## Abstract

This paper presents theory and finite-difference time-domain (FDTD) calculations for a single and arrays of sub-wavelength cylindrical holes in metallic films presenting large transmission. These calculations are in excellent agreement with experimental measurements. This effect has to be understood in terms of the properties exhibited by the dielectric constant of metals which cannot be treated as ideal metals for the purpose of transmission and diffraction of light. We discuss the cases of well-differentiated metals silver and tungsten. It is found that the effect of surface plasmons or other surface wave excitations due to a periodical set of holes or other roughness at the surface is marginal. The effect can enhance but also can depress the transmission of the arrays as shown by theory and experiments. The peak structure observed in experiments is a consequence of the interference of the wavefronts transmitted by each hole and is determined by the surface array period independently of the material. Without large transmission through a single hole there is no large transmission through the array. We found that in the case of Ag which at the discussed frequencies is a metal there are cylindrical plasmons at the wall of the hole, as reported by Economu et al 30 years ago, that enhanced the transmission. But it turns out, as will be explained, that for the case of W which behaves as a dielectric, there is also a large transmission when compared with that of an ideal metal waveguide at large wavelengths. To deal with this problem one has to use the measured dielectric function of the metals. We discuss thoroughly all these cases and compare with the data. We notice that to discuss these data, for a single hole’s transmission, in terms of the Bethe approximation of ideal metals is misleading. Therefore, the extraordinary enhancement of the transmission for the holes arrays versus the single hole does not exist.

©2006 Optical Society of America

## 1. Introduction

Experiments were reported [1] showing that the
transmission of light through sub-wavelength holes drilled periodically in a
metallic film of Ag was large, 1000′s times larger, as compared with the
transmission of one single hole of the same size in the same material. Recent
experiments [2], by part of the same team of
Ref. [1] (Lezec and Thio), appear to
contradict the earlier experiments [1]. The
explanation of the experiments [1] where
based on the existence of surface plasmons polaritons (SPPs) that are excited in the
case of a set of periodic holes. After the initial work, a whole serial of papers
have appeared insisting on the same point (for a review see Ref. [2]). The last one, to our knowledge, appears
recently on the same matter [3]. The recent
paper by Lezec and Thio [2], reviewing the
field and containing many new data, criticized much the whole saga of papers on the
matter of the extraordinary enhancement of the transmission in a periodic array of
holes in metallic films. This paper [2]
claims to dismount the interpretation of SPP to understand the experiments and
disclaims the entire picture in periodic arrays of holes. Also, it showed
experimentally that the transmission enhancement by the periodic array with respect
to that of a single hole is as much as a factor of 7, not a factor of 1000s, so that
it can be a depression of the relative transmission as well. In fact the title of
this paper is: “*Diffracted evanescent wave model for enhanced and suppressed
optical transmission through subwavelengths hole arrays*.” Maybe there
is a point of broken physical argument in Ref. [1] and subsequent papers. Their claims are based on that they compared
the transmission by each hole of the array with the transmission of a single hole
reported in an earlier paper by Bethe [4].
This paper [4] was a theoretical study
showing that the transmission of a sub-wavelength hole drilled in a perfect metal
screen, ideal conductor (dielectric constant being negative infinity, ε→-∞), behaves
as (*r*/*λ*)^{4}, where λ is the wavelength of
the radiation and *r* is the hole radius. However ref.4 makes an
approximation that is only valid for holes much smaller than *λ*, so
that the field is practically constant on the hole. This does not hold for the
experiments reported [1–3] because for the frequencies considered, the hole’s radius
should be smaller than 25 nm but the holes used in the experiments are much larger.
Another point is that an ideal metal has little resemblance regarding optical
propagation with plasmonic metals as Ag or with a dielectric as W for the
experimental frequencies. This has been discussed in part in a recent work [5]. Therefore the Bethe paper [4] has no significance in the problem at hand.
In fact the main result in Ref. 2, in our opinion, is that the single whole
transmission is very large when compared with that reported in ref.4 or with that
resulting from a theory for wave propagation in ideal metals long waveguides.
Therefore it seems reasonable to try to discuss and understand these experiments
using the experimental frequency dependent dielectric constants for the metals Ag
and W. For these two materials we used extensive comparisons with the data.

