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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 22 — Oct. 29, 2007
  • pp: 14396–14403

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Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles

Luca Dal Negro and Ning-Ning Feng  »View Author Affiliations


Optics Express, Vol. 15, Issue 22, pp. 14396-14403 (2007)
http://dx.doi.org/10.1364/OE.15.014396


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Abstract

In this paper we study the spectral, localization and dispersion properties of dipolar modes in quasi-periodically modulated nanoparticle chains based on the Fibonacci sequence. By developing a transfer matrix approach for the calculation of resonant frequencies, oscillation eigenvectors and integrated density of states (IDS) of spatially-modulated dipole chains, we demonstrate the presence of large spectral gaps and calculate the pseudo-dispersion diagram of Fibonacci plasmonic chains. The presence of plasmonic band-gaps and localized states in metal nanoparticle chains based on quasi-periodic order can have a large impact in the design and fabrication of novel nanophotonics devices.

© 2007 Optical Society of America

1. Introduction

Quasi-periodic photonic structures are defined by simple mathematical rules which generate a non-periodic modulation of optical constants, encoding a fascinating complexity.

The Fibonacci sequence is the chief example of long-range order without periodicity, and can be easily generated according to the following inflation rule [1

M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020–1033 (1987). [CrossRef]

]: A → AB, B→ A. The sequence Fn starts with an arbitrary seed element, F0=A for instance, and the inflation rule is repeatedly applied to obtain: F1=AB, F2=ABA, F3=ABAAB, etc, which displays the well known Fibonacci symmetry: Fn+1={Fn, Fn-1} for j≥1.

Materials structures based on the Fibonacci sequence show distinctive diffraction patterns which cannot be found in either periodic or disordered systems [2

D. Levine and P. J. Steinhardt, “Quasicrystals: definition and structure,” Phys. Rev. B 34, 596–616 (1986). [CrossRef]

]. Unlike periodic and random media, the Fourier spectra of Fibonacci structures consist of isolated Bragg peaks with incommensurate periods, which densely fill the reciprocal space [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

]. Remarkably, the Bragg peaks of these characteristic types of spectra, known as singular continuous Fourier spectra, can only be indexed using two integers [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

]. These spectral features, including the appearance of more Miller indices than the dimensionality of the physical system, induce fascinating physical properties in Fibonacci structures, which have been widely investigated in different physical contexts [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

,4

T. Fujiwara and T. Ogawa, Quasicrystals (Springer-Verlag, Berlin, 1990). [CrossRef]

]. In particular, It has been shown that in Fibonacci structures the transport of wave-like excitations (such as electronic [5

R. B. Capaz, B. Koiller, and S. L. A. de Queiroz, “Gap states and localization properties of one-dimensional Fibonacci quasicrystals,” Phys. Rev. B 42, 6402–6406 (1990). [CrossRef]

], optical [6

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

], mechanical [7

C. Benoit, G. Poussigue, and A. Azougarh, “Neutron scattering by phonons in quasi-crystals,” J. Phys.: Condens. Matter 2, 2519–2536 (1990). [CrossRef]

], spin waves [8

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225–337 (2003). [CrossRef]

]) as well as “mixed-waves” (polariton waves [8

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225–337 (2003). [CrossRef]

]) exhibits unique properties described by multi-fractal energy spectra [9

A. Rudinger and F. Piechon, “On the multifractal spectrum of the Fibonacci chain,” J. Phys. A.: Math. Gen. 31, 155–164 (1998). [CrossRef]

], critical wavefunctions with power-law scaling [10

T. Fujiwara, M. Kohmoto, and T. Tokihiro, “Multifractal wavefunctions on a Fibonacci lattice,” Phys. Rev. B 40, 7413–7416 (1989). [CrossRef]

] and anomalous diffusion [11

F. Igloi, L. Turban, and H. Rieger, “Anomalous diffusion in aperiodic environments,” Phys. Rev. E. 59, 1465–1474 (1999). [CrossRef]

].

