## Suppression of long-range collective effects in meta-surfaces formed by plasmonic antenna pairs |

Optics Express, Vol. 19, Issue 22, pp. 22142-22155 (2011)

http://dx.doi.org/10.1364/OE.19.022142

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### Abstract

The collective effects in a periodic array of plasmonic double-antenna meta-molecules are studied. We experimentally observe that the collective behavior in this structure substantially differs from the one observed in their single-antenna counterparts. This behavior is explained using an analytical dipole model. We find that in the double-antenna case the effective dipole-dipole interaction is significantly modified and the transverse long-range interaction is suppressed, giving rise to the disappearance of Wood’s anomalies. Numerical calculations also show that such suppression of long-range interaction results in an anomalous spatial dispersion of the electric-dipolar mode, making it insensitive to the angle of incidence. In contrast, the quadrupolar mode of the antenna pair experiences strong spatial dispersion. These results show that collective effects in plasmonic metamaterials are very sensitive to the design and topology of meta-molecules. Our findings envision the possibility of suppressing the spatial dispersion effects to weaken the dependence of the metamaterials’ response on the incidence angle.

© 2011 OSA

## 1. Introduction

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4. R. Adato, A. A. Yanik, J. J. Amsden, D. L. Kaplan, F. G. Omenetto, M. K. Hong, S. Erramilli, and H. Altug, “Ultra-sensitive vibrational spectroscopy of protein monolayers with plasmonic nanoantenna arrays,” Proc. Natl Acad. Sci. U.S.A. **106**, 19227–19232 (2009). [CrossRef] [PubMed]

5. A. Yanik, A. Cetin, M. Huang, A. Artar, S. H. Mousavi, A. B. Khanikaev, J. Connor, G. Shvets, and H. Altug, “Seeing protein monolayers with naked eye through plasmonic Fano resonances,” Proc. Natl Acad. Sci. U.S.A. **108**, 11784–11789 (2011). [CrossRef] [PubMed]

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9. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B **70**, 125429 (2004). [CrossRef]

10. B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. **101**, 143902 (2008). [CrossRef] [PubMed]

11. G. Vecchi, V. Giannini, and J. Gómez Rivas, “Surface modes in plasmonic crystals induced by diffractive coupling of nanoantennas,” Phys. Rev. B **80**, 201401 (2009). [CrossRef]

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

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14. S. H. Mousavi, A. B. Khanikaev, B. Neuner III, Y. Avitzour, D. Korobkin, G. Ferro, and G. Shvets, “Highly confined hybrid spoof surface plasmons in ultrathin metal-dielectric heterostructures,” Phys. Rev. Lett. **105**, 176803 (2010). [CrossRef]

15. S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. **120**, 10871–10875 (2004). [CrossRef] [PubMed]

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19. V.G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely Narrow Plasmon Resonances Based on Diffraction Coupling of Localized Plasmons in Arrays of Metallic Nanoparticles,” Phys. Rev. Lett. **101**, 087403 (2008). [CrossRef] [PubMed]

9. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B **70**, 125429 (2004). [CrossRef]

10. B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. **101**, 143902 (2008). [CrossRef] [PubMed]

15. S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. **120**, 10871–10875 (2004). [CrossRef] [PubMed]

16. S. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys. **121**, 12606–12612 (2004). [CrossRef] [PubMed]

20. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

21. F. J. García de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**, 1267–1290 (2007). [CrossRef]

13. F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. **95**, 233901 (2005). [CrossRef]

21. F. J. García de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**, 1267–1290 (2007). [CrossRef]

*topology*of the unit cell, on the other hand, has not received significant attention.