The aim of this paper is to study the transmissions of a hole of sub-wavelength size in a flat metallic film with thickness of the order of the hole diameter, as those used experimentally [1–3]. When simplistic estimations are considered, the transmission can be of the order of several thousand times larger than that for the same hole in an ideal metal. The understanding of this phenomenon is due to two terms for Ag: i) surface plasmons excited at the cylinder walls defining the hole, having the same nature that those described by Pfeiffer, Economou and Ngai [6] also Martinos and Economou [7] for metallic cylinders and recently discussed in the contest of the problem at hand [5]. These are surface plasmons rotating at the surface of the cylinders and propagating along its axis. ii) the penetration of the field in the metal. For the dielectric metal W, there may be a question whether cylindrical plasmons exist, but still is the penetration of the field in the metal. The signatures of waves at the cylinder surface are well manifested in the intensity curve depending of frequency for single holes. As will be seen these are effects that are governed in a subtle way by the values of the dielectric function at each frequency.

## 2. Calculations

The structures we calculated in this work are single circular hole or hole arrays in
a flat metallic film with a given diameter *d* and thickness
*t*. Figures 1(a) and
1(b) present a view of the single and
arrays of hole structures with period P. The plane wave impinges perpendicular to
the plane of the holes. Plane pulse wave with a broad band is set as incident wave.
We recorded both the incident and transmitted wave through the structure. The
wavelength of the carrying wave of the incident pulsed is 600nm. The band width of
the incident pulse is set more than 1500nm in wavelength, which covers from 200nm up
to 1700nm in wavelength.

The calculations are performed using 3D-FDTD method. The wave spectrums before and after the structures are achieved by Fourier transform from the recorded wave signal in time domain. The frequency response of the structure is calculated by dividing the spectrum of the transmitted wave by the spectrum of the incident wave.

In the FDTD method we used for different materials, the metals like Ag and W are modeled as frequency dispersive media. The corresponding frequency dependent permittivities are retrieved from experimental data by Johnson and Christy [8] and from Physics Data [9]. For ideal metal, it is treated by means that the electric components of wave are excluded from any part of the metal; i.e. the field is forced to be zero at the metal surface. This is equivalent to set the permittivity as infinitely negative.

The perfect matched layer absorbing boundary condition [10] is applied in FDTD calculations for single hole’s transmission, and periodical boundary condition are applied for hole arrays transmission. The grids and time steps are taken fine enough to obtain convergent solutions. In detail, the space mesh are sized by 200×150×150 with grid size ds set as 10nm or 15nm. Time step is set as 0.2ds/c, where c is the speed of light in vacuum.

#### 2.1 The Ag case for single holes

### 2.1.1 Calculations

Ag is a paradigmatic case for studying surface plasmons and there are a large
amount of literatures on surface polariton plasmons (SPPs) by gratings
[11] in the visible region. The
reason for that is because at these wavelengths the imaginary part of the
permittivity (ε_{2}) is small, then the SPP is well defined [11, 12]. Figure 2(a) shows
the real (ε_{1}) and the imaginary (ε_{2}) part of the
permittivity. It represents a nice Drude plasmonic behaviour with a bulk
plasmon wavelength, λ_{p}=325nm (ω_{p}~3.8ev). It has been
also shown that in a periodic surface with a certain small single Fourier
component the enhancement of the field due to the SPP can be very large
(≈100 times that of the incident field) [12,13]. However when the
Fourier component increases or many Fourier component exist (for example a
step-like profile of the surface) the enhancement is reduced drastically due
to the enlargement of the SPP linewidth. We mention these points to stress
that the existence of SPP in a surface does not imply large enhancements but
many other requisites are needed. In particular the structure for study in
Fig. 1(b) has many Fourier
components and the SPP enhancement due to SPP can not be large.

To model metal Ag in the calculation, a Drude dispersion relation [Eq. (1)] is used to meet the frequency dependent permittivity of Ag.

which is characterized by fitting permittivity ε_{f} in the visible
range, the bulk plasma frequency ω_{p} and the damping constant δ.
These parameters are chosen to fit the experimental data by Johnson and
Christy [8] for Ag. The chosen
parameters are ε_{f}=6.8, ω_{p}~3.8eV and δ~-0.02eV. In the
frequency region that we are interested, the whole visual range, these
values give a satisfactory good fit with the experimental data for both the
real and the imaginary parts of the permittivity, as shown in Fig. 2(a).

