Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires
We consider the two-dimensional scattering of the H-polarized electromagnetic plane waves of the visible range by three types of gratings made of periodically arranged circular cylindrical sub-wavelength wires. Using the field expansions in local coordinates and addition theorems for cylindrical...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2012
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| Cite this: | Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires / D.M. Natarov, R. Sauleau, A.I. Nosich // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 204-208. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1183082017-05-30T03:05:49Z Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires Natarov, D.M. Sauleau, R. Nosich, A.I. We consider the two-dimensional scattering of the H-polarized electromagnetic plane waves of the visible range by three types of gratings made of periodically arranged circular cylindrical sub-wavelength wires. Using the field expansions in local coordinates and addition theorems for cylindrical functions, we obtain a block-type matrix equation for the field expansion coefficients. This equation is of the Fredholm second-kind form that guarantees convergence of numerical solution. The scattering and absorption cross-sections and the near-field patterns are computed. The interplay of plasmon and grating-type resonances is studied for two and three-layer gratings of identical periods, stacked two-period gratings, and in-line two-period gratings made of nano-diameter silver wires. 2012 Article Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires / D.M. Natarov, R. Sauleau, A.I. Nosich // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 204-208. — Бібліогр.: 8 назв. — англ. 1560-8034 PACS 42.79.Dj, 78.35.+c http://dspace.nbuv.gov.ua/handle/123456789/118308 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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We consider the two-dimensional scattering of the H-polarized
electromagnetic plane waves of the visible range by three types of gratings made of
periodically arranged circular cylindrical sub-wavelength wires. Using the field
expansions in local coordinates and addition theorems for cylindrical functions, we
obtain a block-type matrix equation for the field expansion coefficients. This equation is
of the Fredholm second-kind form that guarantees convergence of numerical solution.
The scattering and absorption cross-sections and the near-field patterns are computed.
The interplay of plasmon and grating-type resonances is studied for two and three-layer
gratings of identical periods, stacked two-period gratings, and in-line two-period gratings
made of nano-diameter silver wires. |
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Article |
| author |
Natarov, D.M. Sauleau, R. Nosich, A.I. |
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Natarov, D.M. Sauleau, R. Nosich, A.I. Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Natarov, D.M. Sauleau, R. Nosich, A.I. |
| author_sort |
Natarov, D.M. |
| title |
Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires |
| title_short |
Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires |
| title_full |
Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires |
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Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires |
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Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires |
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resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/118308 |
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Resonance scattering and absorption of light by finite two-period gratings of circular silver nanowires / D.M. Natarov, R. Sauleau, A.I. Nosich // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 204-208. — Бібліогр.: 8 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT natarovdm resonancescatteringandabsorptionoflightbyfinitetwoperiodgratingsofcircularsilvernanowires AT sauleaur resonancescatteringandabsorptionoflightbyfinitetwoperiodgratingsofcircularsilvernanowires AT nosichai resonancescatteringandabsorptionoflightbyfinitetwoperiodgratingsofcircularsilvernanowires |
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2025-07-08T13:42:01Z |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 204-208.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
204
PACS 42.79.Dj, 78.35.+c
Resonance scattering and absorption of light
by finite two-period gratings of circular silver nanowires
D.M. Natarov1, R. Sauleau2, A.I. Nosich1
1Institute of Radio-Physics and Electronics, NAS of Ukraine,
12, Proskury str. 61085 Kharkiv, Ukraine
E-mail: den.natarov@gmail.com
2IETR, University of Rennes 1, Campus Beaulieu, bat 11-D, 35042 Rennes Cedex, France
E-mail: ronan.sauleau@univ-rennes1.fr
Abstract. We consider the two-dimensional scattering of the H-polarized
electromagnetic plane waves of the visible range by three types of gratings made of
periodically arranged circular cylindrical sub-wavelength wires. Using the field
expansions in local coordinates and addition theorems for cylindrical functions, we
obtain a block-type matrix equation for the field expansion coefficients. This equation is
of the Fredholm second-kind form that guarantees convergence of numerical solution.
The scattering and absorption cross-sections and the near-field patterns are computed.
