Abstract

We examine the possibility of optimizing the emission and the near-field signal of apertureless silver and gold tips by using an optimized non-periodic grating. In this context, we consider the emission of a single quantum emitter in close proximity to optimized tips. Additionally, we study the far-field coupling efficiency of a tightly focused beam to the near-field of the tip. The gain in performance is compared with unstructured tips and the comparison with a pure plasmonic excitation of an unstructured tip is discussed. The optimized, structured tips show a significant enhancement of the total decay rate, as a result of standing plasmonic waves between the grating and the tip apex, leading to a resonant behavior. The resonances can be explained well with a Fabry-Pérot model. Furthermore, the total decay rate of an emitter near a structured tip can also be decreased as compared to an unstructured tip, when the grating is shifted from the optimal resonant position. The proposed scheme represents an interesting novel nano-antenna, for which the resonance as well as the directivity can be controlled by the grating.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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2014 (2)

J. Mihaljevic, S. Slama, R. Röpke, and A. J. Meixner, “Geometric tailoring of plasmonic nanotips for atom traps,” Phys. Rev. A 90, 013421 (2014).
[Crossref]

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

2013 (4)

J. Mihaljevic, C. Hafner, and A. J. Meixner, “Simulation of a metallic snom tip illuminated by a parabolic mirror,” Opt. Express 21, 25926–25943 (2013).
[Crossref] [PubMed]

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

2012 (4)

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

F.-A. Fortin, F.-M. De Rainville, M.-A. Gardner, M. Parizeau, and C. Gagné, “DEAP: Evolutionary algorithms made easy,” J. Mach. Learn. Res. 13, 2171–2175 (2012).

A. M. Kern, A. J. Meixner, and O. J. F. Martin, “Molecule-dependent plasmonic enhancement of fluorescence and raman scattering near realistic nanostructures,” ACS Nano 6, 9828–9836 (2012).
[Crossref] [PubMed]

P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75, 024402 (2012).
[Crossref] [PubMed]

2011 (6)

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: Evolution of sub- and super-radiant modes,” Nano Lett. 11, 1020–1024 (2011).
[Crossref] [PubMed]

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

J. S. Lee, S. Han, J. Shirdel, S. Koo, D. Sadiq, C. Lienau, and N. Park, “Superfocusing of electric or magnetic fields using conical metal tips: effect of mode symmetry on the plasmon excitation method,” Opt. Express 19, 12342–12347 (2011).
[Crossref] [PubMed]

T. Antosiewicz, P. Wróbel, and T. Szoplik, “Performance of scanning near-field optical microscope probes with single groove and various metal coatings,” Plasmonics 6, 11–18 (2011).
[Crossref] [PubMed]

D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. 11, 1609–1613 (2011).
[Crossref] [PubMed]

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

2009 (3)

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

F. Baida and A. Belkhir, “Superfocusing and light confinement by surface plasmon excitation through radially polarized beam,” Plasmonics 4, 51–59 (2009).
[Crossref]

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

2008 (2)

C. Stanciu, M. Sackrow, and A. Meixner, “High na particle- and tip-enhanced nanoscale raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229, 247–253 (2008).
[Crossref] [PubMed]

A. Hartschuh, “Tip-enhanced near-field optical microscopy,” Angew. Chem., Int. Ed. 47, 8178–8191 (2008).
[Crossref]

2007 (3)

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[Crossref] [PubMed]

N. A. Issa and R. Guckenberger, “Fluorescence near metal tips: The roles of energy transfer and surface plasmon polaritons,” Opt. Express 15, 12131–12144 (2007).
[Crossref] [PubMed]

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: Comparison of exact theory and an improved gersten and nitzan model,” Phys. Rev. B 76, 115123 (2007).
[Crossref]

2006 (2)

R. Carminati, J.-J. Greffet, C. Henkel, and J. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[Crossref] [PubMed]

2005 (2)

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95, 017402 (2005).
[Crossref] [PubMed]

2004 (1)

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[Crossref] [PubMed]

2002 (1)

E. Moreno, D. Erni, C. Hafner, R. Kunz, and R. Vahldieck, “Modeling and optimization of non-periodic grating couplers,” Opt. Quant. Electron. 34, 1051–1069 (2002).
[Crossref]

2001 (2)

M. Lieb and A. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8, 458–474 (2001).
[Crossref] [PubMed]

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evolutionary Computation 9, 159–195 (2001).
[Crossref] [PubMed]

1981 (1)

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[Crossref]

Accardo, A.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Alabastri, A.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

Albrecht, M.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[Crossref] [PubMed]

Angelis, F. D.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[Crossref] [PubMed]

Antosiewicz, T.

