is downshifted somewhat when the thickness of
stainless steel plate increases from 0.5 mm to 5.5 mm, be-
yond which it stayed constant 4.1 GHz and 17.2 GHz for
E
, 2.4 GHz and 9.0 GHz for E
up to the maximum sample
thickness of 14.5 mm. Transmission through a single fractal
aperture was measured to be 9%. In contrast, standard theory
for a square hole with the same opening area 99.5 mm
2
as
the fractal aperture predicts transmission on the order of
1%.
5
It should be noted that at the lowest peak frequency,
FIG. 1. A schematic picture of the fractal slit array and the unit
cell.
PHYSICAL REVIEW B 72, 153406 2005
1098-0121/2005/7215/1534064/$23.00 ©2005 The American Physical Society153406-1

the incident wavelength 12.5 cm is 12.5 times the longest
slit 1cm on the steel plate. Hence the aperture cross sec-
tion can be significantly subwavelength in both dimensions.
The transmission characteristics of the fractal slit array
were investigated by finite difference time domain FDTD
simulations.
11,12
We consider an infinite plane tiled by a pe-
riodic replica of the five-level fractal slit patterns, and stud-
ied one unit cell with periodic conditions imposed at the
outer boundaries. Perfect conductor boundary conditions, ex-
cellent for microwave frequencies, were applied to the
metal/air interfaces. The simulation results are shown as
white solid lines in Figs. 2a and 2b. Except for the
two thick samples, the agreement is seen to be excellent
1% difference. For the two thick samples some of the
simulated transmission peaks are not seen experimentally, for
reasons not yet clear. However, in all cases the lowest fre-
quency, subwavelength transmission peak was always ob-
served. The fact that the transmission peak persists to thin
sample thicknesses points to localized fractal resonances as
the origin of this transmission mode. It was shown previ-
ously that planar metallic fractal patterns can have multiple
localized resonances, and at the resonance frequencies the
fractal plate can exhibit total reflection.
13,14
Via Babinet’s
principle,
15
the total reflection can become total transmission
for the complimentary structure, accompanied by an inter-
change of electric and magnetic field configurations, plus a
90° rotation of the pattern. The present transmission mode is
simply a direct extension of such localized planar reso-
nances. In particular, it can be regarded simply as the same
resonance repeated at every slice of the sample. It is inter-
esting to observe that for a reference sample consisting of an
array of square holes 18 mm 18 mm opening, the trans-
mission is considerably lower, as shown in Fig. 2c. Thus by
removing metal, the sample becomes less transparent. The
physics indicated here is that it is the contour line of the slit
pattern which can cause the resonances, see below which
matters, not the total size of the openings.
The periodicity and incident angle dependencies were
also investigated, with experimental results shown in Fig. 3.
Here the periodicity variation was achieved by covering 16
of the 25 apertures by metallic films, resulting effectively in
a 3 by 3 array with twice the periodicity. It is seen that
neither the periodicity of the array nor the incident angle of
the EM wave has any noticeable effect on the positions of
the transmission peaks, although the magnitude of the trans-
mittance is affected to some degree. The simulation has con-
firmed this experimental observation,
16
which is very differ-
ent from the extraordinary optical transmission through two-
dimensional 2D periodic array of holes caused by the
FIG. 2. Microwave transmittance through the 5 5 fractal ar-
rays of various thicknesses for a E
polarization and b E
polar-
ization; the thicknesses is denoted on the curves; the dark grey
symbols and dark solid lines are the measured and simulated re-
sults, respectively; one vertical grid represents 50% transmission.
c Comparison of measured transmittances of the two polariza-
tions for the fractal array are indicated as symbols on a 3 mm-thick
metallic plate with the square-hole array solid dark line oname-
tallic plate with the same 3 mm thickness. Fractal array and square
holes have the same periodicity.
FIG. 3. Variation of the measured transmission spectra as a
function of a the incident angle
and b periodicity a. The
sample is 0.5 mm thick and illuminated with E
polarization. The
calculated peak frequencies are shown as arrows; they exhibit no
variation with either the incident angle or the periodicity.
BRIEF REPORTS PHYSICAL REVIEW B 72, 153406 2005
153406-2

