Abstract

Near-field microscopy is widely used for characterizing electromagnetic fields at nanoscale, where nanoprobes afford the opportunity to extract subwavelength optical quantities, including the amplitude, phase, polarization, chirality, etc. However, owing to the complexity of various nanoprobes, a general and intuitive theory is highly desired to assess the vectorial responses of nanoprobes and interpret the mechanism of the probe-field interaction. Here, we develop a general imaging theory based on the reciprocity of electromagnetism and multipole expansion analysis. The proposed theory closely resembles the multipolar Hamiltonian for light-matter interaction energy, revealing the coupling mechanism of the probe-field interaction. Based on this theory, we introduce a new paradigm for the design of functional nanoprobes by analyzing the reciprocal dipole moments, and establish effective design principles for the imaging of vectorial near fields. As application examples of the proposed theory, we numerically analyze the responses of two typical probes, a split-ring probe and a nanoparticle probe, which can quantitatively reproduce and well explain the experimental results of previously reported measurements of the optical magnetism and the transverse spin angular momentum. Our work provides a powerful tool for the design and analysis of new functional probes that may enable the probing of various physical quantities of the vectorial near field.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

M. Neugebauer, J. Eismann, T. Bauer, and P. Banzer, “Magnetic and electric transverse spin density of spatially confined light,” Phys. Rev. X 8(2), 021042 (2018).

2017 (2)

S. Kruk and Y. Kivshar, “Functional meta-optics and nanophotonics govern by mie resonances,” ACS Photonics 4(11), 2638–2649 (2017).
[Crossref]

T. Feng, Y. Xu, W. Zhang, and A. E. Miroshnichenko, “Ideal magnetic dipole scattering,” Phys. Rev. Lett. 118(17), 173901 (2017).
[Crossref] [PubMed]

2016 (5)

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11(1), 23–36 (2016).
[Crossref] [PubMed]

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), 2472 (2016).
[Crossref] [PubMed]

I. V. Kabakova, A. de Hoogh, R. E. C. van der Wel, M. Wulf, B. le Feber, and L. Kuipers, “Imaging of electric and magnetic fields near plasmonic nanowires,” Sci. Rep. 6, 22665 (2016).
[Crossref] [PubMed]

A. Alabastri, X. Yang, A. Manjavacas, H. O. Everitt, and P. Nordlander, “Extraordinary light-induced local angular momentum near metallic nanoparticles,” ACS Nano 10(4), 4835–4846 (2016).
[Crossref] [PubMed]

S. Schmidt, A. E. Klein, T. Paul, H. Gross, S. Diziain, M. Steinert, A. C. Assafrao, T. Pertsch, H. P. Urbach, and C. Rockstuhl, “Image formation properties and inverse imaging problem in aperture based scanning near field optical microscopy,” Opt. Express 24(4), 4128–4142 (2016).
[Crossref] [PubMed]

2015 (11)

T. Neuman, P. Alonso-González, A. García-Etxarri, M. Schnell, R. Hillenbrand, and J. Aizpurua, “Mapping the near fields of plasmonic nanoantennas by scattering-type scanning near-field optical microscopy,” Laser Photonics Rev. 9(6), 637–649 (2015).
[Crossref]

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15(3), 2137–2142 (2015).
[Crossref] [PubMed]

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
[Crossref]

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

D. K. Singh, J. S. Ahn, S. K. Koo, T. H. Kang, J. Y. Kim, S. K. Lee, N. K. Park, and D. S. Kim, “Selective electric and magnetic sensitivity of aperture probes,” Opt. Express 23(16), 20820–20828 (2015).
[Crossref] [PubMed]

M. Kasperczyk, S. Person, D. Ananias, L. D. Carlos, and L. Novotny, “Excitation of magnetic dipole transitions at optical frequencies,” Phys. Rev. Lett. 114(16), 163903 (2015).
[Crossref] [PubMed]

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5(1), 011039 (2015).

A. Aiello, P. Banzer, M. Neugebaueru, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref] [PubMed]

D. Permyakov, I. Sinev, D. Markovich, P. Ginzburg, A. Samusev, P. Belov, V. Valuckas, A. I. Kuznetsov, B. S. Luk’yanchuk, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Probing magnetic and electric optical responses of silicon nanoparticles,” Appl. Phys. Lett. 106(17), 171110 (2015).
[Crossref]

2014 (6)

U. Zywietz, A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Laser printing of silicon nanoparticles with resonant optical electric and magnetic responses,” Nat. Commun. 5, 3402 (2014).
[Crossref] [PubMed]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

N. Rotenberg and L. Kuipers, “Mapping nanoscale light fields,” Nat. Photonics 8(12), 919–926 (2014).
[Crossref]

D. Denkova, N. Verellen, A. V. Silhanek, P. V. Dorpe, and V. V. Moshchalkov, “Lateral magnetic near-field imaging of plasmonic nanoantennas with increasing complexity,” Small 10(10), 1959–1966 (2014).
[Crossref] [PubMed]

F. B. Arango, T. Coenen, and A. F. Koenderink, “Underpinning hybridization intuition for complex nanoantennas by magnetoelectric quadrupolar polarizability retrieval,” ACS Photonics 1(5), 444–453 (2014).
[Crossref]

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nat. Photonics 8(1), 43–46 (2014).
[Crossref]

2013 (4)

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B 30(10), 2589–2598 (2013).
[Crossref]

D. Denkova, N. Verellen, A. V. Silhanek, V. K. Valev, P. V. Dorpe, and V. V. Moshchalkov,“Mapping magnetic near-field distributions of plasmonic nanoantennas,” ACS Nano 7(4), 3168–3176 (2013).
[Crossref] [PubMed]

H. W. Kihm, J. E. Kim, S. M. Koo, J. S. Ahn, K. J. Ahn, K. G. Lee, N. K. Park, and D. S. Kim, “Optical magnetic field mapping using a subwavelength aperture,” Opt. Express 21(5), 5625–5633 (2013).
[Crossref] [PubMed]

F. B. Arango and A. F. Koenderink, “Polarizability tensor retrieval for magnetic and plasmonic antenna design,” New J. Phys. 15(7), 073023 (2013).
[Crossref]

2012 (4)

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012).
[Crossref] [PubMed]

K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012).
[Crossref]

L. Yu, T. Sfez, V. Paeder, P. Stenberg, W. Nakagawa, M. Kuittinen, and H. P. Herzig, “Concurrent polarization retrieval in multi-heterodyne scanning near-field optical microscopy: Validation on silicon form-birefringent grating,” Opt. Express 20(21), 23088–23099 (2012).
[Crossref] [PubMed]