The interplay of plasmon and grating-type resonances is studied for two and three-layer
gratings of identical periods, stacked two-period gratings, and in-line two-period gratings
made of nano-diameter silver wires.
Keywords: plasmon resonance, grating resonance, finite grating, nanowire, scattering,
absorption, cross-section.
Manuscript received 08.05.12; revised version received 08.06.12; accepted for
publication 14.06.12; published online 25.09.12.
1. Introduction
Two fundamental effects are known to influence the
scattering of light by periodically structured metal
scatterers. On the one hand, surface-plasmon resonances
are observed for sub-wavelength noble-metal particles
and wires in the mid-infrared and optical bands [1, 2].
Nanosize objects can exhibit resonance behavior at
certain frequencies for which the object permittivity is
negative. This results in powerful enhancement of
scattered and absorbed light that is used in the design of
optical antennas and biochemical sensors for advanced
applications. In the leading terms, the plasmon
resonance wavelength depends on the object shape but
not on its dimensions.
On the other hand, periodically structured
scatterers, or finite and infinite gratings, arrays or chains
of particles and holes in metallic screens (in D-3 ) or
wires and slots (in D-2 ), are attracting large attention
of researchers in today nano-optics [ 6-3 ]. This is
caused by the effects of extraordinarily large reflection,
transmission, emission, and near-field enhancement that
have been found in the scattering of light by periodic
scatterers. Recently, it has been discovered that these
phenomena are explained by the existence of the so-
called grating resonances or poles of the field function
[ 6-4 ] (a.k.a. geometrical, lattice and Bragg
resonances). Their wavelengths lay near the Rayleigh
wavelengths [7], i.e. near to period being a multiple of
the wavelength, if all elementary scatterers of a grating
are excited in the same phase, and their size is a fraction
of the period. In the wave scattering by infinite gratings,
they lead to almost total reflection of the incident field
by a sparse thin-dielectric-wire grating in narrow
wavelength bands [3, 5].
The goal of our paper is extension of this study to
more complicated periodically structured silver-wire
configurations where both plasmon and grating
resonances are present.
2. Scattering problem and its numerical solution
Consider finite collections of M parallel wires
illuminated by an H-polarized plane wave shown in
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 204-208.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
205
p
2
M
p
x2
M
p
y
p
x
1 M
layer
M
N-layer
1 M
1l
p
y
p
x1
M
1p
1
N-layer grating
Stacked two-period grating
In-line two-period grating
Fig. 1. Scattering problem geometries and notations.
Fig. 1. The wires are assumed to be infinite circular
cylinders, each having the radius a and complex relative
dielectric permittivity ε. For a D-2 problem, one has to
find a scalar function rHz
that is the scattered
magnetic-field z-component.
This function has to satisfy the Helmholtz equation
with corresponding wavenumbers inside and outside the
cylinders, the tangential field components continuity
conditions, the radiation condition, and the condition of
the local power finiteness. The full-wave numerical
solution can be obtained similarly to [7, 8], by
expanding the field function in terms of the azimuth
exponents in the local polar coordinates, using addition
theorems for cylindrical functions and applying the
boundary conditions on the surface of each of M wires.
This leads to an infinite MM block-type matrix
equation where each block is infinite. Still a close
inspection shows that the matrix equations used in the
mentioned above papers did not provide guaranteed
convergence of solutions. Here, the convergence is
understood in mathematical sense, as a possibility of
minimizing the error of computations by solving
progressively larger matrices. We fix this defect by re-
scaling the unknown coefficients as explained in [6].
The obtained in such a way matrix equation is a
block-type Fredholm second kind operator equation. In
this case, the convergence of solution, after truncation of
each block to finite order N, to an exact one if N→∞, is
guaranteed by the Fredholm theorems. The results
presented below were computed with N = 4…5; this
provided 3 correct digits in the far-field characteristics of
the sparse gratings of silver wires with radii a ≤ 75 nm
and periods p ≥ 200 nm. Note that denser gratings may
need larger values of N to achieve the same accuracy.
