Abstract

Rigorous electromagnetic theory is utilized to characterize the partial spatial coherence and partial polarization of a two-mode field consisting of the long-range and the short-range surface-plasmon polariton at a metallic nanofilm. By employing appropriate formulations for the spectral degrees of coherence and polarization, we examine the fundamental limits for these quantities associated with such a superposition field and explore how the degrees are influenced when the media, frequency, and slab thickness are varied. It is in particular shown that coherence lengths extending from subwavelength scales up to thousands of wavelengths are possible and their physical origins are elucidated. In addition, we demonstrate that for ultra-thin films the generally highly polarized two-mode field can be partially polarized in close vicinity of the polariton excitation region. The results could benefit cross-disciplinary electro-optical applications in which near-field interactions between plasmons and nanoparticles are exploited.

© 2015 Optical Society of America

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References

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2015 (1)

2014 (4)

A. Norrman, T. Setälä, and A. T. Friberg, “Surface-plasmon polariton solutions at a lossy slab in a symmetric surrounding,” Opt. Express 22, 4628–4648 (2014).
[Crossref] [PubMed]

A. Norrman, T. Setälä, and A. T. Friberg, “Long-range higher-order surface-plasmon polaritons,” Phys. Rev. A 90, 053849 (2014).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635 (2014).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

2013 (2)

A. Norrman, T. Setälä, and A. T. Friberg, “Exact surface-plasmon polariton solutions at a lossy interface,” Opt. Lett. 38, 1119–1121 (2013).
[Crossref] [PubMed]

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

2012 (3)

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

S. Aberra Guebrou, J. Laverdant, C. Symonds, S. Vignoli, and J. Bellessa, “Spatial coherence properties of surface plasmon investigated by Young’s slit experiment,” Opt. Lett. 37, 2139–2141 (2012).
[Crossref]

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

2011 (3)

2010 (2)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photon. 4, 83–91 (2010).
[Crossref]

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

2009 (2)

2008 (1)

K. Joulain and C. Henkel, “The near field correlation spectrum of a metallic film,” Appl. Phys. B 93, 151–158 (2008).
[Crossref]

2007 (2)

A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express 15, 183–197 (2007).
[Crossref] [PubMed]

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[Crossref]

2006 (3)

2005 (2)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

2004 (2)

2003 (2)

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[Crossref] [PubMed]

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

2002 (3)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[Crossref] [PubMed]

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

2000 (1)

A. V. Shchegrov, K. Joulain, R. Carminati, and J. J. Greffet, “Near-field spectral effects due to electromagnetic surface exitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[Crossref] [PubMed]

1999 (1)

R. Carminati and J. J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[Crossref]

1991 (1)

F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B 44, 5855–5872 (1991).
[Crossref]

1986 (1)

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

1981 (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[Crossref]

1971 (1)

E. Kretschmann, “Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflächen-plasmaschwingungen,” Z. Phys. 241, 313–324 (1971).
[Crossref]

1968 (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[Crossref]

1965 (1)

W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
[Crossref]

Aberra Guebrou, S.

S. Aberra Guebrou, J. Laverdant, C. Symonds, S. Vignoli, and J. Bellessa, “Spatial coherence properties of surface plasmon investigated by Young’s slit experiment,” Opt. Lett. 37, 2139–2141 (2012).
[Crossref]

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

Agranovich, V. M.

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

Baets, R.

Bai, W.

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

Barnes, W. L.

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

Bartoli, F. J.

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

Bellessa, J.

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

S. Aberra Guebrou, J. Laverdant, C. Symonds, S. Vignoli, and J. Bellessa, “Spatial coherence properties of surface plasmon investigated by Young’s slit experiment,” Opt. Lett. 37, 2139–2141 (2012).
[Crossref]

Berini, P.

Bienstman, P.

Bond, W. L.

W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
[Crossref]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photon. 4, 83–91 (2010).
[Crossref]

Bradberry, G. W.

F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B 44, 5855–5872 (1991).
[Crossref]

Brigo, L.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Brosseau, C.

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), vol. 49, pp. 315–380.
[Crossref]

Brusatin, G.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Carminati, R.

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

A. V. Shchegrov, K. Joulain, R. Carminati, and J. J. Greffet, “Near-field spectral effects due to electromagnetic surface exitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[Crossref] [PubMed]

R. Carminati and J. J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[Crossref]

Chen, Y.