M. Esslinger and R. Vogelgesang,“Reciprocity theory of apertureless scanning near-field optical microscopy with point-dipole probes,” ACS Nano 6(9), 8173–8182 (2012).
[Crossref] [PubMed]

2011 (4)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[Crossref]

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5(2–3), 64–73 (2011).
[Crossref]

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun. 2, 451 (2011).
[Crossref] [PubMed]

I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83(24), 245102 (2011).
[Crossref]

2010 (1)

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

2009 (3)

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102(3), 033902 (2009).
[Crossref] [PubMed]

M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009).
[Crossref] [PubMed]

A. García-Etxarri, I. Romero, F. J. G. de Abajo, R. Hillenbrand, and J. Aizpurua, “Influence of the tip in near-field imaging of nanoparticle plasmonic modes: Weak and strong coupling regimes,” Phys. Rev. B 79(12), 125439 (2009).
[Crossref]

2007 (2)

J. Sun, P. S. Carney, and J. C. Schotland,“Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102(10), 103103 (2007).
[Crossref]

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1(1), 53–56 (2007).
[Crossref]

2005 (1)

A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
[Crossref] [PubMed]

2003 (1)

K. Joulain, R. Carminati, J. P. Mulet, and J. J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68(24), 245405 (2003).
[Crossref]

2001 (1)

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 (2001).
[Crossref]

2000 (4)

J. A. Porto, R. Carminati, and J. J. Greffet, “Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy,” J. Appl. Phys. 88(8), 4845–4850 (2000).
[Crossref]

A. Dereux, C. Girard, and J. C. Weeber, “Theoretical principles of near-field optical microscopies and spectroscopies,” J. Chem. Phys. 112(18), 7775–7789 (2000).
[Crossref]

R. Carminati and J. J. Sáenz, “Scattering theory of bardeen’s formalism for tunneling: New approach to near-field microscopy,” Phys. Rev. Lett. 84(22), 5156–5159 (2000).
[Crossref] [PubMed]

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref] [PubMed]

1997 (1)

J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56(3), 133–237 (1997).
[Crossref]

1996 (1)

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59(5), 657–699 (1996).
[Crossref]

1994 (1)

D. Courjon and C. Bainier,“Near field microscopy and near field optics,” Rep. Prog. Phys. 57(10), 989–1028 (1994).
[Crossref]

1961 (1)

J. Bardeen, “Tunnelling from a many-particle point of view,” Phys. Rev. Lett. 6(2), 57–59 (1961).
[Crossref]

Ahn, J. S.

Ahn, K. J.

Aiello, A.

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

A. Aiello, P. Banzer, M. Neugebaueru, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

Aizpurua, J.

T. Neuman, P. Alonso-González, A. García-Etxarri, M. Schnell, R. Hillenbrand, and J. Aizpurua, “Mapping the near fields of plasmonic nanoantennas by scattering-type scanning near-field optical microscopy,” Laser Photonics Rev. 9(6), 637–649 (2015).
[Crossref]

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

A. García-Etxarri, I. Romero, F. J. G. de Abajo, R. Hillenbrand, and J. Aizpurua, “Influence of the tip in near-field imaging of nanoparticle plasmonic modes: Weak and strong coupling regimes,” Phys. Rev. B 79(12), 125439 (2009).
[Crossref]

Alabastri, A.

A. Alabastri, X. Yang, A. Manjavacas, H. O. Everitt, and P. Nordlander, “Extraordinary light-induced local angular momentum near metallic nanoparticles,” ACS Nano 10(4), 4835–4846 (2016).
[Crossref] [PubMed]

Alkorta, J.

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

Alonso-González, P.

T. Neuman, P. Alonso-González, A. García-Etxarri, M. Schnell, R. Hillenbrand, and J. Aizpurua, “Mapping the near fields of plasmonic nanoantennas by scattering-type scanning near-field optical microscopy,” Laser Photonics Rev. 9(6), 637–649 (2015).
[Crossref]

Ananias, D.

M. Kasperczyk, S. Person, D. Ananias, L. D. Carlos, and L. Novotny, “Excitation of magnetic dipole transitions at optical frequencies,” Phys. Rev. Lett. 114(16), 163903 (2015).
[Crossref] [PubMed]

Arango, F. B.

F. B. Arango, T. Coenen, and A. F. Koenderink, “Underpinning hybridization intuition for complex nanoantennas by magnetoelectric quadrupolar polarizability retrieval,” ACS Photonics 1(5), 444–453 (2014).
[Crossref]

F. B. Arango and A. F. Koenderink, “Polarizability tensor retrieval for magnetic and plasmonic antenna design,” New J. Phys. 15(7), 073023 (2013).
[Crossref]

Assafrao, A. C.

Baba, T.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102(3), 033902 (2009).
[Crossref] [PubMed]

Bainier, C.

D. Courjon and C. Bainier,“Near field microscopy and near field optics,” Rep. Prog. Phys. 57(10), 989–1028 (1994).
[Crossref]

Bak, W. S.

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun. 2, 451 (2011).
[Crossref] [PubMed]

Bakker, R. M.

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15(3), 2137–2142 (2015).
[Crossref] [PubMed]

Banzer, P.

M. Neugebauer, J. Eismann, T. Bauer, and P. Banzer, “Magnetic and electric transverse spin density of spatially confined light,” Phys. Rev. X 8(2), 021042 (2018).

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

A. Aiello, P. Banzer, M. Neugebaueru, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

Bao, K.

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun. 2, 451 (2011).
[Crossref] [PubMed]

Bardeen, J.

J. Bardeen, “Tunnelling from a many-particle point of view,” Phys. Rev. Lett. 6(2), 57–59 (1961).
[Crossref]

Bauer, T.

M. Neugebauer, J. Eismann, T. Bauer, and P. Banzer, “Magnetic and electric transverse spin density of spatially confined light,” Phys. Rev. X 8(2), 021042 (2018).

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

Beggs, D. M.

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nat. Photonics 8(1), 43–46 (2014).
[Crossref]

Bekshaev, A. Y.

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5(1), 011039 (2015).

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

Belov, P.

D. Permyakov, I. Sinev, D. Markovich, P. Ginzburg, A. Samusev, P. Belov, V. Valuckas, A. I. Kuznetsov, B. S. Luk’yanchuk, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Probing magnetic and electric optical responses of silicon nanoparticles,” Appl. Phys. Lett. 106(17), 171110 (2015).
[Crossref]

Belov, P. A.

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref] [PubMed]

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5(1), 011039 (2015).

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012).
[Crossref]

Bohren, C.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998).
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Brongersma, M. L.

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), 2472 (2016).
[Crossref] [PubMed]

Brown, T.