We have considered three sparse (p – 2a > 2a)
configurations of finite number of sub-wavelength silver
nanowires: two and three-layer gratings of the same
periods, stacked two-period gratings, and in-line two-
period gratings (Fig. 1). Dense configurations are also
interesting objects, however they deserve a separate
study. To characterize the optical properties of
considered discrete scatterers, we have used the
wavelength dependences of the total scattering (TSCS)
and the absorption (ACS) cross-sections and calculated
the field patterns in the near zone. In Figs 2 to 6, the
wavelength varies from 300 up to 500 nm, and the
complex-valued dielectric function of silver has been
borrowed from the classical paper of Johnson and
Christy.
3. Optical response of two- and three-layer gratings
N-layer gratings are interesting structures for research,
because they are periodic along two axes and can be
considered as finite-size photonic or plasmonic crystals.
How this structuring influences the grating and plasmon
resonances is a question that needs clarification.
Presented in Fig. 2 are per-wire TSCS and ACS as
functions of the wavelength for the H-wave normal
incidence on the gratings of two and three chains of 100
nanowires with radii a = 70 nm, with periods along the
x-axis px = 360 nm and along the y-axis py = 5a. As
known, the plasmon-resonance peak for a single silver
nanowire in free space is near 350 nm, so both periods
are in the vicinity of this value. As one can see, the
resonant peaks of the averaged per-wire cross-sections
are higher for two-layer grating. This can be understood
as a result of shadowing of the lower layers by the upper
(better illuminated) ones. The TSCS wavelength
dependences have several local maxima in the vicinity of
350 nm, and in the case of the two-layer grating there is
one resonance at 350–360 nm wavelengths, but in the
case of the three-layer grating there are two low-quality
resonances in the same range.
At the 383 and 394 nm wavelengths, one can see the
highest resonances for the two- and three-layer gratings,
respectively (see Fig. 3). Note that the ACS dependences
show the peak values at the same wavelengths where
TSCS dependences have their minima.
Presented in Fig. 3 are near-field amplitude patterns
for the central part of the two- and three-layer gratings in
the main resonances. The visualized part of the grating
includes 11 central wires from each layer. One can see
that the most intensive H-field maxima are located at the
illuminated side of the upper-layer wires for the two-
layer grating, and between the layers for the three-layer
grating. Besides, one can see a standing wave above the
illuminated side for each grating.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 204-208.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
206
300 350 400 450 500
0
200
400
600
800
M=100, a=70, p=360, 1 layer
M
layer
=100, a=70, p
x
=360, p
y
=5a, 2 layers
M
layer
=100, a=70, p
x
=360, p
y
=5a, 3 layers
T
SC
S
/M
, [
nm
]
Wavelength, [nm] a
300 350 400 450 500
0
50
100
150
A
C
S
/M
, [
nm
]
Wavelength, [nm]
M=100, a=70, p=360, 1 layer
M
layer
=100, a=70, p
x
=360, p
y
=5a, 2 layers
M
layer
=100, a=70, p
x
=360, p
y
=5a, 3 layers
b
Fig. 2. Normalized per number of wires TSCS (a) and ACS (b)
as functions of the wavelength for the H-wave normal
incidence on the 1-, 2- and 3-layer gratings of identical silver
wires.
a
b
Fig. 3. Near-field amplitude patterns of the central parts of the
two- and three-layer gratings from Fig. 2 in the TSCS maxima
at the wavelengths 383 (a) and 394 nm (b). The incident plane
wave comes from the upper half-space normally to the grating.
4. Optical response of two-period stacked
and in-line gratings
Two-period linear gratings that consist of two chains or
arms with different periods are interesting for applications
because of existence of two different grating resonances,
in addition to the plasmon resonance of each individual
wire. This feature can be useful for electromagnetic
engineering of the novel wideband absorbers for solar
cells. Our analysis of such gratings have shown that it is
indeed possible to combine the resonances and enhance
the per-wire TSCS and ACS, if the wire radius is 50 nm or
larger and their number is at least 100.
