Abstract

The interplay between localized surface plasmon (LSP) resonances and their collective responses, known as surface lattice resonances (SLRs), in metal nanoparticle arrays can lead to resonances with high Q-factors (∼100). These responses have in the past usually been studied for LSP resonances in the plane of the array of the nanoparticles (assumed to be nonmagnetic), thus restricting efficient coupling to particles separated along a specific direction. In the present study, we demonstrate that LSPs oscillating perpendicular to the plane of the surface can lead to stronger inter-particle coupling, which enhances the SLRs. This stronger coupling occurs because the out-of-plane oscillations can couple in all directions within the plane of the array. We study the resulting SLRs for square and hexagonal lattices using the discrete-dipole approximation, and we predict much larger Q-factors in the wavelength range near 650 nm. This prediction suggests that SLRs could be very useful in enhancing various optical processes, and in many applications such as sensing and nonlinear optical wave mixing.

© 2016 Optical Society of America

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2015 (2)

M. Kataja, T. K. Hakala, A. Julku, M. J. Huttunen, S. van Dijken, and P. Törmä, “Surface lattice resonances and magneto-optical response in magnetic nanoparticle arrays,” Nat. Commun. 6, 7072 (2015).
[Crossref] [PubMed]

P. Törmä and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. 78, 013901 (2015).
[Crossref]

2014 (5)

L. Shi, T. K. Hakala, H. T. Rekola, J.-P. Martikainen, R. J. Moerland, and P. Törmä, “Spatial coherence properties of organic molecules coupled to plasmonic surface lattice resonances in the weak and strong coupling regimes,” Phys. Rev. Lett. 112, 153002 (2014).
[Crossref] [PubMed]

A. D. Humphrey and W. L. Barnes, “Plasmonic surface lattice resonances on arrays of different lattice symmetry,” Phys. Rev. B 90, 075404 (2014).
[Crossref]

S.-Q. Li, W. Zhou, D. B. Buchholz, J. B. Ketterson, L. E. Ocola, K. Sakoda, and R. P. H. Chang, “Ultra-sharp plasmonic resonances from monopole optical nanoantenna phased arrays,” Appl. Phys. Lett. 104, 231101 (2014).
[Crossref]

M. J. Huttunen, K. Lindfors, D. Andriano, J. Mäkitalo, G. Bautista, M. Lippitz, and M. Kauranen, “Three-dimensional winged nanocone optical antennas,” Opt. Lett. 39, 3686–3689 (2014).
[Crossref] [PubMed]

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
[Crossref] [PubMed]

2013 (3)

J. Hu, C. Wang, S. Yang, F. Zhou, Z. Li, and C. Kan, “Surface plasmon resonance in periodic hexagonal lattice arrays of silver nanodisks,” Nanotechnology 2013, 838191 (2013).

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110, 106801 (2013).
[Crossref] [PubMed]

A. I. Väkeväinen, R. J. Moerland, H. T. Rekola, A.-P. Eskelinen, J.-P. Martikainen, D.-H. Kim, and P. Törmä, “Plasmonic surface lattice resonances at the strong coupling regime,” Nano Lett. 14(4), 1721–1727 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (4)

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano 5, 5151–5157 (2011).
[Crossref] [PubMed]

L. Novotny and N. van Hulst, “Antennas for light,” Nature Photon. 5, 83–90 (2011).
[Crossref]

R. Kullock, W. R. Hendren, A. Hille, S. Grafström, P. R. Evans, R. J. Pollard, R. Atkinson, and L. M. Eng, “Polarization conversion through collective surface plasmons in metallic nanorod arrays,” Opt. Express 16, 21672–21681 (2011).