In the context of optics, linear and non-linear Fibonacci dielectric multilayer structures have been widely investigated, both theoretically and experimentally, as a model of quasi-periodicity [6

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

]. Light localization and spectral gaps formation in dielectric Fibonacci structures were theoretically investigated by Kohmoto et al [6

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

]. Shortly after, pioneering experiments were performed by Gellermann et al. [12

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

] who observed self-similarity in the transmission spectrum of Fibonacci dielectric multilayers and by Hattori et al. [13

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220–4223 (1994). [CrossRef]

] who measured the Fibonacci dispersion curves. The dynamical transport properties of dielectric Fibonacci multilayers have been recently investigated by Dal Negro et al. [14

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, L. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003). [CrossRef] [PubMed]

], showing sizable field enhancement effects and group velocity reduction at the critically-localized band-edge states.

On the other hand, the present need to control and manipulate sub-wavelength local fields over nanodevices platforms has recently led to a “re-birth” of fundamental studies on surface plasmon propagation and electromagnetic coupling in metal nanoparticle arrays [15

R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The Next Chip-Scale Technology,” Materials Today 9, 20–27 (2006). [CrossRef]

, 16

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics -A Route to Nanoscale Optical Devices,” Adv. Mater. 13, 1501 (2001). [CrossRef]

]. In particular, periodic metal-nanoparticle chains [17

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003). [CrossRef] [PubMed]

] and bi-dimensional arrays [18

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

] have been intensively investigated in the last five years, both theoretically and experimentally, in relation to the possibility of guiding/controlling sub-wavelength resonant fields via engineered near-field electromagnetic coupling [15–17

R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The Next Chip-Scale Technology,” Materials Today 9, 20–27 (2006). [CrossRef]

,19

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. 62, 356–359 (2000). [CrossRef]

]. However, the impact of quasi-periodicity on the plasmonic transport and localization properties of Fibonacci-based nano-devices has not been directly addressed so far.

In this paper we study the spectral, localization and dispersion characteristics of dipolar modes in quasi-periodically modulated nanoparticle chains based on the Fibonacci sequence. Based on a simple transfer matrix approach valid within the dipole approximation, we calculate the resonant frequencies and dipole oscillation modes of a Fibonacci nanoparticles chain. In addition, we calculate the integrated density of states (IDS) and pseudo-dispersion diagrams of Fibonacci chains, and we demonstrate the presence of large spectral gaps in quasi-periodic plasmonic systems.

2. Computational method

In this section, we will introduce a simple and efficient transfer matrix approach for solving the set of coupled-dipole equations describing metal nanoparticle chains with arbitrary non-periodic modulation of the inter-particle spacing.

In particular, we will apply this model to the numerical calculation of the collective oscillations of finite-size, nanoparticle chains with Fibonacci modulation, and contrast our results against the simpler behavior of periodic chains of finite-length. As in Refs. [17

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003). [CrossRef] [PubMed]

, 19

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. 62, 356–359 (2000). [CrossRef]

], we will consider only nearest-neighbors interactions within the dipole approximation, and we will assume Ag nanoparticles with minimum spacing of 25 nm. These assumptions are generally valid as long as higher order multipolar interaction terms do not significantly influence the quasi-static approximation. Even though more general models including all multiple orders have now been successfully developed for periodic chains [20

S. Y. Park and D. Stroud, “Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B. 69, 125418 (2004). [CrossRef]

, 21

C. Girard and R. Quidant, “Near-field optical transmittance of metal particle chain waveguides,” Opt. Express , 12, 6141 (2004). [CrossRef] [PubMed]

], it has been shown experimentally that the dipole approximation is sufficiently accurate for the description of electromagnetic transport in chains of Ag nanoparticles with up to 50 nm diameter and 25 nm inter-particle separation [17

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003). [CrossRef] [PubMed]

]. Additionally, since here we are only interested in comparing the transport properties originating from the geometrical modulation of the particle positions, we will limit our discussion to dipole chains, which have been largely investigated in the context of periodic metal-nanoparticle arrays [15–21

R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The Next Chip-Scale Technology,” Materials Today 9, 20–27 (2006). [CrossRef]

].

Differently from the periodic case, a Fibonacci chain can be simply obtained by a modulation in the particle positions which results from the identification: dA=A and dB=B in the symbolic Fibonacci sequence of any given order Fn [see Fig. 1(a)].

In this paper, without loss of generality, we will discuss the properties of Fibonacci chains with 144 particles (corresponding to F11), which is a good approximation of a “bulk” Fibonacci structure. In addition, we will choose dA=25 nm and dB=30 nm which, as explained above, ensures the validity of our dipole approach [17

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003). [CrossRef] [PubMed]

,19

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. 62, 356–359 (2000). [CrossRef]

,20

S. Y. Park and D. Stroud, “Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B. 69, 125418 (2004). [CrossRef]

].