23. P. W. Kolb, T. D. Corrigan, H. D. Drew, A. B. Sushkov, R. J. Phaneuf, A. Khanikaev, S. H. Mousavi, and G. Shvets, “Bianisotropy and spatial dispersion in highly anisotropic near-infrared resonator arrays,” Opt. Express **18**, 24025–24036 (2010). [CrossRef] [PubMed]

24. J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B **71**, 235420 (2005). [CrossRef]

25. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. **98**, 266802 (2007). [CrossRef] [PubMed]

26. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. **8**, 758–762 (2009). [CrossRef] [PubMed]

27. C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum Fano resonance,” Phys. Rev. Lett. **106**, 107403 (2011). [CrossRef] [PubMed]

15. S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. **120**, 10871–10875 (2004). [CrossRef] [PubMed]

16. S. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys. **121**, 12606–12612 (2004). [CrossRef] [PubMed]

20. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

10. B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. **101**, 143902 (2008). [CrossRef] [PubMed]

21. F. J. García de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**, 1267–1290 (2007). [CrossRef]

28. K. Kempa, R. Ruppin, and J. B. Pendry, “Electromagnetic response of a point-dipole crystal,” Phys.Rev. B **72**, 205103 (2005). [CrossRef]

29. D. H. Dawes, R. C. McPhedran, and L. B. Whitbourn, “Thin capacitive meshes on a dielectric boundary: theory and experiment,” Appl. Opt. **28**, 3498–3510 (1989). [CrossRef] [PubMed]

## 2. Optical response of SAM and DAM structures

*s*-polarized incidence for which DAM and SAM exhibit a strong discrepancy and DAM experiences a quadrupolar resonance. An

*s*-polarized incidence is defined by the wave-vector of the incident wave in the

*xz*-plane (

*k*= 0,

_{y}*k*≠ 0) and the electric field along the antennas (

_{x}*y*direction).

### 2.1. Single-antenna meta-surface

**120**, 10871–10875 (2004). [CrossRef] [PubMed]

**121**, 12606–12612 (2004). [CrossRef] [PubMed]

20. A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

**101**, 143902 (2008). [CrossRef] [PubMed]

**79**, 1267–1290 (2007). [CrossRef]

28. K. Kempa, R. Ruppin, and J. B. Pendry, “Electromagnetic response of a point-dipole crystal,” Phys.Rev. B **72**, 205103 (2005). [CrossRef]

30. A. Alù and N. Engheta, “Guided Propagation along Quadrupolar Chains of Plasmonic Nanoparticles,” Phys. Rev. B **79**, 235412 (2009). [CrossRef]

31. A. Alù and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B **78**, 085112 (2008). [CrossRef]

**p**with a polarizability

*α*relating the local electric field

**E**

_{loc}to the dipole moment,

**p**=

*α*(

*ω*)

**E**

_{loc}. The local electric field polarizing the dipole at the position

**r**is the superposition of the incident electric field and the electric field scattered by the other dipoles in the assembly,

**E**

_{loc}(

**r**) =

**E**

_{ext}(

**r**) + Σ

_{m}_{≠0}

**E**

*(*

_{m}**r**), where the sum excludes the self-interaction,

**E**

*(*

_{m}**r**) =

*ĝ*(

**R**

*–*

_{m}**r**)exp[–

*i*

**k**

_{||}· (

**R**

*–*

_{m}**r**)]

**p**,

*ĝ*(

**r**) = (∇∇+

*k*

^{2})exp(

*ikr*)/

*r*is the dipole-dipole interaction tensor,

*k*=

*nω*/

*c*where

**k**

_{||}= (

*k*,

_{x}*k*) is the in-plane wave-vector, and

_{y}**R**

*is the position of the*

_{m}*m*-th dipole in the array. By assuming that the antennas are polarizable only along the

*y*direction, one can obtain the effective polarizability of the dipole in the array,

*S*

_{0}(

**k**

_{||},

*ω*) is the interaction lattice sum. The 0-th order reflection and transmission coefficients of the dipole array are found to be

*α*

^{−1}(

*ω*) –

*S*

_{0}(

**k**

_{||},

*ω*). In general, this condition can be satisfied only at complex frequencies

*ω*=

*ω*+

_{r}*iω*, showing that the eigenmodes have a finite lifetime due to Ohmic losses or radiative decay [14

_{i}14. S. H. Mousavi, A. B. Khanikaev, B. Neuner III, Y. Avitzour, D. Korobkin, G. Ferro, and G. Shvets, “Highly confined hybrid spoof surface plasmons in ultrathin metal-dielectric heterostructures,” Phys. Rev. Lett. **105**, 176803 (2010). [CrossRef]

32. A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Let. **105**, 126804 (2010). [CrossRef]

*ω*may be ideally achieved only in the limit of lossless meta-surfaces and for

*k*

_{||}>

*k*, which ensures absence of absorption and radiation.

