Abstract

We present an explicit form of the surface plasmon propagator. Its form has the structure of a vectorial Huygens-Fresnel principle. The propagator appears to be a powerful tool to deal with diffraction, interference and focusing of surface plasmons. In contrast with the scalar approximation used so far, the vectorial propagator accounts for near-field and polarization effects. We illustrate the potential of the propagator by studying diffraction of surface plasmons by a slit and focusing of surface plasmons by a Fresnel lens.

© 2009 Optical Society of America

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2009 (1)

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

2008 (1)

A. G. Curto and F. J. Garc’?a de Abajo, "Near-Field Optical Phase Antennas for Long-Range Plasmon Coupling," Nano Lett. 8, 2479-2484 (2008).
[CrossRef] [PubMed]

2007 (5)

2006 (1)

E. Ozbay, "Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions," Science 311, 189-193 (2006).
[CrossRef] [PubMed]

2005 (3)

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing Surface Plasmons with a Plasmonic Lens," Nano Lett. 5, 1726-1729 (2005).
[CrossRef] [PubMed]

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

A. V. Zayats, I. I. Smolyaninov, A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports 408, 131-314 (2005)
[CrossRef]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

2002 (1)

2000 (1)

J. A. Porto, R. Carminati, and J.-J. Greffet, "Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy," J. Appl. Phys. 88, 4845-4850 (2000).
[CrossRef]

1997 (2)

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, "Experimental study of surface-plasmon scattering by individual surface defects," Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

J. J. Greffet and R. Carminati, "Image formation in near-field optics," Progress in Surface Science 56, 133-237, (1997).
[CrossRef]

1982 (1)

M. P. Givens, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 41, 145-148 (1982).
[CrossRef]

1981 (1)

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Agrawal, A.

Aussenegg, F. R.

Barnard, E. S.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

A. B. Evlyukhin, S. I. Bozhevolnyi, A. L. Stepanov, R. Kiyan, C. Reinhardt, S. Passinger, and B. N. Chichkov, "Focusing and directing of surface plasmon polaritons by curved chains of nanoparticles," Opt. Express 15, 667-680 (2007).
[CrossRef]

Brongersma, M. L.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

R. Zia and M. L. Brongersma, "Surface plasmon polariton analogue to Youngs double-slit experiment," Nature Nanotechnol. 2, 426-429 (2007).
[CrossRef]

Brown, D. E.

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

Carminati, R.

J. N. Walford, J. A. Porto, R. Carminati, and J.-J. Greffet, "Theory of near-field magneto-optical imaging," J. Opt. Soc. Am. A 19, 572-583 (2002).
[CrossRef]

J. A. Porto, R. Carminati, and J.-J. Greffet, "Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy," J. Appl. Phys. 88, 4845-4850 (2000).
[CrossRef]

J. J. Greffet and R. Carminati, "Image formation in near-field optics," Progress in Surface Science 56, 133-237, (1997).
[CrossRef]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

Chichkov, B. N.

A. B. Evlyukhin, S. I. Bozhevolnyi, A. L. Stepanov, R. Kiyan, C. Reinhardt, S. Passinger, and B. N. Chichkov, "Focusing and directing of surface plasmon polaritons by curved chains of nanoparticles," Opt. Express 15, 667-680 (2007).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Curto, A. G.

A. G. Curto and F. J. Garc’?a de Abajo, "Near-Field Optical Phase Antennas for Long-Range Plasmon Coupling," Nano Lett. 8, 2479-2484 (2008).
[CrossRef] [PubMed]

Davis, C. C.

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, "Experimental study of surface-plasmon scattering by individual surface defects," Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

Dereux, A.

Devaux, E.

Drezet, A.

Ebbesen, T. W.

Evlyukhin, A. B.

A. B. Evlyukhin, S. I. Bozhevolnyi, A. L. Stepanov, R. Kiyan, C. Reinhardt, S. Passinger, and B. N. Chichkov, "Focusing and directing of surface plasmon polaritons by curved chains of nanoparticles," Opt. Express 15, 667-680 (2007).
[CrossRef]

Fan, S.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

Garc’ia de Abajo, F. J.

A. G. Curto and F. J. Garc’?a de Abajo, "Near-Field Optical Phase Antennas for Long-Range Plasmon Coupling," Nano Lett. 8, 2479-2484 (2008).
[CrossRef] [PubMed]

Genet, C.

Givens, M. P.