T. Antosiewicz, P. Wróbel, and T. Szoplik, “Performance of scanning near-field optical microscope probes with single groove and various metal coatings,” Plasmonics 6, 11–18 (2011).
[Crossref] [PubMed]

Aussenegg, F. R.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

Baida, F.

F. Baida and A. Belkhir, “Superfocusing and light confinement by surface plasmon excitation through radially polarized beam,” Plasmonics 4, 51–59 (2009).
[Crossref]

Baron, A.

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

Becker, S. F.

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

Belkhir, A.

F. Baida and A. Belkhir, “Superfocusing and light confinement by surface plasmon excitation through radially polarized beam,” Plasmonics 4, 51–59 (2009).
[Crossref]

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[Crossref] [PubMed]

Biagioni, P.

P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75, 024402 (2012).
[Crossref] [PubMed]

Brauer, J. H.

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

Braun, K.

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

Brecht, M.

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

Candeloro, P.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Carminati, R.

R. Carminati, J.-J. Greffet, C. Henkel, and J. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

Chang, D. E.

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

Chizhik, A. I.

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

Cingolani, R.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Cojoc, G.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Coluccio, L. M.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Cuda, G.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

D. Fabrizio, E.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

da Cunha, B. B.

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

Das, G.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

De Rainville, F.-M.

F.-A. Fortin, F.-M. De Rainville, M.-A. Gardner, M. Parizeau, and C. Gagné, “DEAP: Evolutionary algorithms made easy,” J. Mach. Learn. Res. 13, 2171–2175 (2012).

Devaux, E.

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

Ditlbacher, H.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

Dorfmüller, J.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

Ebbesen, T. W.

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

Eisler, H.-J.

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95, 017402 (2005).
[Crossref] [PubMed]

Elsaesser, T.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[Crossref] [PubMed]

Erni, D.

E. Moreno, D. Erni, C. Hafner, R. Kunz, and R. Vahldieck, “Modeling and optimization of non-periodic grating couplers,” Opt. Quant. Electron. 34, 1051–1069 (2002).
[Crossref]

Esmann, M.

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

Etrich, C.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

Failla, A. V.

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

Farahani, J. N.

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95, 017402 (2005).
[Crossref] [PubMed]

Fortin, F.-A.

F.-A. Fortin, F.-M. De Rainville, M.-A. Gardner, M. Parizeau, and C. Gagné, “DEAP: Evolutionary algorithms made easy,” J. Mach. Learn. Res. 13, 2171–2175 (2012).

Francardi, M.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

Gagné, C.

F.-A. Fortin, F.-M. De Rainville, M.-A. Gardner, M. Parizeau, and C. Gagné, “DEAP: Evolutionary algorithms made easy,” J. Mach. Learn. Res. 13, 2171–2175 (2012).

Gardner, M.-A.

F.-A. Fortin, F.-M. De Rainville, M.-A. Gardner, M. Parizeau, and C. Gagné, “DEAP: Evolutionary algorithms made easy,” J. Mach. Learn. Res. 13, 2171–2175 (2012).

Genet, C.

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

Gentile, F.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Giessen, H.

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

Giugni, A.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

Greffet, J.-J.

R. Carminati, J.-J. Greffet, C. Henkel, and J. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

Gro, P.

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

Guckenberger, R.

Hafner, C.

J. Mihaljevic, C. Hafner, and A. J. Meixner, “Simulation of a metallic snom tip illuminated by a parabolic mirror,” Opt. Express 21, 25926–25943 (2013).
[Crossref] [PubMed]

E. Moreno, D. Erni, C. Hafner, R. Kunz, and R. Vahldieck, “Modeling and optimization of non-periodic grating couplers,” Opt. Quant. Electron. 34, 1051–1069 (2002).
[Crossref]

C. Hafner, Post-Modern Electromagnetics Using Intelligent MaXwell Solvers (Wiley, 1999).

Han, S.