It is well-known that for a linear chain of Hertzian dipoles, the dipole equations of motion are given by [19

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. 62, 356–359 (2000). [CrossRef]

]:

p̈ i,n= − ω02 p i,n− Γe p˙ i,n + ΓR ω02 p⃛ i,n − γi αn2 ( p i, n−1+ p i, n+1)
(1)

where: pi,n =qxi,n is the individual dipole, xi,n is the distance from equilibrium in the i polarization direction at point n along the chain, ω 0 is the metal plasma frequency, Γ e is the electronic relaxation frequency, Γ R is the relaxation frequency due to radiation into the far-field, γi is a polarization-dependent term (γ = 1 for transverse modes and γ = - 2 for longitudinal) and α2 n ∝ 1/dn 3 is the near-field coupling term [19

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. 62, 356–359 (2000). [CrossRef]

]. All the materials parameters used here are as in Ref. [19

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. 62, 356–359 (2000). [CrossRef]

], with the important exception that for non-periodic chains the dipole-dipole coupling term αn is not constant, because nearest neighbors of the nth dipole do not repeat regularly, but they are now modulated according to the quasi-periodic Fibonacci sequence. Therefore, the positional disorder of the Fibonacci chain induces a nonuniform coupling along the chain. As a result, an analytical solution cannot be found for the eigenmodes of the Fibonacci dipole chain. However, simple transfer matrix approaches exist for the numerical solution of harmonic chains with spatially-modulated coupling [22

F. A. B. F. de Moura, L. P. Viana, and A. C. Frery, “Vibrational modes in aperiodic one-dimensional harmonic chains,” Phys. Rev. B. 73, 212302 (2006). [CrossRef]

, 23

P. K. Datta and K. Kundu, “The absence of localization in one-dimensional disordered harmonic chains,” J. Phys: Condens. Matter 6, 4465–4478 (1994). [CrossRef]

]. In the past, transfer matrix techniques have been extensively applied to the calculation of phonon modes in quasicrystals [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

]. Here, based on a similar approach, we have developed a simple technique for the numerical solution of modulated dipole chains, and we have specifically investigated the Fibonacci case. We will first transform Eq. (1) in the frequency domain, and rearrange the terms to obtain:

u n+1= [ ω2− ω02−i ( Γeω− ΓR ω3 ω02) γi αn2] un − u n−1− u n+1
(2)

where un is the Fourier transform of the oscillation amplitude of dipole n. Equation (2) can now be reduced to a simpler transfer matrix form as:

u n+1 un= Tn un u n−1
(3)

where we have defined the unimodular transfer matrices Tn , relating the un amplitudes to each other, as:

Tn= ω2− ω02−i ( Γeω− ΓR ω3 ω02) γi αn2 −1 1 0
(4)

It is important to notice that the position-dependent coupling terms αn , which depend purely on the geometrical arrangement of the particles, are known “in advance” by the application of the Fibonacci rule. Once all the αn have been found, the displacement of the Nth dipole with frequency ω,uN (ω), can be easily calculated by cascading the individual transfer matrices:

u N+1 uN= TN T N−1 T N−1… T2 T1 u1 u0=Q u1 u0
(5)

Imposing the fixed boundary condition u 0 = u N+1 = 0 , which describes an N-particle chain, we see that the vibration frequencies of the plasmonic chain must satisfy the eigenvalue equation:

Q11 (ω)=0
(6)

which can now be solved numerically. The eigenvectors of Eq. (6) yields the oscillation amplitudes of the dipole chain, while the eigenvalues yield the oscillation frequencies. Here, following Ref. [19

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. 62, 356–359 (2000). [CrossRef]

] we will assume only negligible plasmon damping, and therefore will only deal with the real part of Q 11, restricting the root finding procedure to real numbers. This choice will allow us to directly focus on the plasmonic transport properties uniquely induced by the quasi-periodic geometry of the Fibonacci chain, regardless of the specific materials parameters. However, it remains clear that the described approach is also suited for arbitrary dispersion parameters, as long as complex-plane root finding is allowed. Once the oscillation properties of the plasmonic chain have been obtained, the integrated density of states (IDS) of the chain can be simply obtained as [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

