Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles

In this paper we investigate for the first time the near-field optical behavior of two-dimensional Fibonacci plasmonic lattices fabricated by electron-beam lithography on transparent quartz substrates. In particular, by performing near-field optical microscopy measurements and three dimensional Finite Difference Time Domain simulations we demonstrate that near-field coupling of nanoparticle dimers in Fibonacci arrays results in a quasi-periodic lattice of localized nanoparticle plasmons. The possibility to accurately predict the spatial distribution of enhanced localized plasmon modes in quasi-periodic Fibonacci arrays can have a significant impact for the design and fabrication of novel nano-plasmonics devices. ©2008 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (240.5420) Polaritons; (230.3990) Microstructure devices; References and links 1. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. 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Introduction
The present need to control and manipulate deep sub-wavelength optical fields at the nanoscale has stimulated a fundamental interest in localized plasmon excitations and electromagnetic interactions in metal nanoparticle arrays [1][2][3][4][5][6].In particular, one dimensional periodic chains of metal-nanoparticles [7][8][9] and two-dimensional periodic arrays [4][5][6][10][11][12] have been intensively investigated in the past years, both theoretically and experimentally, in relation to the possibility of guiding and controlling sub-wavelength plasmon resonances [1,7].Proof-of principle functional elements for the control of plasmon-polariton propagation such as Bragg gratings [13], plasmon waveguides, bends and splitters [14], mode couplers [15] and plasmon-polariton bandgap structures have already been demonstrated, potentially enabling the successful convergence of optics and electronics at the nanoscale.However, the impact of quasi-periodic order and deterministic aperiodicity on the transport and localization properties of optical fields in plasmonic nanostructures has not been fully addressed so far.
The study of quasi-periodically modulated coupling in dipolar chains has recently revealed the presence of large spectral band-gaps and localized dipole modes in the pseudo-dispersion diagram of Fibonacci nanoparticle chains [16].However, to the best of our knowledge, the transport and local-field properties of localized surface plasmons in two-dimensional Fibonacci structures have not been previously investigated.
Fibonacci structures are the paradigm of quasi-periodic order.In one spatial dimension, they can be generated by combining two different optical materials A and B according to the inflation rule: [17]: A → AB, B→ A. By construction, this sequence is long-range correlated although non-periodic and displays fascinating fractal properties which have been discussed at length in the recent years [18,19].For instance, the Fibonacci sequence is self-similar with a scaling factor equal to the Golden ratio [19] and arbitrarily-long identical sub-sequences will appear very close to each other infinitely often along the sequence [19].Because of this unique structural complexity, material systems based on the Fibonacci sequence show distinctive diffraction patterns which cannot be found in either periodic or disordered systems [20].Unlike periodic and random media, the Fourier spectrum of Fibonacci structures consists of isolated Bragg peaks, indexed by two integers, which densely fill the reciprocal space with incommensurate periods [17].As a result, the dynamic excitations in Fibonacci structures possess fractal (Cantor-sets) energy spectra with zero Lebesgue measure and can be described by highly-fragmented multi-fractal modes, known as "critical modes", which exhibit powerlaw spatial localization [21].
In the context of classical optics, the linear and non-linear behaviour of dielectric and metal-dielectric Fibonacci multilayer structures have been widely investigated, both theoretically and experimentally [22].Light localization and spectral gaps formation in dielectric Fibonacci structures were theoretically investigated by Kohmoto, et al [22].Shortly after, pioneering experiments were performed by Gellermann et al. [23] who first observed self-similarity in the transmission spectrum of Fibonacci multilayers and by Hattori et al. [24] who measured the Fibonacci fractal dispersion.Recent experiments have investigated the dynamic transport properties of porous silicon Fibonacci multilayers and strongly suppressed group velocity has been observed at the edge modes of the fundamental Fibonacci bandgap [25].These states show sizable field enhancement and characteristic self-similar patterns [26], which are typical of critically localized modes.
However, despite the significant interest in one-dimensional Fibonacci systems, the optical behavior of the more complex two-dimensional Fibonacci structures, especially in relation to plasmonic nanostructures, still remains to be explored.
In this paper we investigate for the first time the spatial distribution of localized plasmon modes in two-dimensional Fibonacci arrays of gold nanoparticles fabricated by electron-beam lithography on a quartz substrate.By performing near-field optical measurements and three dimensional Finite Difference Time Domain (FDTD) simulations we demonstrate that nearfield coupling of nanoparticle dimers in Fibonacci arrays results in a predictable quasiperiodic lattice of localized plasmon modes with enhanced field intensity.These results suggest alternative schemes to design and improve the sensitivity of plasmon sensor arrays and Surface Enhanced Raman Scattering (SERS) substrates.

