We analyze the progressive introduction of disorder in periodic subwavelength hole arrays. Two models of disorder are discussed from their associated Fourier transforms and correlation functions. The optical transmission properties of the corresponding arrays are closely related with the evolutions of structure factors, as experimentally detailed. Remarkably, the optical properties of random arrays are not in general equal to those of the single hole as a result of short-range correlations corresponding to hole-to-hole interactions. These correlations are due to packing constraints that are controlled through the careful generation of random patterns. For high density pattern, short-range order can take over long-range order associated with the periodic array.
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The influence of disorder on the properties of photonic crystals leads to rich discussions spanning from light localization issues [1, 2] to spectral sensitivities [3, 4]. Similar discussions can be carried in the context of metallic arrays of nanoparticles and subwavelength holes, where disorder is introduced starting from a periodic arrangement . In such periodic arrays, a delocalized surface mode can develop and induce a strong resonance in the optical spectrum. This resonance is determined from a momentum transfer from the array to the surface mode , as read from the structure factor associated with the array. From an elementary point of view, the mechanism can be analyzed at the level of the individual scatterers seen as localized sources of surface waves [7, 8]. Looking at a periodic subwavelength hole array for e.g., surface fields originating from the different apertures will constructively interfere as they propagate. This will lead to optical resonances and extraordinary optical transmission (EOT), ultimately related to the presence of long-range order in the array . Recently, specific correlation effects on the optical spectra of disordered arrays have revealed that the scattering properties of random arrays do not simply reflect those of its constituent individual scatterers [5, 10].
In this article, we carefully study the meaning of randomness in the context of subwavelength hole arrays and show that randomness can only be defined by taking into account packing constraints. We confirm that a Fourier-based analysis can unveil specific short-range correlations related to these constraints. Correlations can be quantified from the actual transmission spectrum of the single hole [11, 12] and in such conditions, random arrays are well suited to study the relation between long and short-range orders.
2. From periodic to random arrays
All our structures (single hole and hole arrays) are milled with a Focused Ion Beam through the same 275 nm thick Au film, sputtered on a glass substrate. Prior to optical measurements the structures have been covered with an index matching liquid tuned to the refractive index of the glass substrate (n ≈ 1.5). All the structures were thus in a symmetric configuration (refractive index of the dielectric media on both sides of the film is the same). Measuring the absolute optical transmission spectrum through a single subwavelength hole needs specific techniques that have been detailed elsewhere . We have implemented them both for single holes and random hole distributions. We insist on the fact that the greatest care was taken to keep the geometrical parameters of each hole as much identical as possible in all types of structures studied. This can be checked looking at our scanning electron microscopy (SEM) images on Figs. 1, 4 and 6.
We start with a periodic array as the reference structure. We consider a N = 30 × 30 holes array defined on a square lattice of periodicity p = 450 nm, with hole diameter d = 160 nm (see Fig. 1(a)). We will focus on the lowest energy EOT resonance which is spectrally isolated and far from inter-band transition of Au (located ∼ 500 nm) as seen in Fig. 2(a) (black line). This peak is associated with the fundamental reciprocal vector of the structure G1,0. By analogy with diffraction experiments, the intensity of the resonance can be expected to scale as13,14]. As a consequence, SP can only probe a limited extend of the array and finite size effects on spectral and emission profiles are observed [12, 15]. When strict periodicity is lost, as it is the case for quasi-periodic hole arrays , lower ℓSP are measured as compared to the periodic system. This decrease has been shown to be related to the specific spectrum of the quasi-periodic structure factor [6,16]. It is therefore important to keep in mind that Sk itself can have an effect on ℓSP and therefore reduce the effective long-range order of a system.