]:

g (ω)= ∑kδ ( ω− ωk)
(7)

The character of the eigenstates is described by the participation ratio [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

]:

P ( ωk)= ∑n ∣ en ( ωk)∣2 N ∑n ∣ en ( ωk)∣4
(8)

where en (ωk ) is the eigenvector defining the polarization of the atomic displacement for the mode of frequency ωk on the atom at position n. When the orthogonality relation is imposed on the eigenvectors:

∑n ∣ en ( ωk)∣2=1
(9)

the value of the participation ratio yields a quantitative measure of the localization degree of the eigenstates [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

]. For extended modes P(ωk ) ≈ 1, while for the most localized modes (where |en (ωk )|2 = 0 for all n except one) P(ωk ) ≈ 1/ N , which is almost zero in the limit of long chains. The participation ratio of an arbitrary mode can therefore take a value between 1/N and 1, depending on the degree of localization of the modes.

Fig. 1. (a). Schematics of the aperiodic nanoparticle chain, and real parts of the transfer matrix element Q11 for (b) periodic and (c) aperiodic Fibonacci chains.

3. Results and discussion

In Figs. 1(b) and 1(c) we plot the real part of Q 11 for a periodic and a Fibonacci quasi-periodic chain, respectively. As expected for a general N-particles chain, in both the cases the number of oscillation frequencies (the zeros of Eq. 6) equals the number of particles in the chain. However, in the case of the Fibonacci chain, the distribution of zeros shows non-periodic frequency gaps separating groups of allowed modes. In order to clearly relate this observation with the plasmon transport along the Fibonacci chain, we calculated the Integrated Density of States (IDS) of periodic and Fibonacci plasmon chains, as described in the previous section.

Fig. 2. Integrated density of states (IDS) for the (a) periodic and (b) quasi-periodic Fibonacci chains.

In Fig. 2 we show the longitudinal and transverse IDS branches calculated for a periodic (a) and a Fibonacci chain (b) with the same number of particles. We notice that, contrary to the case of a periodic chain, the Fibonacci quasi-periodicity induces a non-uniform distribution of IDS spectral gaps for both polarizations [Fig. 1(b)]. As seen from Fig. 2(b), the longitudinal IDS gaps are significantly wider than the transverse ones due to the stronger electromagnetic coupling of longitudinal modes. In addition, it is important to notice here that the two central Fibonacci gaps shown in Fig. 2(b) already appear even for shorter Fibonacci chains down to only 21 particles. However, as the particle number is increased, smaller and smaller satellite plasmonic gaps, nested in a self-similar hierarchy emerge across the frequency range. This is the typical behavior of the Fibonacci energy spectra, which in the limit of many particles converge to Cantor set fractals. Interestingly, the Fibonacci IDS gaps shown in Fig. 2(b) appears to follow a non-uniform distribution similar to the multi-fractal Devil’s staircase distribution [24

M. Schroeder, Fractals, Chaos, Power Laws (Freeman, NY, 1991).

], which describes intermittent chaos and quasi-periodicity in many different areas of physics [25

R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, 2000).

, 26

U. Frisch, Turbolence (Cambridge University Press, 2004).

].

In Fig. 3 we show the spatial profiles of the most localized dipole modes (in order of decreasing participation ratio) found in periodic [Fig. 3(a)] and Fibonacci chains [Figs. 3(b), 3(c)]. The mode shown in Fig. 3(a) corresponds to the well known transverse band-edge mode found in periodic chains [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

]. On the other hand, the Fibonacci modes shown in panels (b) and (c) display the characteristic features of critical and gap-edge modes, respectively [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

]. Our findings show that, in the case of negligible plasmon damping, localized states with very low participation ratio exist in coupled-dipole chains based on quasi-periodic Fibonacci order.

However, the Fibonacci frequency gaps (band-gaps) can only be completely understood by studying the band-dispersion diagram of the structures, as shown in Fig. 4. In the case of periodic plasmon chains [Fig. 4(a)], our numerical model accurately reproduces the analytical solutions calculated, in the limit of low damping, in Ref. 19. As discussed in Ref. 19, we notice that for periodic chains the group velocity vanishes at the borders of the Brillouin zone, and that the bandwidth of the longitudinal branch is larger than that of the degenerate transverse branch. Interestingly, the transverse branch shows a negative slope across most of the Brillouin zone, compatible with a negative index of refraction.