Design and fabrication of two-dimensional Fibonacci lattices
Several methods have been recently developed to extend one-dimensional Fibonacci structures into two spatial dimensions while preserving their unique Fourier spectral properties [27][28][29].We have recently shown that a very simple method exists based on the alternation of one-dimensional inflation maps along orthogonal directions [30].Following our approach, a two-dimensional Fibonacci symbolic lattice (quasi-lattice) can simply be obtained from a seed letter A or B by applying two complementary one-dimensional Fibonacci inflation maps f A : A→AB, B→A and f B : A→B, B→BA along the horizontal and the vertical directions depending if the first elements encountered in the matrix expansion are A or B. The direct and reciprocal Fibonacci lattices obtained as explained above agree very well with generalizations based on more complex iterative methods [27][28][29].The application of our method to the generation of a two-dimensional Fibonacci lattice is illustrated in Fig. 1(a), limited to the first three Fibonacci generation steps.The Fibonacci direct lattice obtained after 7 iteration steps is shown in Fig. 1(b), where the black dots represent the positions of arbitrary nanoparticles.The two-dimensional Fourier spectrum of the Fibonacci lattice (reciprocal lattice) is shown in Fig. 1(c) and it displays the unique spectral features and the high degree of rotational symmetry (fourfold) of the quasi-periodic Fibonacci lattice [29].Finally we notice that every vertical or horizontal section (cut) of the two-dimensional Fibonacci lattice defines a one-dimensional sequence whose Fourier spectrum is identical to the well-known spectrum of the Fibonacci sequence, as shown in Fig. 1(d).Therefore, the 2D Fibonacci lattice can also be obtained from the proper combination of two orthogonal onedimensional Fibonacci sequences, and it can be considered a proper two-dimensional generalization of one-dimensional Fibonacci sequences [31].The simple iterative procedure described above enables the fabrication of planar optical devices with two-dimensional Fibonacci spectra.Our fabrication process flow starts with 180nm of PMMA 950 (Poly Methyl MethAcrylate) spin coated on top of the substrates which consist in 10 nm thick indium tin-oxide (ITO) films on quartz.After spin coating, the substrate was soft baked at 180ºC for 20 minutes.The Fibonacci aperiodic patterns were written according to the generation procedure discussed above by using a Zeiss SUPRA 40VP SEM equipped with Raith Beam Blanker and a Nano Pattern Generation System (NPGS) for nanopatterning stage.Subsequently, the resist was developed in a 1:3 solution of MIBK (Methyl Isobutyle Ketone) and isopropanol for 70 s, and rinsed in isopropanol.Finally, a 30nm thin Au film was deposited on the patterned surface by e-beam evaporation, and the liftoff process using acetone resulted in the Fibonacci array of Au cylindrical nanoparticles.The height of each particle, as measured by AFM, was h=30nm, the particles circular diameter d=200nm and a minimum edge-to-edge inter-particle separation a=50nm (only a minimum distance is defined for a Fibonacci due to its lack of periodicity).
Using the same process described above, we also fabricated a reference periodic structure consisting of a square lattice of Au nanoparticles with h=30nm, d=200nm and a lattice constant (center-to-center distance) L=250nm, which results in a=50nm as in the case of the Fibonacci array.An identical series of samples with 100nm inter-particle separation were additionally fabricated for far-field scattering studies.Due to the larger particle separation, this last set of samples is less affected by the structural imperfections localized at the edges of periodic/Fibonacci arrays with 50nm inter-particle separation which causes a strong spurious background (not related to the arrays periodicity) in the near-infrared spectral region (already visible in Fig. 2 (c)).Figures 2(a) and 2(b) shows Scanning Electron Microscopy (SEM) images of the resulting periodic and Fibonacci samples, respectively.The insets are obtained at a higher magnification and highlight details of the respective nanoparticle arrays.From these images we can appreciate the high-degree of uniformity of the nanofabricated structures down to few nanometer length-scales.In addition, we also notice that our choice for geometrical parameters of the arrays resulted in a larger nanoparticles filling fraction (defined as the total area covered by Au particles with respect to the total array area) F=0.502 for the periodic array, compared to F=0.237 for the Fibonacci structure.In order to understand the spectral response of our arrays we performed dark-field scattering measurements.Figures 2(c) and 2(d) shows the experimentally measured extinction spectra of the periodic and Fibonacci arrays, respectively.The spectra have been measured using an upright microscope (Olympus BX51 W1).The nanoparticle arrays were immersed in index-matching oil, and illuminated with unpolarized white light from a 100 W tungsten halogen lamp using an oil dark-field condenser (NA = 1.2 -1.4) in transmission mode.The light scattered by the substrate was collected with a 40x objective (NA =0.65) and imaged using a digital camera with an active area of 620x580 pixels.Spectral analysis of the scattered light was achieved by an imaging spectrometer (Acton Research, InSpectrum 150) and a back-illuminated CCD detector (Hamamatsu INS-122B).All the spectra were corrected be the system's response and source emission.
The spectra shown in Figs.2(c) and 2(d) clearly illustrate the important role of long-range diffractive coupling in the optical response of plasmonic arrays.In particular, the strong multiple scattering in closely-packed plasmonic arrays shifts the scattering response well into the visible spectral range even in the case of 200nm diameter particles, as already shown in [30,32].The insets in Figs.2(c) and 2(d) show the spatially-resolved scattering intensity (scattering maps) experimentally measured for the two arrays.In addition, the inset of Fig. 2 (d) shows that multiple-light scattering in a Fibonacci structure leads to the formation of inhomogeneous scattering modes with highly structured spatial profiles.