The transmission coefficient is presented in Fig. 2(b) for a hole of diameter *d*=270nm in
metal film with thickness *t*=340nm for the permittivity
[Eq. (1)]. The incidence
is in normal direction with respect to the surface of the metal film. We
also present results for a hole in ideal metal film with the same diameter
and with thickness of *t*=340nm and 750nm. The results are
quite illuminating and tell us what is going on in the single hole
transmission. It is clear that the transport using the theory of transport
in waveguides in ideal metals which shows the first wavelength cutoff at
*λ*_{c} =2*πd*/3.68=1.705*d* [14] only applies for large values of
*t*=750nm [see Fig.
2(b)]. For comparison, the waveguide theory for ideal metal with
much larger thickness is also plotted. For the values of *t*
smaller than 340nm which is used in the experiments, a hole in the ideal
metal gives a considerable transmission in long wavelength tail. Even at
λ=700nm, the coefficient is 0.10. Therefore these result show that making
considerations for enhancement of transmission comparing with long waveguide
in ideal metals is unphysical and not realistic.

The striking result is that when calculate the transmission with Ag film
using the experimental values of permittivity for Ag, we found a much higher
transmitivity at larger wavelengths. The cutoff has been moved from 460nm
(ideal waveguide cutoff for *d*=270nm) to ~630nm. Notice the
results presented in Fig. 5(a), at
700nm where a transmitivity peak of 1.4 appears for the holes’ arrays, one
single hole provides a transmitivity of 0.5 in Fig. 2(b). So the transmission enhancement from hole
arrays versus single hole is only a factor of 2.8. Then the question is:
*why has the real Ag the behavior of large transmission at large
wavelengths*? This has to be searched in the effect of surface
modes located at the cylindrical cavity defining the holes. They are similar
to those discussed by Pfeiffer, Economou and Ngai [6] also Martinos and Economou [7] for metallic cylinders. These waves move the cutoff
of transport in the structure to longer wavelengths and are the
responsibility of the large transmissions through the hole arrays for the
observed peak at 700nm for the *t*=340nm as is the case of
the experiments [1, 3].

### 2.1.2 Cylindrical surface waves

The surface plasmons excited in the cylindrical holes propagate the same way
as those that has been studied on the surface of metallic cylinder [6, 7]. The surface plasmons locate along the circumference of the
cylinder with wavelength ${\mathrm{\lambda}}_{\mathrm{n}}^{\mathrm{\theta}}$=2*πr/n*, i.e. circumference length divided
by the index branch *n*. And the SPs also propagate along the
cylinder axis z with a wave *k*_{z} . The possibility of exciting long wavelength modes is given by the
cylindricality α=2*πr*/λ_{p}, where λ_{p} is
the bulk plasmon wavelength (λ_{p}=325nm for Ag [8]). This theory is done for Drude
plasmon dispersion relation with δ=0 in (1). In our case we do not have a
cylinder of infinite length as has been discussed in ref. 5 and 6, but we
have a cylindrical metallic cavity of certain thickness *t*.
However by looking at the boundary conditions the same kind of modes should
exist and our full solution of Maxwell equations shows up in the
transmission. In Fig. 4(a)
modulations in the transmission can be identified, which actually may
correspond to the surface plasmons modes excited in the cavity surface. The
peak with longer wavelength corresponds to the smaller *n,
n*=1 identifies the longest wavelength surface plasmons mode, i.e.
there is a cutoff for the surface plasmons modes.

