We numerically investigated the enhanced optical transmission through sub-wavelength centered-polygonal hole arrays (CPHA) in a thin Ag film deposited on the silica substrate. In octagonal and decagonal-CPHAs, we observed new hybrid transmission characteristics that were inherited from both crystalline and quasi-crystalline hole arrays. This peculiar nature was attributed to the unique arrangement of CPHAs which can be covered with copies of a single unit cell as in crystalline arrays, and their rotational symmetry as observed in quasi-crystalline arrays. Hybrid natures in CPHAs were further investigated in the transmission spectra and Fourier space representations of the arrays. Contributions from the nearest neighbor hole-to-hole distance to enhanced transmission were analyzed in order to quantify the plasmonic contributions from the Air/Ag interface and Silica/Ag interface. We also investigated the impact of layer structure, Air/Ag/Air versus Air/Ag/Silica in the transmissions and found that in CPHAs in Air/Ag/Silica structures, contributions from the Air/Ag interface became dominant in contrast to crystalline hole arrays with lower fold symmetry.
©2011 Optical Society of America
An isolated nanoscale hole in a thin metal film can serve as a point-like surface plasmon polarition (SPP) source, when illuminated at normal incidence of light . If the sub-wavelength holes were structured in periodic arrays on a metal film, peculiar phenomenon known as enhanced optical transmission (EOT) occurs . Theoretical studies to explain EOT have been extensively carried out using various numerical methods such as finite difference time domain (FDTD) method [3,4] and modal expansion formalism [5,6].
Thus far crystalline hole arrays in noble metal films such as square  and hexagonal hole arrays  have been studied and the influence of holes size, period of arrays [7,9], holes depth [10,11], and metal film properties  have been investigated. In recent years quasi-crystalline hole arrays with 10-fold (Penrose) , 8-fold , and 12-fold  rotationally symmetries have been explored. In contrast to crystalline hole arrays composed of successive copies of identical unit cells, quasi-crystalline hole arrays are covered by copies of more than one type of unit cells. For instance, Penrose hole array is made of two basic rhombi (skinny and fat) [12,15,16] and other arrays made of rhombus-square and rhombus-triangular units have been reported [13,14,17].
In this paper, we studied a different arrangement of sub-wavelength hole arrays called centered polygonal hole arrays (CPHA), where the rotational symmetry is preserved while the lattice can be covered by only unit rhombic cell. Note that CPHA carries a hybrid nature such that it shows aperiodicity in the x-y plane showing rotational symmetry about z-axis as in quasi-crystalline hole arrays, but composed of successive copies of identical unit cells similar to crystalline hole arrays. We analyzed this hybrid nature of CPHA in terms of EOT for the first time to the best knowledge of the authors.
2. The hole arrangements used in this study
In this paper, CPHAs with the octagonal and the decagonal symmetries are discussed, which are schematically shown in Fig. 1 . The octagonal and decagonal CPHAs are characterized by the single rhombic unit cells, which are shown in solid red lines in Fig. 1(a) and 1(b). The unit cell of octagonal CPHA is a rhombus with vertex angles of 45° and 135° and the corresponding angles in the decagonal CPHA are 36° and 144°. Conventional square- and hexagonal-crystalline hole arrays are shown in Fig. 1(c) and 1(d), respectively. We will designate them as square and hexagonal arrays in the rest of this paper.
Instead of prior free standing metal film , we discussed the three-layer structure as shown in Fig. 2(a) , where the silver layer of thickness h is deposited on a silica glass substrate and is perforated with holes of diameter d .We will refer this structure as “D-film” and the prior free standing film as “F-film” in the following discussion. D-films would find more practical applications integrated with optical waveguides, in comparison to prior F-films which require delicate handling . Scientifically D-film will also reveal more physics because it provides two different boundaries; Air/Ag and Silica/Ag interfaces. We assumed normal incidence of light from the silica side. We also assume the same hole parameters for the structures in Fig. 2: d and Λ.
