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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 14 — Jul. 2, 2012
  • pp: 15222–15231
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Measurement of surface plasmon correlation length differences using Fibonacci deterministic hole arrays

Tho Duc Nguyen, Ajay Nahata, and Z. Valy Vardeny  »View Author Affiliations


Optics Express, Vol. 20, Issue 14, pp. 15222-15231 (2012)
http://dx.doi.org/10.1364/OE.20.015222


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Abstract

Using terahertz (THz) transmission measurements through two-dimensional Fibonacci deterministic subwavelength hole arrays fabricated in metal foils, we find that the surface plasmon-polariton (SPP) correlation lengths for aperiodic resonances are smaller than those associated with the underlying grid. The enhanced transmission spectra associated with these arrays contain two groups of Fano-type resonances: those related to the two-dimensional Fibonacci structure and those related to the underlying hole grid array upon which the aperiodic Fibonacci array is built. For both groups the destructive interference frequencies at which transmission minima occur closely match prominent reciprocal vectors in the hole array (HA) structure-factor in reciprocal space. However the Fibonacci-related transmission resonances are much weaker than both their calculated Fourier intensity in k space and the grid-related resonances. These differences may arise from the complex, multi-fractal dispersion relations and scattering from the underlying grid arrays. We also systematically studied and compared the transmission resonance strength of Fibonacci HA and periodic HA lattices as a function of the number of holes in the array structure. We found that the Fibonacci-related resonance strengths are an order of magnitude weaker than that of the periodic HA, consistent with the smaller SPP correlation length for the aperiodic structure.

© 2012 OSA

1. Introduction

Recently it has been recognized that reduced transport could occur not only in QCs or disordered systems, but also in deterministic aperiodic systems (DAS). Unlike random media or QCs, DAS are described by simple mathematical prescriptions such as ‘inflation rules’, which encode a fascinating complexity. One such structure, for example is based on the Fibonacci series. DAS belong to a special class of aperiodic structures that have physical properties that are distinct from periodic, disordered, random, or QC structures. For example, they show highly localized excitation states characterized by high field enhancement and low energy transport. In fact, experimental evidence for light localization in DAS formed from dielectric multilayers that follow one-dimensional Fibonacci series has been achieved by several groups [8

8. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987). [CrossRef] [PubMed]

10

10. R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, “Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,” Appl. Phys. Lett. 80(17), 3063–3065 (2002). [CrossRef]

]. Dallapiccola et al. recently reported near-field optical microscopy measurements on two-dimensional (2D) Fibonacci nano-particle structures [11

11. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544–5555 (2008). [CrossRef] [PubMed]

, 12

12. A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17(5), 3741–3753 (2009). [CrossRef] [PubMed]

]. They found that a larger near-field intensity enhancement could be obtained in Fibonacci structures as compared to periodic square arrays of metal nano-particles. We note, however, that evidence for weak localization of SPP excitations in DAS structures is still missing. It is important to study SPP localization in DAS structures in order to show that the suppression of transport in such structures is a universal phenomenon.

2. Experimental details

A one-dimensional Fibonacci series is usually constructed by stacking together two different optical materials, A and B, which are designed using the following deterministic generation scheme: Sj + 1 = {Sj-1, Sj} for j ≥1, where S0 = {B}, S1 = {A} and Sj is a structure obtained after j iterations of the generation rule; here A and B are seed letters. 2D Fibonacci structures have been developed previously by several groups [13

13. R. Lifshitz, “The square Fibonacci tiling,” J. Alloy. Comp. 342(1-2), 186–190 (2002). [CrossRef]

15

15. L. D. Negro, N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A, Pure Appl. Opt. 10(6), 064013 (2008). [CrossRef]

] using an algorithm that generates Fibonacci series in two orthogonal directions (x,y). In this work we generated 2D Fibonacci structures using the method proposed by Dal Negro et al. [15

15. L. D. Negro, N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A, Pure Appl. Opt. 10(6), 064013 (2008). [CrossRef]

], in which we applied two complementary one-dimensional Fibonacci recursive generators: gx: A→AB, B→A and gy: A→B, B→BA in two orthogonal directions (x,y). We designed and fabricated HA structures with circular apertures on 5x5 cm2 area of 75 µm thick free-standing stainless steel foils using this generation method, where A is an aperture, and B is a space lacking an aperture (i.e. a missing hole). The details of the HA generation method can be found elsewhere [11

11. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544–5555 (2008). [CrossRef] [PubMed]

]. An important property of such a structure is that is based on an underlying grid HA that also contributes to the structure factor. We fabricated HA patterns of periodic, 2D Fibonacci, and disordered structures with various numbers of holes in arrays of up to 90x90, with varying hole diameters using a computerized generation scheme.