This cutoff of the surface plasmons is determined by the dispersion relation of the n=1 mode. The dispersion relation was studied with a planar approximation in Ref. [6]. As an approximation the cylindrical surface was treated as semi-infinite plane, the curvature of the cylinder was considered as periodic boundary condition. The resulted dispersion relation in the surface can be written as [6]:

$$Q={[{K}^{2}+{(n\u2044\alpha )}^{2}]}^{\frac{1}{2}}\phantom{\rule{.5em}{0ex}}K=k\u2044{k}_{p}$$

where *k*_{p} =2*π*/λ_{p} and the cylindricality
*α*=*dk*_{p} /*2 play an important role in the dispersion
relation*.. Figure 3(a)
actually is the case for cylinder holes in Ag film with *λ*_{p} =325nm, diameter *d*=270nm, the same as in Fig. 2(b). As shown in Fig. 3(a), with fixed cylindricality
parameter *α*, the possible modes of surface plasmons that
can be excited at the cross points with the photon line are limited in
wavelength.. From Fig. 3(a), the
wavelength for the possible surface plasmons can be excited in the cylinder
holes is in range of 470nm~630nm. The 630nm corresponds approximately with
the cutoff for the transmission using the permittivity in Fig. 2(b). There are also weak
oscillations in the structure of the transmission, in both experiments and
calculations that we may tentatively assign to the different plasmons index
*n*, in Fig. 3(a)
and in Fig. 4 as well.

However, we noticed that the above dispersion relation is based on the
semi-infinite plane approximation for the cylindrical surface for a Drude
metal with δ=0. The peak structures are not expected to be completely
matched with the experiments. In fact the *n* index in our
cylindrical hole structure may depart considerably of those given by Ref.
[2] and plotted in Fig. 3(a) to illustrate the problem.
Still our holes are of finite small thickness while the theory of Ref.
[6] is for infinite cylinders.
The more important point is that this theory result explains the extra
transmission above the waveguide cutoff limit. The cylindrical wave is
excited at the entrance of the hole and it carries the energy through the
other side of the hole which is not allowed in ideal waveguide. It is
assuming that the thickness of the hole is not big enough because the wave
has a propagation decay length. When the thickness of the hole becomes
larger than the decay length, the transmission will be controlled by the
waveguide modes without cylindrical surface waves. We have performed
calculations for Ag with *d*=270nm and changing the thickness
*t*=340nm, 525nm and 735nm to check the propagation decay
length, as shown in Fig. 3(b).
Clearly when *t*=735nm, the cutoff is retracted to 600nm,
however still bigger than that of the ideal metal cutoff (~500nm) for
*t*=750nm in Fig.
2(b). This establishes that the transmission in a hole of Ag at
larger wavelengths (>500nm) is controlled by cylindrical surface waves
with a decay length that we estimated to be more than 1um, which we will
discuss in another work. The decay should depend also of the diameter
because it limits the extension of the cylindrical wave into the vacuum, in
the hole, as well as of the thickness.

### 2.1.3 Comparison with experiments

Reference 2 has presented an ample number of experiments for single holes of
different values of *d* and *t*. These
represent a good set of experimental data to contrast with our calculations.
Figure 4(a) presents the
experimental data [Ref. [2], Fig. 2(c)] as well as the calculations
for the same parameters as in the experiments. The comparison is strikingly
good for all the cases. Also, as is important we plot the enhancement for
the case of *d*=270nm. This is defined as the transmission
for the real Ag divided by the transmission predicted by the ideal waveguide
theory [14]. It is observed that this
enhancement can be up to 1000. Analogously we present similar results in
Fig. 4(b) for the case of
*d*=200nm and the enhancement is of the order of 10000.
This proves that the enhancements by single holes we calculated are already
of the order of those measured for hole arrays in Ref. [1] and claimed to be due to SPPs of the
periodical arrays. Calculations and experiments [2] prove that it is not necessary to have hole arrays
in order to have such enhancements. This tends to rule out the SPP between
the arrays as the physical reason for enhancement from arrays. However it is
due to the cylindrical surface waves in the walls of the cavity drilled on
the metal to have enhancement from single hole. It is clear that the
influence of SPPs, if exist, is marginal for the large values observed in
the transmission from holes arrays.