Near-field optical characterization of Fibonacci arrays
Near-field optical modes of Fibonacci arrays were studied by Near Field Scanning Optical Microscopy (NSOM).The measurements were performed with a MultiView™ 2000 Scanning Probe Microscope/NSOM System (Nanonics Imaging Ltd., Jerusalem, Israel).The SPM head allows for simultaneous Atomic Force Microscopy (AFM) and NSOM imaging.The samples were uniformly illuminated at normal incidence with a 520 nm Ar:Kr laser (Spectra Physics, Innova 70C Spectrum) line through a single mode polarization maintaining optical fiber.With the sample accurately held in position, a 200 nm aperture NSOM probe was scanned above the arrays to collect the transmitted light, which is collected through an NSOM fiber onto a photomultiplier tube (PMT) detector.An AFM image is obtained simultaneously.The independent probe and sample piezo scanners were of great importance for the characterization of light propagation in these structures.In this configuration, the electromagnetic coupling of localized surface plasmons in nanoparticle arrays can be conveniently investigated as shown in [33].All the experiments have been performed with fiber tips coated with chromium.Such coating doesn't exhibit surface plasmon resonance at visible frequencies and therefore does not influence the observations of local fields that will be discussed in the following [34].
Figure 3 shows the collection mode NSOM images and the corresponding AFM topographic images obtained by scanning a 6 µm x 5.9 µm area of the periodic (Figs.3(b), 3(d) and a 7.8 µm x 6.9 µm area of the Fibonacci array (Figs.3(a), 3(c), respectively.The incident light was polarized in the plane of the arrays.
The NSOM images shown in Fig. 3 clearly demonstrate the presence of hot spots corresponding to the resonant excitation of localized nanoparticle plasmons (LNPs) both in the periodic and Fibonacci samples.The direct comparison with the topographic AFM images shown in Fig. 3(a) and Fig. 3(b) allows us to conclude that the LNPs are localized at interstitial positions with respect to the nanoparticles sites.However, while for the periodic structures the observed LNPs are periodically arranged into a simple square lattice, the question arises on how the LNPs are arranged in a Fibonacci lattice.A closer look at Fig. 3(c) suggests that the LNPs are localized in between vertically-coupled dimer structures in the Fibonacci array, which are efficiently excited by the vertically polarized incident field (x direction).In order to experimentally measure the enhancement of the transmitted intensity in the near-field we performed line-scan analysis [33] directly on the NSOM images, considering both Fibonacci and periodic structures.Images for the two structures were acquired one after the other rapidly and care was taken to maintain identical conditions in both imaging sessions.The results obtained are still not free of topographic effects [35], but we notice that these would spuriously enhance the fields probed for the periodic structure with respect to the Fibonacci one.Hence, our measured intensity enhancement for the Fibonacci array should be considered as a conservative estimate of the maximum field intensity enhancement.In order to clarify the mechanism of the enhancement and the geometrical distribution of hot spots we will discuss in the following section our computational results based on threedimensional Finite Difference Time Domain (FDTD) electromagnetic simulations [36].Based on this knowledge, we will show that LNPs in a Fibonacci structure form a characteristic quasi-periodic pattern which results from the composition of a regular Fibonacci sequence and a "perturbed" Fibonacci sequence.

