The phenomenon of extraordinary transmission in the optical regime for circular hole arrays in optically thick metal films is studied as a function of hole size and depth. In the limit of small holes compared to the depth, the transmission properties follow a waveguide type behavior. By describing the transmission process as resulting from the interference between a resonant and a non-resonant contribution, a transition is clearly revealed through the specific spectral variations of the resonance at a given hole depth. This transition is associated to a change in the attenuation through the hole as its size increases, and corresponds to the optimal condition for surface plasmon excitation.
© 2009 Optical Society of America
Since the first observations of the extraordinary optical transmission (EOT) phenomenon , hole arrays have generated wide interest as they offer unique features and possibilities which have led to a variety of applications  in different fields from sensors [3, 4] to photonic devices .
Hole arrays are characterized by transmission spectra that display resonances for which the transmission can be larger than unity (i.e. extraordinary) when normalized to the area occupied by the holes. This enhancement is due to the excitation of surface plasmons (SP). The resonance positions λ(i, j) can be related to the period P of a square array of holes through
Here, εm and εd are the dielectric constants of the metal and the dielectric medium forming the interfaces of the film, and (i, j) are the scattering orders of the square array. Note that the SP modes can be excited on both interfaces of the hole array, each giving rise to a set of transmission peaks which are offset by the difference in the εd of the dielectric media in contact with the metal. The actual peak positions are typically redshifted as compared to the prediction of Eq. (1) which can be explained by a Fano-type analysis [6, 7, 8]. The EOT is sensitive to geometrical parameters such as the dimensions of the individual apertures [9, 10, 11, 12, 13] and the optical properties of metal [14, 15]. The role of the hole shape has for instance been studied extensively [16, 17] and it is clear that both the presence of localized modes and a cutoff function at level of the individual apertures can strongly modulate the transmission spectrum of an array.
Typically, the EOT phenomenon has been studied in the limit of hole sizes small compared to the resonance wavelength. In this subwavelength regime, the electromagnetic fields decay exponentially inside the hole with the hole depth [9, 11]. Here we are particularly interested in analyzing the EOT in the transition through this cutoff limit. This is simply done by increasing the hole diameter d relative to the period and therefore to the transmission peak wavelength of the array. By doing this for different hole depths h, we can access in our analysis different ratios of h/d and h/λ.
We focus on the influence of hole diameter and depth and compare the measured peak intensities to waveguide theory and corresponding spectra to a Fano-type analysis. We find a clear evidence for a transition which influences the EOT phenomenon.
2. Spectral evolution as a function of hole size and depth
Circular hole arrays (30×30 holes), like those shown in Fig. 1, were milled on Au films using a focused ion beam (FIB). Different thicknesses (thickness h varying from 140–560nm) were prepared by sputtering Au on glass substrates. Array period was fixed at 460nm and the hole diameter for each array was gradually increased from 100 to 400nm and transmission spectra were recorded using a microscope coupled to a spectrometer, using a white collimated light beam and are shown in Fig. 2. It should be noted that the holes are not completely circular but have a slightly conical form and the consequent error on the hole diameters is estimated at ±5%.
The transmission spectra are characterized by a set of peaks and we will focus on the most isolated one, which is also at the longest wavelength, around 800nm, corresponding to the (i, j) = (1, 0) mode of the glass-metal interface, see Eq. (1). The fact that our analysis is carried out for hole arrays milled in an asymmetric dielectric environment (i.e. the dielectric media on both sides of the film are different, air and glass) enables us to isolate one SP mode (one transmission peak) associated to a specific interface and avoid the additional variable introduced by coupling between modes of the two interfaces. Such coupling which can result in peak broadening/splitting and enhancement has been carefully studied [9, 18, 19].
The (1, 0) transmission intensity increases with the hole size and the peak position shifts first to the red. For the hole sizes larger than 250nm, we observe that the resonances broaden and no longer shift to larger wavelengths.