It is also worth to notice that the transmission of a single hole shows weak
oscillations as we tentative assign to the different cylindrical surface
waves index *n* indicated in Fig. 3(a). However, there cannot be full agreement with Refs.
[6] and [7] that are calculations for cylinders and for an
idealized plasmonic metal. The oscillations tend to agree with the data
although there is some mismatching between our calculations and the
experimental data. One may argue that these oscillations change with the
cell used in calculations. This gives slightly displacements of the
oscillations but we have not been able to match the data better. In any
case, the validity of the operational cylindrical surface waves is not based
on these oscillations but on the larger wavelength cut-off produced in the
transmission. Figure 4 is clear in
this sense: the ideal metal has a shorter cut-off as indicated in Fig. 2(b). If there is more
experimental data available for different *t* and
*d* values, we shall be able to study better the
substructure of the hole’s transmission. More work in this direction may be
worth to tackle to the last of the problem. Also, we have assumed idealized
geometries with no roughness at the cylindrical walls of the cavities. That
may not be so, and in fact is not. When writing with an ion focused beam,
there are irregularities as well as some ion implantation, from the beam, in
the samples. These may introduce some structure in the hole that in turn may
provide structure in the transmission. Therefore, we consider that at the
present experimental situation it is not worth to tackle more the problem
from the theory side.

### 2.2 The Ag case for arrays of holes

We now proceed by discussing the transmission of light through an array of
holes following the same procedures discussed above. Since a single hole
gives such an enhancement beyond the cut-off wavelength, it will not be a
surprise that a periodical array will give also a very large enhancement.
The result for an array will be produced by the interference of the waves
merging from the holes. Therefore the transmission will have for some
frequencies enhancements over the single hole’s transmission and for other
frequencies depressions. Same ideas have been described in Ref. 2, however
our FDTD calculations will prove all at once. In order to prove this we have
performed such calculations for periodical arrays of holes to compare with
existing experimental results [2,
3]. Figure 5(a) shows the transmission results for the
periodical array with P=600nm, *d*=270nm and
*t*=225nm, the same parameters corresponding to the data
presented in Fig. 1 of Ref. [3]. The agreement is again strikingly
remarkable and without fitting any parameter, just taking the permittivity
of Ag [8]. The three experimental
peaks at λ≈700nm, 550nm and 430nm are excellently described not only the
peaks positions but also the measured amplitude. For comparison we also
presented the existing theory performed in Ref. [3] in which a rather good agreement is claimed. In
fact, there is not such claimed agreement because the calculations only show
two peaks at 630nm and 460nm which are shifted from the three experimental
peaks. Or to be more explicit, the peaks of the theory in Ref. [3] correspond, not with the
experimental peaks, but with the minima. *Where is the agreement
then*?

Figure 5(b) shows comparison for the
experiments in Ref. [2] for P=600nm,
*d*=250nm and *t*=340nm and again the
agreement is remarkable in the peak positions, the intensities and the
enhancement with respect to a single hole intensities. Our calculations are
for an infinity array of holes, while in the experiment the hole arrays are
finite. However the experiments [2]
showed that arrays of N×N holes yield practically the same results for
N>9, so the infinite arrays give, as shown by the calculations,
practically the same answer at normal incidence.

We would like to explain a little bit on the appearance of the peaks
positions in the periodical arrays as following. Once the cylindrical
surface plasmons are excited, there will be a comparative large transmission
per hole. These plasmons radiate waves at the surface and then interfere.
The peaks positions and their intensity are given by the value of the period
P. The hole’s diameter intervenes in the peak intensities. Because when, for
a given frequency, a large value of a single hole’s transmission intensity
falls at the same position as that of the ideal interference peak, this
interference peak shows a pronounced maximum. However if these conditions do
not match, the peak of the array is much smaller. As an illustration we
present calculations in Fig. 5(c)
for *d*=200nm and P=600nm. It is clearly seen that the
enhanced peak at around λ≈690nm is strongly reduced (compared with Figs. 5(a), 5(b) for the same P
value), because the single hole at this wavelength has little intensity as
shown in Fig. 4(b) (compared with
Fig. 4(a) for different
*d* value). This is also in excellent agreement with the
data of Ref. [2]. To provide further
information as a prediction result, we present in Fig. 6(a) a set of calculations for the values of
P=750nm, 870nm and 1050nm with same *d*=270nm and
*t*=340nm. In agreement with the discussion above, the
intensity peaks move according to the produced interferences. This shifts
their peak wavelengths with periodic parameter P. Another series of
transmission are also calculated [Fig.
6(b)] fixing P=1200nm and *t*=340nm but for
different *d*=270nm, 300nm and 360nm. This time, the peak
structure is always at the same position because P is fixed. However, the
peaks change their intensity because *d* is varied. Therefore
P and *d* determined the peak position and intensities
respectively. Moreover, the thickness *t* also counts,
because the material has absorption and the plasmons have certain decay
length. Actually, the propagation length, the plasmons speed and the
retardations, etc [15] all play a
role and show up in the experiments.