M. P. Givens, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 41, 145-148 (1982).
[CrossRef]

Greffet, J. J.

J. J. Greffet and R. Carminati, "Image formation in near-field optics," Progress in Surface Science 56, 133-237, (1997).
[CrossRef]

Greffet, J.-J.

J. N. Walford, J. A. Porto, R. Carminati, and J.-J. Greffet, "Theory of near-field magneto-optical imaging," J. Opt. Soc. Am. A 19, 572-583 (2002).
[CrossRef]

J. A. Porto, R. Carminati, and J.-J. Greffet, "Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy," J. Appl. Phys. 88, 4845-4850 (2000).
[CrossRef]

Hiller, J. M.

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

Hohenau, A.

Hua, J.

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

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Kimball, C. W.

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

Kiyan, R.

A. B. Evlyukhin, S. I. Bozhevolnyi, A. L. Stepanov, R. Kiyan, C. Reinhardt, S. Passinger, and B. N. Chichkov, "Focusing and directing of surface plasmon polaritons by curved chains of nanoparticles," Opt. Express 15, 667-680 (2007).
[CrossRef]

Koller, D.

Krenn, J. R.

Laluet, J.-Y.

Leitner, A.

Li, Y.

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Liu, Z.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing Surface Plasmons with a Plasmonic Lens," Nano Lett. 5, 1726-1729 (2005).
[CrossRef] [PubMed]

Mait, J.

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, "Experimental study of surface-plasmon scattering by individual surface defects," Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports 408, 131-314 (2005)
[CrossRef]

Mazzoni, D. L.

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, "Experimental study of surface-plasmon scattering by individual surface defects," Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

Nahata, Ajay

Ozbay, E.

E. Ozbay, "Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions," Science 311, 189-193 (2006).
[CrossRef] [PubMed]

Passinger, S.

A. B. Evlyukhin, S. I. Bozhevolnyi, A. L. Stepanov, R. Kiyan, C. Reinhardt, S. Passinger, and B. N. Chichkov, "Focusing and directing of surface plasmon polaritons by curved chains of nanoparticles," Opt. Express 15, 667-680 (2007).
[CrossRef]

Pearson, J.

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

Pikus, Y.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing Surface Plasmons with a Plasmonic Lens," Nano Lett. 5, 1726-1729 (2005).
[CrossRef] [PubMed]

Porto, J. A.

J. N. Walford, J. A. Porto, R. Carminati, and J.-J. Greffet, "Theory of near-field magneto-optical imaging," J. Opt. Soc. Am. A 19, 572-583 (2002).
[CrossRef]

J. A. Porto, R. Carminati, and J.-J. Greffet, "Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy," J. Appl. Phys. 88, 4845-4850 (2000).
[CrossRef]

Reinhardt, C.

A. B. Evlyukhin, S. I. Bozhevolnyi, A. L. Stepanov, R. Kiyan, C. Reinhardt, S. Passinger, and B. N. Chichkov, "Focusing and directing of surface plasmon polaritons by curved chains of nanoparticles," Opt. Express 15, 667-680 (2007).
[CrossRef]

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports 408, 131-314 (2005)
[CrossRef]

I. I. Smolyaninov, D. L. Mazzoni, J. Mait, and C. C. Davis, "Experimental study of surface-plasmon scattering by individual surface defects," Phys. Rev. B 56, 1601-1611 (1997).
[CrossRef]

Srituravanich, W.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing Surface Plasmons with a Plasmonic Lens," Nano Lett. 5, 1726-1729 (2005).
[CrossRef] [PubMed]

Steele, J. M.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing Surface Plasmons with a Plasmonic Lens," Nano Lett. 5, 1726-1729 (2005).
[CrossRef] [PubMed]

Stepanov, A. L.

A. B. Evlyukhin, S. I. Bozhevolnyi, A. L. Stepanov, R. Kiyan, C. Reinhardt, S. Passinger, and B. N. Chichkov, "Focusing and directing of surface plasmon polaritons by curved chains of nanoparticles," Opt. Express 15, 667-680 (2007).
[CrossRef]

Sun, C.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing Surface Plasmons with a Plasmonic Lens," Nano Lett. 5, 1726-1729 (2005).
[CrossRef] [PubMed]

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

Vlasko-Vlasov, V. K.

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

Walford, J. N.

Weeber, J.-C.

Welp, U.

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

White, J. S.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

Wolf, E.