*α*

_{eff}= 0) and the array is effectively invisible and transparent to the incident radiation (

*r*=0,

*t*=1). This happens at the onset of the diffraction orders, for which the lattice sum diverges

*S*

_{0}(

**k**,

_{||}*ω*) → ∞. The origin of this divergence can be understood by considering a simpler model of a linear chain [9

9. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B **70**, 125429 (2004). [CrossRef]

**74**, 205436 (2006). [CrossRef]

33. A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express **17**, 5723–5730 (2009). [CrossRef] [PubMed]

*x*direction. A divergent term in this sum originates in the long-range interactions (

*g*(

_{yy}*x*→ ∞,

*y*= 0) ≈

*k*

^{2}exp(

*ikx*)/

*x*) between dipoles. At the onset of

*l*-th diffraction order, where

*k*= ±(2

*πl*/

*P*+

_{x}*k*), the lattice sum

_{x}*S*

_{0}(

*k*,

_{x}*ω*) grows as

*S*

_{0}(

**k**

_{||},

*ω*) in the reciprocal space [21

**79**, 1267–1290 (2007). [CrossRef]

*l*,

*q*) diffraction order when

*p*-polarized (0,1) diffraction order, which, from the physical point of view, can be explained as a result of transverse character of the long-range interaction among dipoles, which do not radiate in the far-field along the direction of their moments (

*g*(

_{yy}*x*= 0,

*y*→ ∞) ∼ 1/

*y*

^{2}while

*g*(

_{yy}*x*→ ∞,

*y*= 0) ∼ 1/

*x*).

### 2.2. Double-antenna meta-surface

*p*

_{1}and

*p*

_{2}aligned along the

*y*-direction and separated by a distance

*d*along the

*x*-direction under an

*s*-polarized illumination with

*k*=0. Assuming the same polarizability

_{y}*α*for both dipoles, we obtain a relation for the dipole moments in the array: where

*E*

_{ext}is the external electric field at the center of the 0th unit cell,

*S*

_{0}= Σ

_{m}_{≠0}

*g*(

_{yy}**R**

*) exp(−*

_{m}*i*

**k**

_{||}·

**R**

*) and*

_{m}*S*

_{±}= Σ

*(*

_{m}g_{yy}**R**

*±*

_{m}*x̂d*) exp(−

*i*

**k**

_{||}·

**R**

*). The lattice sums*

_{m}*S*

_{0}and

*S*

_{+}/

*S*

_{−}characterize the interaction strength within and between two sub-lattices formed by

*p*

_{1}’s and

*p*

_{2}’s, respectively. The basis of the two electric dipoles can be changed to the more instructive basis of sub-radiant (quadrupolar) and super-radiant (dipolar) modes by the unitary transformation

*p*

_{sub}=

*p*

_{1}exp(

*ik*/2) –

_{x}d*p*

_{2}exp(−

*ik*/2) and

_{x}d*p*

_{sup}=

*p*

_{1}exp(

*ik*/2) +

_{x}d*p*

_{2}exp(−

*ik*/2): where Δ = −1/2[

_{x}d*S*

_{+}exp(

*ik*) +

_{x}d*S*

_{−}exp(−

*ik*)] accounts for the splitting caused by the interaction between the two sub-lattices. This splitting makes the sub- and super-radiant modes red-shifted and blue-shifted, respectively, with respect to the SAM’s electric-dipolar resonance. The disparity in the radiative coupling of the modes, which is reflected in their names, can be seen from the rhs of Eq. (3) which shows that only the super-radiant mode can be excited by the incident field. By reciprocity, the subradiant mode is dark and cannot radiate. Note that at normal incidence the coupling between the modes

_{x}d*κ*= −

*i*/2[

*S*

_{+}exp(

*ik*) −

_{x}d*S*

_{−}exp(−

*ik*)] vanishes, implying that in this particular case the sub-radiant component

_{x}d*p*

_{sub}cannot be excited, either by the external field or through the coupling with the super-radiant mode

*p*

_{sup}. For oblique incidence, however, the sub-radiant mode couples with the super-radiant mode and therefore it is indirectly coupled to the external radiation.