In this paper, we investigated two key issues that have not been addressed thus far:
- 1) What is the impact of centered polygonal hole arrays (CPHAs) in Fig. 1 in optical transmission? How would it differ from well-known crystalline and quasi-crystalline arrays?
- 2) What is the impact of the layer structure-Air/Ag/Silica in Fig. 2? How the Plasmonic resonances in the two different interfaces-Air/Ag, Silica/Ag would contribute to output optical transmission?
In the following discussions, we employed two physical analyses tools 1) Fourier transformation to find our appropriate reciprocal vectors that excites the surface plasmons, 2) Finite difference time domain (FDTD) analysis to find out the optical transmission through the proposed CPHAs in Air/Ag/Silica structures. Based on the plasmonic dispersion relations we then assigned the transmission peaks in terms of SPPs and corresponding reciprocal vectors, which can answer the above questions.
3.1. Surface plasmon polarition (SPP) in crystalline hole arraysFig. 3 i and j are integers for the corresponding reciprocal vector:
The index pairs (i, j) also indicate the order of SPP modes. For the square array: η=0 and α=1 and for the hexagonal array: η=1 and α=. The SPP waves resonantly guided in dielectric/metal interface of crystalline hole arrays called surface plasmon polarition Bloch waves (SPP-BWs) .
In addition to SPP-BWs, Wood’s anomalies also occur in transmission at the wavelengths given by:22].
3.2 Surface plasmon polarition (SPP) in qausi -crystalline hole arrays
The plasmonic dispersion equation in quasi-crystalline hole arrays  can be expressed as;
is one of reciprocal vectors, , which are linear superposition of the basis reciprocal vectors () in the reciprocal space. In a 2N-fold rotationally symmetric quasi-crystalline hole array, reciprocal vectors are given by :
Figure 4 shows the reciprocal space representation for octagonal CPHA, decagonal CPHA and 10-fold rotationally symmetric Penrose array, where reciprocal vectors are indicated by bright spots and the basis reciprocal vectors, , are shown in arrows. It is noted that quasi-crystalline hole arrays such as Penrose array has reciprocal vectors in discrete points due to its spatial aperiodicity (Fig. 4(c)). In contrast, the octagonal and decagonal CPHAs have not only the discrete points (points marked by red cross signs in Fig. 4(a) and 4(b)), but also the back ground space filled by scattering patterns, which exhibits their semi-crystalline property  These reciprocal vectors in Fourier space and corresponding hole-to-hole distances in the real space were utilized to assign appropriate SPP modes for EOT peaks calculated by FDTD, which is discussed in the following sections.
4. Numerical analysis
The permittivity of air was assumed to be εa=1. The permittivity of silver and silica glass was obtained from data book . To solve the electromagnetic wave equations in sub-wavelength structure as shown in Fig. 2, FDTD method was applied using a commercially available program . The total simulation size for FDTD analysis was 20×20× 2 μm3 along with the mesh-grid resolution of 15nm, which were specifically chosen large enough to diminish the diffraction effect at the Ag film edges. We assumed that the white light source is a normally incident plane wave with a linear polarization covering the area of 10 × 10 μm2 and placed at 1μm from the film inside the silica glass. The monitor was positioned in the air, 5nm from the metal surface to detect the near field. Note that the spectral position of EOT peaks is independent of the incident light polarization at normal incidence of light (Eq. (1) and (4)), and therefore the simulation routine was designed for an arbitrary polarization in the plane wave.
All the structures in Fig. 2 have the identical geometrical parameters: holes size d=200nm, hole pitch Λ=600nm, film thickness h=340nm. Finite-size arrays covered the area of 7.2×7.2 μm2 ~8.5 × 7.2 μm2 on the surface of a 20×20μm2 Ag film. We assumed that the silica substrate is extended to the infinity. Experiments showed that a N×N square array resulted in the asymptotic transmission spectra of an infinite array for N≥9 . Here we will assume the hole number N=169, except decagonal CPHA of 211 holes, which is sufficiently large to provide the asymptotic transmission spectra.