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

Chulkov, E. V.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[Crossref]

Cittadini, M.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Collier, R. J.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Debackere, P.

Dereux, A.

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

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

Dogariu, A.

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), vol. 49, pp. 315–380.
[Crossref]

Ebbesen, T. W.

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

Echenique, P. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[Crossref]

Fan, B.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Friberg, A. T.

T. Voipio, T. Setälä, and A. T. Friberg, “Statistical similarity and complete coherence of electromagnetic fields in time and frequency domains,” J. Opt. Soc. Am. A 32, 741–750 (2015).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635 (2014).
[Crossref]

A. Norrman, T. Setälä, and A. T. Friberg, “Long-range higher-order surface-plasmon polaritons,” Phys. Rev. A 90, 053849 (2014).
[Crossref]

A. Norrman, T. Setälä, and A. T. Friberg, “Surface-plasmon polariton solutions at a lossy slab in a symmetric surrounding,” Opt. Express 22, 4628–4648 (2014).
[Crossref] [PubMed]

A. Norrman, T. Setälä, and A. T. Friberg, “Exact surface-plasmon polariton solutions at a lossy interface,” Opt. Lett. 38, 1119–1121 (2013).
[Crossref] [PubMed]

A. Norrman, T. Setälä, and A. T. Friberg, “Partial spatial coherence and partial polarization in random evanescent fields on lossless interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
[Crossref] [PubMed]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[Crossref] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[Crossref] [PubMed]

Gan, Q.

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

Gao, Y.

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

Gartstein, Yu. N.

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

Gazzola, E.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photon. 4, 83–91 (2010).
[Crossref]

Greffet, J. J.

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

A. V. Shchegrov, K. Joulain, R. Carminati, and J. J. Greffet, “Near-field spectral effects due to electromagnetic surface exitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[Crossref] [PubMed]

R. Carminati and J. J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[Crossref]

Guglielmi, M.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).
[Crossref]

Henkel, C.

K. Joulain and C. Henkel, “The near field correlation spectrum of a metallic film,” Appl. Phys. B 93, 151–158 (2008).
[Crossref]

Homeyer, E.

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

Homola, J.

R. Slavíc and J. Homola, “Simultaneous excitation of long and short range surface plasmons in an asymmetric structure,” Opt. Commun. 259, 507–512 (2006).
[Crossref]

Hu, S.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Huang, Y.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Jiang, S.

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

Joulain, K.

K. Joulain and C. Henkel, “The near field correlation spectrum of a metallic film,” Appl. Phys. B 93, 151–158 (2008).
[Crossref]

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

A. V. Shchegrov, K. Joulain, R. Carminati, and J. J. Greffet, “Near-field spectral effects due to electromagnetic surface exitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[Crossref] [PubMed]

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[Crossref] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Kretschmann, E.

E. Kretschmann, “Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflächen-plasmaschwingungen,” Z. Phys. 241, 313–324 (1971).
[Crossref]

Laverdant, J.

Leppänen, L.-P.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635 (2014).
[Crossref]

Levy, U.

Li, H.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Li, W.

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

Li, Y.

Y. Li, H. Zhang, N. Zhu, T. Mei, D. H. Zhang, and J. Teng, “Short-range surface plasmon propagation supported by stimulated amplification using electrical injection,” Opt. Express 19, 22107–22112 (2011).
[Crossref] [PubMed]

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Lin, L. H.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

Liu, F.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Mainguy, S.

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Mansuripur, M.

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

Martucci, A.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Mei, T.

Miura, Y.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Moloney, J. V.

Mulet, J. P.

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

Norrman, A.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).
[Crossref]

Ohnishi, D.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Otto, A.

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[Crossref]

Pitarke, J. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[Crossref]

Plenet, J. C.

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

Polman, A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

Romanato, F.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Saastamoinen, K.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Sambles, J. R.

F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B 44, 5855–5872 (1991).
[Crossref]

Sarid, D.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[Crossref]

Scheerlinck, S.

Setälä, T.