We used THz time-domain spectroscopy (THz-TDS) to measure the optical transmission spectra, t(ω) of the perforated metal films, where the THz frequency ν = ω/2π. Photoconductive devices were utilized for both emission and coherent detection of the THz electric field. Two off-axis paraboloidal mirrors were used to collect and collimate the THz radiation beam from the emitter and focus the beam to the detector. The samples were attached to a solid metal plate with a 5 cm x 5 cm opening that is significantly larger than the THz beam size and placed in the path of the collimated THz beam. The detected transient photocurrent, PC(τ) was recorded as a function of the translation stage path that determined the time delay, τ between the ‘pump’ beam that hits the emitter and the ‘probe’ beam that arrives at the detector. PC(τ) was subsequently Fourier transformed and normalized to a reference transmission, yielding both the electric field transmission magnitude and phase, t(ω) in the range ~0.1 THz to 0.5 THz. The resulting Fourier transformed data may be described by the relation:

t(ω)=|t(ω)|exp[iφ(ω)]=Etransmitted(ω)Eincident(ω),
(1)

In this expression, Eincident and Etransmitted are the incident and transmitted THz electric fields, respectively, and |t(ω)| and φ(ω) are the magnitude and phase of the amplitude transmission coefficient, respectively. The THz-TDS technique is unique in that it allows for a direct measurement of the transient THz electric field transmitted through the structures, yielding both amplitude and phase information. From these spectra both real and imaginary components of the dielectric response, ε(ω) can be directly obtained without the need for Kramers-Kronig transformations, where somewhat arbitrary assumptions about asymptotic behavior are typically made.

3. Experimental results and discussion

For our studies we also fabricated corresponding HA structures in which the holes are randomly generated on a square grid having the same number of holes and nearest neighbor distance, a as the 2D Fibonacci HA structures; these structures are denoted ‘random on grid’, or RG. Figure 1(c) shows an example of such an RG aperture structure in which 173 holes are randomly placed on a square grid with 0.8 mm spacing. The 2D FFT spectrum of this RG aperture structure contains only two sets of the G reciprocal vectors (G1 and G2, respectively) that arise from the grid structure, with no other prominent reciprocal vector visible; this confirms that the aperture distribution in this sample is indeed random.

Another method of enhancing the F-resonances in the EOT spectrum is to change the hole diameter, d [4

4. T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007). [CrossRef] [PubMed]

]. Figure 2(c) shows t(ω) of Fibonacci HA structures having the same nearest neighbor distance, a but different d, ranging from 0.3 mm to 0.68 mm. These diameters correspond to the aperture waveguide cut-off frequency, fc ranging from 0.58 THz to 0.26 THz, respectively [18

18. A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express 16(13), 9601–9613 (2008). [CrossRef] [PubMed]

]. The variation of d strengthens the F-resonances at frequencies close to fc, since the Fano resonance coupling between a discrete EOT feature and the transmission continuum due to the individual holes becomes stronger when the discrete frequency is close to fc [16

16. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

, 17

17. A. Miroshnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010). [CrossRef]

]. In Fig. 2(d) we calculated the real and imaginary components of the dielectric constant, ε(ω) of the Fibonacci HA from the transmission amplitude and phase (not shown) given in Fig. 2(a) (d = 0.68 mm). We have previously shown that ε(ω) can be well described by a superposition of a non-resonant (continuous) and resonant (discrete) components given by the formula proposed by Agrawal et al. [18

18. A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express 16(13), 9601–9613 (2008). [CrossRef] [PubMed]

]. In comparison to ε(ω) of periodic HAs [18

18. A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express 16(13), 9601–9613 (2008). [CrossRef] [PubMed]

], ε(ω) of Fibonacci HAs is much more complicated because of the multitude of resonances associated with the G-vectors as well as the F-vectors. We note the enhanced intensity of the F2 resonance close to fc of the individual holes. We also note that the F-resonance at 0.15 THz more clearly appears in this HA structure in agreement with our finding in the tF/tRG spectrum of Fig. 2(b).