Discussion
In order to gain a deeper insight into the mechanism of plasmon localization in closely-spaced metal nanoparticles Fibonacci arrays we have performed three dimensional finite difference time-domain (FDTD) simulations with non-uniform gridding [37] and compared with the near-field intensity profiles directly measured by NSOM.In FDTD methods, one directly solves Maxwell's dynamics equations on a discrete spatio-temporal grid.However, particular care must always be taken in extracting local field enhancement values from FDTD calculations, which can suffer from significant numerical dispersion and staircasing errors, preventing the accessibility of steady-state solutions [38,39].In order to closely approximate our experimental geometry, the nanoparticle arrays have been excited by a plane wave (520nm wavelength) incident from the bottom.The input wave polarization was chosen in the plane of the arrays and polarized along the x direction, and the material gold dispersion was modeled as described in [40].A grid size step of 5 nm was used in the finely meshed region around the Au nanoparticles, while a 20 nm grid was used in the principal mesh elsewhere.Perfectly Matching Layers (PML) boundary conditions were imposed in order to avoid reflections from the edges of the computational window.Finally, because of memory limitations, we limit our numerical simulations to arrays with 80 (Fibonacci) and 64 (periodic) particles.
Figure 5 shows the x component of the electric field calculated for a periodic (Fig. 5(b)) and a Fibonacci (Fig. 5(d)) structure.This polarization has been found to yield an excellent agreement with the experimentally measured near-field profiles obtained by controlling the input light polarization in the plane of the arrays.The observation of negligible dimer coupling along the horizontal direction (z-direction) confirms that the polarization of the incident field was predominantly aligned along the x-direction.The NSOM images corresponding to the same portion of the periodic/Fibonacci arrays are also displayed in Fig. 5(a) and Fig. 5(c).We can see from Fig. 5 that the positions of the LSPs in periodic and Fibonacci arrays are in excellent agreement with the results from FDTD simulations.In particular, as qualitatively anticipated in the previous section, we can see in Fig. 5(b) that when we excite the square periodic array using an x-polarized electric field, the surface plasmons localized in the x direction between neighboring particles can also weakly couple in the transverse z direction.This additional coupling effect is observed in the periodic square array due to its higher filling fraction (F=0,502) with respect to the Fibonacci array.The horizontal (z-direction) coupling of vertically-localized (x-direction) surface plasmons observed in the periodic structure gives rise to delocalized plasmon states which extend across the entire periodic sample, therefore reducing the intensity enhancement (Fig. 5(b)).On the opposite, the absence of periodicity of the Fibonacci structure results in a non-uniform arrangement of particle dimers which are spatially more separated due to the smaller filling fraction of the Fibonacci lattice.This prevents the formation of extended plasmon states by weakening the near-field coupling of neighboring dimers, and results in larger values of field enhancement (Fig. 5(d)).Therefore, the combination of our experimental and computational results demonstrates that nanoparticles plasmons are predominantly localized by the inhomogeneous distribution of dimers associated to the Fibonacci geometry.This experimental result enables a very simple picture for the understanding of the geometrical hot-spot distribution in the Fibonacci array under consideration.In fact, according to our findings, the LNPs in a Fibonacci structure will be arranged in the non-periodic sub-lattice of particles dimers contained in the Fibonacci lattice.This sub-lattice will depend on the in-plane polarization of the applied field, which selects the dimers which are effective in plasmon localization.For example, for vertically polarized (x-axis) excitation, the hot-spot sub-lattice will consist of all the vertically aligned dimers contained in the Fibonacci structure.Interestingly this sub-lattice is non-periodic, and must be computed starting from the original Fibonacci array.In Fig. 6(a) we show the dimer sub-lattice (red dots) together with the original Fibonacci lattice (black dots).The nonperiodic distribution of red dots shown in Fig. 6(a) corresponds exactly to the position of the Fibonacci localized plasmons, or hot-spots, that we have experimentally investigated by NSOM measurements.