T. Voipio, T. Setälä, and A. T. Friberg, “Statistical similarity and complete coherence of electromagnetic fields in time and frequency domains,” J. Opt. Soc. Am. A 32, 741–750 (2015).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635 (2014).
[Crossref]

A. Norrman, T. Setälä, and A. T. Friberg, “Surface-plasmon polariton solutions at a lossy slab in a symmetric surrounding,” Opt. Express 22, 4628–4648 (2014).
[Crossref] [PubMed]

A. Norrman, T. Setälä, and A. T. Friberg, “Long-range higher-order surface-plasmon polaritons,” Phys. Rev. A 90, 053849 (2014).
[Crossref]

A. Norrman, T. Setälä, and A. T. Friberg, “Exact surface-plasmon polariton solutions at a lossy interface,” Opt. Lett. 38, 1119–1121 (2013).
[Crossref] [PubMed]

A. Norrman, T. Setälä, and A. T. Friberg, “Partial spatial coherence and partial polarization in random evanescent fields on lossless interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
[Crossref] [PubMed]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[Crossref]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[Crossref] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[Crossref] [PubMed]

Shchegrov, A. V.

A. V. Shchegrov, K. Joulain, R. Carminati, and J. J. Greffet, “Near-field spectral effects due to electromagnetic surface exitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[Crossref] [PubMed]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Silkin, V. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[Crossref]

Slavíc, R.

R. Slavíc and J. Homola, “Simultaneous excitation of long and short range surface plasmons in an asymmetric structure,” Opt. Commun. 259, 507–512 (2006).
[Crossref]

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Stockman, M. I.

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

Symonds, C.

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

S. Aberra Guebrou, J. Laverdant, C. Symonds, S. Vignoli, and J. Bellessa, “Spatial coherence properties of surface plasmon investigated by Young’s slit experiment,” Opt. Lett. 37, 2139–2141 (2012).
[Crossref]

Tamir, T.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Teng, J.

Tervo, J.

Vignoli, S.

Voipio, T.

Wan, R.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Wu, W.

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

Xia, Y.

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

Yanai, A.

Yang, F.

F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B 44, 5855–5872 (1991).
[Crossref]

Zacco, G.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Zakharian, A. R.

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

Zhang, D. H.

Zhang, H.

Zhu, N.

Zilio, P.

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Adv. Opt. Photon. (1)

Appl. Phys. B (1)

K. Joulain and C. Henkel, “The near field correlation spectrum of a metallic film,” Appl. Phys. B 93, 151–158 (2008).
[Crossref]

Appl. Phys. Lett. (1)

R. Wan, F. Liu, Y. Huang, S. Hu, B. Fan, Y. Miura, D. Ohnishi, Y. Li, H. Li, and Y. Xia, “Excitation of short range surface plasmon polariton mode based on integrated hybrid coupler,” Appl. Phys. Lett. 97, 141105 (2010).
[Crossref]

J. Appl. Phys. (1)

W. L. Bond, “Measurement of the refractive indices of several crystals,” J. Appl. Phys. 36, 1674–1677 (1965).
[Crossref]

J. Opt. Soc. Am. A (4)

Nanotechnology (1)

L. Brigo, E. Gazzola, M. Cittadini, P. Zilio, G. Zacco, F. Romanato, A. Martucci, M. Guglielmi, and G. Brusatin, “Short and long range surface plasmon polariton waveguides for xylene sensing,” Nanotechnology 24, 155502 (2013).
[Crossref] [PubMed]

Nature (2)

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416, 61–64 (2002).
[Crossref] [PubMed]

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

Nature Photon. (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nature Photon. 4, 83–91 (2010).
[Crossref]

New J. Phys. (1)

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Opt. Commun. (1)

R. Slavíc and J. Homola, “Simultaneous excitation of long and short range surface plasmons in an asymmetric structure,” Opt. Commun. 259, 507–512 (2006).
[Crossref]

Opt. Express (7)

Opt. Lett. (4)

Phys. Rep. (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

Phys. Rev. A (1)

A. Norrman, T. Setälä, and A. T. Friberg, “Long-range higher-order surface-plasmon polaritons,” Phys. Rev. A 90, 053849 (2014).
[Crossref]

Phys. Rev. B (3)

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B 44, 5855–5872 (1991).
[Crossref]

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[Crossref]

Phys. Rev. Lett. (5)

R. Carminati and J. J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[Crossref]

A. V. Shchegrov, K. Joulain, R. Carminati, and J. J. Greffet, “Near-field spectral effects due to electromagnetic surface exitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[Crossref] [PubMed]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[Crossref] [PubMed]