The localization properties of the SPP excitations in the Fibonacci HA structure can explain the weak F-resonances in the EOT spectrum. To show this, we compare the F-resonance with the G-resonance intensities in Fibonacci HA structures for different numbers of holes, N, while keeping constant the lattice constant a = 0.8 mm and diameter d = 0.44 mm; and periodic HA having the same number of holes, d, and a. Figure 3(a)
Fig. 3 The dependence of THz transmission resonance strength on the number of holes for the Fibonacci and periodic HA structures. (a) THz transmission spectrum of a typical Fibonacci HA structure, and (c) periodic HA lattice, compared to the spectrum of the corresponding random HAs (all structures have 800 holes). The Fibonacci resonance F2 and periodic resonance G1 strengths are shaded. The integration of (b) F2 and (d) G2 resonance strengths is plotted versus the number of holes in the structure. The corresponding strengths in the structure factor in k-space are calculated (blue symbols). (b) The inset shows an intensity profile which is formed by cutting the reciprocal lattice shown in Fig. 1(b) along the line that connects the origin with the F2 reciprocal vector.
shows t(ω) of a Fibonacci HA structure with 800 holes; the characteristic G- and F-resonances are clearly seen. The spectrum t(ω) of the corresponding random HA having the same N and d is also shown in Fig. 3(a). The F2-resonance is chosen for studying the resonance intensity, I, defined as the area under the transmission of the F2 peak with respect to the transmission of the corresponding random HA structure. The F2 resonance intensities defined in this way are plotted vs. N in Fig. 3(b). We also calculated the resonance peak intensity from the FT intensity associated with the corresponding reciprocal vectors in the structure factor; Fig. 3(b) inset demonstrates this calculation for the FT F2 intensity. We found that the intensity of both resonances increases nonlinearly with N. In addition it is clear that the experimental intensities are smaller than the calculated intensities from the FT. For the G1-resonance we need a multiplication factor of ~4 to match the experimental to the calculated intensities. A factor of 2 is justified because the polarized THz beam that subtends only two G-vectors out of possible four; another factor of two shows that in reality there are some SPP losses in the HA structure. However for the F2 resonance we need a factor of 24(!) to match the theoretical calculation. We therefore conclude that the F-resonances are much weaker than the G-resonances in the Fibonacci HA structure, and do not fit the FT calculation even relative to the G1 resonance. This shows that the F-resonances are inherently weak, and it therefore requires a more sophisticated explanation.

It is clear that the experimental periodic G1 transmission strength is nearly two orders of magnitude stronger than the Fibonacci F2 transmission strength, and the Fibonacci experimental strength is an order of magnitude smaller than its calculation value. A possible reason that might lead to this large difference between the G- and F-resonant strength is the proximity of the different resonances to the cut-off frequency, fc. While the G1 resonance frequency at 0.375 THz is close to fc ~0.4 THz, the F2 resonance frequency at 0.27 THz is further away (see Fig. 2(c)). The strongest transmission strength for the F2 resonance occurs for HA structures having d = 0.68 mm, which is ~5 times stronger than it is for structures with d = 0.44 mm. Nevertheless, the transmission strength F2 at d = 0.68 mm is still an order of magnitude weaker than the transmission strength of the G1 resonance at d = 0.44 mm. We therefore conclude that the G-resonances are much stronger than the F-resonances even if the proximity to the cut-off frequency fc is taken into account. This is in contrast with the behavior of hot electromagnetic spot intensity in Fibonacci structures using nano-particle arrays [11

11. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544–5555 (2008). [CrossRef] [PubMed]

].