On the other hand, both the definition and the calculation of the dispersion diagrams for a Fibonacci quasi-periodic chain [Figs. 4(b) and 4(c)] certainly deserve more attention. In fact, the lack of periodicity in a Fibonacci chain prevents proper definition of a reciprocal space, and modes cannot be considered as simple plane waves with well-defined wavevectors [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

].

Fig. 3. Dipole moments of the (a) extended state of a periodic chain, (b) critical state (ω/ω0 = 0.986) and (c) the most localized states (ω/ω0 = 0.984) of a quasi-periodic Fibonacci chain. The participation ratios for cases (b) and (c) are 0.201 and 0.044.

Furthermore, the description of excitations in terms of a dispersion law ω= f(k) becomes inappropriate in non-periodic chains. In the case of a quasi-periodic chain, the spatial disorder (Fibonacci quasi-periodicity) induces spatially localized modes which can be described by a superposition of numerous Fourier components with different relative intensities. This situation is reflected in the Fibonacci mode profiles shown in Fig. 3. However, by determining the spatial frequencies associated with Fibonacci modes of different angular frequencies, it is possible to define dispersion diagrams, known as pseudo-Brillouin zones, even in the case of non-periodic structures [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

].

The pseudo-dispersion curves of the Fibonacci plasmon chain, for the two different polarizations, are shown in Figs. 4(b) and 4(c). The intensity of the Fourier components of each mode is suggested by the dot sizes and colors. We notice that Fibonacci pseudo-dispersion diagrams of coupled-dipole chains show a remarkable similarity with pseudo-Brillouin zones widely investigated in the context of quasicrystals phonons [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

, 4

T. Fujiwara and T. Ogawa, Quasicrystals (Springer-Verlag, Berlin, 1990). [CrossRef]

, 7

C. Benoit, G. Poussigue, and A. Azougarh, “Neutron scattering by phonons in quasi-crystals,” J. Phys.: Condens. Matter 2, 2519–2536 (1990). [CrossRef]

].

Indeed, the pseudo-dispersion diagrams shown in Figs. 4(b) and 4(c) are characteristic of monatomic chains with a modulation period incommensurate with the lattice constant [3

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

] and, as for any general linear chain system, mass modulation and gap openings in the energy spectrum are closely related phenomena.

Our results clearly demonstrate that the Fibonacci modulation of the electromagnetic coupling in plasmonic chains is responsible for the gap formation in the pseudo-Brillouin zone, for both longitudinal and transverse modes.

The presence of plasmonic band-gaps and localized states in metal nanoparticle chains based on quasi-periodic order can have a large impact in the design and fabrication of novel nanophotonics devices.

Fig. 4. Dispersion diagrams for (a) periodic chain: (b) quasi-periodic Fibonacci chain, transverse polarization; (c) quasi-periodic Fibonacci chain, longitudinal polarization.

4. Summary

In this paper we have investigated the spectral, localization and dispersion properties of dipolar modes in non-periodically modulated nanoparticle chains based on the Fibonacci sequence. We have developed a simple transfer matrix approach for the calculation of resonant frequencies, oscillation eigenvectors and integrated density of states (IDS) of arbitrarily modulated dipole chains, and we have demonstrated the presence of plasmonic band-gaps in the pseudo-dispersion diagram of Fibonacci chains.

Acknowledgments

This work was supported by the College of Engineering Dean’s Catalyst Award and the Photonics Center at Boston University.

References and links

1.

M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020–1033 (1987). [CrossRef]

2.

D. Levine and P. J. Steinhardt, “Quasicrystals: definition and structure,” Phys. Rev. B 34, 596–616 (1986). [CrossRef]

3.

C. Janot, Quasicrystals: A Primer (Oxford University Press, NY, 1997).

4.

T. Fujiwara and T. Ogawa, Quasicrystals (Springer-Verlag, Berlin, 1990). [CrossRef]

5.

R. B. Capaz, B. Koiller, and S. L. A. de Queiroz, “Gap states and localization properties of one-dimensional Fibonacci quasicrystals,” Phys. Rev. B 42, 6402–6406 (1990). [CrossRef]

6.