Presented in Fig. 4 are per-wire TSCS and ACS as
functions of wavelength for the normal incidence of the
H-wave on the stacked and in-line double-periodic
gratings of silver nanowires with radii a = 70 nm and
periods p1 = 360 nm (180 or 179 wires) and p2 = 450 nm
(100 wires); the distance between the layers in the
stacked grating is 210 nm. One can see that the cross-
sections for the in-line configuration have more
intensive and sharper resonances of both types,
apparently because such a configuration has no part
shaded by other wires. The TSCS reaches its maximum
value at 374 and 372 nm for the stacked and in-line two-
period gratings, respectively. This resonance is
associated with the smaller period and, in part, with the
plasmon resonance of each wire. The other grating
resonance in the vicinity of the 450 nm wavelength (the
larger period value) has low intensity, especially for
ACS because the bulk losses in silver are smaller there.
300 350 400 450 500
0
100
200
300
400
500
600
700
T
S
C
S/
M
, [
nm
]
Wavelength, [nm]
M
1
=100, M=180, a=70, p
x1
=360, p
x2
=450, p
y
=5a, stacked 2-period gr.
M
1
=100, M=179, a=70, p
x1
=360, p
x2
=450, in-line 2-period gr.
a
300 350 400 450 500
0
50
100
150
200
M
1
=100, M=180, a=70, p
x1
=360, p
x2
=450, p
y
=5a, stacked 2-period gr.
M
1
=100, M=179, a=70, p
x1
=360, p
x2
=450, in-line 2-period gr.
A
C
S/
M
, [
nm
]
Wavelength, [nm] b
Fig. 4. Normalized TSCS (a) and ACS (b) as functions of the
wavelength for the H-wave normally incident on the stacked
and in-line two-period gratings of identical silver nanowires.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 204-208.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
207
a
b
Fig. 5. Near-field amplitude patterns of the central parts of the
stacked two-period grating from Fig. 4 in the grating
resonances at the wavelengths 374 (a) and 449 nm (b). The
incident plane wave comes from the upper half-space normally
to the grating.
a
b
Fig. 6. Near-field amplitude patterns of the central parts of the
in-line two-period grating from Fig. 4 in the grating resonances
at the wavelengths 372 (a) and 449 nm (b). The incident plane
wave comes from the upper half-space normally to the grating.
Presented in Figs. 5 and 6 are the near-field
amplitude patterns for the central parts of the stacked
and in-line two-period gratings, respectively, in the
grating resonances. In Fig. 5, we show the near-field
patterns at 374 and 449 nm in the grating resonances
associated with the smaller and larger periods,
respectively. Note that in the second case the field
pattern demonstrates that the top grating (tuned into
resonance) efficiently screens the bottom grating, which
remains in the deep shadow. In Fig. 6, we show near-
field patterns at 372 and 449 nm in the grating
resonances for the smaller and larger periods of the in-
line grating. In each case, the standing waves are formed
along the x and y axes, however only near the arm of the
grating that is tuned to the resonance.
5. Conclusions
We have presented results of accurate calculations of the
scattering and absorption spectra for several periodically
structured configurations made of silver nanowires, in
the visible range. They demonstrate co-existence of
plasmon resonances related to each silver wire and
several grating-type resonances in structures with more
than one period. At the wavelength of each grating
resonance, the scattering and absorption demonstrate
maxima. Visualization of in-resonance near fields shows
that the contribution to these maxima comes from the
corresponding sub-gratings whose period is tuned to the
incoming wavelength. Presented results can be useful for
the modelling of optical response of “photonic-
plasmonic molecules” and for engineering the novel
wideband absorbers for solar cells.
Acknowledgments
This work has been partially supported by the National
Academy of Sciences of Ukraine via the State Target
Program “Nanotechnologies and Nanomaterials” and the
European Science Foundation via the Research
Networking Programme “Newfocus”.
References
1. V. Giannini and J.A. Sànchez-Gil, Calculations of
light scattering from isolated and interacting
metallic nanowires of arbitrary cross-section by
means of Green’s theorem surface integral
equations in parametric form // J. Opt. Soc. Am. A,
24(9), p. 241-248 (2007).
2. D.R. Fredkin, I. Mayergoyz, Resonant behavior of
dielectric objects (electrostatic resonances) // Phys.
Rev. Lett. 91, p. 3902-3905 (2003).