3. Comparing the evolution of the EOT intensity with waveguide theory
Measurements such as those in Fig. 2 were repeated for different film thicknesses (hole depths). In Fig. 3, the (1, 0) transmission intensity has been plotted as a function of the area of the hole for each hole depth. These curves show a sigmoidal shape with 0 and 1 as the natural limits when the hole area becomes respectively very small and very large as compared to the period. For thin films (140–220nm) the transmission can exceed unity before dropping back to one as the holes size approaches the period. Some modulation is apparent in the curves and this will be discussed in part 4. As the films become thicker, the rise in intensity with hole area becomes slower as expected for such small holes.
A simple waveguide approach already helps in understanding the evolution of peak intensities of Fig 3. Within this frame, the transmission through a hole is a function of both the hole depths and the propagation constants of the waveguide modes. Considering each hole as a cylinder in a perfect metal conductor (PEC), we determine the associated wavevectors considering guided modes . For thick enough films and circular holes, the transmission is mainly governed by the fundamental TE 11 mode with associated wavevector:
The value a is set by the geometry of the guide and in the case of a cylindrical structure is given by the first root of the first order Bessel function as a = 1.841.
From the ratio between the incident wavelength λ and hole radius r set by the geometry of the guide, the modes are either propagating, when k 0 is imaginary or evanescent when k 0 turns real. In our work, we will essentially follow a given resonance λres and in the simple waveguide picture, the limit rc = λresa/2π below which all the guided modes become evanescent, is generally referred to as the cutoff radius.
At this point, we will solely concentrate on the attenuation that is induced by such a sub-wavelength waveguide. We define the transmission T as by the ratio between the input and output fluxes of the Poynting vector of a plane wave through such a waveguide. By doing this, we consider the holes as isolated and knowingly neglect the implications of the fact that the holes form an array. The transmission for radii smaller than rc is mainly governed by how the electromagnetic field is attenuated inside the guide as a function of depth, whereas above rc all the light that enters the holes is transmitted. To account for the penetration of the electromagnetic field into the metal, we increase the radius r of each waveguide by the corresponding skin depth (sd) evaluated at λres, while we still use the real hole size to determine the input flux in order to have the same normalization as for our experimental data (in such a way that for r>rc, the normalized transmission can be larger than 1). The transmission T is then given by:
In Fig. 4, the transmission peak intensities that were measured for different hole sizes and depths are plotted on a logarithmic scale together with the calculated transmissions through subwavelength cylindrical waveguides following Eq. (3). All the calculated curves meet for a specific hole size (~ 410nm hole diameter), associated with the cutoff beyond which light can propagate freely through the hole. This means that taken as classical waveguides, none of the studied holes is large enough to guide waves.
If we would consider the metal without skin depth (sd = 0), all the curves from Fig. 4 would be shifted to larger hole sizes, as each hole would appear smaller to the incoming light. For instance the cutoff radius would in this case fall around 470nm
The general evolution of the curves calculated from Eq. (3) follows well the experimental data, especially in the limit of films which are thick compared to the hole size. The transmission intensities found from the calculation are however 2–3 orders of magnitude smaller than what is measured by the experiments. More sophisticated models than Eq. (3) have been proposed to describe the optical behavior of single apertures but they consider either lossless metals [21,22] or square apertures ([23,24] which do not directly apply to the present data. In addition, collective effects need to be taken into account which is precisely why arrays of holes, having SP resonances, generate far higher transmissions than single apertures as the comparison above shows.