Random arrays are generated by positional disorder as holes are moved from their initial lattice positions on the initial periodic array. The displacement is restricted to a specified area χ2, homothetic with the array unit cell, the size of which defines a maximal shift χ/2. A minimal hard spheres-type separation distance σ between the holes (from center to center) is simultaneously set, in order to avoid adjacent holes overlapping. Fabrication issues fix edge to edge distances at a minimum of 75 nm which we slightly increase to 90 nm for precaution. For apertures of diameter d = 160 nm, this then corresponds to a minimal center to center distance σ = 250 nm. Random arrays are thus characterized by the two relevant parameters, the minimal distance σ between the holes and the density of holes per unit of area ρ.
3. Models of disorder
Two different models of positional disorder are implemented. In a first local model, we fix a maximum shift to χ/2 by which we move R holes in a random orientation an increasing way. We go from a defect-free periodic array (period fixed at p = 450 nm) with R = 0 to a fully shifted array with R = N, where N is the total number of holes. For the case studied, we choose χ = 300 nm. As a second model, corresponding to global disorder, we move all the holes by increasing values of χ. Note that in these conditions the hole density is ρ ≈ 9.9 % and σ is fixed to 250 nm for all the structures. Generating the arrays, we make sure that the distribution of hole displacements giving disorder is uniform through the displacement cell.
SEM images of the fabricated structures with local disorder (top row) and global disorder (bottom row) are shown in Fig. 1. From the known positions of each hole, we numerically evaluate Fourier transforms (FT) associated with the arrays. The progressive reduction of the amplitude of the Fourier components displayed in Fig. 1 is a measure of the increase of disorder. In the case of local disorder we note that the information of periodicity is not totally lost, as a Fourier peak remains. This is not surprising since in this case we did not allow the holes to go outside of their initial unit cell (χ < p). For global disorder in contrast, as soon as χ = p, periodicity is lost and no peak appear anymore on the Fourier spectrum.
That these two models induce different types of disorder cannot be assessed from a sole Fourier analysis, as it is seen from Fig. 1. But the difference becomes clear using hole-to-hole correlation functions. These functions can be directly evaluated from the holes coordinates as
In the case of the periodic lattice (Fig. 1(a)), pair distances are limited to the lattice vectors, and the correlation function is a sum of peaks at the lattice sites. When introducing disorder in a local model, the averaging over all directions reduces the peak amplitudes and induces a uniform continuum that increases towards one as R is increased. At R = N, any pair distance becomes equally probable and the memory of the periodicity only remains as a weak modulation of the continuum. In the case of global disorder, the evolution is very different with a dramatic broadening of the correlation peaks and an extended continuum, smearing out the peaks of the correlation function at large distance, as expected for a truly random system.
The transmission spectra associated with each disordered arrays are measured and gathered in Fig. 2. All the spectra presented in this paper are normalized to area occupied by the holes. We will compare them together by defining the degree of disorder D as a surface ratio with D = (R/N) × (χ/p)2 in order to comply with the bi-dimensionality of our problem.
As clear from Figs. 2(a) and 2(b), both types of disorder lead to similar spectral evolutions regarding peak intensity reduction and peak width increase. Note that a progressively less pronounced spectral dip together with a slight blue shift of the resonance position is a typical signature for a global reduction of the SP contribution in the transmission process with respect to the direct contribution through the holes. Looking at the intensity evolution of the first FT peak (located at k = Gi=1,j=0) as a function of D (Fig. 2(c)) reveals how close both models of disorder are in terms of Fourier transforms. Similarly close evolutions are observed in Fig. 2(d) for the measured resonance intensities and widths, for both types of experiments. In order to get consistent values, we normalized each spectrum T of each array by the spectrum TSH of the single hole. This ratio corresponds to the relative efficiency of EOT  and we extracted the full width at half maxima (FWHM) of these sets of data by considering T/TSH = 1 as the base line. Propagation length ℓSP for the SP modes are evaluated from the FWHM considering an exponential decay of SP on the structure. As it clearly appears from the data, local and global models are indistinguishable, reinforcing the relation between structure factor and optical resonances of subwavelength hole arrays [6, 16].