3. M. Laroche, S. Albaladejo, R. Gomez-Medina, and
J.J. Saenz, Tuning the optical response of
nanocylinder arrays: an analytical study // Phys.
Rev. B, 74(9), p. 245422-10 (2006).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 204-208.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
208
4. F.J.G. Garcia de Abajo, Colloquium: Light
scattering by particle and hole arrays // Rev. Mod.
Phys. 79(4), p. 1267-1289 (2007).
5. V.O. Byelobrov, J. Ctyroky, T.M. Benson, et al.,
Low-threshold lasing eigenmodes of an infinite
periodic chain of quantum wires // Opt. Lett.
35(21), p. 3634-3636 (2010).
6. D.M. Natarov, V.O. Byelobrov, R. Sauleau,
T.M. Benson, and A.I. Nosich, Periodicity-induced
effects in the scattering and absorption of light by
infinite and finite gratings of circular silver
nanowires // Opt. Express, 19(22), p. 22176-22190
(2011).
7. H.A. Ragheb, M. Hamid, Scattering by N parallel
conducting circular cylinders // Int. J. Electronics,
59(4), p. 407-421 (1985).
8. D. Felbacq, G. Tayeb, and D. Maystre, Scattering
by a random set of parallel cylinders // J. Opt. Soc.
Am. A, 11(9), p. 2526-2538 (1994).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 204-208.
PACS 42.79.Dj, 78.35.+c
Resonance scattering and absorption of light
by finite two-period gratings of circular silver nanowires
D.M. Natarov1, R. Sauleau2, A.I. Nosich1
1Institute of Radio-Physics and Electronics, NAS of Ukraine,
12, Proskury str. 61085 Kharkiv, Ukraine
E-mail: den.natarov@gmail.com
2IETR, University of Rennes 1, Campus Beaulieu, bat 11-D, 35042 Rennes Cedex, France
E-mail: ronan.sauleau@univ-rennes1.fr
Abstract. We consider the two-dimensional scattering of the H-polarized electromagnetic plane waves of the visible range by three types of gratings made of periodically arranged circular cylindrical sub-wavelength wires. Using the field expansions in local coordinates and addition theorems for cylindrical functions, we obtain a block-type matrix equation for the field expansion coefficients. This equation is of the Fredholm second-kind form that guarantees convergence of numerical solution. The scattering and absorption cross-sections and the near-field patterns are computed. The interplay of plasmon and grating-type resonances is studied for two and three-layer gratings of identical periods, stacked two-period gratings, and in-line two-period gratings made of nano-diameter silver wires.
Keywords: plasmon resonance, grating resonance, finite grating, nanowire, scattering, absorption, cross-section.
Manuscript received 08.05.12; revised version received 08.06.12; accepted for publication 14.06.12; published online 25.09.12.
1. Introduction
Two fundamental effects are known to influence the scattering of light by periodically structured metal scatterers. On the one hand, surface-plasmon resonances are observed for sub-wavelength noble-metal particles and wires in the mid-infrared and optical bands [1, 2]. Nanosize objects can exhibit resonance behavior at certain frequencies for which the object permittivity is negative. This results in powerful enhancement of scattered and absorbed light that is used in the design of optical antennas and biochemical sensors for advanced applications. In the leading terms, the plasmon resonance wavelength depends on the object shape but not on its dimensions.
On the other hand, periodically structured scatterers, or finite and infinite gratings, arrays or chains of particles and holes in metallic screens (in
D
-
3
) or wires and slots (in
D
-
2
), are attracting large attention of researchers in today nano-optics [
6
-
3
]. This is caused by the effects of extraordinarily large reflection, transmission, emission, and near-field enhancement that have been found in the scattering of light by periodic scatterers. Recently, it has been discovered that these phenomena are explained by the existence of the so-called grating resonances or poles of the field function [
6
-
4
] (a.k.a. geometrical, lattice and Bragg resonances). Their wavelengths lay near the Rayleigh wavelengths [7], i.e. near to period being a multiple of the wavelength, if all elementary scatterers of a grating are excited in the same phase, and their size is a fraction of the period. In the wave scattering by infinite gratings, they lead to almost total reflection of the incident field by a sparse thin-dielectric-wire grating in narrow wavelength bands [3, 5].