S. Aberra Guebrou, C. Symonds, E. Homeyer, J. C. Plenet, Yu. N. Gartstein, V. M. Agranovich, and J. Bellessa, “Coherent emission from a disordered organic semiconductor induced by strong coupling with surface plasmons,” Phys. Rev. Lett. 108, 066401 (2012).
[Crossref] [PubMed]

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[Crossref]

Plasmonics (1)

Q. Gan, W. Bai, S. Jiang, Y. Gao, W. Li, W. Wu, and F. J. Bartoli, “Short-range surface plasmon polaritons for extraordinary low transmission through ultra-thin metal films with nanopatterns,” Plasmonics 7, 47–52 (2012).
[Crossref]

Rep. Prog. Phys. (1)

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007).
[Crossref]

Z. Phys. (2)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[Crossref]

E. Kretschmann, “Die Bestimmung optischer Konstanten von Metallen durch Anregung von Oberflächen-plasmaschwingungen,” Z. Phys. 241, 313–324 (1971).
[Crossref]

Other (7)

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1998).

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2006), vol. 49, pp. 315–380.
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).
[Crossref]

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Figures (14)

Fig. 1
Fig. 1 Illustration of the directions of phase-front propagation (black arrows) and field attenuation (solid-red curves) for SPPs at a lossy metal film of thickness d in a non-absorptive dielectric surrounding, possessing relative permittivities εr1 (complex) and εr2 (real), respectively. The boundaries of the slab are at z = ±d/2.
Fig. 2
Fig. 2 Degree of coherence μr, ω) for the LRSPP-SRSPP superposition above a 50 nm (left) and 20 nm (right) thick Ag slab in vacuum, as a function of the longitudinal and transverse distances Δx/λ0 and Δz/λ0, respectively, at the free-space wavelength λ0 = 632.8 nm. The modes are mutually uncorrelated and have the same intensities at the excitation point Δx = Δz = 0. The relative permittivity of Ag is εr1 = −15.87 + i1.07 [46].
Fig. 3
Fig. 3 Density plots corresponding to the 3D graphs of μr, ω) in Fig. 2. The dashed and dotted lines illustrate the directions in which the degree of coherence is solely oscillating and solely decaying, determined by Δk″ and Δk′, respectively. The dotted lines are located at the positions of local coherence minima. The vectors are drawn with correct relative magnitudes in each panel (but not among the panels).
Fig. 4
Fig. 4 Degree of coherence μr, ω) for the LRSPP-SRSPP field along the x (left) and z (right) axes above Ag slabs of thickness d in vacuum at the free-space wavelength λ0 = 632.8 nm. The modes are mutually uncorrelated and have the same intensities at the excitation point Δx = Δz = 0. The relative permittivity of Ag is εr1 = −15.87 + i1.07 [46].
Fig. 5
Fig. 5 Maxima μmax and minima μmin of the degree of coherence for the LRSPP-SRSPP superposition as a function of the film thickness d for a Ag slab in vacuum at various free-space wavelengths λ0 (left) and in different surroundings at λ0 = 632.8 nm (right). The solid-blue, dashed-green, and dotted-red lines in the left (right) panel correspond to λ0 = 400 nm (vacuum), λ0 = 550 nm (SiO2), and λ0 = 700 nm (ZnO), respectively. The modes are mutually uncorrelated and have the same intensities at the excitation point. The relative permittivity for Ag is εr1 = −3.77 + i0.67 (λ0 = 400 nm), εr1 = −11.13 + i0.83 (λ0 = 550 nm), εr1 = −15.87+ i1.07 (λ0 = 632.8 nm), and εr1 = −20.44+ i1.29 (λ0 = 700 nm) [46], and those for SiO2 and ZnO are εr2 = 2.12 [46] and εr2 = 3.96 [47], respectively.
Fig. 6