Interesting too is the fact that as the disorder increases, the FWHM increases and ℓSP decreases. The broadening stems from the fact that as disorder is introduced from the periodic situation, the structure factor becomes non-zero for wave vectors smaller than the minimal reciprocal vector of the initial periodic lattice (this can be seen on the FT displayed in Fig. 1), opening new diffractive channels that correspond to additional scattering loss for the surface wave excited over the array . In other words, the structure factor itself influences the propagation length of the SP modes which indeed decreases from c.a. 2 μm to less than 1 μm for the D = 0.45 disordered arrays (Fig. 2(d)). This reduction in the propagation length should not be confused with some inhomogeneous broadening , since here the individual scatterer, the single subwavelength hole, is intrinsically a poorly resonant element of the system, in contrast to nanoparticles in random arrays .
4. Debye-Waller factor
The close relation between the structure factors and the optical transmission spectra can be further illustrated by following the evolution of the ratio between the relative efficiency of EOT and the peak amplitude in the calculated FT. This compared evolution is presented in Fig. 3(a). For small amounts of disorder, namely D < 0.3, the ratio is nearly constant: the intensity of the transmission peak, which is a measure of the efficiency of the EOT, is directly proportional to the intensity of the corresponding peak in the calculated FT.
Interestingly, we note that in the case of a global model of disorder with R = N, both the optical resonance and the Fourier peak appear to follow a Debye-Waller type evolution. As disorder is introduced, the intensities decrease as the mean-squared displacement 〈δ2〉 of the holes from their lattice sites increases. The decrease in intensity scales asFig. 3(b) between the numerical results obtained from the Debye-Waller law and both the experimental data and the FT data clearly reveals the Debye-Waller evolution of the transmission resonance, at least for small values of D related with small positional fluctuations. This analysis stresses that the effect of disorder in the global model only affects the amplitudes of the diffracted peaks, and it is consistent with measuring disorder through an increasing effective temperature of the array.
These elements show that for low values of D, the optical properties of the arrays are dominated by EOT. As we see on Fig. 3(a), the EOT global transmission efficiency is determined by the shape and strength of the structure factor, and therefore related to long-range order. Such order allows delocalized SP modes, coherently launched at the level of the holes, to propagate over ℓSP on the array. However, as D increases, a new regime is clearly identifiable for both types of disorders. On Fig. 3(a) indeed, data points corresponding to large values of D do not fall anymore along the constant ratio value with transmission maxima decreasing more slowly than the FT peak amplitude.
As shown in Fig. 3(b), the transmission intensities decrease more slowly, deviating from the thermal Debye-Waller evolution for small levels of disorder (〈δ2〉 < 10 · 10−3μm2). This deviation will correspond to the onset of a new regime of transmission to be discussed further below. At the same time, the calculated FT peak starts diverging from the Debye-Waller law for 〈δ2〉 > 10·10−3μm2, as expected since the Debye-Waller law is based on a small displacement approximation. In contrast with transmitted intensities, the FT peak decreases more rapidly than the Debye-Waller law. Note that for the highest values of 〈δ2〉, this deviation also reflects the constraint induced by nearest neighbours (measured by the parameter σ) which biases the random displacement of each hole as discussed in the next section.
This regime of high disorder (high values of D and 〈δ2〉) deserves more attention. In the case of a local model of disorder, as it can be seen in Fig. 1(d) and Fig. 2(c), full disorder is not reached with an upper limit of D = 0.45. But for a global model, higher values of D are possible. As seen on the FT plot of Figs. 1(e)–1(h), it is possible to reach a level of disorder (χ ≥ p) beyond which the memory of the periodicity is completely lost. The associated correlation function takes exactly the shape of an ideal gas correlation function with any pair distances equiprobable. But interestingly, the optical transmission spectrum is not identical to what would be expected from the smeared out FT and correlation functions. The random array shown in Fig. 1(h) corresponds to D = 1.0 and to a maximally disordered array, while its actual transmission spectrum is not equal to the sum of each independent contribution of single isolated holes. This is clearly seen in Fig. 2(b) when the normalized transmission through the array is compared to the transmission measured through a single hole. The difference explicitly points towards another transmission mechanism which enhances the transmission with respect to single subwavelength hole. The nature of this mechanism can be related to localized SP modes, hole-to-hole interactions mediated by propagating SP or, as pointed out recently if the spacing between the holes is sufficiently small, quasi-cylindrical waves [10, 17].