**k**

_{||}=0), this is not the case for oblique incidence. In this case there is a coupling between the super- and sub-radiant modes

*p*

_{sup}and

*p*

_{sub}which no longer represent the eigenmodes of the system. Eq. (3) can be diagonalized through an eigen-decomposition procedure using the transformation

*D*= 1/(1+

*δ*

^{2})[

*p*

_{sup}+

*iδp*

_{sub}] and

*Q*= 1/(1 +

*δ*

^{2})[

*iδp*

_{sup}+

*p*

_{sub}]: where

*δ*= (

*X*– Δ)/

*κ*and

*X*= (Δ

^{2}+

*κ*

^{2})

^{1/2}.

*δ*| ≪ 1), the modes

*D*and

*Q*are still dominated by super-radiant and sub-radiant components, respectively, and also have very disparate radiative coupling. It would be reasonable for these modes to be referred to as quasi super- and sub-radiant modes, however, for the sake of convenience we refer them to as dipolar and quadrupolar modes. In the special case of normal incidence, the quadrupolar mode

*Q*coincides with the sub-radiant mode

*p*

_{sub}(

*δ*= 0) and is completely decoupled from the incident light having its lifetime limited only by the Ohmic losses. However, at finite angles, as a result of hybridization with the super-radiant component

*p*

_{sup}, it acquires a finite electric-dipolar moment and its radiative coupling and bandwidth gradually increase. The dipolar mode

*D*, in contrast, is always strongly radiatively coupled and hence it is spectrally broad at any incidence angle.

*S*=

_{D}*S*

_{0}–

*X*and

*S*=

_{Q}*S*

_{0}+

*X*as effective lattice sums for the modified modal coefficients

*D*and

*Q*of the double dipole meta-molecules. Just as in the case of SAM, for DAM the modal dispersion can be obtained by tracing the poles of the denominators in Eqs. (4).

*D*and

*Q*modes as

*S*=

_{D}*S*

_{0}–

*X*cancel out, but they are still present in

*S*=

_{Q}*S*

_{0}+

*X*, implying that at the Wood’s anomaly, the quadrupolar component vanishes (

*Q*=0) but the dipolar component (

*D*) remains finite, hence the reflection does not vanish as in the SAM case.

*d*→ 0), DAM reduces to a periodic array of single electric dipoles with a polarizability 2

*α*, thus confirming the self-consistency of the generalized dipole model. As

*d*→ 0, the Wood’s anomalies in DAM reappear and the quadrupolar resonance disappears. At the position of the Wood’s anomaly,

*p*

_{2}= −

*p*

_{1}and

*p*

_{sup}=

*p*

_{1}+

*p*

_{2}=0, implying that we restore the well-know result

*r*=0 and

*t*=1, which also follows from Eq. (9).

### 2.3. Comparison between SAM and DAM

*ω*(

_{c}**k**

_{||}) defined by the equation: Another one appears as a maximum of transmission, corresponding to the Wood’s anomalies (shown by the color arrows in Fig. 2a and the dashed lines in Fig. 2c) of the array at the onset of the diffraction orders (

34. X. M. Bendana and F. J. García de Abajo, “Confined collective excitations of self-standing and supported planar periodic particle arrays,” Opt. Express **17**, 18826–18835 (2009). [CrossRef]

35. B. Auguié, X. M. Bendaña, W. L. Barnes, and F. J. García de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B **82**, 155447 (2010). [CrossRef]

29. D. H. Dawes, R. C. McPhedran, and L. B. Whitbourn, “Thin capacitive meshes on a dielectric boundary: theory and experiment,” Appl. Opt. **28**, 3498–3510 (1989). [CrossRef] [PubMed]

*s*-polarized incidence (

*k*= 0,

_{y}*k*≠ 0), i.e., when the wave-vector of the incident wave lies in the

_{x}*xz*-plane. For

*p*-polarized incidence (

*k*=0,

_{x}*k*≠ 0), when the wave-vector lies in the

_{y}*yz*-plane, the dragging effect does not appear within the framework of the dipole model.