5.1 Optical transmission of square and hexagonal arrays in D- and F-films
Figure 5 renders transmission spectra of square and hexagonal arrays for both F- and D-film structures. We could confirm the accuracy and consistency of our numerical analysis by verifying the reported EOT peaks in the F-film structure. In Table.1 we summarized the orders of SPP-BWs, Wood’s anomalies, and their corresponding free space resonant wavelengths calculated by FDTD in the D-film structures. We referred the SPP-BWs traveling at the air/Ag and the silica/Ag interfaces as (i,j)a and (i,j)s, respectively. Resonant wavelengths of SPP-BWs were estimated utilizing Plasmonic dispersion equation in Eq. (1), (2) and their values are shown in the parentheses. The location of Wood’s anomalies was estimated using Eq. (3). The difference in the wavelengths obtained by plasmonic dispersion equation and FDTD were 1~14%, which is attributed to the missing radiative damping term in plasmonic dispersion equation in Eq. (1) .
In contrast to F-film, the D-film has asymmetric boundaries  and the resonance wavelengths were grouped by the Air/Ag interface (peak 1,2) and the silica/Ag interface (peak 3~7) in Table 1. The minimum adjacent to the peak ‘4’ of square array and minima adjacent to peaks ‘1’ and ‘3′ of hexagonal array are assigned to Wood’s anomalies using Eq. (3) .
It is note that in Fig. 5(a), the dominant transmission peak in F-film of Square array is ‘2’ originated from the Air/Ag interface. In contrast, the dominant peaks are ‘5′, ‘6’, and ‘7’ in D-film of Square array, which are originated from the Silica/Ag interface. In fact the peak ‘2’ is still present between dominant ‘6’ and ‘7’ peaks in D-film, and its intensity is about 1/2 of the peak ‘2’ in F-film. This is consistent with well-known physics reported in reference , where the symmetric layer structure Air/Ag/Air in F-film provides stronger transmission peaks. This phenomena is also observed in the case of hexagonal arrays, Fig. 5(b), where two peaks ‘1’, ‘2’ originated from the Air/Ag interface have weaker intensities in D-film.
5.2 Optical transmission of octagonal and decagonal CPHA in D- and F-films
The real space and reciprocal space representations of octagonal and decagonal CPHAs are shown in Fig. 6 , and Fig. 7 , respectively. In real space representations we specified the nearest-neighbor hole-to-hole distances: a~e in Fig. 6(a) and a´~d´ in Fig. 7(a) along with Λ. The corresponding reciprocal vectors in the reciprocal space are represented in the concentric circles in Fig. 6(b) and Fig. 7(b). The magnitudes of these reciprocal vectors were calculated by using Eq. (5) for the nearest-neighbor hole-to-hole distance (L), or its fractions (L/2, L/3,..). The crossings of the bright spots and the concentric circles in the Fourier space will define the Gspp, which is used to determine the resonance wavelength in the Plasmonic dispersion relation in Eq. (4).
Transmission spectra were obtained using FDTD and the results are summarized in Fig. 8 . Here we considered both D-film (solid line) and F-film (dashed line). The EOT peaks calculated by FDTD were assigned using the Plasmonic dispersion equation of quasi-crystalline hole arrays in Eq. (4) and the results are summarized in Table 2 and 3 .