Fig. 6 Local coherence length lcoh,x with respect to the free-space wavelength λ0 (left) and the surface propagation distance lx (right) for the LRSPP-SRSPP superposition along (Δz = 0) a Ag slab in vacuum, as a function of the film thickness d at selected wavelengths: λ0 = 400 nm (solid-blue lines), λ0 = 550 nm (dashed-green lines), and λ0 = 700 nm (dotted-red lines). The modes are mutually uncorrelated and have the same intensities at the excitation point. The insets demonstrate the behavior of lcoh,x at very thin films. The relative permittivity of Ag is εr1 = −3.77 + i0.67 (λ0 = 400 nm), εr1 = −11.13 + i0.83 (λ0 = 550 nm), and εr1 = −20.44 + i1.29 (λ0 = 700 nm) [46].
Fig. 7
Fig. 7 Local coherence length lcoh,x with respect to the wavelength in the surrounding medium λ2 (left) and the surface propagation distance lx (right) for the LRSPP-SRSPP field along (Δz = 0) a Ag slab as a function of the film thickness d at λ0 = 632.8 nm in different surroundings: vacuum (solid-blue lines), SiO2 (dashed-green lines), and ZnO (dotted-red lines). The modes are mutually uncorrelated and have the same intensities at the excitation point. The insets demonstrate the behavior of lcoh,x at very thin films. The relative permittivities of Ag, SiO2, and ZnO are εr1 = −15.87 + i1.07 [46], εr2 = 2.12 [46], and εr2 = 3.96 [47], respectively.
Fig. 8
Fig. 8 Global coherence length Lcoh,x with respect to the free-space wavelength λ0 (left) and the surface propagation length lx (right) for the fields of Fig. 6 as a function of the film thickness d. The labeling of the curves (λ0 values) is as in Fig. 6.
Fig. 9
Fig. 9 Global coherence length Lcoh,x with respect to the wavelength in the surrounding medium λ2 (left) and the surface propagation distance lx (right) for the fields of Fig. 7 as a function of the film thickness d. The labeling of the curves (εr2 values) is as in Fig. 7.
Fig. 10
Fig. 10 Global coherence length Lcoh,z with respect to the free-space wavelength λ0 (left) and the penetration depth lz (right) for the fields of Fig. 6 and 8 as a function of the film thickness d. The labeling of the curves (λ0 values) is as in Fig. 6.
Fig. 11
Fig. 11 Global coherence length Lcoh,z with respect to the wavelength in the surrounding medium λ2 (left) and the penetration depth lz (right) for the fields of Fig. 7 and 9 as a function of the film thickness d. The labeling of the curves (εr2 values) is as in Fig. 7.
Fig. 12
Fig. 12 Degree of polarization P(r, ω) for the LRSPP-SRSPP field above a 50 nm (left) and 20 nm (right) thick Ag slab in vacuum, as a function of the longitudinal and transverse distances x/λ0 and z/λ0, respectively, at the free-space wavelength λ0 = 632.8 nm. The modes are mutually uncorrelated and have the same intensities at the excitation point x = z = 0. The straight lines illustrate the directions in which P(r, ω) is constant, determined by Δk″ (the vectors are drawn with correct relative magnitudes). The relative permittivity of Ag is εr1 = −15.87 + i1.07 [46].
Fig. 13
Fig. 13 Degree of polarization P(r, ω) for the LRSPP-SRSPP field along the x (left) and z (right) axes above ultra-thin Ag slabs of thickness d in vacuum at the free-space wavelength λ0 = 632.8 nm. The modes are mutually uncorrelated and have the same intensities at the excitation point x = z = 0. The relative permittivity of Ag is εr1 = −15.87 + i1.07 [46].
Fig. 14
Fig. 14 Minimum Pmin of the degree of polarization for the LRSPP-SRSPP field above a Ag slab for different wavelengths in vacuum (left), and various surroundings at the free-space wavelength λ0 = 632.8 nm (right), as a function of the film thickness d. The solid-blue, dashed-green, and dotted-red lines in the left (right) panel correspond to λ0 = 400 nm (vacuum), λ0 = 550 nm (SiO2), and λ0 = 700 nm (ZnO), respectively. The modes are mutually uncorrelated and have the same intensities at the excitation point. The relative permittivity of Ag is εr1 = −3.77 + i0.67 (λ0 = 400 nm), εr1 = −11.13 + i0.83 (λ0 = 550 nm), εr1 = −15.87+ i1.07 (λ0 = 632.8 nm), and εr1 = −20.44+ i1.29 (λ0 = 700 nm) [46], and those of SiO2 and ZnO are εr2 = 2.12 [46] and εr2 = 3.96 [47], respectively.