The goal of our paper is extension of this study to more complicated periodically structured silver-wire configurations where both plasmon and grating resonances are present.
2. Scattering problem and its numerical solution
Consider finite collections of M parallel wires illuminated by an H-polarized plane wave shown in Fig. 1. The wires are assumed to be infinite circular cylinders, each having the radius a and complex relative dielectric permittivity ε. For a
D
-
2
problem, one has to find a scalar function
(
)
r
H
z
r
that is the scattered magnetic-field z-component.
This function has to satisfy the Helmholtz equation with corresponding wavenumbers inside and outside the cylinders, the tangential field components continuity conditions, the radiation condition, and the condition of the local power finiteness. The full-wave numerical solution can be obtained similarly to [7, 8], by expanding the field function in terms of the azimuth exponents in the local polar coordinates, using addition theorems for cylindrical functions and applying the boundary conditions on the surface of each of M wires. This leads to an infinite
M
M
´
block-type matrix equation where each block is infinite. Still a close inspection shows that the matrix equations used in the mentioned above papers did not provide guaranteed convergence of solutions. Here, the convergence is understood in mathematical sense, as a possibility of minimizing the error of computations by solving progressively larger matrices. We fix this defect by re-scaling the unknown coefficients as explained in [6].
The obtained in such a way matrix equation is a block-type Fredholm second kind operator equation. In this case, the convergence of solution, after truncation of each block to finite order N, to an exact one if N→∞, is guaranteed by the Fredholm theorems. The results presented below were computed with N = 4…5; this provided 3 correct digits in the far-field characteristics of the sparse gratings of silver wires with radii a ≤ 75 nm and periods p ≥ 200 nm. Note that denser gratings may need larger values of N to achieve the same accuracy.
p
2
M
p
x
2
M
p
y
p
x
1
M
layer
M
N-
layer
1
M
1
l
p
y
p
x
1
M
1
p
1
N-layer grating
Stacked two-period grating
In-line two-period grating
We have considered three sparse (p – 2a > 2a) configurations of finite number of sub-wavelength silver nanowires: two and three-layer gratings of the same periods, stacked two-period gratings, and in-line two-period gratings (Fig. 1). Dense configurations are also interesting objects, however they deserve a separate study. To characterize the optical properties of considered discrete scatterers, we have used the wavelength dependences of the total scattering (TSCS) and the absorption (ACS) cross-sections and calculated the field patterns in the near zone. In Figs 2 to 6, the wavelength varies from 300 up to 500 nm, and the complex-valued dielectric function of silver has been borrowed from the classical paper of Johnson and Christy.
3. Optical response of two- and three-layer gratings
N-layer gratings are interesting structures for research, because they are periodic along two axes and can be considered as finite-size photonic or plasmonic crystals. How this structuring influences the grating and plasmon resonances is a question that needs clarification. Presented in Fig. 2 are per-wire TSCS and ACS as functions of the wavelength for the H-wave normal incidence on the gratings of two and three chains of 100 nanowires with radii a = 70 nm, with periods along the x-axis px = 360 nm and along the y-axis py = 5a. As known, the plasmon-resonance peak for a single silver nanowire in free space is near 350 nm, so both periods are in the vicinity of this value. As one can see, the resonant peaks of the averaged per-wire cross-sections are higher for two-layer grating. This can be understood as a result of shadowing of the lower layers by the upper (better illuminated) ones. The TSCS wavelength dependences have several local maxima in the vicinity of 350 nm, and in the case of the two-layer grating there is one resonance at 350–360 nm wavelengths, but in the case of the three-layer grating there are two low-quality resonances in the same range.
At the 383 and 394 nm wavelengths, one can see the highest resonances for the two- and three-layer gratings, respectively (see Fig. 3). Note that the ACS dependences show the peak values at the same wavelengths where TSCS dependences have their minima.
