Increased cut-off wavelength for a
subwavelength hole in a real metal
Reuven Gordon and Alexandre G. Brolo*
Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, V8W 3P6
*Department of Chemistry, University of Victoria, BC, Canada, V8W 3V6
[email protected]
Abstract: A waveguide mode of a subwavelength rectangular hole in a real
metal is analyzed. Due to coupling between surface plasmons on the long
edges of the hole, the cut-off wavelength increases as the hole-width is
reduced. The cut-off wavelength is found to be much larger than Rayleigh’s
criterion for perfect metals – 2.3 times as large for a 15 nm wide hole. The
analytical results are verified by finite-difference calculations. The finite
difference calculations also show the influence of including material loss.
©2005 Optical Society of America
OCIS codes: (240.6680) Surface plasmons; (230.7370) Waveguides
References and links
1.
2.
3.
4.
5.
6.
7.
8.
9.
Lord Rayleigh, “On the Passage of Electric Waves Through Tubes,” Philos. Mag. 43, 125-132 (1897).
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T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary Optical Transmission
through Sub-Wavelength Hole Arrays,” Nature 391, 667-669 (1998).
H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal,
T. W. Ebbesen, “Beaming Light from a Subwavelength Aperture,” Science 297, 820-822 (2002).
R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong Polarization
in the Optical Transmission through Elliptical Nanohole Arrays,” Phys. Rev. Lett. 92, 037401 (2004).
K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong Influence of Hole
Shape on Extraordinary Transmission through Periodic Arrays of Subwavelength Holes,” Phys. Rev. Lett.
92, 183901 (2004).
A. Degiron, H. J. Lezec, N. Yamamoto, T. W. Ebbesen, “Optical Transmission Properties of a Single
Subwavelength Aperture in a Real Metal,” Opt. Commun. 239, 61-66 (2004).
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R. C. Booton Jr., Computational Methods for Electromagnetics and Microwaves, (John Wiley & Sons, New
York, 1992).
1. Introduction
The cut-off condition, for which there can be no propagation of light through a hole in a
perfect metal, occurs when the wavelength of light is more than twice the hole-length across
[1]. Below cut-off, Bethe showed that the transmission of light through an aperture rapidly
decreases as the fourth power of the ratio of the aperture length to the optical wavelength [2].
Subwavelength arrays of metal holes [3] and single metal holes flanked with gratings [4]
allow for extraordinary transmission as compared with Bethe’s theory, which has been
explained in terms of resonant coupling to surface plasmon (SP) modes on the surface of the
metal. Recently there has been an interest in the influence of the hole-shape on the
transmission properties [5-7]. It was shown that in randomly distributed rectangular holes the
transmission depends upon the aspect ratio of the hole, which was called the “shape-effect” in
that paper [6]. It was postulated that the “shape-effect” was the result of localized SP
resonances. A recent work has investigated the transmission through a single rectangular
aperture, while systematically varying the aspect ratio of the rectangle [7]. That work found
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Received 9 February 2005; revised 4 March 2005; accepted 6 March 2005
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the surprising result that the maximum transmission through the hole was red-shifted as the
hole was made smaller.
In this paper, we explain the shift to longer wavelengths when decreasing the aperture
width in real metals. By considering the influence of a coupled SP mode from the long edges
of the aperture and penetration of the electric field into the metal, we show that the Rayleigh’s
cut-off condition is increased significantly for real metals. To verify the applicability of our
analytic theory, we calculate numerically the mode profiles and the waveguide dispersion for
rectangular holes in a metal.
2. Cut-off increase for a subwavelength hole in a real metal
Figure 1 shows the geometry of the problem under consideration. We break the 2D problem
down into two 1D effective-index problems. This allows us to identify two separate
contributions to the extended cut-off wavelength: penetration of the field into the metal along
the x-direction, and coupling between SP modes along the y-direction. The latter effect
explains how the cut-off wavelength actually increases as the hole size is reduced.
l
y
εair = 1
εm
E
w
x
l
εm
εair = 1
w
H
εd
εm
TM mode
E
TE mode
Fig. 1. Schematic of hollow rectangular waveguide in a metal, with co-ordinates shown. The
lowest-order mode is analyzed by considering the TM mode of a slab of width w, to derive an
effective dielectric constant, and then solving for the TE mode component in a slab of
separation l filled with the effective dielectric. This provides the effective dielectric constant
for the lowest-order mode of the rectangular waveguide.
Although the effective index approach works for a perfect electric conductor (PEC), it is
only an approximation for real metals because the boundary conditions cannot be consistently
matched along orthogonal direction when the electric field penetrates into the metal.
Nevertheless, for good metals, the field components that do not match at the boundaries are
small, and the approximation gives good quantitative results. The validity of this approach is
verified by numerical calculation with the finite-difference method. These calculations also
show that material loss has a small influence on these results.
