Abstract

We study the scattering of surface plasmons from sub-wavelength holes and find that it exhibits a stronger wavelength dependence than the traditional λ−4 scaling found for Rayleigh scattering of light from small particles. This experimental observation is consistent with recent theoretical work and linked to the two-dimensional nature of the surface plasmon and the wavelength dependence of its spatial extent in the third dimension. The scattering cross sections are obtained with a frequency-correlation technique, which compares intensity speckle patterns observed behind various random structures of holes and recorded at different wavelengths. This powerful technique even allows us to distinguish between scattering of surface plasmons into photons and scattering into other surface plasmons.

© 2014 Optical Society of America

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2013 (1)

N. Rotenberg, T. L. Krijger, B. L. Feber, M. Spasenović, F. J. G. de Abajo, L. Kuipers, “Magnetic and electric response of single subwavelength holes,” Phys. Rev. B 88, 241408 (2013).
[CrossRef]

2012 (8)

A. G. Brolo, “Plasmonics for future biosensors,” Nature Photon. 6, 709–713 (2012).
[CrossRef]

K. Vynck, M. Burresi, F. Riboli, D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nature Mater. 11, 1017–1022 (2012).

J. M. Yi, A. Cuche, de León Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. M. Moreno, T. W. Ebbesen, “Diffraction Regimes of Single Holes,” Phys. Rev. Lett. 109, 023901 (2012).
[CrossRef] [PubMed]

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

N. Rotenberg, M. Spasenović, T. L. Krijger, B. L. Feber, F. J. G. de Abajo, L. Kuipers, “Plasmon Scattering from Single Subwavelength Holes,” Phys. Rev. Lett. 108, 127402 (2012).
[CrossRef] [PubMed]

F. van Beijnum, C. Retif, C. B. Smiet, H. Liu, P. Lalanne, M. P. van Exter, “Quasi-cylindrical wave contribution in experiments on extraordinary optical transmission,” Nature 492, 411–414 (2012).
[CrossRef] [PubMed]

F. van Beijnum, J. Sirre, C. Rétif, M. P. van Exter, “Speckle correlation functions applied to surface plasmons,” Phys. Rev. B 85, 035437 (2012).
[CrossRef]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[CrossRef] [PubMed]

2011 (1)

2010 (3)

A. Y. Nikitin, F. J. García-Vidal, L. Martín-Moreno, “Surface Electromagnetic Field Radiated by a Subwavelength Hole in a Metal Film,” Phys. Rev. Lett. 105, 073902 (2010).
[CrossRef] [PubMed]

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, P. Lodahl, “Cavity Quantum Electrodynamics with Anderson-Localized Modes,” Science 327, 1352–1355 (2010).
[CrossRef] [PubMed]

F. J. García-Vidal, L. Martín-Moreno, T. W. Ebbesen, L. Kuipers, “Light passing through sub-wavelength apertures,” Rev. of Mod. Phys. 82, 729–787 (2010).
[CrossRef]

2009 (5)

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nature Mater. 8, 568–571 (2009).
[CrossRef]

A. Y. Nikitin, S. G. Rodrigo, F. J. García-Vidal, L. Martín-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New Journal of Physics 11, 123020 (2009).
[CrossRef]

W. Dai, C. M. Soukoulis, “Theoretical analysis of the surface wave along a metal-dielectric interface,” Phys. Rev. B 80, 155407 (2009).
[CrossRef]

J. Li, H. Iu, D. Y. Lei, J. T. K. Wan, J. B. Xu, H. P. Ho, M. Y. Waye, H. C. Ong, “Dependence of surface plasmon lifetimes on the hole size in two-dimensional metallic arrays,” Appl. Phys. Lett. 94, 183112 (2009).
[CrossRef]

S. Faez, P. M. Johnson, A. Lagendijk, “Varying the Effective Refractive Index to Measure Optical Transport in Random Media,” Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef] [PubMed]

2008 (1)

H. T. Liu, P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[CrossRef] [PubMed]

2007 (2)

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

J. W. Lee, M. A. Seo, D. H. Kang, K. S. Khim, S. C. Jeoung, D. S. Kim, “Terahertz Electromagnetic Wave Transmission through Random Arrays of Single Rectangular Holes and Slits in Thin Metallic Sheets,” Phys. Rev. Lett. 99, 137401 (2007).
[CrossRef] [PubMed]

2006 (1)