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

2009 (3)

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102, 123904 (2009).
[Crossref] [PubMed]

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]

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]

2008 (5)

Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. 93, 181108 (2008).
[Crossref]

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

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]

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photon. Rev. 2, 136–159 (2008).
[Crossref]

Y.-R. Zhen, K. H. Fung, and C. T. Chan, “Collective plasmonic modes in two-dimensional periodic arrays of metal nanoparticles,” Phys. Rev. B 78, 035419 (2008).
[Crossref]

2006 (1)

S. Eustis and M. A. El-Sayed, “Why gold nanoparticles are more precious than pretty gold: noble metal surface plasmon resonance and its enhancement of the radiative and nonradiative properties of nanocrystals of different shapes,” Chem. Soc. Rev. 35, 209–217 (2006).
[Crossref] [PubMed]

2005 (3)

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[Crossref] [PubMed]

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett. 5, 1065–1070 (2005).
[Crossref] [PubMed]

W. M. Saj, “FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice,” Opt. Express 13, 4818–4827 (2005).
[Crossref] [PubMed]

2004 (2)

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]

2003 (1)

C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, “Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays,” J. Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

1997 (1)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

1994 (1)

1993 (1)

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40(11), 2281–2291 (1993).
[Crossref]

1991 (1)

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

1986 (1)

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Aizpurua, J.

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photon. Rev. 2, 136–159 (2008).
[Crossref]

Andriano, D.

Atkinson, R.

R. Kullock, W. R. Hendren, A. Hille, S. Grafström, P. R. Evans, R. J. Pollard, R. Atkinson, and L. M. Eng, “Polarization conversion through collective surface plasmons in metallic nanorod arrays,” Opt. Express 16, 21672–21681 (2011).

Auguié, B.

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

Barnes, W. L.

P. Törmä and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. 78, 013901 (2015).
[Crossref]

A. D. Humphrey and W. L. Barnes, “Plasmonic surface lattice resonances on arrays of different lattice symmetry,” Phys. Rev. B 90, 075404 (2014).
[Crossref]

G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110, 106801 (2013).
[Crossref] [PubMed]

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

Bautista, G.

Berger, V.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Boyd, R. W.

Briggs, D. P.

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
[Crossref] [PubMed]

Brongersma, S. H.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano 5, 5151–5157 (2011).
[Crossref] [PubMed]

Bryant, G.

M. Pelton, J. Aizpurua, and G. Bryant, “Metal-nanoparticle plasmonics,” Laser Photon. Rev. 2, 136–159 (2008).
[Crossref]

Buchholz, D. B.

S.-Q. Li, W. Zhou, D. B. Buchholz, J. B. Ketterson, L. E. Ocola, K. Sakoda, and R. P. H. Chang, “Ultra-sharp plasmonic resonances from monopole optical nanoantenna phased arrays,” Appl. Phys. Lett. 104, 231101 (2014).
[Crossref]

Carron, K. T.

Chan, C. T.

D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102, 123904 (2009).
[Crossref] [PubMed]

Y.-R. Zhen, K. H. Fung, and C. T. Chan, “Collective plasmonic modes in two-dimensional periodic arrays of metal nanoparticles,” Phys. Rev. B 78, 035419 (2008).
[Crossref]

Chang, R. P. H.

S.-Q. Li, W. Zhou, D. B. Buchholz, J. B. Ketterson, L. E. Ocola, K. Sakoda, and R. P. H. Chang, “Ultra-sharp plasmonic resonances from monopole optical nanoantenna phased arrays,” Appl. Phys. Lett. 104, 231101 (2014).
[Crossref]

Chu, Y.

Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. 93, 181108 (2008).
[Crossref]

Costard, E.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Crego-Calama, M.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano 5, 5151–5157 (2011).
[Crossref] [PubMed]

Crozier, K. B.

Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. 93, 181108 (2008).
[Crossref]

Dolgaleva, K.

Draine, B. T.

Eisler, H.-J.

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[Crossref] [PubMed]

El-Sayed, M. A.