Hansen, N.

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evolutionary Computation 9, 159–195 (2001).
[Crossref] [PubMed]

Hartschuh, A.

A. Hartschuh, “Tip-enhanced near-field optical microscopy,” Angew. Chem., Int. Ed. 47, 8178–8191 (2008).
[Crossref]

Hecht, B.

P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75, 024402 (2012).
[Crossref] [PubMed]

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95, 017402 (2005).
[Crossref] [PubMed]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Henkel, C.

R. Carminati, J.-J. Greffet, C. Henkel, and J. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

Hentschel, M.

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

Hofer, F.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

Hohenau, A.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

Huang, J.-S.

P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75, 024402 (2012).
[Crossref] [PubMed]

Hugonin, J.-P.

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

Issa, N. A.

Jäger, R.

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

Jäger, S.

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

Kern, A. M.

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

A. M. Kern, A. J. Meixner, and O. J. F. Martin, “Molecule-dependent plasmonic enhancement of fluorescence and raman scattering near realistic nanostructures,” ACS Nano 6, 9828–9836 (2012).
[Crossref] [PubMed]

Kern, K.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

Kim, D.-S.

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

Kleppner, D.

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[Crossref]

Koenderink, A. F.

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: Comparison of exact theory and an improved gersten and nitzan model,” Phys. Rev. B 76, 115123 (2007).
[Crossref]

Koo, S.

Kreibig, U.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

Krenn, J. R.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

Kunz, R.

E. Moreno, D. Erni, C. Hafner, R. Kunz, and R. Vahldieck, “Modeling and optimization of non-periodic grating couplers,” Opt. Quant. Electron. 34, 1051–1069 (2002).
[Crossref]

Lalanne, P.

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

Lederer, F.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

Lee, J. S.

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. 11, 1609–1613 (2011).
[Crossref] [PubMed]

J. S. Lee, S. Han, J. Shirdel, S. Koo, D. Sadiq, C. Lienau, and N. Park, “Superfocusing of electric or magnetic fields using conical metal tips: effect of mode symmetry on the plasmon excitation method,” Opt. Express 19, 12342–12347 (2011).
[Crossref] [PubMed]

Liberale, C.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Lieb, M.

Lienau, C.

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. 11, 1609–1613 (2011).
[Crossref] [PubMed]

J. S. Lee, S. Han, J. Shirdel, S. Koo, D. Sadiq, C. Lienau, and N. Park, “Superfocusing of electric or magnetic fields using conical metal tips: effect of mode symmetry on the plasmon excitation method,” Opt. Express 19, 12342–12347 (2011).
[Crossref] [PubMed]

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[Crossref] [PubMed]

Lukin, M. D.

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

Malerba, M.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

Martin, O. J. F.

A. M. Kern, A. J. Meixner, and O. J. F. Martin, “Molecule-dependent plasmonic enhancement of fluorescence and raman scattering near realistic nanostructures,” ACS Nano 6, 9828–9836 (2012).
[Crossref] [PubMed]

Mecarini, F.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Meixner, A.

C. Stanciu, M. Sackrow, and A. Meixner, “High na particle- and tip-enhanced nanoscale raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229, 247–253 (2008).
[Crossref] [PubMed]

M. Lieb and A. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8, 458–474 (2001).
[Crossref] [PubMed]

Meixner, A. J.

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

J. Mihaljevic, S. Slama, R. Röpke, and A. J. Meixner, “Geometric tailoring of plasmonic nanotips for atom traps,” Phys. Rev. A 90, 013421 (2014).
[Crossref]

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

J. Mihaljevic, C. Hafner, and A. J. Meixner, “Simulation of a metallic snom tip illuminated by a parabolic mirror,” Opt. Express 21, 25926–25943 (2013).
[Crossref] [PubMed]

A. M. Kern, A. J. Meixner, and O. J. F. Martin, “Molecule-dependent plasmonic enhancement of fluorescence and raman scattering near realistic nanostructures,” ACS Nano 6, 9828–9836 (2012).
[Crossref] [PubMed]

Mertens, H.

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: Comparison of exact theory and an improved gersten and nitzan model,” Phys. Rev. B 76, 115123 (2007).
[Crossref]

Mihaljevic, J.