excitation of SP resonance, where the transmission spectra
are sensitive to the periodicity and strongly influenced by the
incident angle.
1–4
The insensitivity of the present transmis-
sion spectra to the plate thickness also distinguishes it from
the FP resonances.
6–9
In Figs. 2a and 2b there are also
calculated FP modes e.g., 10.7 GHz and 12.8 GHz in the E
case. However, the measured transmissions are weaker than
predicted. In fact, whereas the FP resonance mechanism
works only when the plate is sufficiently thick, here the total
transmission can persist to very thin thicknesses as seen from
Fig. 2.
To obtain a physical picture of the subwavelength trans-
mission mode, we calculated the E field distributions inside
the fractal slits at the four resonance frequencies: 2.4 GHz,
4.1 GHz, 9.0 GHz, and 17.2 GHz. The field in the slits forms
a transverse-electric TE mode with significantly enhanced
amplitude. The intensity distributions, shown in Fig. 4, illus-
trate these four resonant modes on the plane transverse to the
propagation direction. It is seen that the different resonances
are excited at the different levels of the fractal pattern. The
lowest one 2.4 GHz has the highest intensity localized in
the first level of the fractal pattern, while the next ones are
localized at the higher levels of the fractal pattern, etc. The
same behavior was seen in the complementary structure.
13,16
In Fig. 5 we focus on the 2.4 GHz transmission mode in
the case of 6.0 mm plate thickness. From the xy-plane distri-
bution pattern of the Poynting vector S= E H
*
shown in
Fig. 5a for an arbitrary transverse plane inside the sample,
we find the field to be strongly enhanced inside the first-level
fractal slit regions, but not confined just to the first level. We
also find that the wave transmits across the metal mainly in
the form of electric energy—while the E field is strongly
enhanced, there is no obvious enhancement in the H field.
Figure 5b shows the E
x
field pattern on the yz-plane in the
central slit, where the field is approximately 23 times higher
than the incident field. An important feature for the transmis-
sion process is that there is no phase change when an EM
wave transmits across the metallic plate, that is to say, k =0
k being the axial wave number. The localized nature of the
resonances also implies independence on the incidence
angle and the periodicity.
The high transmission also implies very high coupling
between the incident wave and the waveguide. This is some-
FIG. 4. The calculated field magnitude, E, inside the fractal
slits at four frequencies: a 2.4 GHz, b 4.1 GHz, c 9.0 GHz, and
d 17.2 GHz. In the grey scale, dark grey represents the strongest
frequency. The sample thickness is taken to be 6 mm.
FIG. 5. a The calculated magnitude of the Poynting vector
component S
z
for the 2.4 GHz mode normalized to the incident
wave S
z
0
E
-polarized with a frequency of 2.4 GHz in an arbi-
trary xy-plane inside the metallic plate. b The yz-plane distribution
of E
x
for the same mode normalized to the same incident wave as
a兲兴 inside the central slit. The sample thickness is taken to be
6 mm.
FIG. 6. Simulation result for a 5 5 fractal array with 50 cm
thickness for the E
polarization. The 2.4 GHz peak the arrow
indicates the k =0 peak becomes extremely narrow, so that it can-
not be resolved completely and thereby appears lower than 100%.
BRIEF REPORTS PHYSICAL REVIEW B 72, 153406 2005
153406-3

what puzzling at first sight because one expects a large im-
pedance mismatch between the aperture and the air. Suppose
the aperture is characterized by effective impedance Z and a
typical dispersion relation k =
2
c
2
/c in which
c
is the
cutoff frequency or resonance frequency in our case. The
transmission coefficient across such an aperture with thick-
ness W can be written down in analogy to transmission lines,
i.e.,
t =
4Z
0
Ze
ikW
Z + Z
0
2
Z Z
0
2
e
i2kW
, 1
where Z
0
is the impedance for air. When the impedance con-
trast is high Z Z
0
or Z Z
0
, it is easy to check that t0.
However, Eq. 1 predicts perfect transmission t =1 for
k 0. The latter implies sample thickness independence. It
should be noted that Eq. 1 was derived originally for non-
zero k, finite Z transmission lines, and the above conclusion
i.e., t =1 for k 0 holds for every frequency component.
Hence even if Z is frequency dependent or the interface be-
tween the structure and the air is reactive, t= 1 still holds
provided k 0.
In Fig. 6 we show the simulation result for the 5 5 array
of fractal slits that are 50 cm in thickness. In the simulation,
periodic boundary condition and perfect conductor approxi-
mation were assumed just as above. It is seen that increased
thickness means many more FP transmission peaks. How-
ever, the 2.4 GHz transmission peak is preserved with a nar-
rower width. As the thickness is now many wavelengths, the
lowest frequency transmission may be viewed as a k=0
waveguide mode.
In summary, we show that by using fractal cross sectional
geometry, subwavelength total transmission through metallic
slits can be achieved. The effect is robust and should be
applicable to infrared frequencies as well.
This work is supported by Hong Kong RGC through
CA02/03.SC01.
1
T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A.
Wolff, Nature London 391, 667 1998.
2
H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J.
Lezec, Phys. Rev. B 58, 6779 1998.
3
L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin,
T. Thio, J. B. Pendry, and T. W. Ebbesen, Phys. Rev. Lett. 86,
1114 2001.
4
W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W.
Ebbesen, Phys. Rev. Lett. 92, 107401 2004.
5
H. A. Bethe, Phys. Rev. 66, 163 1944.
6
J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, Phys. Rev. Lett.
83, 2845 1999.
7
Y. Takakura, Phys. Rev. Lett. 86, 5601 2001.
8
F. Yang and J. R. Sambles, Phys. Rev. Lett. 89, 063901 2002.
9
H. E. Went, A. P. Hibbins, J. R. Sambles, C. R. Lawrence, and A.
P. Crick, Appl. Phys. Lett. 77, 2789 2000.
10
E. Popov, M. Nevière, S. Enoch, and R. Reinisch, Phys. Rev. B
62, 16100 2000.
11
K. S. Yee, IEEE Trans. Antennas Propag. 14, 302 1966.
12
Simulations were performed using the package CONCERTO 3.1,
developed by Vector Fields Limited, England, 2002.
13
W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, Phys.
Rev. Lett. 89, 223901 2002.
14
L. Zhou, W. Wen, C. T. Chan, and P. Sheng, Appl. Phys. Lett. 82,
1012 2003.
15
B. A. Munk, Frequency Selective Surfaces, Theory and Design
Wiley, New York, 2000.
16
The calculation of transmission under off-normal incidence fol-
lows the method described in L. Zhou, C. T. Chan, and P. Sheng,
J. Phys. D 37, 368 2004.
BRIEF REPORTS PHYSICAL REVIEW B 72, 153406 2005
153406-4
Figures (6)
Citations
More filters