S. Eustis and M. A. El-Sayed, “Why gold nanoparticles are more precious than pretty gold: noble metal surface plasmon resonance and its enhancement of the radiative and nonradiative properties of nanocrystals of different shapes,” Chem. Soc. Rev. 35, 209–217 (2006).
[Crossref] [PubMed]

Eng, L. M.

R. Kullock, W. R. Hendren, A. Hille, S. Grafström, P. R. Evans, R. J. Pollard, R. Atkinson, and L. M. Eng, “Polarization conversion through collective surface plasmons in metallic nanorod arrays,” Opt. Express 16, 21672–21681 (2011).

Eskelinen, A.-P.

A. I. Väkeväinen, R. J. Moerland, H. T. Rekola, A.-P. Eskelinen, J.-P. Martikainen, D.-H. Kim, and P. Törmä, “Plasmonic surface lattice resonances at the strong coupling regime,” Nano Lett. 14(4), 1721–1727 (2013).
[Crossref] [PubMed]

Eustis, S.

S. Eustis and M. A. El-Sayed, “Why gold nanoparticles are more precious than pretty gold: noble metal surface plasmon resonance and its enhancement of the radiative and nonradiative properties of nanocrystals of different shapes,” Chem. Soc. Rev. 35, 209–217 (2006).
[Crossref] [PubMed]

Evans, P. R.

R. Kullock, W. R. Hendren, A. Hille, S. Grafström, P. R. Evans, R. J. Pollard, R. Atkinson, and L. M. Eng, “Polarization conversion through collective surface plasmons in metallic nanorod arrays,” Opt. Express 16, 21672–21681 (2011).

Flatau, P. J.

Fluhr, W.

Fung, K. H.

Y.-R. Zhen, K. H. Fung, and C. T. Chan, “Collective plasmonic modes in two-dimensional periodic arrays of metal nanoparticles,” Phys. Rev. B 78, 035419 (2008).
[Crossref]

Gauthier-Lafaye, O.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps and holography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Giannini, V.

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]

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]

Gómez Rivas, J.

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano 5, 5151–5157 (2011).
[Crossref] [PubMed]

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]

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]

Goodman, J. J.

Grafström, S.

R. Kullock, W. R. Hendren, A. Hille, S. Grafström, P. R. Evans, R. J. Pollard, R. Atkinson, and L. M. Eng, “Polarization conversion through collective surface plasmons in metallic nanorod arrays,” Opt. Express 16, 21672–21681 (2011).

Grigorenko, A. N.

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]

Gunnarsson, L.

E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett. 5, 1065–1070 (2005).
[Crossref] [PubMed]

C. L. Haynes, A. D. McFarland, L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikulis, B. Kasemo, and M. Käll, “Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays,” J. Phys. Chem. B 107, 7337–7342 (2003).
[Crossref]

Hakala, T. K.

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G. Weick, C. Woollacott, W. L. Barnes, O. Hess, and E. Mariani, “Dirac-like plasmons in honeycomb lattices of metallic nanoparticles,” Phys. Rev. Lett. 110, 106801 (2013).
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Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. 93, 181108 (2008).
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[Crossref]

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D. Han, Y. Lai, J. Zi, Z.-Q. Zhang, and C. T. Chan, “Dirac spectra and edge states in honeycomb plasmonic lattices,” Phys. Rev. Lett. 102, 123904 (2009).
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E. M. Hicks, S. Zou, G. C. Schatz, K. G. Spears, R. P. Van Duyne, L. Gunnarsson, T. Rindzevicius, B. Kasemo, and M. Käll, “Controlling plasmon line shapes through diffractive coupling in linear arrays of cylindrical nanoparticles fabricated by electron beam lithography,” Nano Lett. 5, 1065–1070 (2005).
[Crossref] [PubMed]

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ACS Nano (1)

P. Offermans, M. C. Schaafsma, S. R. K. Rodriguez, Y. Zhang, M. Crego-Calama, S. H. Brongersma, and J. Gómez Rivas, “Universal scaling of the figure of merit of plasmonic sensors,” ACS Nano 5, 5151–5157 (2011).
[Crossref] [PubMed]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (2)