We can understand the weak transmission of F-resonances in the Fibonacci HA by introducing a correlation length, R to the SPP waves. The transmission resonances are formed by SPP waves that are launched from the individual apertures and subsequently interfere. With no attenuation, the interference would be as strong as the intensity of the reciprocal vectors in the Fourier space. However attenuation may reduce the interference strength because the electric field amplitude decreases with the distance. In solid state physics this effect has been studied by introducing a ‘correlation length’, R to the Bloch wave function, ΨBk(r)~exp(-ikr); so that the ‘attenuated’ Bloch function in real materials is Ψatk(r)~exp(-ikr)exp(-r/R), where the second exponential decay term describes the amplitude attenuation with the distance, r. To see the effect of the ‘attenuation term’ on the strength of the reciprocal vectors in k-space due to smaller interference, we calculated the structure factor of a periodic square HA using Ψatk(r) rather than ΨBk(r), as depicted in Fig. 4
Fig. 4 Calculation of the correlation length, R. (a) DFT spectra calculated at two different correlation lengths around (1,0) peak of a periodic structure with a period of 1 mm. (b) The integration of (1,0) peaks as a function of R in a log-log scale. The fit line shows that the DFT intensity increases quadratic with R. (c) Transmission spectra of a Fibonacci HA with d = 0.68 mm hole diameter (dash blue line) and a periodic HA with d = 0.4 mm (red line). The blue solid line shows a Fibonacci spectrum normalized to the same hole area with the periodic HA. The dash dark line is a cut-off line above which we calculate the transmission strength of the Fibonacci F2 and periodic G1 peaks. (d) Hole-to-hole correlation function calculation, g of periodic, RG and Fibonacci HAs with 800 holes and the nearest neighbor distance of 0.8 mm.
. It is clearly seen that the intensity of the FT peak above the background diminishes substantially at small R (Fig. 4(a)). To further study this effect we calculated the integrated intensity of the G1 transmission resonance in a periodic HA as a function of R (Fig. 4(b)). It is seen that the integrated intensity of the transmission G1-resonance increases quadratic with R.

For completeness, we also analyzed our data using the concept of hole-to-hole correlation functions, as was recently suggested by Przybilla et al. [19

19. F. Przybilla, C. Genet, and T. W. Ebbesen, “Long vs. short-range orders in random subwavelength hole arrays,” Opt. Express 20(4), 4697–4709 (2012). [CrossRef] [PubMed]

]. Figure 4(d) shows the hole-to-hole correlation function calculation of three different HAs with 800 holes. The spectra are discretely distributed at certain distances. We found that these structures give the same pair distances, which are defined by the lattice vectors of the grid. We therefore conclude that the concept of ‘local’ and ‘global’ disorder, as introduced in ref [19

19. F. Przybilla, C. Genet, and T. W. Ebbesen, “Long vs. short-range orders in random subwavelength hole arrays,” Opt. Express 20(4), 4697–4709 (2012). [CrossRef] [PubMed]

]. cannot be applied to aperiodic structures.

4. Conclusion

Acknowledgments

This work was supported by the NSF MRSEC program at the University of Utah under grant # DMR 1121252.

References and links

1.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

2.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002). [CrossRef] [PubMed]

3.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004). [CrossRef] [PubMed]

4.

T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature 446(7135), 517–521 (2007). [CrossRef] [PubMed]

5.

A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett. 102(11), 113901 (2009). [CrossRef] [PubMed]

6.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: Can one state have both characteristics?” Phys. Rev. Lett. 87(16), 167401 (2001). [CrossRef] [PubMed]

7.

D. Mayou, C. Berger, F. Cyrot-Lackmann, T. Klein, and P. Lanco, “Evidence for unconventional electronic transport in quasicrystals,” Phys. Rev. Lett. 70(25), 3915–3918 (1993). [CrossRef] [PubMed]

8.

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987). [CrossRef] [PubMed]

9.

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994). [CrossRef] [PubMed]

10.

R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, “Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,” Appl. Phys. Lett. 80(17), 3063–3065 (2002). [CrossRef]

11.

R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express 16(8), 5544–5555 (2008). [CrossRef] [PubMed]

12.

A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17(5), 3741–3753 (2009). [CrossRef] [PubMed]

13.