J. Mihaljevic, S. Slama, R. Röpke, and A. J. Meixner, “Geometric tailoring of plasmonic nanotips for atom traps,” Phys. Rev. A 90, 013421 (2014).
[Crossref]

J. Mihaljevic, C. Hafner, and A. J. Meixner, “Simulation of a metallic snom tip illuminated by a parabolic mirror,” Opt. Express 21, 25926–25943 (2013).
[Crossref] [PubMed]

Moreno, E.

E. Moreno, D. Erni, C. Hafner, R. Kunz, and R. Vahldieck, “Modeling and optimization of non-periodic grating couplers,” Opt. Quant. Electron. 34, 1051–1069 (2002).
[Crossref]

Moretti, M.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Neacsu, C. C.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[Crossref] [PubMed]

Novotny, L.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
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L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Ostermeier, A.

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evolutionary Computation 9, 159–195 (2001).
[Crossref] [PubMed]

Parizeau, M.

F.-A. Fortin, F.-M. De Rainville, M.-A. Gardner, M. Parizeau, and C. Gagné, “DEAP: Evolutionary algorithms made easy,” J. Mach. Learn. Res. 13, 2171–2175 (2012).

Park, H.

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

Park, N.

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. 11, 1609–1613 (2011).
[Crossref] [PubMed]

J. S. Lee, S. Han, J. Shirdel, S. Koo, D. Sadiq, C. Lienau, and N. Park, “Superfocusing of electric or magnetic fields using conical metal tips: effect of mode symmetry on the plasmon excitation method,” Opt. Express 19, 12342–12347 (2011).
[Crossref] [PubMed]

Perozziello, G.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Pertsch, T.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

Petit, R.

R. Petit, Electromagnetic Theory of Gratings, vol. 22 of Topics in Current Physics (Springer, 1980).
[Crossref]

Piglosiewicz, B.

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

Pohl, D. W.

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95, 017402 (2005).
[Crossref] [PubMed]

Polman, A.

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: Comparison of exact theory and an improved gersten and nitzan model,” Phys. Rev. B 76, 115123 (2007).
[Crossref]

Raschke, M. B.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[Crossref] [PubMed]

Rockstuhl, C.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

Rodier, J.-C.

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

Rogers, M.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

Ropers, C.

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[Crossref] [PubMed]

Röpke, R.

J. Mihaljevic, S. Slama, R. Röpke, and A. J. Meixner, “Geometric tailoring of plasmonic nanotips for atom traps,” Phys. Rev. A 90, 013421 (2014).
[Crossref]

Rousseau, E.

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

Sackrow, M.

C. Stanciu, M. Sackrow, and A. Meixner, “High na particle- and tip-enhanced nanoscale raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229, 247–253 (2008).
[Crossref] [PubMed]

Sadiq, D.

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. 11, 1609–1613 (2011).
[Crossref] [PubMed]

J. S. Lee, S. Han, J. Shirdel, S. Koo, D. Sadiq, C. Lienau, and N. Park, “Superfocusing of electric or magnetic fields using conical metal tips: effect of mode symmetry on the plasmon excitation method,” Opt. Express 19, 12342–12347 (2011).
[Crossref] [PubMed]

Schmidt, S.

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

Selishcheva, E.

D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. 11, 1609–1613 (2011).
[Crossref] [PubMed]

Shirdel, J.

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. 11, 1609–1613 (2011).
[Crossref] [PubMed]

J. S. Lee, S. Han, J. Shirdel, S. Koo, D. Sadiq, C. Lienau, and N. Park, “Superfocusing of electric or magnetic fields using conical metal tips: effect of mode symmetry on the plasmon excitation method,” Opt. Express 19, 12342–12347 (2011).
[Crossref] [PubMed]

Slama, S.

J. Mihaljevic, S. Slama, R. Röpke, and A. J. Meixner, “Geometric tailoring of plasmonic nanotips for atom traps,” Phys. Rev. A 90, 013421 (2014).
[Crossref]

Stanciu, C.