**79**, 1267–1290 (2007). [CrossRef]

4. R. Adato, A. A. Yanik, J. J. Amsden, D. L. Kaplan, F. G. Omenetto, M. K. Hong, S. Erramilli, and H. Altug, “Ultra-sensitive vibrational spectroscopy of protein monolayers with plasmonic nanoantenna arrays,” Proc. Natl Acad. Sci. U.S.A. **106**, 19227–19232 (2009). [CrossRef] [PubMed]

36. G. Vecchi, V. Giannini, and J. Gómez Rivas, “Shaping the fluorescent emission by lattice resonances in plasmonic crystals of nanoantennas,” Phys. Rev. Lett. **102**, 146807 (2009). [CrossRef] [PubMed]

37. V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett. **105**, 266801 (2010). [CrossRef]

*k*

_{||}>

*k*), the imaginary parts of

*α*

^{−1}and the lattice sum

*S*

_{0}exactly cancel out giving rise to the formation of guided surface waves [9

**70**, 125429 (2004). [CrossRef]

**74**, 205436 (2006). [CrossRef]

**101**, 143902 (2008). [CrossRef] [PubMed]

11. G. Vecchi, V. Giannini, and J. Gómez Rivas, “Surface modes in plasmonic crystals induced by diffractive coupling of nanoantennas,” Phys. Rev. B **80**, 201401 (2009). [CrossRef]

**79**, 1267–1290 (2007). [CrossRef]

**120**, 10871–10875 (2004). [CrossRef] [PubMed]

**121**, 12606–12612 (2004). [CrossRef] [PubMed]

17. W. Zhou and T. W. Odom, “Tunable subradiant lattice plasmons by out-of-plane dipolar interactions,” Nat Nano **6**, 423–427 (2011). [CrossRef]

19. V.G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely Narrow Plasmon Resonances Based on Diffraction Coupling of Localized Plasmons in Arrays of Metallic Nanoparticles,” Phys. Rev. Lett. **101**, 087403 (2008). [CrossRef] [PubMed]

**79**, 1267–1290 (2007). [CrossRef]

*S*

_{0}(

**k**

_{||},

*ω*) diverges at the frequency corresponding to the onset of diffraction orders. This gives rise to a Wood’s anomaly which appears as a peak in the transmission spectrum. In the DAM case, one would expect Wood’s anomalies due to the divergence of the effective lattice sum for the electric-dipolar moment of Eq. (4),

*S*(

_{D}**k**

_{||},

*ω*) =

*S*

_{0}(

**k**

_{||},

*ω*) –

*X*(

**k**

_{||},

*ω*). However in DAM the diverging terms in

*S*

_{0}(

**k**

_{||},

*ω*) and

*X*(

**k**

_{||},

*ω*), as can be seen from Eq. (8), cancel out exactly, resulting in the disappearance of the Wood’s anomaly and eliminating the expected spectral features.

*s*-polarized diffraction order. We consider one row of double-antenna meta-molecules arranged along the

*x*direction. From the first equality in Eq. (2), we obtain Now we will show that at the onset of the (−1,0) diffraction order, the effective lattice sum for the

*p*

_{1}’s dipoles,

*r*

^{−1}) among meta-molecules. Let us consider a DAM meta-molecule (labeled

*m*) located in the far-field from the origin. As can be seen from the expression for

*g*given in Eq. (1), in the far-field limit, the contribution to the lattice sum

_{yy}*S*

_{1}due to the interaction with the

*m*-th meta-molecule becomes At the (−1,0) Wood’s anomaly

*m*-th unit cell to the lattice sum

*S*

_{1}decays as Thus, the interaction among the double-antenna meta-molecules decays too fast to result in a divergence in the lattice sum, which gives rise to the observed disappearance of the Wood’s anomaly.