R. Lifshitz, “The square Fibonacci tiling,” J. Alloy. Comp. 342(1-2), 186–190 (2002). [CrossRef]

14.

X. Fu, Y. Liu, B. Cheng, and D. Zheng, “Spectral structure of two-dimensional Fibonacci quasilattices,” Phys. Rev. B Condens. Matter 43(13), 10808–10814 (1991). [CrossRef] [PubMed]

15.

L. D. Negro, N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A, Pure Appl. Opt. 10(6), 064013 (2008). [CrossRef]

16.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

17.

A. Miroshnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. 82(3), 2257–2298 (2010). [CrossRef]

18.

A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express 16(13), 9601–9613 (2008). [CrossRef] [PubMed]

19.

F. Przybilla, C. Genet, and T. W. Ebbesen, “Long vs. short-range orders in random subwavelength hole arrays,” Opt. Express 20(4), 4697–4709 (2012). [CrossRef] [PubMed]

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(240.6680) Optics at surfaces : Surface plasmons
(260.3090) Physical optics : Infrared, far
(160.1245) Materials : Artificially engineered materials

ToC Category:
Optics at Surfaces

History
Original Manuscript: March 14, 2012
Revised Manuscript: June 13, 2012
Manuscript Accepted: June 14, 2012
Published: June 22, 2012

Citation
Tho Duc Nguyen, Ajay Nahata, and Z. Valy Vardeny, "Measurement of surface plasmon correlation length differences using Fibonacci deterministic hole arrays," Opt. Express 20, 15222-15231 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15222


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References

  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
  2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science297(5582), 820–822 (2002). [CrossRef] [PubMed]
  3. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004). [CrossRef] [PubMed]
  4. T. Matsui, A. Agrawal, A. Nahata, and Z. V. Vardeny, “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature446(7135), 517–521 (2007). [CrossRef] [PubMed]
  5. A. Agrawal, T. Matsui, W. Zhu, A. Nahata, and Z. V. Vardeny, “Terahertz spectroscopy of plasmonic fractals,” Phys. Rev. Lett.102(11), 113901 (2009). [CrossRef] [PubMed]
  6. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: Can one state have both characteristics?” Phys. Rev. Lett.87(16), 167401 (2001). [CrossRef] [PubMed]
  7. D. Mayou, C. Berger, F. Cyrot-Lackmann, T. Klein, and P. Lanco, “Evidence for unconventional electronic transport in quasicrystals,” Phys. Rev. Lett.70(25), 3915–3918 (1993). [CrossRef] [PubMed]
  8. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett.58(23), 2436–2438 (1987). [CrossRef] [PubMed]
  9. W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett.72(5), 633–636 (1994). [CrossRef] [PubMed]
  10. R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, “Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,” Appl. Phys. Lett.80(17), 3063–3065 (2002). [CrossRef]
  11. R. Dallapiccola, A. Gopinath, F. Stellacci, and L. Dal Negro, “Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles,” Opt. Express16(8), 5544–5555 (2008). [CrossRef] [PubMed]
  12. A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. Dal Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express17(5), 3741–3753 (2009). [CrossRef] [PubMed]
  13. R. Lifshitz, “The square Fibonacci tiling,” J. Alloy. Comp.342(1-2), 186–190 (2002). [CrossRef]
  14. X. Fu, Y. Liu, B. Cheng, and D. Zheng, “Spectral structure of two-dimensional Fibonacci quasilattices,” Phys. Rev. B Condens. Matter43(13), 10808–10814 (1991). [CrossRef] [PubMed]
  15. L. D. Negro, N. Feng, and A. Gopinath, “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” J. Opt. A, Pure Appl. Opt.10(6), 064013 (2008). [CrossRef]
  16. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124(6), 1866–1878 (1961). [CrossRef]
  17. A. Miroshnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys.82(3), 2257–2298 (2010). [CrossRef]
  18. A. Agrawal, Z. V. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express16(13), 9601–9613 (2008). [CrossRef] [PubMed]
  19. F. Przybilla, C. Genet, and T. W. Ebbesen, “Long vs. short-range orders in random subwavelength hole arrays,” Opt. Express20(4), 4697–4709 (2012). [CrossRef] [PubMed]

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