Journal ArticleDOI
Abstract: We report on a planar metamaterial, the resonant transmission frequency of which does not depend on the polarization and angle of incidence of electromagnetic waves. The resonance results from the excitation of high-Q antisymmetric trapped current mode and shows sharp phase dispersion characteristic to Fano-type resonances of the electromagnetically induced transparency phenomenon.

253 citations


Journal ArticleDOI
Shulin Sun1, Qiong He1, Jiaming Hao2, Shiyi Xiao3  +1 moreInstitutions (3)
Abstract: Metasurfaces are ultrathin metamaterials consisting of planar electromagnetic (EM) microstructures (e.g., meta-atoms) with pre-determined EM responses arranged in specific sequences. Based on careful structural designs on both meta-atoms and global sequences, one can realize homogenous and inhomogeneous metasurfaces that can possess exceptional capabilities to manipulate EM waves, serving as ideal candidates to realize ultracompact and highly efficient EM devices for next-generation integration-optics applications. In this paper, we present an overview on the development of metasurfaces, including both homogeneous and inhomogeneous ones, focusing particularly on their working principles, the fascinating wave-manipulation effects achieved both statically and dynamically, and the representative applications so far realized. Finally, we also present our own perspectives on possible future directions of this fast-developing research field in the conclusion.

143 citations


Journal ArticleDOI
Joong Wook Lee1, Minah Seo1, Doo Jae Park1, Dai-Hong Kim1  +4 moreInstitutions (4)
TL;DR: The results show that plasmonic structures with different geometric shaped holes are extremely versatile, dependable, easy to control and easy to make terahertz filters.
Abstract: Terahertz transmission filters have been manufactured by perforating metal films with various geometric shapes using femtosecond laser machining. Two dimensional arrays of square, circular, rectangular, c-shaped, and epsilon-shaped holes all support over 99% transmission at specific frequencies determined by geometric shape, symmetry, polarization, and lattice constant. Our results show that plasmonic structures with different geometric shaped holes are extremely versatile, dependable, easy to control and easy to make terahertz filters.

107 citations


Journal ArticleDOI
Abstract: A fractal-featured metallic thin film with Sierpinski Carpet pattern is fabricated on silicon wafer by microfabrication techniques. Transmission infrared spectroscopy indicates that there exists extraordinary high transmission at specific wavelengths, which can be ascribed to the effect of surface plasmon resonance, and are determined by hierarchy of apertures of different sizes in the fractal structure. This patterned film provides a unique system to achieve enhanced transmission simultaneously at different selected frequencies of electromagnetic wave.

47 citations


Journal ArticleDOI
Abstract: Isotropic frequency selective surface (FSS) made of cubic arrangements of split ring resonators (SRRs) is proposed and analyzed. For this purpose, a suitable isotropic modification of the SRR was used in the design of a cubic unit element invariant under the tetrahedral point symmetry group. It was experimentally demonstrated that the transmission through such a FSS is angle and polarization independent. For comparison, another FSS, whose unit elements do not satisfy necessary symmetries, was measured, showing clearly anisotropic behavior. We feel then that symmetries play an important role. Potential device applications are envisioned for antenna technology at microwave and terahertz frequencies.