We will now discuss more in detail the geometry of the hot-spot array, and we will show that the distribution of hot-spots in a Fibonacci structure can be generated by a simple deterministic rule, different from the Fibonacci one.We will start considering the distribution of hot-spots (red dots) along the horizontal (z) direction (the rows of red dots in Fig. 6 (a)).By comparing any of the hot-spots horizontal lines with immediately neighboring rows of Fibonacci particles (black dots), we notice that each red particle, representing the presence of a dimer, corresponds to a missing black dot in the neighboring rows of the Fibonacci lattice (black dots), and the absence of a red dot corresponds to the presence of a particle in the original Fibonacci lattice.Therefore, we deduce that along the horizontal direction the sequence of hot-spots is a Fibonacci sequence where the A and B in the Fibonacci generation rule defined is section 2 are simply interchanged.In Fig. 6(c) we show the calculated Fourier spectrum of the horizontal distribution of hot-spots (any of the horizontal lines of red particles), which confirms the Fibonacci nature of the horizontal sequence.In particular, it is well-known [17][18][19]  .Consistently, as shown in Fig. 6(c) these two frequencies are located at approximately 61.8% and 38.2% of the normalized sampling frequency, respectively.In addition, the Fourier spectrum of the Fibonacci sequence is also characterized by strong diffraction peaks, which scale with the golden ratio, at frequencies ,... , , We will now discuss the distribution of hot-spots along the vertical direction (x) direction (the columns of red dots in Fig. 6 (a)).We start by noticing that along this direction the hotspots are not arranged according to a Fibonacci structure.In fact, as shown in Fig. 6(b), the calculated Fourier spectrum of hot-spots along the x direction deviates significantly from the Fibonacci spectrum.All the central frequencies of a Fibonacci spectrum are now missing and the intensities of the diffraction peaks do not scale as for the Fibonacci case.Therefore we need to understand the aperiodic spatial distribution of hot-spots along the vertical (x) dimension.The experimental demonstration of near-field dimer coupling as the origin of plasmon localization in a Fibonacci structure enables the complete solution of this problem.As a result, we will have to find the vertical distribution of particle pairs (dimers) in a Fibonacci lattice in order to solve this problem.Symbolically, we remind here that a Fibonacci sequence is generated by the iteration of the inflation f: 0 → 01, 1→ 0. Therefore, since in our correspondence a particle dimer is associated to a pair of 0 digits we need to find a new inflation map which generates the sequence of pairs of 0s in a Fibonacci chain.This can easily be done by induction, and the map is: where the seeds are 0 0 = g and 00 1 = g , and at any other generation we append the two previous ones separated by the last digit of the preceding order, which is: (regardless of the iteration number, the first digit of the Fibonacci sequence will always be 0 while the last digit will oscillate between 0 and 1 depending if the generation number is even or odd).
In the representation of Eq. ( 1) we choose to represent the presence of a particle pair (dimer) by a one and the absence of a dimer by a zero.The iteration of the sequence (1) reproduces exactly the particle pair distribution in a Fibonacci chain, or the hot-spot distribution along the vertical x direction.Since the sequence generated by Eq. ( 1) differs from a standard Fibonacci sequence by the presence of an alternating single digit (0 or 1), or "point defect" at each generation step, we decide to call this new aperiodic sequence a "perturbed Fibonacci sequence".The calculated Fourier spectrum of the perturbed Fibonacci sequence obtained with Eq. ( 1) agrees exactly with the one shown in Fig. 6(b), which is calculated on the dimer lattice shown in Fig. 6(a).
We therefore conclude that the quasi-periodic hot-spot distribution experimentally observed in a Fibonacci plasmonic array represents an example of anisotropic quasi-periodic distribution, since it is a Fibonacci one when decomposed along the horizontal axis and a perturbed Fibonacci sequence when analyzed along the vertical direction.It is also clear that due the very same picture will apply when exciting the Fibonacci array with the opposite (horizontal) field polarization, due to the fourfold symmetry of the Fibonacci lattice.The more general case of unpolarized excitation simply follows from decomposition in the two orthogonal components, which will follow the inflations previously discussed in this paper.