C. Stanciu, M. Sackrow, and A. Meixner, “High na particle- and tip-enhanced nanoscale raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229, 247–253 (2008).
[Crossref] [PubMed]

Stefani, F. D.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: Evolution of sub- and super-radiant modes,” Nano Lett. 11, 1020–1024 (2011).
[Crossref] [PubMed]

Stockman, M. I.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[Crossref] [PubMed]

Szoplik, T.

T. Antosiewicz, P. Wróbel, and T. Szoplik, “Performance of scanning near-field optical microscope probes with single groove and various metal coatings,” Plasmonics 6, 11–18 (2011).
[Crossref] [PubMed]

Taminiau, T. H.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: Evolution of sub- and super-radiant modes,” Nano Lett. 11, 1020–1024 (2011).
[Crossref] [PubMed]

Thompson, J. D.

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

Tirinato, L.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Toma, A.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Torre, B.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

Vahldieck, R.

E. Moreno, D. Erni, C. Hafner, R. Kunz, and R. Vahldieck, “Modeling and optimization of non-periodic grating couplers,” Opt. Quant. Electron. 34, 1051–1069 (2002).
[Crossref]

van Hulst, N. F.

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: Evolution of sub- and super-radiant modes,” Nano Lett. 11, 1020–1024 (2011).
[Crossref] [PubMed]

Vasa, P.

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

Vigoureux, J.

R. Carminati, J.-J. Greffet, C. Henkel, and J. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

Vogelgesang, R.

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

Vuletic, V.

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

Wackenhut, F.

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

Wagner, D.

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

Weitz, R. T.

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

Wróbel, P.

T. Antosiewicz, P. Wróbel, and T. Szoplik, “Performance of scanning near-field optical microscope probes with single groove and various metal coatings,” Plasmonics 6, 11–18 (2011).
[Crossref] [PubMed]

Zaccaria, P.

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

Zaccaria, P. R.

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Zhang, D.

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

Zibrov, A. S.

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

Zoller, P.

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

ACS Nano (2)

S. Schmidt, B. Piglosiewicz, D. Sadiq, J. Shirdel, J. S. Lee, P. Vasa, N. Park, D.-S. Kim, and C. Lienau, “Adiabatic nanofocusing on ultrasmooth single-crystalline gold tapers creates a 10-nm-sized light source with few-cycle time resolution,” ACS Nano 6, 6040–6048 (2012).
[Crossref] [PubMed]

A. M. Kern, A. J. Meixner, and O. J. F. Martin, “Molecule-dependent plasmonic enhancement of fluorescence and raman scattering near realistic nanostructures,” ACS Nano 6, 9828–9836 (2012).
[Crossref] [PubMed]

Angew. Chem., Int. Ed. (1)

A. Hartschuh, “Tip-enhanced near-field optical microscopy,” Angew. Chem., Int. Ed. 47, 8178–8191 (2008).
[Crossref]

Beilstein J. Nanotechnol. (1)

M. Esmann, S. F. Becker, B. B. da Cunha, J. H. Brauer, R. Vogelgesang, P. Gro, and C. Lienau, “k-space imaging of the eigenmodes of sharp gold tapers for scanning near-field optical microscopy,” Beilstein J. Nanotechnol. 4, 603–610 (2013).
[Crossref] [PubMed]

Chem. Soc. Rev. (1)

A. M. Kern, D. Zhang, M. Brecht, A. I. Chizhik, A. V. Failla, F. Wackenhut, and A. J. Meixner, “Enhanced single-molecule spectroscopy in highly confined optical fields: from λ/2-fabry–pérot resonators to plasmonic nano-antennas,” Chem. Soc. Rev. 43, 1263–1286 (2014).
[Crossref]

Evolutionary Computation (1)

N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolution strategies,” Evolutionary Computation 9, 159–195 (2001).
[Crossref] [PubMed]

J. Mach. Learn. Res. (1)

F.-A. Fortin, F.-M. De Rainville, M.-A. Gardner, M. Parizeau, and C. Gagné, “DEAP: Evolutionary algorithms made easy,” J. Mach. Learn. Res. 13, 2171–2175 (2012).