Buchler, B. C.

A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
[Crossref] [PubMed]

Burresi, M.

M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009).
[Crossref] [PubMed]

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102(3), 033902 (2009).
[Crossref] [PubMed]

Carlos, L. D.

M. Kasperczyk, S. Person, D. Ananias, L. D. Carlos, and L. Novotny, “Excitation of magnetic dipole transitions at optical frequencies,” Phys. Rev. Lett. 114(16), 163903 (2015).
[Crossref] [PubMed]

Carminati, R.

K. Joulain, R. Carminati, J. P. Mulet, and J. J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68(24), 245405 (2003).
[Crossref]

R. Carminati and J. J. Sáenz, “Scattering theory of bardeen’s formalism for tunneling: New approach to near-field microscopy,” Phys. Rev. Lett. 84(22), 5156–5159 (2000).
[Crossref] [PubMed]

J. A. Porto, R. Carminati, and J. J. Greffet, “Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy,” J. Appl. Phys. 88(8), 4845–4850 (2000).
[Crossref]

J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56(3), 133–237 (1997).
[Crossref]

Carney, P. S.

J. Sun, P. S. Carney, and J. C. Schotland,“Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102(10), 103103 (2007).
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Chichkov, B. N.

U. Zywietz, A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Laser printing of silicon nanoparticles with resonant optical electric and magnetic responses,” Nat. Commun. 5, 3402 (2014).
[Crossref] [PubMed]

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B 30(10), 2589–2598 (2013).
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Choi, S. B.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1(1), 53–56 (2007).
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Choi, W. J.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1(1), 53–56 (2007).
[Crossref]

Coenen, T.

F. B. Arango, T. Coenen, and A. F. Koenderink, “Underpinning hybridization intuition for complex nanoantennas by magnetoelectric quadrupolar polarizability retrieval,” ACS Photonics 1(5), 444–453 (2014).
[Crossref]

Courjon, D.

D. Courjon and C. Bainier,“Near field microscopy and near field optics,” Rep. Prog. Phys. 57(10), 989–1028 (1994).
[Crossref]

Das, T.

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

de Abajo, F. J. G.

A. García-Etxarri, I. Romero, F. J. G. de Abajo, R. Hillenbrand, and J. Aizpurua, “Influence of the tip in near-field imaging of nanoparticle plasmonic modes: Weak and strong coupling regimes,” Phys. Rev. B 79(12), 125439 (2009).
[Crossref]

de Hoogh, A.

I. V. Kabakova, A. de Hoogh, R. E. C. van der Wel, M. Wulf, B. le Feber, and L. Kuipers, “Imaging of electric and magnetic fields near plasmonic nanowires,” Sci. Rep. 6, 22665 (2016).
[Crossref] [PubMed]

DeCrescent, R. A.

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
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Denisyuk, A. I.

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
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Denkova, D.

D. Denkova, N. Verellen, A. V. Silhanek, P. V. Dorpe, and V. V. Moshchalkov, “Lateral magnetic near-field imaging of plasmonic nanoantennas with increasing complexity,” Small 10(10), 1959–1966 (2014).
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D. Denkova, N. Verellen, A. V. Silhanek, V. K. Valev, P. V. Dorpe, and V. V. Moshchalkov,“Mapping magnetic near-field distributions of plasmonic nanoantennas,” ACS Nano 7(4), 3168–3176 (2013).
[Crossref] [PubMed]

Dereux, A.

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 (2001).
[Crossref]

A. Dereux, C. Girard, and J. C. Weeber, “Theoretical principles of near-field optical microscopies and spectroscopies,” J. Chem. Phys. 112(18), 7775–7789 (2000).
[Crossref]

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59(5), 657–699 (1996).
[Crossref]

Diziain, S.

Dorpe, P. V.

D. Denkova, N. Verellen, A. V. Silhanek, P. V. Dorpe, and V. V. Moshchalkov, “Lateral magnetic near-field imaging of plasmonic nanoantennas with increasing complexity,” Small 10(10), 1959–1966 (2014).
[Crossref] [PubMed]

D. Denkova, N. Verellen, A. V. Silhanek, V. K. Valev, P. V. Dorpe, and V. V. Moshchalkov,“Mapping magnetic near-field distributions of plasmonic nanoantennas,” ACS Nano 7(4), 3168–3176 (2013).
[Crossref] [PubMed]

Eah, S. H.

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun. 2, 451 (2011).
[Crossref] [PubMed]

Eismann, J.

M. Neugebauer, J. Eismann, T. Bauer, and P. Banzer, “Magnetic and electric transverse spin density of spatially confined light,” Phys. Rev. X 8(2), 021042 (2018).

Engelen, R. J. P.

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102(3), 033902 (2009).
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M. Esslinger and R. Vogelgesang,“Reciprocity theory of apertureless scanning near-field optical microscopy with point-dipole probes,” ACS Nano 6(9), 8173–8182 (2012).
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Everitt, H. O.

A. Alabastri, X. Yang, A. Manjavacas, H. O. Everitt, and P. Nordlander, “Extraordinary light-induced local angular momentum near metallic nanoparticles,” ACS Nano 10(4), 4835–4846 (2016).
[Crossref] [PubMed]

Evlyukhin, A. B.

U. Zywietz, A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Laser printing of silicon nanoparticles with resonant optical electric and magnetic responses,” Nat. Commun. 5, 3402 (2014).
[Crossref] [PubMed]

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B 30(10), 2589–2598 (2013).
[Crossref]

Evlyukhin, E.

Feng, T.

T. Feng, Y. Xu, W. Zhang, and A. E. Miroshnichenko, “Ideal magnetic dipole scattering,” Phys. Rev. Lett. 118(17), 173901 (2017).
[Crossref] [PubMed]

Fu, Y. H.

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012).
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Garcia-Etxarri, A.

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

García-Etxarri, A.

T. Neuman, P. Alonso-González, A. García-Etxarri, M. Schnell, R. Hillenbrand, and J. Aizpurua, “Mapping the near fields of plasmonic nanoantennas by scattering-type scanning near-field optical microscopy,” Laser Photonics Rev. 9(6), 637–649 (2015).
[Crossref]

A. García-Etxarri, I. Romero, F. J. G. de Abajo, R. Hillenbrand, and J. Aizpurua, “Influence of the tip in near-field imaging of nanoparticle plasmonic modes: Weak and strong coupling regimes,” Phys. Rev. B 79(12), 125439 (2009).
[Crossref]

Ginzburg, P.