P. Lalanne, J. P. Hugonin, “Interaction between optical nano-objects at metallo-dielectric interfaces,” Nature Phys. 2, 551–556 (2006).
[CrossRef]

2005 (3)

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, L. Kuipers, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory,” Phys. Rev. B 72, 045421 (2005).
[CrossRef]

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength Focusing and Guiding of Surface Plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

2003 (3)

D. S. Kim, S. C. Hohng, V. Malyarchuk, Y. C. Yoon, Y. H. Ahn, K. J. Yee, J. W. Park, J. Kim, Q. H. Park, C. Lienau, “Microscopic Origin of Surface-Plasmon Radiation in Plasmonic Band-Gap Nanostructures,” Phys. Rev. Lett. 91, 143901 (2003).
[CrossRef] [PubMed]

R. E. Thomas, G. A. Haas, “Diffusion measurements in thin films utilizing work function changes: Cr into au,” J. of Appl. Phys. 43, 4900–4907 (2003).
[CrossRef]

M. Rost, D. Quist, J. Frenken, “Grains, growth, and grooving,” Phys. Rev. Lett. 91, 026101(2003).
[CrossRef]

1998 (1)

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

1997 (1)

A. V. Shchegrov, I. V. Novikov, A. A. Maradudin, “Scattering of Surface Plasmon Polaritons by a Circularly Symmetric Surface Defect,” Phys. Rev. Lett. 78, 4269–4272 (1997).
[CrossRef]

1994 (1)

R. Berkovits, S. Feng, “Correlations in coherent multiple scattering,” Phys. Reports 238, 135–172 (1994).
[CrossRef]

1991 (1)

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

1988 (2)

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and Fluctuations of Coherent Wave Transmission through Disordered Media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, S. Feng, “Memory Effects in Propagation of Optical Waves through Disordered Media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

1972 (1)

P. B. Johnson, R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1944 (1)

H. A. Bethe, “Theory of Diffraction by Small Holes,” Phys. Rev. Online Archive (Prola) 66, 163–182 (1944).

1871 (1)

J. W. Strutt, “XV. On the light from the sky, its polarization and colour,” Phil. Mag. Series 4 41, 107–120 (1871).

Agrawal, A.

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

Ahn, Y. H.

D. S. Kim, S. C. Hohng, V. Malyarchuk, Y. C. Yoon, Y. H. Ahn, K. J. Yee, J. W. Park, J. Kim, Q. H. Park, C. Lienau, “Microscopic Origin of Surface-Plasmon Radiation in Plasmonic Band-Gap Nanostructures,” Phys. Rev. Lett. 91, 143901 (2003).
[CrossRef] [PubMed]

Alegret, J.

J. M. Yi, A. Cuche, de León Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. M. Moreno, T. W. Ebbesen, “Diffraction Regimes of Single Holes,” Phys. Rev. Lett. 109, 023901 (2012).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, X. Zhang, “An optical cloak made of dielectrics,” Nature Mater. 8, 568–571 (2009).
[CrossRef]

Berkovits, R.

R. Berkovits, S. Feng, “Correlations in coherent multiple scattering,” Phys. Reports 238, 135–172 (1994).
[CrossRef]

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[CrossRef] [PubMed]

Bethe, H. A.

H. A. Bethe, “Theory of Diffraction by Small Holes,” Phys. Rev. Online Archive (Prola) 66, 163–182 (1944).

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[CrossRef] [PubMed]

Brolo, A. G.

A. G. Brolo, “Plasmonics for future biosensors,” Nature Photon. 6, 709–713 (2012).
[CrossRef]

Brown, D. E.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, C. W. Kimball, “Subwavelength Focusing and Guiding of Surface Plasmons,” Nano Lett. 5, 1399–1402 (2005).
[CrossRef] [PubMed]

Brueck, S. R. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Burresi, M.

K. Vynck, M. Burresi, F. Riboli, D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nature Mater. 11, 1017–1022 (2012).

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Cuche, A.

J. M. Yi, A. Cuche, de León Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. M. Moreno, T. W. Ebbesen, “Diffraction Regimes of Single Holes,” Phys. Rev. Lett. 109, 023901 (2012).
[CrossRef] [PubMed]

Dai, W.

W. Dai, C. M. Soukoulis, “Theoretical analysis of the surface wave along a metal-dielectric interface,” Phys. Rev. B 80, 155407 (2009).
[CrossRef]

de Abajo, F. J. G.