J. Microsc. (1)

C. Stanciu, M. Sackrow, and A. Meixner, “High na particle- and tip-enhanced nanoscale raman spectroscopy with a parabolic-mirror microscope,” J. Microsc. 229, 247–253 (2008).
[Crossref] [PubMed]

Nano Lett. (6)

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabrypérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9, 2372–2377 (2009).
[Crossref]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Optical nanorod antennas modeled as cavities for dipolar emitters: Evolution of sub- and super-radiant modes,” Nano Lett. 11, 1020–1024 (2011).
[Crossref] [PubMed]

A. Baron, E. Devaux, J.-C. Rodier, J.-P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
[Crossref] [PubMed]

D. Sadiq, J. Shirdel, J. S. Lee, E. Selishcheva, N. Park, and C. Lienau, “Adiabatic nanofocusing scattering-type optical nanoscopy of individual gold nanoparticles,” Nano Lett. 11, 1609–1613 (2011).
[Crossref] [PubMed]

S. Jäger, A. M. Kern, M. Hentschel, R. Jäger, K. Braun, D. Zhang, H. Giessen, and A. J. Meixner, “Au nanotip as luminescent near-field probe,” Nano Lett. 13, 3566–3570 (2013).
[Crossref] [PubMed]

C. Ropers, C. C. Neacsu, T. Elsaesser, M. Albrecht, M. B. Raschke, and C. Lienau, “Grating-coupling of surface plasmons onto metallic tips: A nanoconfined light source,” Nano Lett. 7, 2784–2788 (2007).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

A. Giugni, B. Torre, A. Toma, M. Francardi, M. Malerba, A. Alabastri, P. Zaccaria, M. I. Stockman, and E. D. Fabrizio, “Hot-electron nanoscopy using adiabatic compression of surface plasmons,” Nat. Nanotechnol. 8, 845–852 (2013).
[Crossref] [PubMed]

Nat. Photon. (1)

F. D. Angelis, F. Gentile, F. Mecarini, G. Das, M. Moretti, P. Candeloro, L. M. Coluccio, G. Cojoc, A. Accardo, C. Liberale, P. R. Zaccaria, G. Perozziello, L. Tirinato, A. Toma, G. Cuda, R. Cingolani, and E. D. Fabrizio, “Breaking the diffusion limit with super-hydrophobic delivery of molecules to plasmonic nanofocusing sers structures,” Nat. Photon. 5, 682–687 (2011).
[Crossref]

Opt. Commun. (1)

R. Carminati, J.-J. Greffet, C. Henkel, and J. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[Crossref]

Opt. Express (4)

Opt. Quant. Electron. (1)

E. Moreno, D. Erni, C. Hafner, R. Kunz, and R. Vahldieck, “Modeling and optimization of non-periodic grating couplers,” Opt. Quant. Electron. 34, 1051–1069 (2002).
[Crossref]

Phys. Rev. A (1)

J. Mihaljevic, S. Slama, R. Röpke, and A. J. Meixner, “Geometric tailoring of plasmonic nanotips for atom traps,” Phys. Rev. A 90, 013421 (2014).
[Crossref]

Phys. Rev. B (1)

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: Comparison of exact theory and an improved gersten and nitzan model,” Phys. Rev. B 76, 115123 (2007).
[Crossref]

Phys. Rev. Lett. (6)

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[Crossref]

H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver nanowires as surface plasmon resonators,” Phys. Rev. Lett. 95, 257403 (2005).
[Crossref] [PubMed]

J. N. Farahani, D. W. Pohl, H.-J. Eisler, and B. Hecht, “Single quantum dot coupled to a scanning optical antenna: A tunable superemitter,” Phys. Rev. Lett. 95, 017402 (2005).
[Crossref] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[Crossref] [PubMed]

D. E. Chang, J. D. Thompson, H. Park, V. Vuletić, A. S. Zibrov, P. Zoller, and M. D. Lukin, “Trapping and manipulation of isolated atoms using nanoscale plasmonic structures,” Phys. Rev. Lett. 103, 123004 (2009).
[Crossref] [PubMed]

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[Crossref] [PubMed]

Plasmonics (2)

F. Baida and A. Belkhir, “Superfocusing and light confinement by surface plasmon excitation through radially polarized beam,” Plasmonics 4, 51–59 (2009).
[Crossref]

T. Antosiewicz, P. Wróbel, and T. Szoplik, “Performance of scanning near-field optical microscope probes with single groove and various metal coatings,” Plasmonics 6, 11–18 (2011).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75, 024402 (2012).
[Crossref] [PubMed]

Other (4)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

R. Petit, Electromagnetic Theory of Gratings, vol. 22 of Topics in Current Physics (Springer, 1980).
[Crossref]

C. Hafner, “Openmaxwell: Graphic platform for computational electromagnetics and computational optics,” http://openmax.ethz.ch/ ,ETH Zürich (2013).