D. Permyakov, I. Sinev, D. Markovich, P. Ginzburg, A. Samusev, P. Belov, V. Valuckas, A. I. Kuznetsov, B. S. Luk’yanchuk, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Probing magnetic and electric optical responses of silicon nanoparticles,” Appl. Phys. Lett. 106(17), 171110 (2015).
[Crossref]

Girard, C.

A. Dereux, C. Girard, and J. C. Weeber, “Theoretical principles of near-field optical microscopies and spectroscopies,” J. Chem. Phys. 112(18), 7775–7789 (2000).
[Crossref]

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59(5), 657–699 (1996).
[Crossref]

Gonzaga, L.

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15(3), 2137–2142 (2015).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), 3rd ed.

Greffet, J. J.

K. Joulain, R. Carminati, J. P. Mulet, and J. J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68(24), 245405 (2003).
[Crossref]

J. A. Porto, R. Carminati, and J. J. Greffet, “Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy,” J. Appl. Phys. 88(8), 4845–4850 (2000).
[Crossref]

J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56(3), 133–237 (1997).
[Crossref]

Gross, H.

Guzhva, M. E.

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
[Crossref]

Halas, N. J.

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun. 2, 451 (2011).
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Hecht, B.

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

M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009).
[Crossref] [PubMed]

Herzig, H. P.

Hillenbrand, R.

T. Neuman, P. Alonso-González, A. García-Etxarri, M. Schnell, R. Hillenbrand, and J. Aizpurua, “Mapping the near fields of plasmonic nanoantennas by scattering-type scanning near-field optical microscopy,” Laser Photonics Rev. 9(6), 637–649 (2015).
[Crossref]

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

A. García-Etxarri, I. Romero, F. J. G. de Abajo, R. Hillenbrand, and J. Aizpurua, “Influence of the tip in near-field imaging of nanoparticle plasmonic modes: Weak and strong coupling regimes,” Phys. Rev. B 79(12), 125439 (2009).
[Crossref]

Huffman, D.

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998).
[Crossref]

Iyer, P. P.

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
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Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), 3rd ed.

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S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11(1), 23–36 (2016).
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Jahani, S.

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11(1), 23–36 (2016).
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Joulain, K.

K. Joulain, R. Carminati, J. P. Mulet, and J. J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68(24), 245405 (2003).
[Crossref]

Kabakova, I. V.

I. V. Kabakova, A. de Hoogh, R. E. C. van der Wel, M. Wulf, B. le Feber, and L. Kuipers, “Imaging of electric and magnetic fields near plasmonic nanowires,” Sci. Rep. 6, 22665 (2016).
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M. Kasperczyk, S. Person, D. Ananias, L. D. Carlos, and L. Novotny, “Excitation of magnetic dipole transitions at optical frequencies,” Phys. Rev. Lett. 114(16), 163903 (2015).
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Pertsch, T.

Porto, J. A.

J. A. Porto, R. Carminati, and J. J. Greffet, “Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy,” J. Appl. Phys. 88(8), 4845–4850 (2000).
[Crossref]

Reinhardt, C.

U. Zywietz, A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Laser printing of silicon nanoparticles with resonant optical electric and magnetic responses,” Nat. Commun. 5, 3402 (2014).
[Crossref] [PubMed]

A. B. Evlyukhin, C. Reinhardt, E. Evlyukhin, and B. N. Chichkov, “Multipole analysis of light scattering by arbitrary-shaped nanoparticles on a plane surface,” J. Opt. Soc. Am. B 30(10), 2589–2598 (2013).
[Crossref]

Rockstuhl, C.

Romero, I.

A. García-Etxarri, I. Romero, F. J. G. de Abajo, R. Hillenbrand, and J. Aizpurua, “Influence of the tip in near-field imaging of nanoparticle plasmonic modes: Weak and strong coupling regimes,” Phys. Rev. B 79(12), 125439 (2009).
[Crossref]

Ropers, C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1(1), 53–56 (2007).
[Crossref]

Rotenberg, N.

N. Rotenberg and L. Kuipers, “Mapping nanoscale light fields,” Nat. Photonics 8(12), 919–926 (2014).
[Crossref]

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nat. Photonics 8(1), 43–46 (2014).
[Crossref]

Sáenz, J. J.

R. Carminati and J. J. Sáenz, “Scattering theory of bardeen’s formalism for tunneling: New approach to near-field microscopy,” Phys. Rev. Lett. 84(22), 5156–5159 (2000).
[Crossref] [PubMed]

Samusev, A.

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15(3), 2137–2142 (2015).
[Crossref] [PubMed]

D. Permyakov, I. Sinev, D. Markovich, P. Ginzburg, A. Samusev, P. Belov, V. Valuckas, A. I. Kuznetsov, B. S. Luk’yanchuk, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Probing magnetic and electric optical responses of silicon nanoparticles,” Appl. Phys. Lett. 106(17), 171110 (2015).
[Crossref]

Samusev, A. K.

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
[Crossref]

Sandoghdar, V.

A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
[Crossref] [PubMed]

Schmidt, S.

Schnell, M.

T. Neuman, P. Alonso-González, A. García-Etxarri, M. Schnell, R. Hillenbrand, and J. Aizpurua, “Mapping the near fields of plasmonic nanoantennas by scattering-type scanning near-field optical microscopy,” Laser Photonics Rev. 9(6), 637–649 (2015).
[Crossref]

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

Schoenmaker, H.

M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009).
[Crossref] [PubMed]

Schotland, J. C.

J. Sun, P. S. Carney, and J. C. Schotland,“Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102(10), 103103 (2007).
[Crossref]

Schröter, U.

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 (2001).
[Crossref]

Schuller, J. A.

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

Sersic, I.

I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83(24), 245102 (2011).
[Crossref]

Sfez, T.

Silhanek, A. V.

D. Denkova, N. Verellen, A. V. Silhanek, P. V. Dorpe, and V. V. Moshchalkov, “Lateral magnetic near-field imaging of plasmonic nanoantennas with increasing complexity,” Small 10(10), 1959–1966 (2014).
[Crossref] [PubMed]

D. Denkova, N. Verellen, A. V. Silhanek, V. K. Valev, P. V. Dorpe, and V. V. Moshchalkov,“Mapping magnetic near-field distributions of plasmonic nanoantennas,” ACS Nano 7(4), 3168–3176 (2013).
[Crossref] [PubMed]

Simovski, C. R.

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
[Crossref]

Sinev, I.

D. Permyakov, I. Sinev, D. Markovich, P. Ginzburg, A. Samusev, P. Belov, V. Valuckas, A. I. Kuznetsov, B. S. Luk’yanchuk, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Probing magnetic and electric optical responses of silicon nanoparticles,” Appl. Phys. Lett. 106(17), 171110 (2015).
[Crossref]

Sinev, I. S.