45 citations


References
More filters

Journal ArticleDOI
Kane Yee1Institutions (1)
Abstract: Maxwell's equations are replaced by a set of finite difference equations. It is shown that if one chooses the field points appropriately, the set of finite difference equations is applicable for a boundary condition involving perfectly conducting surfaces. An example is given of the scattering of an electromagnetic pulse by a perfectly conducting cylinder.

13,304 citations


Journal ArticleDOI
12 Feb 1998-Nature
Abstract: The desire to use and control photons in a manner analogous to the control of electrons in solids has inspired great interest in such topics as the localization of light, microcavity quantum electrodynamics and near-field optics1,2,3,4,5,6. A fundamental constraint in manipulating light is the extremely low transmittivity of apertures smaller than the wavelength of the incident photon. While exploring the optical properties of submicrometre cylindrical cavities in metallic films, we have found that arrays of such holes display highly unusual zero-order transmission spectra (where the incident and detected light are collinear) at wavelengths larger than the array period, beyond which no diffraction occurs. In particular, sharp peaks in transmission are observed at wavelengths as large as ten times the diameter of the cylinders. At these maxima the transmission efficiency can exceed unity (when normalized to the area of the holes), which is orders of magnitude greater than predicted by standard aperture theory. Our experiments provide evidence that these unusual optical properties are due to the coupling of light with plasmons — electronic excitations — on the surface of the periodically patterned metal film. Measurements of transmission as a function of the incident light angle result in a photonic band diagram. These findings may find application in novel photonic devices.

7,135 citations


Book
26 Apr 2000
Abstract: General Overview. Element Types: A Comparison. Evaluating Periodic Structures: An Overview. Spectral Expansion of One- and Two-Dimensional Periodic Structures. Dipole Arrays in a Stratified Medium. Slot Arrays in a Stratified Medium. Band-Pass Filter Designs: The Hybrid Radome. Band-Stop and Dichroic Filter Designs. Jaumann and Circuit Analog Absorbers. Power Handling of Periodic Surfaces. Concluding Remarks and Future Trends. Appendices. References. Index.

3,631 citations


Journal ArticleDOI
Hans A. Bethe1Institutions (1)
Abstract: The diffraction of electromagnetic radiation by a hole small compared with the wave-length is treated theoretically. A complete solution is found satisfying Maxwell's equations and the boundary conditions everywhere (Section 4). The solution holds for a circular hole in a perfectly conducting plane screen, but it is believed that the method will be applicable to much more general problems (Section 8). The method is based on the use of fictitious magnetic charges and currents in the diffracting hole which has the advantage of automatically satisfying the boundary conditions on the conducting screen. The charges and currents are adjusted so as to give the correct tangential magnetic, and normal electric, field in the hole. The result (Section 5) is completely different from that of Kirchhoff's method, giving for the diffracted electric and magnetic field values which are smaller in the ratio (radius of the hole/wave-length) (Section 6). The diffracted field can be considered as caused by a magnetic moment in the plane of the hole, and an electric moment perpendicular to it (Section 6). The theory is applied to the problem of mutual excitation of cavities coupled by small holes (Section 9). This leads to equations very similar to those for ordinary coupled circuits. The phase and amplitude relations of two coupled cavities are not uniquely determined, but there are two modes of oscillation, of slightly different frequency, for which these relations are opposite (Section 10). The problem of stepping up the excitation from one cavity to another is treated (Section 11).

2,473 citations


Journal ArticleDOI
TL;DR: A fully three-dimensional theoretical study of the extraordinary transmission of light through subwavelength hole arrays in optically thick metal films shows that the enhancement of transmission is due to tunneling through surface plasmons formed on each metal-dielectric interface.
Abstract: We present a fully three-dimensional theoretical study of the extraordinary transmission of light through subwavelength hole arrays in optically thick metal films. Good agreement is obtained with experimental data. An analytical minimal model is also developed, which conclusively shows that the enhancement of transmission is due to tunneling through surface plasmons formed on each metal-dielectric interface. Different regimes of tunneling (resonant through a ``surface plasmon molecule,'' or sequential through two isolated surface plasmons) are found depending on the geometrical parameters defining the system.

1,552 citations


Network Information
Related Papers (5)
12 Feb 1998-Nature

Thomas W. Ebbesen, Thomas W. Ebbesen +5 more

14 Aug 2003-Nature

William L. Barnes, Alain Dereux +1 more

06 Aug 2004-Science

John B. Pendry, Luis Martín-Moreno +1 more

04 Jan 2007-Nature

Cyriaque Genet, Thomas W. Ebbesen

Performance
Metrics
No. of citations received by the Paper in previous years
YearCitations
20191
20184
20171
20161
20156
20146