*D*in DAM. This can be clearly seen by comparing the spectral position of the minima corresponding to the collective dipolar modes in SAM and DAM transmission spectra in Figs. 2c and 2d. One can see that in the case of SAM, the dipolar mode is strongly dispersive because of “dragging” by the Wood’s anomaly while in the DAM it is rather flat. In the case of SAM, the spectral position of the dipolar mode is strongly affected by the presence of the diverging lattice sum in the denominator of Eq. (1),

*α*(

*ω*)

^{–1}–

*S*

_{0}(

**k**

_{||},

*ω*), and the mode acquires strong angular dependence. In contrast, in the case of DAM, the spatial dispersion of the mode is largely reduced, due to the cancellation of the diverging terms in the effective lattice sum

*S*(

_{D}**k**

_{||},

*ω*).

*because the effective “lattice sum” S*(

_{Q}**k**

_{||},

*ω*) =

*S*

_{0}(

**k**

_{||},

*ω*) +

*X*(

**k**

_{||},

*ω*)

*for the quadrupolar moment of the meta-molecules does diverge*. As a result we see that the spectral position of the collective quadrupolar mode is affected by the Wood’s anomaly and the resonance is “dragged” to the long wavelength range at large angles (Fig. 2d). Just as in the case of the dipolar resonance in SAM, in DAM the quadrupolar resonance follows the Wood’s anomaly and experiences a strong spatial dispersion.

## 3. Experimental results

### 3.2. Mid-IR spectroscopy of SAM and DAM

*λ*= 6

*μm*, however it is not as pronounced as in the theoretical calculations, which is probably the result of the imperfections of the structure. Indeed, it is expected that the quadrupolar mode is more sensitive to metal roughness and disorder as compared to the dipolar mode due to its higher quality factor. Nevertheless, the experimental data clearly reveal theoretically predicted strong interaction among the quadrupolar moments of the meta-molecules. This is manifested in the transmission spectra as a dragging of the quadrupolar resonance by the Wood’s anomaly. One can see that, as the frequency corresponding to the onset of the

*s*-polarized substrate-side (−1,0) diffraction order approaches the quadrupolar resonance, its spectral position red-shifts due to ”repulsion” from the Wood’s anomaly.

### 3.3. Near-IR spectroscopy of SAM and DAM

23. P. W. Kolb, T. D. Corrigan, H. D. Drew, A. B. Sushkov, R. J. Phaneuf, A. Khanikaev, S. H. Mousavi, and G. Shvets, “Bianisotropy and spatial dispersion in highly anisotropic near-infrared resonator arrays,” Opt. Express **18**, 24025–24036 (2010). [CrossRef] [PubMed]

23. P. W. Kolb, T. D. Corrigan, H. D. Drew, A. B. Sushkov, R. J. Phaneuf, A. Khanikaev, S. H. Mousavi, and G. Shvets, “Bianisotropy and spatial dispersion in highly anisotropic near-infrared resonator arrays,” Opt. Express **18**, 24025–24036 (2010). [CrossRef] [PubMed]

_{2}substrate) confirm the theoretically predicted behavior. The experimental results are presented in Fig. 4b, and clearly show the disappearance of the Wood’s anomalies and anomalously flat dispersion of the dipolar mode caused by the cancellation of the long-range interaction between the double-antenna meta-molecules.

## 4. Conclusion

## Acknowledgments

## References and links

1. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London)424, 824–830 (2003). [CrossRef] |

2. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Mater. |

3. | N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater. |

4. | R. Adato, A. A. Yanik, J. J. Amsden, D. L. Kaplan, F. G. Omenetto, M. K. Hong, S. Erramilli, and H. Altug, “Ultra-sensitive vibrational spectroscopy of protein monolayers with plasmonic nanoantenna arrays,” Proc. Natl Acad. Sci. U.S.A. |

5. | A. Yanik, A. Cetin, M. Huang, A. Artar, S. H. Mousavi, A. B. Khanikaev, J. Connor, G. Shvets, and H. Altug, “Seeing protein monolayers with naked eye through plasmonic Fano resonances,” Proc. Natl Acad. Sci. U.S.A. |