5. Single hole and random arrays: short-range order effects
We will focus on the influence of short-range order over these interactions rather than their nature. This influence is observed in the transmission spectra of random arrays as deviations from the spectral signature of a single hole.
We first characterize the transmission spectrum of a single hole (see SEM images in Fig. 4(d)), following our previous study . As expected from the high-index symmetric environment of the hole described above, the transmission displayed in Fig. 5 is very broad, without any well defined resonances. The transparency window centered at 540 nm is due to the inter-band transition of Au. In the long wavelength regime, the subwavelength character of the hole is dominating, with a transmission intensity decrease that fits well the prediction of Bethe. Our fit indeed gives a λ−5 dependence, close to the predicted −4 exponent originally calculated by Bethe .
In order to probe the influence of short-range order, we perform two sets of experiment in two very different ranges of averaged hole density ρ. A first set aims at minimizing the effects of short-range order by generating random arrays with low densities of holes. In contrast, a second set aims at looking at random arrays with high-densities of holes. The maximal hole density ρmax achievable on a random hole array in a given window is fixed by the minimal hole separation σ, which acts as the essential packing constraint. We study systematically the influence of σ on the spectral response of the structures to reveal the influence of short-range order. Therefore, for each value of σ we generate an array with ρ = ρmax/3 (first set of experiment: low density) and an array with ρ = ρmax (second set of experiment: high density).
The first set consists in generating arrays by positioning randomly apertures in a square window of about 13 × 13 μm2 (see SEM images in Fig. 4). This is done in such a way that no particular spatial frequency dominates the spectrum, as it can be checked from the uniformity of the calculated FT associate with each pattern (see Fig. 4). Transmission spectra are given in Fig. 5, for increasing values of σ from 250 to 850 nm (steps of 50 nm) and thus decreasing values of ρ. For clarity, only a half number of the spectra are displayed, the other ones showing the same trends. From the reference provided by the single hole spectrum, we immediately observe that the hole array spectra are close to the single hole spectrum, particularly for the largest values of σ (σ = 750 nm in Fig. 5). In this case, the transmission properties of a random hole array can be considered as being close to the sum of the independent contribution of single isolated holes. Note however that even at this large σ value, small ripples on the random hole array spectrum reveal deviations from the spectrum of the single hole. These differences increase as σ decreases and witness interactions between the holes within the random pattern. Similar ripples are also visible on the transmission spectrum presented in other studies [19,20]. As discussed earlier  and recently measured [10,22–24], these modulations can be attributed to local resonances between adjacent apertures.
The array with σ = 250 nm shows the largest deviation. It corresponds also to the structure with the highest hole density (ρ ≃ 7.5%), high enough to put constraints on the random draw of the positions of holes in the given area of the square window. The second set of experiments explicitly probes such constraints by generating random hole arrangements with a maximum value of ρ within the constraints limit imposed by σ. Apart from the hole density parameter, all arrays of this second set are generated using the same protocol as the first set.
This time, the hole density is so high that packing constraints are immediately observed in the numerical FT shown in Fig. 6. These spectra bring immediately out the order induced by the high value of ρ as concentric circles. As can be seen on the averaged radial cross sections of the FT, the corresponding spatial frequencies are related to σ, typically 2π/σ. This is expected since σ defines the nearest neighbor distance. Note however that the FT are peaked at slightly larger (c.a. 15 nm) values than σ.