Equations (29)

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E 2 ( ± ) ( r , ω ) = E 2 e i k x x | k 2 | { e + i k 2 z z ( k 2 z e ^ x + k x e ^ z ) , z d / 2 , e i k 2 z z ( ± k 2 z e ^ x ± k x e ^ z ) , z d / 2 ,
E 1 ( ± ) ( r , ω ) = E 1 e i ( k x x + k 1 z z ) | k 1 | [ k 1 z ( 1 e 2 i k 1 z z ) e ^ x + k x ( 1 ± e 2 i k 1 z z ) e ^ z ] , | z | < d / 2 ,
Symmetric ( + ) : ε r 1 ε r 2 k 2 z k 1 z = tanh ( 1 2 i k 1 z d ) ,
Antisymmetric ( ) : ε r 1 ε r 2 k 2 z k 1 z = coth ( 1 2 i k 1 z d ) ,
k x k 0 ε r z , k 1 z k 0 ε r 1 ε r 2 , k 2 z 0 ,
W ( r 1 , r 2 , ω ) = E ( r 1 , ω ) E ( r 2 , ω ) .
Φ ( r , ω ) W ( r , r , ω ) ,
μ ( r 1 , r 2 , ω ) W ( r 1 , r 2 , ω ) F tr Φ ( r 1 , ω ) tr Φ ( r 2 , ω ) ,
0 μ ( r 1 , r 2 , ω ) 1 ,
P ( r , ω ) 2 tr Φ 2 ( r , ω ) tr 2 Φ ( r , ω ) 1 ,
0 P ( r , ω ) 1 .
W ( r 1 , r 2 , ω ) = | E ( + ) | 2 e i [ k ( + ) r 2 k ( + ) * r 1 ] p ^ ( + ) p ^ ( + ) + | E ( ) | 2 e i [ k ( ) r 2 k ( ) * r 1 ] p ^ ( ) p ^ ( ) + E ( + ) * E ( ) e i [ k ( ) r 2 k ( + ) * r 1 ] p ^ ( + ) p ^ ( ) + E ( ) * E ( + ) e i [ k ( + ) r 2 k ( ) * r 1 ] p ^ ( ) p ^ ( + ) ,
p ^ ( a ) p ^ ( b ) = 1 | k ( a ) | | k ( b ) | ( k z ( a ) * k z ( b ) k z ( a ) * k x ( b ) k x ( a ) * k z ( b ) k x ( a ) * k x ( b ) ) , a , b { + , } .
μ ( r 1 , r 2 , ω ) = μ ( Δ r , ω ) = 1 2 1 + κ cos ( Δ k Δ r ) cos ( Δ k Δ r ) ,
κ [ p ^ ( + ) * p ^ ( ) ] 2 = | k x ( + ) k x ( ) * + k z ( + ) k z ( ) * | 2 | k ( + ) | 2 | k ( ) | 2 ,
Δ k [ k x ( + ) k x ( ) ] e ^ x + [ k z ( + ) k z ( ) ] e ^ z ,
Δ k [ k x ( + ) k x ( ) ] e ^ x + [ k z ( + ) k z ( ) ] e ^ z .
1 / 2 κ 1 ,
μ ( Δ r , ω ) { 1 / 2 , | Δ r | , ( 1 + κ ) / 2 , | Δ r | 0 .
μ ( Δ r , ω ) { 1 , d , 1 / 2 , d 0 ,
μ max = ( 1 + κ ) / 2 , μ min = ( 1 κ ) / 2 ,
3 / 2 μ max 1 , 0 μ min 1 / 2 ,
| Δ r | L coh | μ ( Δ r , ω ) μ | ξ , ξ 1 .
l coh = π / | Δ k | .
Φ ( r , ω ) = | E ( + ) | 2 e 2 k ( + ) r p ^ ( + ) p ^ ( + ) + | E ( ) | 2 e 2 k ( ) r p ^ ( ) p ^ ( ) + 2 e [ k ( + ) + k ( ) ] r [ E ( + ) * E ( ) e i Δ k r p ^ ( + ) p ^ ( ) ] .
P ( r , ω ) = 1 1 κ cosh 2 ( Δ k r ) ,
P ( r , ω ) { 1 , | r | , κ , | r | 0 ,
1 / 2 1 / κ P ( r , ω ) 1 .
P ( r , ω ) { 1 , d 1 , d 0 ,

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