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
[Crossref]

Singh, D. K.

Smirnova, D.

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref] [PubMed]

Steinert, M.

Stenberg, P.

Sun, J.

J. Sun, P. S. Carney, and J. C. Schotland,“Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102(10), 103103 (2007).
[Crossref]

Tuambilangana, C.

I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83(24), 245102 (2011).
[Crossref]

Urbach, H. P.

Valev, V. K.

D. Denkova, N. Verellen, A. V. Silhanek, V. K. Valev, P. V. Dorpe, and V. V. Moshchalkov,“Mapping magnetic near-field distributions of plasmonic nanoantennas,” ACS Nano 7(4), 3168–3176 (2013).
[Crossref] [PubMed]

Valuckas, V.

D. Permyakov, I. Sinev, D. Markovich, P. Ginzburg, A. Samusev, P. Belov, V. Valuckas, A. I. Kuznetsov, B. S. Luk’yanchuk, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Probing magnetic and electric optical responses of silicon nanoparticles,” Appl. Phys. Lett. 106(17), 171110 (2015).
[Crossref]

van der Wel, R. E. C.

I. V. Kabakova, A. de Hoogh, R. E. C. van der Wel, M. Wulf, B. le Feber, and L. Kuipers, “Imaging of electric and magnetic fields near plasmonic nanowires,” Sci. Rep. 6, 22665 (2016).
[Crossref] [PubMed]

van Hulst, N.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[Crossref]

van Oosten, D.

M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009).
[Crossref] [PubMed]

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102(3), 033902 (2009).
[Crossref] [PubMed]

Verellen, N.

D. Denkova, N. Verellen, A. V. Silhanek, P. V. Dorpe, and V. V. Moshchalkov, “Lateral magnetic near-field imaging of plasmonic nanoantennas with increasing complexity,” Small 10(10), 1959–1966 (2014).
[Crossref] [PubMed]

D. Denkova, N. Verellen, A. V. Silhanek, V. K. Valev, P. V. Dorpe, and V. V. Moshchalkov,“Mapping magnetic near-field distributions of plasmonic nanoantennas,” ACS Nano 7(4), 3168–3176 (2013).
[Crossref] [PubMed]

Vogelgesang, R.

M. Esslinger and R. Vogelgesang,“Reciprocity theory of apertureless scanning near-field optical microscopy with point-dipole probes,” ACS Nano 6(9), 8173–8182 (2012).
[Crossref] [PubMed]

Voroshilov, P. M.

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
[Crossref]

Weeber, J. C.

A. Dereux, C. Girard, and J. C. Weeber, “Theoretical principles of near-field optical microscopies and spectroscopies,” J. Chem. Phys. 112(18), 7775–7789 (2000).
[Crossref]

Woo, D. H.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1(1), 53–56 (2007).
[Crossref]

Wulf, M.

I. V. Kabakova, A. de Hoogh, R. E. C. van der Wel, M. Wulf, B. le Feber, and L. Kuipers, “Imaging of electric and magnetic fields near plasmonic nanowires,” Sci. Rep. 6, 22665 (2016).
[Crossref] [PubMed]

Xu, Y.

T. Feng, Y. Xu, W. Zhang, and A. E. Miroshnichenko, “Ideal magnetic dipole scattering,” Phys. Rev. Lett. 118(17), 173901 (2017).
[Crossref] [PubMed]

Yang, X.

A. Alabastri, X. Yang, A. Manjavacas, H. O. Everitt, and P. Nordlander, “Extraordinary light-induced local angular momentum near metallic nanoparticles,” ACS Nano 10(4), 4835–4846 (2016).
[Crossref] [PubMed]

Yoon, Y. C.

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1(1), 53–56 (2007).
[Crossref]

Youngworth, K.

Yu, L.

Yu, Y. F.

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15(3), 2137–2142 (2015).
[Crossref] [PubMed]

Zhang, J.

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012).
[Crossref] [PubMed]

Zhang, W.

T. Feng, Y. Xu, W. Zhang, and A. E. Miroshnichenko, “Ideal magnetic dipole scattering,” Phys. Rev. Lett. 118(17), 173901 (2017).
[Crossref] [PubMed]

Zywietz, U.

U. Zywietz, A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Laser printing of silicon nanoparticles with resonant optical electric and magnetic responses,” Nat. Commun. 5, 3402 (2014).
[Crossref] [PubMed]

ACS Nano (3)

D. Denkova, N. Verellen, A. V. Silhanek, V. K. Valev, P. V. Dorpe, and V. V. Moshchalkov,“Mapping magnetic near-field distributions of plasmonic nanoantennas,” ACS Nano 7(4), 3168–3176 (2013).
[Crossref] [PubMed]

M. Esslinger and R. Vogelgesang,“Reciprocity theory of apertureless scanning near-field optical microscopy with point-dipole probes,” ACS Nano 6(9), 8173–8182 (2012).
[Crossref] [PubMed]

A. Alabastri, X. Yang, A. Manjavacas, H. O. Everitt, and P. Nordlander, “Extraordinary light-induced local angular momentum near metallic nanoparticles,” ACS Nano 10(4), 4835–4846 (2016).
[Crossref] [PubMed]

ACS Photonics (2)

S. Kruk and Y. Kivshar, “Functional meta-optics and nanophotonics govern by mie resonances,” ACS Photonics 4(11), 2638–2649 (2017).
[Crossref]

F. B. Arango, T. Coenen, and A. F. Koenderink, “Underpinning hybridization intuition for complex nanoantennas by magnetoelectric quadrupolar polarizability retrieval,” ACS Photonics 1(5), 444–453 (2014).
[Crossref]

Appl. Phys. Lett. (1)

D. Permyakov, I. Sinev, D. Markovich, P. Ginzburg, A. Samusev, P. Belov, V. Valuckas, A. I. Kuznetsov, B. S. Luk’yanchuk, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Probing magnetic and electric optical responses of silicon nanoparticles,” Appl. Phys. Lett. 106(17), 171110 (2015).
[Crossref]

J. Appl. Phys. (2)

J. Sun, P. S. Carney, and J. C. Schotland,“Strong tip effects in near-field scanning optical tomography,” J. Appl. Phys. 102(10), 103103 (2007).
[Crossref]

J. A. Porto, R. Carminati, and J. J. Greffet, “Theory of electromagnetic field imaging and spectroscopy in scanning near-field optical microscopy,” J. Appl. Phys. 88(8), 4845–4850 (2000).
[Crossref]

J. Chem. Phys. (1)

A. Dereux, C. Girard, and J. C. Weeber, “Theoretical principles of near-field optical microscopies and spectroscopies,” J. Chem. Phys. 112(18), 7775–7789 (2000).
[Crossref]