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211-215 (1981).
[CrossRef]

Yin, L.

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

Yu, Z.

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Reports 408, 131-314 (2005)
[CrossRef]

Zhang, X.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing Surface Plasmons with a Plasmonic Lens," Nano Lett. 5, 1726-1729 (2005).
[CrossRef] [PubMed]

Zhu, W.

Zia, R.

R. Zia and M. L. Brongersma, "Surface plasmon polariton analogue to Youngs double-slit experiment," Nature Nanotechnol. 2, 426-429 (2007).
[CrossRef]

J. Appl. Phys. (1)

J. A. Porto, R. Carminati, and J.-J. Greffet, "Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy," J. Appl. Phys. 88, 4845-4850 (2000).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nano Lett. (4)

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, "Focusing Surface Plasmons with a Plasmonic Lens," Nano Lett. 5, 1726-1729 (2005).
[CrossRef] [PubMed]

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

L. Verslegers, P. B. Catrysse, Z. Yu, J. S. White, E. S. Barnard, M. L. Brongersma, and S. Fan, "Planar Lenses Based on Nanoscale Slit Arrays in a Metallic Film," Nano Lett. 9, 235-238 (2009).
[CrossRef]

A. G. Curto and F. J. Garc’?a de Abajo, "Near-Field Optical Phase Antennas for Long-Range Plasmon Coupling," Nano Lett. 8, 2479-2484 (2008).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1.
Fig. 1.

Diffraction pattern of a slit with width w=6λsp illuminated from the left by a homogeneous surface plasmon at normal incidence. λsp =400 nm. The intensity is normalized by the intensity of the z-component of the incident surface plasmon field at x=0. Black is zero intensity. Yellow indicates the maximum intensity and corresponds to 0.25 in (a) 0.01 in (b), and 1.2 in (c).

Fig. 2.
Fig. 2.

The intensity of z-component of surface plasmon field normalized to the intensity of the z-component of the incident surface plasmon field calculated at y=0 using the general Huygens-Fresnel principle Eq. (3) (black curve) and the asymptotic solution Eq. (5) (red curve). The others parameters are the same as on Fig. 1.

Fig. 3.
Fig. 3.

(a) Schematically the metal surface and the diffraction of surface plasmons by a Fresnel lens. (b-e) The intensity of the diffracted surface plasmon field I=|ESP x |2+|ESP y |2+|ESP z |2 normalized to the intensity of the z-component of the incident surface plasmon field. The homogeneous surface plasmon propagating along the glass/gold interface with the wavelength (b) λsp =400 nm, (c) 375 nm (d) 350 nm (e) 325 nm impinges the lens from the left. The Fresnel lens has the focal distance f=3λsp and composed from non-periodical array of slits. The number of slits is N=2n+1, n=13. The inset in figure (b) shows a detail of focal spot. (f) the diffraction of surface plasmon λsp =325 nm by a single slit of the width w=2λsp . (g) the same as (e) with no losses. The color scale is different for each pattern and indicates as black the zero intensity and yellow the maximum intensity (b) 3.21, (c) 2.22, (d) 1.31, (e) 1.07, (f) 1.11 (g) 5.53.

Fig. 4.
Fig. 4.

(a) Normalized intensity of surface plasmon field diffracted by a Fresnel lens made of an array of N=2n+1 slits with a focal distance f=3λsp versus n. (b) Normalized intensity of the surface plasmon field diffracted by a Fresnel lens made of a nonperiodical array of N=2n+1 slits, n=13, as a function of the focal distance f. The calculation has been performed for the different surface plasmon wavelengths: λsp =435 nm (black circles (a), black curve (b)), 400 nm (red squares (a), red curve (b)) 375 nm (blue triangles (a) and blue curve (b)). The lines in figure (a) are for eyes guidance.

Equations (6)

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

kSP2=kx2+ky2=ω2c2ε1ε2(ω)ε1+ε2(ω),
ESP(x,y)= dky2π ESP (ky) eikSP2ky2x+ikyy .
ESP(x,y)= d y' EzSP (x=0,y')K(x,yy'),
K(x,yy')=[KxKyKz]=12[kzkSP22x2H0(1)(kSPρ)kzkSP22xyH0(1)(kSPρ)ixH0(1)(kSPρ)],
EzSP(x,y)=
=iλSPdy'cosθEzSP(x=0,y')eikSPρρ eiπ4 ,

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