### 2.3 The W case for single holes

In the frequency region we discussed, Ag is considered as a special case because of its ability to support surface wave for extra transmission with respect to the ideal metal. We would like to see how the holes in different real metals transmit wave from the same hole’s structure. For W, in the whole visible frequency range, the real part of the permittivity is positive and approximately constant. It behaves as a dielectric. The SPP waves cannot be supported by this metal. Let us be no so definitive because this may need further discussion. But let us accept, at least, that the SPP waves, as those existing in Ag owning to permittivity [Eq. (1)] cannot be hold at the surface. Then, the question is should the transmission of the holes in W be very different from Ag case?

To study the case of W in the calculation, its frequency dependent permittivity is set as:

a model used for a conductor, with *ε*_{r} >0 as the real part of the permittivity and *σ* as
the conductivity. In the sense of optical transmission, the material
governed by this model behaves as a dielectric but with big attenuation. It
means that the wave will penetrate the material and meanwhile loss the
energy because of the attenuation. It should be noticed that in an ideal
metal there is no attenuation and neither penetration, which actually plays
an important role in transmission from the hole. The experimental
permittivity data [9] is shown in
Fig. 7, together with the fitted
data by Eq. (3) with
*ε*_{r} =4 and σ=6.46×10^{5}s/m.

The calculation result of transmission from a single hole
(*d*=300nm, *t*=400nm, the profile of the
holes used in Ref. [2], Fig. 3(a) for arrays) in W is
presented in Fig. 8, together with
the transmission for the same hole in an ideal metal. Their transmission
profiles are similar but very different from Ag case in Figs. 2(b) and 4. The strong extra transmission beyond the cutoff in Ag case
does not exist, which is consistent with the fact that there are not surface
cylindrical plasmons for W in the large wavelengths to transmit the wave.
The other fact should be noticed is that the transmission is smaller than
that of an ideal metal for λ<700nm. It is understandable, as we mentioned
above, the wave in the hole will constantly penetrate into the metal, and
the energy will be constantly killed because of the attenuation nature of W.
We verified this with more calculations the transmission becomes smaller and
smaller till totally dies for bigger thickness of W film. However for
λ>700nm the transmittivity for W overpass that of the ideal metal. This
is due to penetration, which makes the effective hole size bigger. Therefore
losses and penetration of the wave interplay during the transmission.

On the other hand, there is similarity between the W case and the ideal metal
case. We have calculated for a metal with the same model in Eq. (3), but increasing σ by
200 times, the resulted transmission is almost identical as ideal metal
case. To understand this result we should turn to the complex reflect index * n*=

*n*+

*ik*, where the imaginary part of the permittivity makes

*ε*

_{i}»

*ε*

_{r}, so that

*n*≈

*k*»1. The decay length of the wave penetration into the metal is therefore greatly reduced and the reflectivity is ~1. That is the reason that this model gives the same transmission as an ideal metal. In principle, the transmitted light through a hole in W film can be treated the same as waveguide but with large attenuation. All this is a qualitative discussion. However for fixed

*d*and

*t*of the order of the experiments, ideal waveguides are not well defined. Then ideal metal and W provide more transmission than expected by long ideal waveguides. This needs further discussions. We stress that the thickness of the holes used are too short to consider comparisons with waveguides.

### 2.4 The W case for arrays of holes

We also perform calculations for the transmission of hole arrays in W film.
The result is compared with the experiment in ref. 2 for the same structure,
*d*=300nm, *t*=400nm, P=600nm. As shown in
Fig. 9, the calculation fits
well to the experiments, for the whole profile, the peak positions and
intensities. The peak position at λ~700nm is the same as those in Fig. 5(a) and Fig. 5(b), with the same periodical parameter P, but
with totally different material, hole size and thickness. This strongly
confirm the regularity we discussed above in Ag case, that the transmission
peaks of hole arrays are determined only by arrays periodic, the same
conclusion that is mentioned in Ref. [2]. However, the intensity of the peaks in W case is 6 times
smaller than those of Ag. It is merely because the transmittivity of single
hole in W is smaller, by a similar factor, than that of Ag case in Fig. 2(b).