N. Rotenberg, T. L. Krijger, B. L. Feber, M. Spasenović, F. J. G. de Abajo, L. Kuipers, “Magnetic and electric response of single subwavelength holes,” Phys. Rev. B 88, 241408 (2013).
[CrossRef]

N. Rotenberg, M. Spasenović, T. L. Krijger, B. L. Feber, F. J. G. de Abajo, L. Kuipers, “Plasmon Scattering from Single Subwavelength Holes,” Phys. Rev. Lett. 108, 127402 (2012).
[CrossRef] [PubMed]

de León Pérez,

J. M. Yi, A. Cuche, de León Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. M. Moreno, T. W. Ebbesen, “Diffraction Regimes of Single Holes,” Phys. Rev. Lett. 109, 023901 (2012).
[CrossRef] [PubMed]

Degiron, A.

J. M. Yi, A. Cuche, de León Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. M. Moreno, T. W. Ebbesen, “Diffraction Regimes of Single Holes,” Phys. Rev. Lett. 109, 023901 (2012).
[CrossRef] [PubMed]

Devaux, E.

J. M. Yi, A. Cuche, de León Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. M. Moreno, T. W. Ebbesen, “Diffraction Regimes of Single Holes,” Phys. Rev. Lett. 109, 023901 (2012).
[CrossRef] [PubMed]

Ebbesen, T. W.

J. M. Yi, A. Cuche, de León Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. M. Moreno, T. W. Ebbesen, “Diffraction Regimes of Single Holes,” Phys. Rev. Lett. 109, 023901 (2012).
[CrossRef] [PubMed]

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

F. J. García-Vidal, L. Martín-Moreno, T. W. Ebbesen, L. Kuipers, “Light passing through sub-wavelength apertures,” Rev. of Mod. Phys. 82, 729–787 (2010).
[CrossRef]

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

Enoch, S.

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, L. Kuipers, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: Experiment and theory,” Phys. Rev. B 72, 045421 (2005).
[CrossRef]

Faez, S.

S. Faez, P. M. Johnson, A. Lagendijk, “Varying the Effective Refractive Index to Measure Optical Transport in Random Media,” Phys. Rev. Lett. 103, 053903 (2009).
[CrossRef] [PubMed]

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Feber, B. L.

N. Rotenberg, T. L. Krijger, B. L. Feber, M. Spasenović, F. J. G. de Abajo, L. Kuipers, “Magnetic and electric response of single subwavelength holes,” Phys. Rev. B 88, 241408 (2013).
[CrossRef]

N. Rotenberg, M. Spasenović, T. L. Krijger, B. L. Feber, F. J. G. de Abajo, L. Kuipers, “Plasmon Scattering from Single Subwavelength Holes,” Phys. Rev. Lett. 108, 127402 (2012).
[CrossRef] [PubMed]

Feng, S.

R. Berkovits, S. Feng, “Correlations in coherent multiple scattering,” Phys. Reports 238, 135–172 (1994).
[CrossRef]

S. Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and Fluctuations of Coherent Wave Transmission through Disordered Media,” Phys. Rev. Lett. 61, 834–837 (1988).
[CrossRef] [PubMed]

I. Freund, M. Rosenbluh, S. Feng, “Memory Effects in Propagation of Optical Waves through Disordered Media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Frenken, J.

M. Rost, D. Quist, J. Frenken, “Grains, growth, and grooving,” Phys. Rev. Lett. 91, 026101(2003).
[CrossRef]

Freund, I.

I. Freund, M. Rosenbluh, S. Feng, “Memory Effects in Propagation of Optical Waves through Disordered Media,” Phys. Rev. Lett. 61, 2328–2331 (1988).
[CrossRef] [PubMed]

Garcia, P. D.

L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, P. Lodahl, “Cavity Quantum Electrodynamics with Anderson-Localized Modes,” Science 327, 1352–1355 (2010).
[CrossRef] [PubMed]

García-Vidal, F. J.

F. J. García-Vidal, L. Martín-Moreno, T. W. Ebbesen, L. Kuipers, “Light passing through sub-wavelength apertures,” Rev. of Mod. Phys. 82, 729–787 (2010).
[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

H. T. Liu, P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
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K. Vynck, M. Burresi, F. Riboli, D. S. Wiersma, “Photon management in two-dimensional disordered media,” Nature Mater. 11, 1017–1022 (2012).