The spectra of these compact random hole arrays are gathered in Fig. 7 together with the transmission spectrum of a single hole. Just like in Fig. 5 for low density random patterns, ripples are observed on the spectra for arrays with σ > 450 nm. In contrast, transmission spectra of compact random hole arrays display a broad resonance which is absent from the spectrum of the single hole. Through the structure factor, i.e. from a momentum transfer argument, we naturally relate this resonance to the spatial frequencies emerging on the FT as the hole density is increased. We are thus extending to random arrays the Fourier-based analysis implemented in periodic and quasi-crystalline arrays [6, 16]. The important observation that the position of the (broad) peaks evolves as σ is varied from 250 to 850 nm is detailed in the inset of Fig. 7 where the positions of the transmission resonance are compared to the spatial frequencies dominating the FT of the random patterns (black dots). These data are also compared to the surface plasmon dispersion relation on a smooth metal-dielectric interface (continuous line). In the small wavelength range studied, this confirms that the resonance follows the same tendency as the one observed for periodic arrays . Typical red-shifts are measured, associated with the interference between the direct and indirect contributions to the transmission process. Also, due to the extension of the spatial spectrum of the structure factor, the resonances observed on the transmission spectrum of the compact random hole arrays are very broad and do not show characteristic strong minima and dissymmetric peak profiles of enhanced transmission peaks of periodic hole arrays.
Hence associated to short-range order, the broad resonances observed in Fig. 7 must involve rather localized surface waves. This can be indirectly checked by studying the evolution of the transmission spectrum as a function of the number of holes N, for fixed values of σ and ρ. For this purpose we fabricated and characterized random hole arrays covering a square window of increasing size (composed by an increasing number of holes). As can be seen on Fig. 8, the spectra of the random hole arrangements do not evolve as N increases. The only small fluctuations of intensity are attributed to local variation of the density of hole within the generated random pattern. These fluctuations have not been taken into account in our normalization. Indeed our measurement scheme only analyzes a fraction of the light transmitted by the structures, limited by the entrance slit of the spectrometer which intercepts a slice of the structure. As a consequence, the size of the investigated zone is proportional to the size of the structure. Therefore, the intensities of the transmission spectra of the smallest structures are particularly sensitive to the local variation of the hole density as these local variations will not be averaged on a large area. As N increases, the measured spectral intensities converge to an average value: the two largest pattern (N = 900 and 1600 holes) have globally the same transmission spectra. Given that these two different structures of identical hole density have been generated independently, the spectral similarity confirms again that the spectral response of compact random hole arrangements is essentially determined by the minimal hole separation σ.
This converged situation can be compared with what happens at the level of arrays displaying long-range orders, such as periodic  and quasi-crystalline arrays . There, the amplitude of the spectral resonance associated with the periodicity increases with the number of holes, and saturates in the limit of large arrays. This evolution is directly related to the delocalized nature of the SP modes excited at the surface of the periodic hole arrays. In contrast, for random hole arrays, the spectral resonances associated to short-range order do not vary with the number of holes, indicating a more localized nature of the excited surface wave at the level of the individual hole.
In the case of periodic arrays, the long-range order becomes better and better defined as the number of periods increases (i.e. the number of holes). In the case of random hole arrays, it is the short-range order that improves as hole density increases. Nevertheless, this short-range order is limited by the minimal hole separation σ.
To summarize, we have generated randoms arrays of subwavelength holes using two different random models. For both, the optical transmission spectra evolve in close relation with the structure factors associated with the arrays, as calculated from the Fourier transforms of the generated random patterns. This confirms yet again the generality of a Fourier analysis in order to predict optical properties of any kind of subwavelength hole distributions. Interestingly, the transmission spectra of highly disorder arrays are not equal in general to the sum of the independent single hole contributions, revealing hole-to-hole interactions not usually accounted for [20, 25]. These interactions correspond to short-range order induced on random arrays due to packing constraints related to the generation of a random pattern. As these constraints can be carefully controlled, we have shown that it is possible, in specific conditions, to prepare random arrays where short-range order can be either minimized or maximized. We believe the discussion is important for characterizing the optical properties of random arrays. Our work clearly stresses up to which point a random array can be considered as representing the optical properties of single subwavelength holes. It also shows that random arrays are appropriate structures to discuss and unravel the relations between short- and long-range orders.
The authors gratefully acknowledge financial support from the European Research Council (ERC grant no. 227577).
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