## 3. Discussion and conclusion

From the calculations and observations made above we reach the following conclusions:

1) One obvious point, that yet many people overlooked, is that one has to use the
experimental data of the dielectric properties of the metal in theoretical
consideration. As well as the parameters (*d* and *t*)
of the hole used.

2) Because of the previous point, the analysis of enhancements in terms of ideal calculations with approximations, like that used with Bethe theory or ideal metal long waveguides, cannot be used because it produces mislead conclusions even if the experiments are interesting and right. The Bethe approximation obviously has no role in the problem at hand as discussed above; the holes used are too large in diameter. Therefore, arguments on enhanced transmittivities of arrays based on the transmittivity of a hole using Bethe approximation are misleading. This has been done in many papers (see references in ref. 2), driving the interpretation of data in an incorrect direction.

3) Plasmonic metals, in particular the paradigmatic Ag, has long cutoff wavelength in
the transmission because of cylindrical surface plasmons, as discussed above and in
earlier references 5–7. This is true as far as the *t* values in the
experiments are shorter than the decay length of the plasmons. For larger values of
*t*, the transmission is drastically reduced and only remains the
waveguide modes with losses. Figure 3(b) is
a clear illustration of this. More experiments should be performed with longer
*t* to clear up this point. Moreover we have made predictions of
what may happen.

4) Dielectric metals (metals that for given frequencies have a positive real part of the dielectric constant), as W in the visible, with the permittivity as given in Fig. 7 behave in a particular way. There is absorption and penetration of the wave function in the metal which reduced the transmission for short wavelengths, but increased it at large wavelength with respect to ideal metal (see Fig. 8). Notice that these metals have transmittivity three times smaller than that for Ag but nevertheless their transmittivity is still much much larger than the obtained with Bethe approximation. 5) If the SPPs in the Ag cases we have discussed, have an influence in the transmission of the holes arrays, it seems to be marginal; i.e. is a small factor. The experiments prove this as well as our calculations. Notice that the transmission per hole in single hole is of the same order as that in the arrays. In fact sometime it is larger. This dependence of the interference of waves by holes is explained in the text.

6) The transmission peak positions of the arrays are given by the period P, and are material independent. Their intensity amplitude is large only if the single hole’s transmission is large. A paradigmatic effect of this is given in Figs. 5(a), 5(c) and Fig. 6.

7) It appears that for the frequency discussed, Ag holds SPPs and W does not. In the present experiments the effect of these SPP’s is marginal, however the existence of other possible surface waves that could enhance the field at the surface, for other geometries and materials, it is not clear yet. We need more experiments to launch a larger scale calculation regarding this point, but there could be surprises. We are studying theoretically this problem.

We would like to manifest that even if in fact that SPPs enhancements play a non
substantial role, in the experiments at hand, does not mean that in other cases or
geometries these SPPs’ waves cannot play a significant role. We believe that the
reason for a marginal role, in the case discussed here, is because the hole’s
geometry has many Fourier components that reduce the enhancement (see Refs. [11–13])
of the field amplitude at the surface due to SPP’s. Optimized enhancement is
produced when the surface geometry is described by small number of Fourier
components. However, there are surface profiles of the hole’s structure that may
induce the desired effect of enhancing the field at the surface of the arrays. If
this is possible, there could be a multiplicative effect: one due to SPPs and the
other to the cylindrical waves. In this sense, to work out with Ag, W and Cr
choosing carefully the structures and the frequencies may give new surprising
results. More experimental data are also needed changing the geometries and the
values *d* and *t* of the holes. For example what
would be the enhancements for squared holes?

## Acknowledgments

We thank the European EU-FP6 Project Molecular Imaging LSHG-CT-2003-503259 for support. N. García thanks M. Nieto-Vesperinas for discussions and for bringing up Ref. [2].

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