Summary
In this paper we have experimentally investigated the field enhancement and the spatial distribution of localized plasmon modes in periodic and quasi-periodic Fibonacci arrays of gold nanoparticles.By performing near-field optical measurements in collection mode and three dimensional FDTD simulations we showed that dimer coupling in a Fibonacci lattice results in a deterministic quasi-periodic lattice of localized plasmon modes which follow a perturbed Fibonacci sequence.In addition, we have shown that stronger field enhancement can be achieved in Fibonacci structures compared to periodic arrays.The possibility to accurately predict the spatial distribution of enhanced localized plasmon modes in quasiperiodic Fibonacci arrays can have a significant impact for the design and fabrication of nanophotonics devices such as plasmon sensor arrays with larger field enhancement/sensitivity, metal-dielectric quasi-periodic photonic crystals structures for label-free multi-parametric sensing, broad-band light extractors for LEDs applications and ultimately engineered SERS substrates with predictable hot-spot distributions.

Fig. 1 .
Fig. 1. (a).First two generations of 2D Fibonacci sequence shown along with the inflation rules, (b) Generation 7 of the 2D Fibonacci lattice, (c) Calculated diffraction spectra of the 2D Fibonacci lattice shown in (b), (d) Cut along the vertical arrow shown in (c) showing the preservation of 1D Fibonacci Fourier characteristics.

Fig. 2 .
Fig. 2. SEM images of (a) Periodic and (b) Fibonacci Au nanoparticle array.The insets show the dimension of the particles (200nm) and the inter-particle separation (50nm); (c) Normalized extinction of periodic arrays with 100 nm particle spacing.The inset shows the true-color image of the array under white light illumination.The reddish edge region of the array, which is responsible for the near-infrared background scattering, arises from structural imperfections due to partial lift-off at the array edge; (d) Normalized extinction of the Fibonacci array (100 nm particle spacing).The inset shows the true-color image of the Fibonacci array under white light illumination.

Fig. 3 .
Fig. 3. (a) and (b) show topographical AFM images on the Fibonacci and Periodic arrays respectively, (c) and (d) are NSOM images obtained from Fibonacci and Periodic nanoparticle arrays obtained under identical conditions.

Figures 4 (
a), 4(b) show 5.6µm x 3.3µm NSOM image of the periodic sample and 5.6 µm x 3.8 µm NSOM image of the Fibonacci sample, respectively.The horizontal white lines have been selected for the line-scan profile, which gives rough estimate of the local field intensity due to the large aperture tip used in our experiments.The resulting horizontal field-intensity profiles are shown in Figs.4(c) for the periodic and Fibonacci structures, respectively.The results (Fig.4(c)) demonstrate that a larger field intensity enhancement, due to an inhomogeneous dimer distribution, can be obtained in Fibonacci structures compared to a periodic square array of metal nanoparticles.

Fig. 4 .
Fig. 4. (a) and (b) are NSOM images obtained from Fibonacci and Periodic nanoparticle arrays obtained under identical conditions, (c) shows the intensity profile along the horizontal cut in the NSOM images along the white lines in (a) and (b).

Fig. 5 .
Fig. 5. (a) and (c) are NSOM images obtained from Fibonacci and Periodic nanoparticle arrays obtained under identical conditions.(b) and (d) are FDTD simulations of periodic and Fibonacci Au nanoparticle arrays respectively, the particles were 200nm and min.interparticle separation was 50nm.Te number of particles for the Fibonacci array is 80 while it is 64 for the periodic lattice.The Au was modeled using the Drude parameters collision frequency = 250 THz, and Plasma Frequency =6790 THz.The illumination was with a CW excitation polarized in the plane of the particle at 520nm.
are the ratio of successive Fibonacci numbers for the truncated sequence, and have Fourier amplitudes which scale as 2 γ .

Fig. 6 .
Fig. 6. (a).Calculated hot-spot locations are shown (red) along with the Fibonacci nanoparticle array (black).(b) Fourier spectrum of the hot-spots sequence along the vertical direction.(c) Fourier spectrum of the hot-spots sequence along the horizontal direction.This spectrum coincides with the one of a one-dimensional Fibonacci sequence.
that the dominant Fourier components in a Fibonacci sequence appear at