In comparison to crystalline hole arrays in Fig. 5, the differences in the intensity of EOT peaks between D-film and F-film were significantly larger in CPHAs in Fig. 8. In the CPHAs, the EOT peak intensity in D-film was reduced by about ~1/5 of F-film. As assigned in Table 2 and 3, two types of reciprocal vectors mainly contribute to SPP in CPHAs; 1) reciprocal vectors directly related to the hole-to-hole distances in real space, such as GΛ(Α) and GΛ(S) corresponding to the length Λ in real space, 2) reciprocal vectors which are equal to integer multiple of inner reciprocal vectors such as Gc/3(A)=3Gc(A). Note that the first type of reciprocal vectors contribution is characteristics of quasi-crystalline hole arrays, and in contrast the second type is characteristics of crystalline hole arrays, which clearly demonstrate the hybrid nature of CPHAs in EOT spectra.
In Table 2 and 3, we referred the reciprocal vectors of SPPs traveling at the Air/Ag and the Silica/Ag interfaces as GL(A) and GL(S), respectively. For each EOT peak in Fig. 8, we assigned the reciprocal vector (GL) and its magnitude (|GL|), corresponding hole-to-hole distances (L) in the tables. The spectral positions of peaks (λr) in Fig. 8 were also listed in nanometer (nm) and the estimated wavelengths from the plasmonic dispersion equation in Eq. (4) are shown in the parentheses. In the square and hexagonal arrays, the resonant wavelengths, λr, predicted by plasmonic dispersion equation were ~14% shorter than those numerically obtained by FDTD analysis based on Fano interpretation . See Table 1. On the while, in octagonal and decagonal CPHAs the plasmonic dispersion predictions were ~18% longer than the FDTD results, which has been similarly observed in quasi-crystalline hole arrays . See Table 2, 3. The reason of this discrepancy is not fully understood yet but this behavior represents the quasi-crystalline nature in the CPHAs.
5.3 Impact of silver D-film parameters over EOT peaks
The impacts of the silver film parameters over the EOT were analyzed and the results for D-film structures are summarized in Fig. 9 . Here we changed the Ag film thickness h, and the hole fraction d/Λ. It is noted that in both octagonal and decagonal CPHAs the major EOT peaks are related with the SPP modes on the Air/Ag interface. This is quite contrasting to the case of square array where the major EOT peaks are from the SPP modes on the silica/Ag interface. The origin of this contrast is not fully understood and is being investigated by the authors. By comparing EOTs of four hole arrays in Fig. 5 and Fig. 8, we found there exist a consistent trend; as the order of symmetry increases from 4-fold (in Square array) to 10-fold (in decagonal-CPHA), the relative intensity of the EOT peaks for the Plasmons from the Air/Ag interface increases significantly stronger than to those from the Silica/Ag interface in D-films. See the peaks ‘1’, ‘2’ in Fig. 5 and ‘iii’~‘vi’ in Fig. 8 for example. This phenomenological observation might lead to an assumption that the rotational symmetry of the hole arrays can selectively influence the enhancement of EOT peaks from the Air/Ag interface.
It is experimentally reported that quasi-crystalline hole arrays showed major EOT peaks contributed from SPP modes on the Air/Ag interface and we could confirm that octagonal and decagonal CPHAs share the nature of quasi-crystalline hole arrays in this regard . It has been shown that by increasing the hole fraction (d/Λ) in square arrays, peaks are enhanced as well as broadened . In addition, as either the hole diameter, or the period of hole is increased, transmission maxima are shifted to longer wavelengths. By increasing the hole fraction from d/Λ=0.36 at Λ=700 nm (black line in Fig. 9(a)–9(d)) to d/Λ=0.40 at Λ=600 nm (green line in Fig. 9(a)–9(d)), maxima in CPHAs (Fig. 9(c) and 9(d)) as well as those of crystalline hole arrays (Fig. 9(a), 9(b)) broadened in such a way that adjacent maxima overlap each other.
As expected for quasi-crystalline hole arrays, decreasing the holes size allows enhancement of higher orders SPP modes . In Fig. 9 by decreasing the hole size from d=250 nm(green lines) to d=200 nm(red lines), higher order peaks appear in the spectrum, such as peaks ‘iii’ and ‘iv’ in octagonal CPHA ; peaks ‘v” and ‘iii” in decagonal CPHA ; peak ‘2’ in hexagonal array ;and peak ‘7’ in square array appears.