6. | S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. |

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

8. | L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. |

9. | W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B |

10. | B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett. |

11. | G. Vecchi, V. Giannini, and J. Gómez Rivas, “Surface modes in plasmonic crystals induced by diffractive coupling of nanoantennas,” Phys. Rev. B |

12. | J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

13. | F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. |

14. | S. H. Mousavi, A. B. Khanikaev, B. Neuner III, Y. Avitzour, D. Korobkin, G. Ferro, and G. Shvets, “Highly confined hybrid spoof surface plasmons in ultrathin metal-dielectric heterostructures,” Phys. Rev. Lett. |

15. | S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys. |

16. | S. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys. |

17. | W. Zhou and T. W. Odom, “Tunable subradiant lattice plasmons by out-of-plane dipolar interactions,” Nat Nano |

18. | Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. |

19. | V.G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely Narrow Plasmon Resonances Based on Diffraction Coupling of Localized Plasmons in Arrays of Metallic Nanoparticles,” Phys. Rev. Lett. |

20. | A. Alù and N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B |

21. | F. J. García de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. |

22. | R. W. Wood, “Films of minute metallic particles,” Phil. Mag. |

23. | P. W. Kolb, T. D. Corrigan, H. D. Drew, A. B. Sushkov, R. J. Phaneuf, A. Khanikaev, S. H. Mousavi, and G. Shvets, “Bianisotropy and spatial dispersion in highly anisotropic near-infrared resonator arrays,” Opt. Express |

24. | J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B |

25. | L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. |

26. | N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. |

27. | C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum Fano resonance,” Phys. Rev. Lett. |

28. | K. Kempa, R. Ruppin, and J. B. Pendry, “Electromagnetic response of a point-dipole crystal,” Phys.Rev. B |

29. | D. H. Dawes, R. C. McPhedran, and L. B. Whitbourn, “Thin capacitive meshes on a dielectric boundary: theory and experiment,” Appl. Opt. |

30. | A. Alù and N. Engheta, “Guided Propagation along Quadrupolar Chains of Plasmonic Nanoparticles,” Phys. Rev. B |

31. | A. Alù and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B |

32. | A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Let. |

33. | A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express |

34. | X. M. Bendana and F. J. García de Abajo, “Confined collective excitations of self-standing and supported planar periodic particle arrays,” Opt. Express |

35. | B. Auguié, X. M. Bendaña, W. L. Barnes, and F. J. García de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B |

36. | G. Vecchi, V. Giannini, and J. Gómez Rivas, “Shaping the fluorescent emission by lattice resonances in plasmonic crystals of nanoantennas,” Phys. Rev. Lett. |

37. | V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(160.3918) Materials : Metamaterials

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(250.5403) Optoelectronics : Plasmonics

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Plasmonics

**History**

Original Manuscript: July 22, 2011

Revised Manuscript: September 15, 2011

Manuscript Accepted: September 16, 2011

Published: October 24, 2011

**Virtual Issues**

Collective Phenomena (2011) *Optics Express*

**Citation**

S. Hossein Mousavi, Alexander B. Khanikaev, Burton Neuner, David Y. Fozdar, Timothy D. Corrigan, Paul W. Kolb, H. Dennis Drew, Raymond J. Phaneuf, Andrea Alù, and Gennady Shvets, "Suppression of long-range collective effects in meta-surfaces formed by plasmonic antenna pairs," Opt. Express **19**, 22142-22155 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-22142