2.1 Cut-off increase from penetration of the electric field into the metal
The propagation constant of the TE01 mode of a rectangular hole in a PEC is given by:
β =π
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⎛2⎞
⎜ ⎟
⎝λ ⎠
2
⎛1⎞
−⎜ ⎟
⎝a⎠
2
(1)
Received 9 February 2005; revised 4 March 2005; accepted 6 March 2005
21 March 2005 / Vol. 13, No. 6 / OPTICS EXPRESS 1934
where a is the length of the rectangle and λ is the wavelength of light in vacuum [1]. The
cut-off wavelength occurs when the propagation constant of the TE01 mode is zero, or
when λ = 2a . For longer wavelengths, the mode decays exponentially with a decay length
that can be found from Eq. (1). This decay results because light at these wavelengths cannot
propagate within the hole, and so it is reflected. For comparison with recent experiments [7],
if the length of the rectangle is 270 nm, and the wavelength of light is 750 nm, the intensity of
light in the mode is attenuated by 99.2% upon transmission through a 300 nm distance. Light
with wavelength shorter than 540 nm should receive no attenuation in the mode at all.
In real metals at optical frequencies larger than the plasma frequency, the Drude model
may be used to estimate the metal’s relative permittivity. Typically, the metal’s relative
permittivity has a negative real part and a small imaginary part (which is neglected for this
analysis, but reconsidered in the numerical model presented below). The electric field can
penetrate into the metal by the skin-depth, which makes the hole appear larger. The
propagation constant, β TE , of a TE mode between two parallel plates of a real metal can be
found from the characteristic equation:
tan⎛⎜ k o 2ε d − β TE 2 l 2 ⎞⎟ =
⎝
where
εm
⎠
is the relative permittivity of the metal, and
εd
β 2 − k o 2ε m
k o ε d − βTE
2
2
(2)
is the relative permittivity of the
dielectric between the metal sides, l is the length between the metal sides, and k o = 2π / λ
is the free-space wave-vector. By setting the propagation to zero, we find the cut-off
wavelength:
λcut-off =
πl ε d
(3)
arctan − ε m / ε d
At 750 nm in silver, the real part of the dielectric constant is -27.5 [8], which will give an
increase in the cut-off wavelength of 14% over the PEC condition for a 270 nm wide hole.
2. 2 Cut-off increase from coupled surface-plasmons
An even larger increase in the cut-off condition is found by considering coupled SP waves on
the long edges of the aperture. This leads to the interesting result that the cut-off wavelength
actually increases as the hole-size becomes smaller.
If we again consider the case of two parallel plates, we can formulate the TM mode of
this configuration as coming from the sum of SP modes on the top and bottom plates. In the
region between the plates, the field has a hyperbolic cosine dependence from the sum of two
exponential decaying SP modes on the top and bottom plates. Not only does the field
penetrate into the metal, as was found in the previous section, but the mode-shape within the
hole is altered. The characteristic equation for this configuration is modified from the
dielectric case to be:
tanh⎛⎜ βTM 2 − k o 2ε air w 2 ⎞⎟ =
⎝
⎠
− ε air
εm
β TM 2 − k o 2ε m
βTM 2 − k o 2ε air
.
(4)
For a 105 nm aperture in silver at 750 nm, this TM mode has a propagation constant that
is 1.2 times the free-space wave-vector. As a result, the effective index squared of this mode is
1.46. Clearly, the effective index increases as the width of the hole is reduced.
If we assume, as is the case in a PEC rectangular waveguide, that the TE01 mode may be
approximated by the TM mode in the transverse direction and the TE mode along the lateral
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Received 9 February 2005; revised 4 March 2005; accepted 6 March 2005
21 March 2005 / Vol. 13, No. 6 / OPTICS EXPRESS 1935
direction, we can estimate the cut-off wavelength by using the effective index squared from
Eq. (4) as the relative permittivity of the dielectric in Eq. (3):
ε d = ( βTM k o ) 2 .
(5)
effective index squared
ε d = ( β TM / k o )
2
As an example, for a 270 nm by 105 nm rectangular waveguide, we find that the cut-off
wavelength for silver is 761 nm. This is 41% larger than the cut-off of a PEC waveguide with
the same geometry.
1.5
1.3
0.8
0.6
PEC (270 nm)
225 nm x 270 nm
185 nm x 270 nm
145 nm x 270 nm
105 nm x 270 nm
0.4
0.2
500
600
700
wavelength (nm)
800
Fig. 2. The top graph shows the effective relative permittivity of the dielectric as calculated
from Eq. (4). The bottom graph shows the TE01 mode for rectangular holes in silver and for a
perfect electric conductor (PEC), as calculated using the effective index method. +, ×, , and
○ are the results from numerical simulations.
Figure 2 shows the dispersion of ε d = ( βTM k o ) 2 (as calculated from Eq. (4)), and the
effective index squared, ( βTE k o ) 2 (as calculated using Eq. (2)). The Drude response was
used to calculate the relative permittivity for silver [8], with an effective mass of 0.96 the free
electron mass, a scattering time of 31 fs, and a background dielectric constant of 4.15. In a
perfect metal, the hole-width does not influence the propagation constant for the TE01 mode,
and the cut-off wavelength is 540 nm. The shift for a TE mode of the 1D problem was found
to be only 14% in section 2.1, which results in a cut-off around 620 nm. Therefore, reducing
the hole-size to increase the SP-mode coupling increases the cut-off wavelength significantly.