Reduction in thickness of the film leads to coupling between sets of SP modes excited on two sides of the metal film and lifting modes degeneracy . Therefore, a SPP mode in one side show sharp high energy peak and its counterpart in the either side shows broad low energy peak. In Fig. 9, by decreasing the film thickness from h=340nm (red line) to h=200nm (blue line), (1,0)s and (1,0)a in hexagonal array (Fig. 9(b)) are excited as a weak broad peak (peak ‘3′), and a strong sharp peak (peak ‘1’) respectively. For two CPHAs (Fig. 9(c), 9(d)), by reducing the film thickness from h=340nm (red lines) to h=200 nm (Blue lines), maxima corresponding to reciprocal vectors of GΛ(A) and GΛ(S) gave sharp intense peaks (peaks ‘iv’ and ‘iv'’) and broadened low energy peaks (peak ‘i’ and ‘i'’) respectively.
Figure 10 illustrates the variation of averaged spectral transmission (AST) against the variation of hole fraction(d/Λ) for D-film structure. Here, AST defined the mean value of transmission in its spectrum over spectral range of 350~1000nm which can indicate the averaged optical throughput of hole arrays. By increasing the hole fraction, the AST of square array surpasses that of the other structures. However, at hole fractions less than0.3 the AST of square array is dominated by that of other arrays.
Figure 11(a) illustrates the transmission variation of EOT peaks: iii, iv, v, and vi (as listed in Table 2) versus the variation of the film thickness for octagonal CPHA. Figure 11(a) quantitatively shows that by increasing the film thickness the transmission of EOT peaks decrease due to the decoupling between SPP modes excited on both sides of the film . However, this trend was not evident for the modes corresponding to relatively outer reciprocal vectors such as ‘v’ and ‘vi’ (see Fig. 6(b)). Similarly, as shown in Fig. 11(b) in decagonal CPHA among EOT peaks: iii', iv', vi', and vii' (as listed in Table 3), transmission of peak ‘vii'’ shows oscillating variations. In inner reciprocal vectors, on the other hand, the transmission gently reduces by increasing the film thickness as expected for lower order SPP modes in crystalline hole arrays . Once again these transmission variations as a function of film thickness indicate hybrid nature of CPHAs.
In this study the optical transmission characteristics of octagonal and decagonal CPHA were numerically investigated. It has been shown that CPHA carries a hybrid nature such that it shows aperiodicity and rotational symmetry of hole arrays as seen in quasi-crystalline hole arrays, but composed of successive copies of identical unit cells similar to crystalline hole arrays. By utilizing plasmonic dispersion equation as well as FDTD analysis we made consistent assignment of EOT peaks in transmission spectra of both CPHAs. In octagonal and decagonal CPHAs the plasmonic dispersion predictions were ~18% longer than the FDTD results, which are similarly observed in quasi-crystalline hole arrays. It has been confirmed that two types of reciprocal vectors mainly contribute to SPP in CPHAs; 1) reciprocal vectors directly related to the hole-to-hole distances in real space similar to quasi-crystalline hole arrays, 2) reciprocal vectors which are equal to integer multiple of inner reciprocal vectors similar to crystalline hole arrays. Finally the impact of silver film parameters on EOT peaks of CPHAs such as film thickness, and hole fraction (d/Λ) was explored to confirm hybrid nature of crystalline hole arrays and quasi-crystalline hole arrays.
This work was supported in part by the Brain Korea 21 Project, in part by the NRF grant funded by the MEST (Nos. 2010-0018442, 2009-00479 EC-FP7/2007-2013 219299 GOSPEL, R15-2004-024-00000-0, F01-2009-000-10200-0, and 2009-00541), in part by the ITEP (Nos. 2009-8-0809 and 2010-8-1415).
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