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

Laser Photonics Rev. (1)

T. Neuman, P. Alonso-González, A. García-Etxarri, M. Schnell, R. Hillenbrand, and J. Aizpurua, “Mapping the near fields of plasmonic nanoantennas by scattering-type scanning near-field optical microscopy,” Laser Photonics Rev. 9(6), 637–649 (2015).
[Crossref]

Metamaterials (1)

S. Mühlig, C. Menzel, C. Rockstuhl, and F. Lederer, “Multipole analysis of meta-atoms,” Metamaterials 5(2–3), 64–73 (2011).
[Crossref]

Nano Lett. (2)

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15(3), 2137–2142 (2015).
[Crossref] [PubMed]

M. Schnell, A. Garcia-Etxarri, J. Alkorta, J. Aizpurua, and R. Hillenbrand, “Phase-resolved mapping of the near-field vector and polarization state in nanoscale antenna gaps,” Nano Lett. 10(9), 3524–3528 (2010).
[Crossref] [PubMed]

Nanoscale (1)

I. S. Sinev, P. M. Voroshilov, I. S. Mukhin, A. I. Denisyuk, M. E. Guzhva, A. K. Samusev, P. A. Belov, and C. R. Simovski, “Demonstration of unusual nanoantenna array modes through direct reconstruction of the near-field signal,” Nanoscale 7(2), 765–770 (2015).
[Crossref]

Nat. Commun. (3)

H. W. Kihm, S. M. Koo, Q. H. Kim, K. Bao, J. E. Kihm, W. S. Bak, S. H. Eah, C. Lienau, H. Kim, P. Nordlander, N. J. Halas, N. K. Park, and D. S. Kim, “Bethe-hole polarization analyser for the magnetic vector of light,” Nat. Commun. 2, 451 (2011).
[Crossref] [PubMed]

U. Zywietz, A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Laser printing of silicon nanoparticles with resonant optical electric and magnetic responses,” Nat. Commun. 5, 3402 (2014).
[Crossref] [PubMed]

K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11(1), 23–36 (2016).
[Crossref] [PubMed]

Nat. Photonics (5)

A. Aiello, P. Banzer, M. Neugebaueru, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photonics 9(12), 789–795 (2015).
[Crossref]

K. G. Lee, H. W. Kihm, J. E. Kihm, W. J. Choi, H. Kim, C. Ropers, D. J. Park, Y. C. Yoon, S. B. Choi, D. H. Woo, J. Kim, B. Lee, Q. H. Park, C. Lienau, and D. S. Kim, “Vector field microscopic imaging of light,” Nat. Photonics 1(1), 53–56 (2007).
[Crossref]

N. Rotenberg and L. Kuipers, “Mapping nanoscale light fields,” Nat. Photonics 8(12), 919–926 (2014).
[Crossref]

B. le Feber, N. Rotenberg, D. M. Beggs, and L. Kuipers, “Simultaneous measurement of nanoscale electric and magnetic optical fields,” Nat. Photonics 8(1), 43–46 (2014).
[Crossref]

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[Crossref]

New J. Phys. (1)

F. B. Arango and A. F. Koenderink, “Polarizability tensor retrieval for magnetic and plasmonic antenna design,” New J. Phys. 15(7), 073023 (2013).
[Crossref]

Opt. Express (5)

Phys. Rev. A (1)

K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 061801 (2012).
[Crossref]

Phys. Rev. B (5)

I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83(24), 245102 (2011).
[Crossref]

A. García-Etxarri, I. Romero, F. J. G. de Abajo, R. Hillenbrand, and J. Aizpurua, “Influence of the tip in near-field imaging of nanoparticle plasmonic modes: Weak and strong coupling regimes,” Phys. Rev. B 79(12), 125439 (2009).
[Crossref]

K. Joulain, R. Carminati, J. P. Mulet, and J. J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68(24), 245405 (2003).
[Crossref]

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B 64(12), 125420 (2001).
[Crossref]

Phys. Rev. Lett. (7)

R. Carminati and J. J. Sáenz, “Scattering theory of bardeen’s formalism for tunneling: New approach to near-field microscopy,” Phys. Rev. Lett. 84(22), 5156–5159 (2000).
[Crossref] [PubMed]

J. Bardeen, “Tunnelling from a many-particle point of view,” Phys. Rev. Lett. 6(2), 57–59 (1961).
[Crossref]

M. Kasperczyk, S. Person, D. Ananias, L. D. Carlos, and L. Novotny, “Excitation of magnetic dipole transitions at optical frequencies,” Phys. Rev. Lett. 114(16), 163903 (2015).
[Crossref] [PubMed]

M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, “Measuring the transverse spin density of light,” Phys. Rev. Lett. 114(6), 063901 (2015).
[Crossref] [PubMed]

M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102(3), 033902 (2009).
[Crossref] [PubMed]

A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
[Crossref] [PubMed]

T. Feng, Y. Xu, W. Zhang, and A. E. Miroshnichenko, “Ideal magnetic dipole scattering,” Phys. Rev. Lett. 118(17), 173901 (2017).
[Crossref] [PubMed]

Phys. Rev. X (2)

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5(1), 011039 (2015).

M. Neugebauer, J. Eismann, T. Bauer, and P. Banzer, “Magnetic and electric transverse spin density of spatially confined light,” Phys. Rev. X 8(2), 021042 (2018).

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J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56(3), 133–237 (1997).
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D. Courjon and C. Bainier,“Near field microscopy and near field optics,” Rep. Prog. Phys. 57(10), 989–1028 (1994).
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[Crossref]

Sci. Rep. (2)

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012).
[Crossref] [PubMed]

I. V. Kabakova, A. de Hoogh, R. E. C. van der Wel, M. Wulf, B. le Feber, and L. Kuipers, “Imaging of electric and magnetic fields near plasmonic nanowires,” Sci. Rep. 6, 22665 (2016).
[Crossref] [PubMed]

Science (3)

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin hall effect of light,” Science 348(6242), 1448–1451 (2015).
[Crossref] [PubMed]

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), 2472 (2016).
[Crossref] [PubMed]

M. Burresi, D. van Oosten, T. Kampfrath, H. Schoenmaker, R. Heideman, A. Leinse, and L. Kuipers, “Probing the magnetic field of light at optical frequencies,” Science 326(5952), 550–553 (2009).
[Crossref] [PubMed]

Small (1)

D. Denkova, N. Verellen, A. V. Silhanek, P. V. Dorpe, and V. V. Moshchalkov, “Lateral magnetic near-field imaging of plasmonic nanoantennas with increasing complexity,” Small 10(10), 1959–1966 (2014).
[Crossref] [PubMed]

Other (5)

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

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1998).
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J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), 3rd ed.

E. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999), 3rd ed.

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Figures (5)

Fig. 1
Fig. 1 Schematic of reciprocity in near-field optics. (a) Experimental scenario: a source Iexp located at ra illuminates the tip-sample system, and a detector at rb obtains the far-field signal Eexp. Near fields {Eexp, Hexp} in an evaluation plane Σ are determined by the probe and sample simultaneously. (b) Reciprocal scenario: a source Irec at rb emits radiation, and a detector at ra gets the far-field signal Erec. Near fields {Erec, Hrec} in Σ are solely determined by the probe, which could be approximated by a series of multipolar moments (e.g., the electric dipole moment ptip, the magnetic dipole moment mtip, etc.).
Fig. 2
Fig. 2 Schematic of the design paradigm for the nanoprobes. Depending on the basic vectorial features of the optical antennas, a nanoprobe with preliminary parameters of material mat and geometry geo is selected. According to the SNOM scheme in the experimental scenario, the incident angle (θ, ϕ) and polarization state pol of the reciprocal source are set and this source excites the volume of the nanoprobe (i.e., the gray box). The reciprocal dipole moments p, m and the FOM BE(ω) are calculated by introducing multipole expansion to the nanoprobe. These FOMs can be regarded as feedbacks to optimize the design of the nanoprobes.
Fig. 3
Fig. 3 Simulations of the optical magnetism detection with the split-ring probe. (a) Schematic of the tip-sample system. The split-ring probe is approximated by a split-ring resonator (SRR). (b) Distributions of the reciprocal dipole moments of the SRR. The solid and dashed curves are for circumstances where the electric field E is parallel and perpendicular to the facets of the split, respectively. (c,d) and (e,f) are distributions of the complex amplitudes along the x-axis with the split in the +x and +y direction, respectively, where (c,e) for channel x (Chx) and (d,f) for channel y (Chy). RR, CM and CMM represent the rigorous reciprocity, the convolution model and the coupled moment model. CMM2 in (f) stands for CMM that contains the electric quadrupole component.
Fig. 4
Fig. 4 BE(ω) of gold and silicon nanoparticles. (a) BE(ω) of gold nanoparticles (GNPs) with different radii. The solid white line locates electric dipole (ED) resonances. (b) BE(ω) of silicon nanoparticles (SiNPs) with different radii. The dashed white line and solid white line locate the magnetic dipole (MD) and ED resonances, respectively. (c,d) Scattering cross section (SCS) for different multipole components calculated by Mie theory for SiNP with radius 78 nm and SiNP with radius 112 nm, respectively. The dotted line in (c) indicates location of solid black square in MD branch in (b). The dashed orange circle in (d) shows multipole distributions at the location of open black square in fake branch in (b).
Fig. 5
Fig. 5 Simulations of intensity and transverse spin angular momentum (tSAM) detection with nanoparticles (NPs) based on CMM. (a) Optical set-up. NP approaches to the focal plane of the cylindrical-vector beams and scatters light to the far-field detector. (b) BE(ω) of the selected GNP (solid red line) and SiNP (dashed blue line). Dotted lines in (b) locate the wavelengths using in (c–r) where λ1 = 530 nm for Row 1 [(c,g,k,o)] and Row 2 [(d,h,l,p)], λ2 = 590 nm for Row 3 [(e,i,m,q)] and λ3 = 630 nm for Row 4 [(f,j,n,r)]. (c–f) and (g–j) depict the intensity Iz and corresponding tSAM for radially polarized light, while (k–n) and (o–r) show the counterparts for azimuthally polarized light. Insets represent the theoretical distributions of intensity and tSAM.

Equations (48)