We may consider the extreme example of a hole-width of 15 nm (which would be challenging
to fabricate), for which the cut-off wavelength increases to 2.3 times the PEC value.
The red-shift in the cut-off wavelength is mediated by the SP-coupling between the longedges of the hole. Therefore, it depends on the hole-width rather than the aspect ratio of the
hole; as the width is reduced, the coupling is increased.
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The PEC limit is recovered as the real part of the metal permittivity, ε m , approaches
negative infinity. In this limit, the left-hand side of Eq. (4) vanishes, which means that in
order to have the right-hand side vanish as well, βTM = k o ε air = k o ; so no red-shift is seen
in the cut-off wavelength as the width of the hole is reduced. The physical interpretation here
is that a PEC cannot support SP modes at the metal-air interface, and so no SP-SP coupling is
allowed.
2.3 Numerical simulations
An analytic solution does not exist for the rectangular waveguide, except for the special case
of the PEC. For this reason, we resorted to numerical simulation using the finite difference
method to validate the effective index model of the previous sections. A detailed discussion of
the finite difference method can be found elsewhere [9]. Convergence was ensured by
reducing the grid-size and extending the artificial PEC boundaries. The smallest grid-size
attempted was 0.5 nm. The largest PEC artificial boundary was set to a width of 1 micron,
where all the field components are negligible. To speed up the calculations, symmetries in the
x and y directions were exploited.
Figure 2 shows the simulation results with the symbols +, ×, , and ○ for the 105 nm,
145 nm, 185 nm and 225 nm wide holes. We performed numerical calculations for holewidths down to 15 nm, where the cut-off wavelength is extended to 1260 nm. In all cases, the
numerically calculated effective index squared was found to be within 0.025 of the value
found by the analytical model and the cut-off wavelength agreed to within 6 nm.
E
H
x
y
z
Fig. 3. E, H fields polarized along the x, y and z directions for the lowest order mode in a
105 nm by 270 nm rectangular waveguide in silver at the wavelength of 750 nm. These mode
profiles were calculated numerically by the finite difference technique. Each figure shows
220 nm along the y-direction, and 380 nm along the x-direction. The graphs were scaled in the
ratios Ex:Ey:Ez given by 0.48:1.0:0.012, and Hx:Hy:Hz given by 1.0:0.016:13.4.
Figure 3 shows the field distribution for the E and H components as calculated
numerically. Within the hole, the H-field along the x-direction and the E-field along the ydirection closely resemble those of the TE01 modes of a PEC rectangular guide. Along the
other directions, and inside the metal, there are differences from the PEC case. These figures
show the existence of E and H components normal to those found for the TE01 mode in a PEC.
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21 March 2005 / Vol. 13, No. 6 / OPTICS EXPRESS 1937
There is a penetration of the various field components into the metal at all boundaries. The
field distributions along the top and bottom boundaries are clearly indicative of SP modes on
these surfaces.
2.4 Influence of material loss
Figure 4 shows the separate contributions of the cut-off attenuation and material losses, as
calculated numerically when the imaginary part of the dielectric constant was added. Material
loss plays a negligible role for wavelengths a few nanometers above the cut-off wavelength.
For example, for a 300 nm film, at optical wavelength of 764 nm, the cut-off attenuation is
12000 times the material absorption. It should be noted that these two effects are distinct;
material loss absorbs the photons, whereas cut-off attenuation occurs when the incident
photons are reflected without absorption because they cannot propagate inside the hole.
attenuation (dB/ µm)
30
material losses
cut-off attenuation
20
10
0
700
750
wavelength (nm)
800
Fig. 4. Attenuation from material losses and cut-off attenuation for a 105 nm by 270 nm
rectangular hole in silver.
3. Conclusions
We presented an analytic theory which showed a dramatic increase in the cut-off wavelength
of real metals as compared to a perfect conductor. The cut-off wavelength was enhanced by
41% for the hole-sizes that were used in recent experiments, and it was more than doubled for
hole-widths at the limits of present fabrication capability. This shows that care should be
taken when using the perfect-metal approximation to decide if these holes are actually below
cut-off. Furthermore, the analytic theory explains how the cut-off wavelength increases as the
holes are made smaller; this enhancement results from coupling between the surface plasmons
on the top and bottom edges of the hole. The results of the analytic theory agreed well with
finite-difference numerical calculations. The numerical calculations also showed that the
influence of material loss did not influence the cut-off wavelength and were negligible for
wavelengths a few nanometers above the cut-off wavelength.
Acknowledgments
The authors acknowledge financial support for this work from an NSERC Special Research
Opportunity grant, and an in-kind contribution of the MODE Solutions software package from
Lumerical Solutions Inc.
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(C) 2005 OSA
Received 9 February 2005; revised 4 March 2005; accepted 6 March 2005
21 March 2005 / Vol. 13, No. 6 / OPTICS EXPRESS 1938