Presented in Fig. 3 are near-field amplitude patterns for the central part of the two- and three-layer gratings in the main resonances. The visualized part of the grating includes 11 central wires from each layer. One can see that the most intensive H-field maxima are located at the illuminated side of the upper-layer wires for the two-layer grating, and between the layers for the three-layer grating. Besides, one can see a standing wave above the illuminated side for each grating.
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Fig. 2. Normalized per number of wires TSCS (a) and ACS (b) as functions of the wavelength for the H-wave normal incidence on the 1-, 2- and 3-layer gratings of identical silver wires.
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Fig. 3. Near-field amplitude patterns of the central parts of the two- and three-layer gratings from Fig. 2 in the TSCS maxima at the wavelengths 383 (a) and 394 nm (b). The incident plane wave comes from the upper half-space normally to the grating.
4. Optical response of two-period stacked
and in-line gratings
Two-period linear gratings that consist of two chains or arms with different periods are interesting for applications because of existence of two different grating resonances, in addition to the plasmon resonance of each individual wire. This feature can be useful for electromagnetic engineering of the novel wideband absorbers for solar cells. Our analysis of such gratings have shown that it is indeed possible to combine the resonances and enhance the per-wire TSCS and ACS, if the wire radius is 50 nm or larger and their number is at least 100.
Presented in Fig. 4 are per-wire TSCS and ACS as functions of wavelength for the normal incidence of the H-wave on the stacked and in-line double-periodic gratings of silver nanowires with radii a = 70 nm and periods p1 = 360 nm (180 or 179 wires) and p2 = 450 nm (100 wires); the distance between the layers in the stacked grating is 210 nm. One can see that the cross-sections for the in-line configuration have more intensive and sharper resonances of both types, apparently because such a configuration has no part shaded by other wires. The TSCS reaches its maximum value at 374 and 372 nm for the stacked and in-line two-period gratings, respectively. This resonance is associated with the smaller period and, in part, with the plasmon resonance of each wire. The other grating resonance in the vicinity of the 450 nm wavelength (the larger period value) has low intensity, especially for ACS because the bulk losses in silver are smaller there.
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Fig. 4. Normalized TSCS (a) and ACS (b) as functions of the wavelength for the H-wave normally incident on the stacked and in-line two-period gratings of identical silver nanowires.
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Fig. 5. Near-field amplitude patterns of the central parts of the stacked two-period grating from Fig. 4 in the grating resonances at the wavelengths 374 (a) and 449 nm (b). The incident plane wave comes from the upper half-space normally to the grating.
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Fig. 6. Near-field amplitude patterns of the central parts of the in-line two-period grating from Fig. 4 in the grating resonances at the wavelengths 372 (a) and 449 nm (b). The incident plane wave comes from the upper half-space normally to the grating.
Presented in Figs. 5 and 6 are the near-field amplitude patterns for the central parts of the stacked and in-line two-period gratings, respectively, in the grating resonances. In Fig. 5, we show the near-field patterns at 374 and 449 nm in the grating resonances associated with the smaller and larger periods, respectively. Note that in the second case the field pattern demonstrates that the top grating (tuned into resonance) efficiently screens the bottom grating, which remains in the deep shadow. In Fig. 6, we show near-field patterns at 372 and 449 nm in the grating resonances for the smaller and larger periods of the in-line grating. In each case, the standing waves are formed along the x and y axes, however only near the arm of the grating that is tuned to the resonance.
5. Conclusions
We have presented results of accurate calculations of the scattering and absorption spectra for several periodically structured configurations made of silver nanowires, in the visible range. They demonstrate co-existence of plasmon resonances related to each silver wire and several grating-type resonances in structures with more than one period. At the wavelength of each grating resonance, the scattering and absorption demonstrate maxima. Visualization of in-resonance near fields shows that the contribution to these maxima comes from the corresponding sub-gratings whose period is tuned to the incoming wavelength. Presented results can be useful for the modelling of optical response of “photonic-plasmonic molecules” and for engineering the novel wideband absorbers for solar cells.
Acknowledgments
This work has been partially supported by the National Academy of Sciences of Ukraine via the State Target Program “Nanotechnologies and Nanomaterials” and the European Science Foundation via the Research Networking Programme “Newfocus”.
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Fig. 1. Scattering problem geometries and notations.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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