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

7.

C. Benoit, G. Poussigue, and A. Azougarh, “Neutron scattering by phonons in quasi-crystals,” J. Phys.: Condens. Matter 2, 2519–2536 (1990). [CrossRef]

8.

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225–337 (2003). [CrossRef]

9.

A. Rudinger and F. Piechon, “On the multifractal spectrum of the Fibonacci chain,” J. Phys. A.: Math. Gen. 31, 155–164 (1998). [CrossRef]

10.

T. Fujiwara, M. Kohmoto, and T. Tokihiro, “Multifractal wavefunctions on a Fibonacci lattice,” Phys. Rev. B 40, 7413–7416 (1989). [CrossRef]

11.

F. Igloi, L. Turban, and H. Rieger, “Anomalous diffusion in aperiodic environments,” Phys. Rev. E. 59, 1465–1474 (1999). [CrossRef]

12.

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

13.

T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B 50, 4220–4223 (1994). [CrossRef]

14.

L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, L. Colocci, and D. S. Wiersma, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. 90, 055501 (2003). [CrossRef] [PubMed]

15.

R. Zia, J. A. Schuller, and M. L. Brongersma, “Plasmonics: The Next Chip-Scale Technology,” Materials Today 9, 20–27 (2006). [CrossRef]

16.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics -A Route to Nanoscale Optical Devices,” Adv. Mater. 13, 1501 (2001). [CrossRef]

17.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232 (2003). [CrossRef] [PubMed]

18.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

19.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B. 62, 356–359 (2000). [CrossRef]

20.

S. Y. Park and D. Stroud, “Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B. 69, 125418 (2004). [CrossRef]

21.

C. Girard and R. Quidant, “Near-field optical transmittance of metal particle chain waveguides,” Opt. Express , 12, 6141 (2004). [CrossRef] [PubMed]

22.

F. A. B. F. de Moura, L. P. Viana, and A. C. Frery, “Vibrational modes in aperiodic one-dimensional harmonic chains,” Phys. Rev. B. 73, 212302 (2006). [CrossRef]

23.

P. K. Datta and K. Kundu, “The absence of localization in one-dimensional disordered harmonic chains,” J. Phys: Condens. Matter 6, 4465–4478 (1994). [CrossRef]

24.

M. Schroeder, Fractals, Chaos, Power Laws (Freeman, NY, 1991).

25.

R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, 2000).

26.

U. Frisch, Turbolence (Cambridge University Press, 2004).

OCIS Codes
(230.3990) Optical devices : Micro-optical devices
(240.5420) Optics at surfaces : Polaritons
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 17, 2007
Revised Manuscript: September 26, 2007
Manuscript Accepted: October 1, 2007
Published: October 17, 2007

Citation
Luca Dal Negro and Ning-Ning Feng, "Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles," Opt. Express 15, 14396-14403 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14396


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References

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  16. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, "Plasmonics - A Route to Nanoscale Optical Devices," Adv. Mater. 13, 1501 (2001). [CrossRef]
  17. S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. G. Requicha, "Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides," Nat. Mater. 2, 229-232 (2003). [CrossRef] [PubMed]
  18. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
  19. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, "Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit," Phys. Rev. B. 62, 356-359 (2000). [CrossRef]
  20. S. Y. Park and D. Stroud, "Surface-plasmon relations in chains of metallic nanoparticles: an exact quasistatic calculation," Phys. Rev. B. 69, 125418 (2004). [CrossRef]
  21. C. Girard and R. Quidant, "Near-field optical transmittance of metal particle chain waveguides," Opt. Express,  12, 6141 (2004). [CrossRef] [PubMed]
  22. F. A. B. F. de Moura, L. P. Viana, A. C. Frery, "Vibrational modes in aperiodic one-dimensional harmonic chains," Phys. Rev. B. 73, 212302 (2006). [CrossRef]
  23. P. K. Datta and K. Kundu, "The absence of localization in one-dimensional disordered harmonic chains," J. Phys: Condens. Matter 6, 4465-4478 (1994). [CrossRef]
  24. M. Schroeder, Fractals, Chaos, Power Laws (Freeman, NY, 1991).
  25. R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, 2000).
  26. U. Frisch, Turbolence (Cambridge University Press, 2004).

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