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Figures (7)

Fig. 1
Fig. 1

(a–c) Our experiments probe three scattering processes: (a) Coupling of a surface plasmon to free space via scattering at a single hole; (b) Surface-plasmon-mediated transmission, where a surface plasmon is first excited at one hole and then transmitted at another hole; (c) Direct transmission through a sub-wavelength hole. (d) Random patterns of sub-wavelength holes are illuminated by a spectrally filtered supercontinuum laser source, of which we scan the wavelength. The change of the speckle pattern as a function of wavelength difference Δλ is quantified by calculating the correlation Cλ).

Fig. 2
Fig. 2

The measured correlation functions Cλ) have a wavelength-dependent contribution, caused by surface plasmons propagating on the gold-glass interface, and a wavelength-independent contribution resulting from light that is directly transmitted through the holes. The correlation functions depend strongly on hole density: the width increases with density while the background decreases. For the clarity of the figure, the plots for ρ = 1.6 μm2 and ρ = 2.5 μm2 are offset by −0.1 and −0.2 respectively.

Fig. 3
Fig. 3

(a, left) The inverse propagation length L tot 1 as a function of density for three different wavelength ranges. Both the axis cutoff, i.e. the absorption, and the slope decrease with wavelength. (b, right) The density dependence of the intensity ratio 〈Is〉/〈Id〉. For each density the intensity ratio decreases with wavelength. In the low density regime the intensity ratio increases linearly.

Fig. 4
Fig. 4

Inverse absorption length L abs 1 as a function of wavelength, as extracted from our experiments. The obtained absorption length is in good agreement with theory, showing both the validity of our experiment and the quality of the gold layer. The error in wavelength corresponds to the characteristic spectral width of the correlation function of the low density samples, and is therefore smaller for larger wavelengths

Fig. 5
Fig. 5

(a, left) The scattering cross section σ, which describes the radiative loss of a surface plasmon at a single hole, decreases almost a factor 10 in the measured wavelength range. (b, right) The intensity-ratio cross section A quantifies the transmission of light via a surface plasmon, by excitation at one hole and scattering and transmission at another hole. This parameter also decreases almost a factor 10 in the measured wavelength range and is comparable in magnitude to the scattering cross section σ. The error in wavelength corresponds to the spectral width of the correlation functions for high density samples

Fig. 6
Fig. 6

(a, left) The measured values of the scattering cross section σ for circular and square holes. For both types, the predicted wavelength dependence reproduces the data accurately. The pre-factor for the round holes is smaller however. (b, right) The measured values of the intensity-ratio cross section A for round and square holes. Also for this parameter the predicted wavelength dependence describes the data of both round and square holes.

Fig. 7
Fig. 7

Sketch of the power flow in our sample. An incident plane wave with power Pin induces a dipole moment. This dipole radiates into three channels: through the hole (P′1, not shown), into the substrate (Pd) and to a surface plasmon mode Pspp. The surface plasmon field is then either absorbed or scattered as photons, into the substrate (power P2) or into the waveguide (power P1).

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

C ( Δ λ ) = 1 I d + I s 2 | I d + I s 1 i L tot Re [ Δ k spp ] | 2 .
L tot 1 ( ρ ) = L abs 1 + ρ σ ,
I s I d ( ρ ) = A ρ ρ σ + L abs 1 ,
σ = ξ k 4 a 6 d s p p
σ = P out P / L 32 π k 0 5 3 | ε | ( | α E | 2 + | α M | 2 )
σ s p p = P s p p P / L 8 π 2 k 0 5 | ε | ( 2 | α E | 2 + | α M | 2 )
σ s p p σ = ( λ / d s p p 2 ) π k 0 4 ( 2 | α E | 2 + | α M | 2 ) ( 1 / d s p p ) ( 16 / 3 ) π k 0 4 ( | α E | 2 + | α M | 2 ) = 3 λ 16 d s p p 2 | α E | 2 + | α M | 2 | α E | 2 + | α M | 2 ,
I s I d P 2 P d = P s p p P d σ ρ L a b s 1 + σ ρ P 2 P 1 + P 2 = η σ s p p ρ L a b s 1 + σ ρ ,
A σ = η σ s p p σ = η 3 λ 0 / n 16 d s p p ,
C = I d 2 I d + I s 2 | 1 + A ρ σ ρ + L abs 1 i Re [ Δ k spp ] | 2 ,

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