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### References

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- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nature Mater.9, 205–213 (2010). [CrossRef]
- N. Liu, M. L. Tang, M. Hentschel, H. Giessen, and A. P. Alivisatos, “Nanoantenna-enhanced gas sensing in a single tailored nanofocus,” Nat. Mater.10, 631–636 (2011). [CrossRef] [PubMed]
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- S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon.1, 641–648 (2007). [CrossRef]
- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London)391, 667–669 (1998). [CrossRef]
- L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett.86, 1114–1117 (2001). [CrossRef] [PubMed]
- W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B70, 125429 (2004). [CrossRef]
- B. Auguié and W. L. Barnes, “Collective resonances in gold nanoparticle arrays,” Phys. Rev. Lett.101, 143902 (2008). [CrossRef] [PubMed]
- G. Vecchi, V. Giannini, and J. Gómez Rivas, “Surface modes in plasmonic crystals induced by diffractive coupling of nanoantennas,” Phys. Rev. B80, 201401 (2009). [CrossRef]
- J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004). [CrossRef] [PubMed]
- F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett.95, 233901 (2005). [CrossRef]
- S. H. Mousavi, A. B. Khanikaev, B. Neuner, Y. Avitzour, D. Korobkin, G. Ferro, and G. Shvets, “Highly confined hybrid spoof surface plasmons in ultrathin metal-dielectric heterostructures,” Phys. Rev. Lett.105, 176803 (2010). [CrossRef]
- S. Zou, N. Janel, and G. C. Schatz, “Silver nanoparticle array structures that produce remarkably narrow plasmon lineshapes,” J. Chem. Phys.120, 10871–10875 (2004). [CrossRef] [PubMed]
- S. Zou and G. C. Schatz, “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticle arrays,” J. Chem. Phys.121, 12606–12612 (2004). [CrossRef] [PubMed]
- W. Zhou and T. W. Odom, “Tunable subradiant lattice plasmons by out-of-plane dipolar interactions,” Nat Nano6, 423–427 (2011). [CrossRef]
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- V.G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely Narrow Plasmon Resonances Based on Diffraction Coupling of Localized Plasmons in Arrays of Metallic Nanoparticles,” Phys. Rev. Lett.101, 087403 (2008). [CrossRef] [PubMed]
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- P. W. Kolb, T. D. Corrigan, H. D. Drew, A. B. Sushkov, R. J. Phaneuf, A. Khanikaev, S. H. Mousavi, and G. Shvets, “Bianisotropy and spatial dispersion in highly anisotropic near-infrared resonator arrays,” Opt. Express18, 24025–24036 (2010). [CrossRef] [PubMed]
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- L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007). [CrossRef] [PubMed]
- N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater.8, 758–762 (2009). [CrossRef] [PubMed]
- C. Wu, A. B. Khanikaev, and G. Shvets, “Broadband slow light metamaterial based on a double-continuum Fano resonance,” Phys. Rev. Lett.106, 107403 (2011). [CrossRef] [PubMed]
- K. Kempa, R. Ruppin, and J. B. Pendry, “Electromagnetic response of a point-dipole crystal,” Phys.Rev. B72, 205103 (2005). [CrossRef]
- D. H. Dawes, R. C. McPhedran, and L. B. Whitbourn, “Thin capacitive meshes on a dielectric boundary: theory and experiment,” Appl. Opt.28, 3498–3510 (1989). [CrossRef] [PubMed]
- A. Alù and N. Engheta, “Guided Propagation along Quadrupolar Chains of Plasmonic Nanoparticles,” Phys. Rev. B79, 235412 (2009). [CrossRef]
- A. Alù and N. Engheta, “Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B78, 085112 (2008). [CrossRef]
- A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Let.105, 126804 (2010). [CrossRef]
- A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express17, 5723–5730 (2009). [CrossRef] [PubMed]
- X. M. Bendana and F. J. García de Abajo, “Confined collective excitations of self-standing and supported planar periodic particle arrays,” Opt. Express17, 18826–18835 (2009). [CrossRef]
- B. Auguié, X. M. Bendaña, W. L. Barnes, and F. J. García de Abajo, “Diffractive arrays of gold nanoparticles near an interface: critical role of the substrate,” Phys. Rev. B82, 155447 (2010). [CrossRef]
- G. Vecchi, V. Giannini, and J. Gómez Rivas, “Shaping the fluorescent emission by lattice resonances in plasmonic crystals of nanoantennas,” Phys. Rev. Lett.102, 146807 (2009). [CrossRef] [PubMed]
- V. Giannini, G. Vecchi, and J. Gómez Rivas, “Lighting up multipolar surface plasmon polaritons by collective resonances in arrays of nanoantennas,” Phys. Rev. Lett.105, 266801 (2010). [CrossRef]

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