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram showing the dominant nearest neighbors for in-plane [(a) and (b)] and out-of-plane [(c) and (d)] dipole oscillations in square [(a) and (c)] and hexagonal [(b) and (d)] lattices. The light grey shaded region in each panel shows the scattering pattern of the nanoparticle at the center of the array [shown as a black arrow in (a) and (b) and a black circled dot in (c) and (d)]. Red arrows [(a) and (b)] and red circled dots [(c) and (d)] indicate the most strongly coupled nearest neighbors, and solid lines show the directions in which inter-particle coupling is strongest. Grey arrows [(a) and (b)] and grey circled dots [(c) and (d)] show neighbors for which the coupling is weak. For in-plane oscillations [(a) and (b)], strong coupling occurs only in one specific direction (indicated by solid lines), whereas for out-of-plane oscillations [(c) and (d)], strong coupling occurs in all directions within the plane of the array (as shown by solid lines). The insets schematically show the relevant polarizability components for in-plane (αx) and out-of-plane (αz) oscillations.

Fig. 2
Fig. 2

Extinction cross-section as a function of the incident photon energy and the inverse array periodicity q = 2π/a, plotted for (a) in-plane square, (b) in-plane hexagonal, (c) out-of-plane square, and (d) out-of-plane hexagonal arrays. Dark-red filled circles follow the local maxima of the extinction and display the dispersion relation of the resonances. For the out-of-plane resonances, the stronger inter-particle coupling results in larger avoided crossings between the modes, visualized by red arrows. The special cases treated in Fig. 3 are indicated by colored dashed lines.

Fig. 3
Fig. 3

The extinction spectra for the array periods of a = 430 nm, 500 nm, 435 nm, and 505 nm, calculated using the lattice sum approach (a) and DDA (b). The slightly different periods were chosen to separate the peaks on the graph for better clarity. The calculated Q-factors of the SLRs are shown next to their corresponding peaks in (a) and (b). These results can be understood in terms of the lattice sums, shown in (c). Strong SLRs occur when the real part of the lattice sum Si and 1/α (shown with the dotted line) cross. Therefore, larger lattice sums can give rise to stronger SLRs. The enhancement of the mesoscopic local field acting on an individual nanoparticle by the array and its resonances is shown in (d).

Fig. 4
Fig. 4

More efficient coupling schemes for (a) square and (b) hexagonal arrays with out-of-plane oscillations. (a) If the incident fields are out-of-phase, the in-plane (out-of-plane) field component of the incident waves interfere destructively (constructively), thus making the coupling into out-of-plane oscillations dominant. (b) Similar to square lattices, hexagonal lattices can be efficiently excited using three incident fields.

Equations (12)

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E inc , j = E 0 exp ( i k r j i ω t ) ,
E j = E inc , j k j A j k p k ,
A j k = exp ( i k r j k ) r j k [ k 2 ( r ^ j k r ^ j k I 3 ) 1 i k r j k r j k 2 ( 3 r ^ j k r ^ j k I 3 ) ] , j k ,
E inc , j = A j j p j + k j A j k p k ,
E inc , j = k = 1 N A j k p k ,
α = A 0 ( ω res ω ) + i γ ,
σ ext = 4 π k | E 0 | 2 j = 1 N Im ( E inc , j * p j ) .
E inc , i = ( 1 / α i S ) p i ,
α i * = 1 1 / α i S .
S in = j = 1 N exp ( i k r j ) r j [ k 2 sin 2 ϑ j + ( 1 i k r j ) ( 3 cos 2 ϑ j 1 ) r j 2 ] ,
S out = j = 1 N exp ( i k r j ) r j [ k 2 1 i k r j r j 2 ] .
σ ext = 4 π k N i Im ( α i * ) ,

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