C. Hafner, Post-Modern Electromagnetics Using Intelligent MaXwell Solvers (Wiley, 1999).

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Figures (7)

Fig. 1
Fig. 1 (a) Geometric model parameters of the tip. R1, R2, R3 and θc were set to a constant value, while the other shown parameters were subject to variations during the optimization. (b) Boundaries used for the evaluation of the power emission integrals of a dipole p below the tip.
Fig. 2
Fig. 2 Imaginary vs. real part of the relative propagation constant of the fundamental TM0 mode for the considered materials, wavelengths and for a sequence of different wire radii (50, 100, 200,..., 6400nm, indicated by the markers).
Fig. 3
Fig. 3 Emission power ( sin θ d P ( θ , ϕ ) d Ω) vs. polar angle (θ) of the optimized tips (normalized to the maximum power).
Fig. 4
Fig. 4 Normalized decay rates vs. grating position. The dashed lines indicate the values of the normalized decay rates of a similar tip, but without a grating.
Fig. 5
Fig. 5 Normalized decay rates vs. wavelength. The dashed lines indicate the values of the normalized decay rates of a similar tip, but without a grating.
Fig. 6
Fig. 6 Cross section of different excitation schemes of near-field tips, whereby for all schemes the geometry and the excitation are radially symmetric. (a) Tip without a grating is excited by a tightly focused beam, where the focus is placed directly on the tip apex. (b) Tip with an optimized grating is excited by a tightly focused beam, where the focus is placed slightly below the grating. (c) Tip without a grating, excited by a TM0 mode that propagates along the shaft.
Fig. 7
Fig. 7 The left column shows the SBR of the considered optimized tips versus the focus shift Δz along the central tip-axis. The right column shows the corresponding field intensity at a position 10nm below the apex normalized by the value obtained for Δz = 0. This value can be used for estimating the coupling efficiency of the excitation field to the tip. The dashed vertical line indicates the position of the lower edge of the grating [i.e. b1 in Fig. 1(a)].

Tables (2)

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Table 1 Geometric parameters of the optimized tips (see Fig. 1(a) for parameter definition). The right columns show the enhancement of the decay rates compared to a tip without a grating, e.g. F + = γ + / γ wg + = P + / P wg + where wg indicates the values for the tip without a grating.

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Table 2 Signal-to-Background-Ratio for the excitation schemes shown in Fig. 6 for different tip materials and wavelengths.

Equations (16)

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k spp = n k 0 sin ϕ ± m 2 π P , m
γ tot = 2 ω 0 3 ε 0 | p | 2 ρ L ( r e , ω 0 ) ,
ρ L ( r e , ω 0 ) = 6 ω 0 π c 2 [ n p Im { G ( r e , r e ; ω 0 ) } n p ] ,
ρ L , 0 ( r e , ω 0 ) = ω 0 2 π 2 c 3
γ 0 = ω 0 2 | p | 2 3 π ε 0 c 3 .
γ tot γ 0 = P tot P 0 .
γ tot = γ r + γ nr .
γ nr = γ spp + γ LET ,
γ rad = γ + + γ .
E ρ = A l i k z κ l Z 1 ( κ l ρ ) e i k z z E z = A l κ l 2 Z 0 ( κ l ρ ) e i k z z H ϕ = A l i ω ε l κ l Z 1 ( κ l ρ ) e i k z z
P P 0 = 1 + 6 π ε 0 ε | p | 2 1 k 3 Im { p * E s ( r e ) }
Δ f = c 2 n L
Δ L = c n 1 2 f = 1 2 2 π k ,
Δ L = 1 2 2 π Re ( k z spp ) cos θ c .
S nf = 2 π 0 R 1 ρ | E ( ρ ) | 2 d ρ
S ff = 2 π R 1 λ 0 ρ | E ( ρ ) | 2 d ρ .

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