4. Analysis of the evolution of the transmission peaks
Although the general evolution of the intensities at resonance seems to follow a waveguide behavior, it is obvious that it is not possible to explain this complex phenomenon thoroughly without considering the influence of SP modes excited at the surface of the array. This is particularly clear for the small depths (see Fig. 3 and 4), which has led us to chose for this analysis a specific depth of h = 190nm. The SP modes that are induced by the periodicity of the holes array according to Eq. (1), need to be considered together with the direct scattering generated by each hole. In fact, the transmission process through an array of holes can be seen as an interference between a non-resonant contribution given by the transmission through each hole, taken as isolated, and a resonant contribution stemming from the excitation of a SP mode. This contribution amounts to a coupling between the holes in the extended array [6, 7, 8]. This picture clearly oversimplifies the problem as supplementary contribution do affect the phenomenon (see [25,26] for a more realistic theoretical description), but yet, it can provide some interesting insight into the transmission process as we will see below.
As in our study we concentrated on the most isolated mode (1, 0), we only need to consider one SP resonance mode. This interference broadens the resonance and shifts its spectral position in the transmission spectrum. The resonance width Γ is related to the lifetime of the SP mode and as such to the propagation length of the surface plasmon on the array. As the scattering of the excited SP mode on the holes induces radiative loss and limits the SP propagation, Γ is thus a direct measure of the coupling strength between the resonant SP contribution and the non-resonant one. The stronger the SP scatters on the holes, the shorter its lifetime and the smaller its propagation length.
Here, ω 0 corresponds to the natural (uncoupled) SP resonance, as given by Eq. (1) and Δ to the shift from this position. The parameter ρ corresponds to the strength ratio between both contributions. The global prefactor α is proportional to the transmittance of the isolated hole. When accounting for the interplay between the two contributions, the shift Δ and the width (at FWHM) Γ of the resonance are related through a dispersion relation:
We use Eq. (4) to fit the transmission spectra that were measured for hole arrays defined with different hole sizes at a given hole depth (h = 190nm), taking α, ρ,Γ and Δ as independent parameters. As can be seen in Fig. 5, the experimental curves can be fitted for the whole range of holes sizes. Nevertheless at high frequencies, Eq. (5) doesn’t provide such a good fit of the data. We relate this to the emergence of other SP resonances related to higher scattering orders on the array, which are not accounted for in our fitting formula limited to a single resonance.
Figure 6 displays the variation of the transmission peak intensity as a function of the hole diameter. The peak intensity increases with the hole diameter until a diameter of about 250nm, from where it starts to saturate. First considering the peak positions as globally stationary as the holes get larger, the data shown on Fig. 6 is grossly reminiscent of what is expected for a waveguide with a cutoff diameter (at a fixed wavelength) around 250nm
This can be refined, by analyzing the evolution of all the fitting parameters. Their specific variations provides further evidence for a transition that occurs for hole diameters around 250nm for the chosen depth.
First, starting from the smallest holes, the resonance peaks start shifting towards the red part of the spectrum. As clearly seen on Fig. 7b, these shifts Δ are reversed towards the blue part of the spectrum when the hole diameter cross 250nm. Simultaneously, the resonance peak broadens with increasing the hole size. This is largely expected, as the resonance width Γ measures the coupling strength between the two contributions. Widths broaden when the non-resonant contribution takes on as the attenuation inside the waveguide is rapidly reduced when hole diameters approach the cutoff limit. Thus, a clear inflexion point of Γ around a diameter of 250nm is seen on Fig. 7a, which corresponds to a minimum of Δ. Interestingly, while these two parameters are adjusted independently from each other, they turn out to be related in a dispersive way. Indeed by injecting the Γ data points shown in Fig. 7a into the Kramers-Koenig relation (5), one ends up with a curve which ressembles the observed evolution of Δ shown in Fig. 7b. This coherence in our results validates both the analysis and the observation of this cutoff transition. It is interesting to note that the theoretical and experimental analysis of the EOT in the THz regime by Bravo et al.  observed very similar peak shifts.
The ratio between the SP contribution and the non-resonant transmission through the holes characterized by the ρ parameter, is plotted in Fig. 7c. It also has a clear maximum at 250nm If the holes are small, the diffraction of the incoming light is not sufficient to effectively excite surface plasmons. With increasing hole size, the coupling between the incoming light and the SP waves increases much faster than the direct transmission channel related to each hole. However beyond 250nm, the situation changes: holes scatter more efficiently the propagating SP mode into free space. The aperture becomes propagative and the transmission is dominated by the non-resonant contribution. It thus turns out that the best SP excitation condition is achieved at the transition, although the highest transmission efficiency can still occur for larger holes (see Fig. 6 for large diameters).
Finally, the fitting results for the global prefactor α are plotted in Fig. 8. This prefactor reflects the transmittance of the isolated hole. Bethe has suggested that for subwavelength holes in perfect metal films infinitely thin, this prefactor should be associated with an effective (magnetic) dipole . The non-resonant contribution is thus expected to follow globally the dipolar (d/λres)4 scaling. This seems to be actually verified for the smallest of our holes, as seen on Fig. 8. To understand this, we note that in this case both the hole sizes (d= 100 – 250nm) and the hole depth (h = 190nm) are much smaller than the resonance wavelength λres ~ 800nm. Within these ranges, the exponential decay of the field through a hole in an array is hence a rather slow varying function of the hole diameter, exactly as in Bethe’s idealized picture. This scaling fails for larger holes and is an additional signature of the differences in the transmission process between small and large holes, the qualitative change being again located around a diameter of 250nm. In a recent study, Nikitin et al.  have addressed this problem theoretically in the context of isolated holes milled in a perfect metal films of various thickness. The optical properties of very small holes milled through thin films are essentially dictated by a large induced dipole moment. For larger holes however, multipole moments must be taken into account. As these higher terms can be associated to higher waveguide modes for a given finite hole depth, it is not surprising to see an abrupt increase in the global prefactor α at the cutoff transition.
It is interesting to note that another spectral change occurs around the cutoff diameter. By taking a close look at Fig. 2, it can be seen that for holes just above this cutoff condition (d = 300nm and beyond), the transmission peak ceases to have one clearly defined peak and appears to double. Similar behavior has been related to coupling between different SP modes  formed on the two opposite interfaces of a hole array. This cannot be the case here as there is no mode from the air-metal interface at the wavelength of the (1, 0) peak of metal - glass interface. One likely explanation is the possibility that the (1, 0) mode is coupling to a localized mode located around the cutoff wavelength [24, 30, 31, 32].
It should be noted that this transition does not only have repercussions on the transmission process, but also strongly modifies nonlinear effects, such as increasing second harmonic generation (SHG) . Working at a wavelength of 800nm, Nieuwstadt et al.  also report evidence for the cutoff transition for hole widths around 250nm, in good agreement with our results. The reason that this transition is defined for much smaller holes than what is calculated from Eq. (2), lies in the properties of real metals, neglected in the PEC approximation. Different work [23, 24, 32, 34] have addressed this issue in the context of nanoscale holes, and confirmed that the interaction of the electromagnetic field with the metal enables the propagation of much longer wavelengths inside a metal waveguide than what can be expected in the PEC approximation.
In conclusion, hole size and depth are not independent parameters in the transmission through circular holes arrays. By analyzing the transmission as a function of hole diameter, a waveguide behavior is clearly seen although the transmission intensity can only be accounted for by the presence of SPs. When the holes start to sustain propagating modes, a transition in the transmission process is revealed, which defines an effective cutoff diameter (for a given wavelength) of the hole. This transition is characterized in particular by a strong broadening of the resonance peak and by a maximum shift in the resonance position. It is worth noting that such a transition appears to be linked to the optimal SP excitation condition on the hole array which is important for sensing applications and more generally speaking for coupling molecules and surface plasmons.
The author would like to thank Frederic Przybilla for fruitful discussions. This work has been funded by the French Agence Nationale de la Recherche under contract ANR 06-BLAN-0275.
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