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E exp I rec | r b E rec I exp | r a = 1 i ω μ 0 d S ( E 2 , j z E 1 , j E 1 , j z E 2 , j ) .
pol ( r tip ) = V d 3 r S E ( + ) ( r ) J tip , P ( r ) ,
S Ψ ( r ) = ξ = ± 1 1 2 π d 2 K S Ψ ξ ( K , z 0 ) exp [ i K R + i ξ γ ( z z 0 ) ] ,
S Ψ ξ ( K , z ) = ξ S Ψ ( K , z + Δ z ) S Ψ ( K , z ) exp ( i ξ γ Δ z ) 2 sinh ( i γ Δ z ) .
pol ( r tip ) = i ω p tip S E ( + ) ( r tip ) + i ω m tip S B ( + ) ( r tip ) 1 6 i ω Q tip : S E ( + ) ( r tip ) + ,
pol exp ( i ω t ) = t [ H ^ int exp ( i ω t ) ] ,
pol ( r tip ) i ω p tip E sample ( + ) ( r tip ) + i ω m tip B sample ( + ) ( r tip ) 1 6 i ω Q tip : E sample ( + ) ( r tip ) + ,
pol ( r tip ) i ω p tip E sample ( + ) ( r tip ) + i ω 1 c m tip Z 0 H sample ( + ) ( r tip ) ,
BE ( ω ) = 10 log 10 { 1 c | m tip ( ω ) | | p tip ( ω ) | }
{ s x Im [ E y * E z + Z 0 2 H y * H z ] s y Im [ E z * E x + Z 0 2 H z * H x ] , s z Im [ E x * E y + Z 0 2 H x * H y ]
{ x i ω p E x + i ω Z 0 c m H z z i ω p E z i ω Z 0 c m H x ,
Im { z * x } + Im { E z * E x } + 10 0.2 B E Z 0 2 Im { H z * H x } + 10 0.1 B E Z 0 Im { e i θ H x * E x } 10 0.1 B E Z 0 Im { e i θ E z * H z } ,
Im { z * x } + Im { E z * E x } + 10 0.2 B E Z 0 2 Im { H z * H x } + 10 0.1 B E Z 0 Im { e i θ H x * E x } .
Im { z * x } Im { E z * E x } s E y
Im { z * x } Z 0 2 Im { H z * H x } s H y .
Im { z * x } s y + Z 0 Im { e i θ H x * E x } .
× × E + k 0 2 E = ω 2 μ 0 ε 0 ( ε ˜ r I ) E ,
× × E ( r ) + k 0 2 E ( r ) = i ω μ 0 J P ( r ) ,
× × G ( r , r ) + k 0 2 G ( r , r ) = I δ ( r r )
{ E s ( r ) = i ω μ 0 V d 3 r G ( r , r ) J P ( r ) H s ( r ) = V d 3 r [ × G ( r , r ) ] J P ( r ) .
G ( r , r ) = ( I + 1 k 0 ) G ( r , r ) = ( I + 1 k 0 ) exp ( i k 0 | r r | ) 4 π | r r | .
G ( r , r ) = i 8 π 2 d 2 K { 1 γ exp [ i γ ( z z ) + i K ( R R ) ] } ,
( E 2 × H 1 E 1 × H 2 ) = 1 i ω μ 0 [ i ( i j k E 2 , j k m n m E 1 , n ) i ( i j k E 1 , j k m n m E 2 , n ) ] = 1 i ω μ 0 i ( E 2 , j i E 1 , j E 1 , j i E 2 , j ) ,
E 2 I 1 | r b E 1 I 2 | r a = V d 3 ( E 2 × H 1 E 1 × H 2 ) = d S [ e ^ z ( E 2 × H 1 E 1 × H 2 ) ] = 1 i ω μ 0 d S ( E 2 , j z E 1 , j E 1 , j z E 2 , j ) .
T ( r ) = k 0 2 V d 3 r G ( r , r ) χ ( r ) T ( r ) = k 0 2 V d 3 r e ^ j T m ( r ) χ m i ( r ) ( δ i j + i j k 0 2 ) G ( r , r ) .
1 i ω μ 0 d S ( E 2 , j z E 1 , j E 1 , j z E 2 , j ) = i ω ε 0 d 3 r { T m ( r ) χ m i ( r ) ( δ i j + i j k 0 2 ) S E , j ( + ) ( r ) }
1 i ω μ 0 d S ( E 2 , j z E 1 , j E 1 , j z E 2 , j ) = i ω ε 0 d 3 r { T m ( r ) χ m j ( r ) S E , j ( + ) ( r ) } = V d 3 r S E ( + ) ( r ) J tip , P ( r ) .
1 i ω μ 0 d S ( E 2 , j z E 1 , j E 1 , j z E 2 , j ) = 1 2 π d 2 K { exp ( i K R tip ) S E ( + ) ( K , z tip ) [ d 3 r J tip , P ( r ) exp ( i k r ) ] } .
d 3 r J tip , P ( r ) exp ( i k r ) = d 3 r J tip , P ( r ) + i k d 3 r r J tip , P ( r ) + m 2 i m m ! d 3 r J tip , P ( r ) ( k r ) m .
p tip = 1 i ω V d 3 r J tip , P ( r ) ,
J tip , P ( r ) + t ρ tip , P ( r ) = 0 ,
k V d 3 r { 1 2 [ r J tip , P ( r ) J tip , P ( r ) r ] } = 1 2 V d 3 r { m n s m j k e ^ s r j J tip , P , k ( r ) a n } = 1 2 k × V d 3 r { r × J tip , P ( r ) } = k × m tip ,
V d 3 r { 1 2 [ r J tip , P ( r ) + J tip , P ( r ) r ] } = i ω 6 ( Q tip + I V d 3 r r 2 ρ ( r ) ) ,
d 3 r J tip , P ( r ) exp ( i k r ) = i ω p tip i k × m tip + 1 6 ω k Q tip = 1 6 ω k I V d 3 r r 2 ρ ( r ) + m 2 i m m ! d 3 r J tip , P ( r ) ( k r ) m .
i ω p tip 1 2 π d 2 K { S E ( + ) ( K , z tip ) exp ( i K R tip ) } = i ω p tip S E ( + ) ( r tip ) .
i 2 π d 2 K { S E ( + ) ( K , z tip ) ( k × m tip ) exp ( i K R tip ) } = i ω μ 0 m tip 1 2 π d 2 K { S H ( + ) ( K , z tip ) exp ( i K R tip ) } = i ω μ 0 m tip S H ( + ) ( r tip ) ,
ω 12 π d 2 K { ( k Q tip ) S E ( + ) ( K , z tip ) exp ( i K R tip ) } = ω 12 π d 2 K { ( Q tip : k S E ( + ) ( K , z tip ) exp ( i K R tip ) } = 1 6 ω Q tip : 1 2 π d 2 K { ( i ) S E ( + ) ( K , z tip ) exp ( i K R tip ) } = 1 6 ω Q tip : S E ( + ) ( r tip ) .
ω 6 d 2 K { ( k I ) S E ( + ) ( K , z tip ) exp ( i K R tip ) } = ω 6 d 2 K { k S E ( + ) ( K , z tip ) exp ( i K R tip ) } = 0 .
1 i ω μ 0 d S ( E 2 , j z E 1 , j E 1 , j z E 2 , j ) = ω p tip S E ( + ) ( r tip ) + i ω m tip S B ( + ) ( r tip ) 1 6 i ω Q tip : S E ( + ) ( r tip ) + ,
E 2 I 1 | r b E 1 I 2 | r a = [ ( E 2 × H 1 E 1 × H 2 ) ] d S ,
pol = [ E 2 , x H 1 , y E 2 , y H 1 , x E 1 , x H 2 , y + E 1 , y H 2 , y ] d S .
{ ψ exp ( R , z 0 ; R tip , z tip ) ψ sample ( R ) | z = z 0 ψ rec ( R , z 0 ; R tip , z tip ) = ψ probe ( R R tip ) | z = z 0 z tip = ψ probe ( i ) ( R tip R ) | z = z 0 z tip ,
d S ψ exp ( R , z 0 ; R tip , z tip ) ψ rec ( R , z 0 ; R tip , z tip ) d S ψ sample ( R ) | z = z 0 ψ probe ( i ) ( R tip R ) | z = z 0 z tip .
d S ψ sample ( R ) ψ probe ( i ) ( R tip R ) = d x d y ψ sample ( x , y ) ψ probe ( i ) ( x tip x , y tip y ) = g ( x tip , y tip ) .
d S ψ exp ( R , z 0 ; R tip , z tip ) ψ rec ( R , z 0 ; R tip , z tip ) ψ sample ( R tip ) | z = z 0 * ψ probe ( i ) ( R tip ) | z = z 0 z tip .
pol [ E sample , x * H probe , y ( i , ) + H sample , x * H probe , y ( i , ) ] [ E sample , y * H probe , x ( i , ) + H sample , y * H probe , x ( i , ) ] ,
{ E ρ = A α 1 α 2 cos 1 2 θ sin 2 θ J 1 ( k ρ sin θ ) e i k z cos θ d θ E z 2 i A α 1 α 2 cos 1 2 θ sin 2 θ J 0 ( k ρ sin θ ) e i k z cos θ d θ , H ϕ 2 Z A α 1 α 2 cos 1 2 θ sin θ J 1 ( k ρ sin θ ) e i k z cos θ d θ
{ E ϕ = 2 A α 1 α 2 cos 1 2 θ sin θ J 1 ( k ρ sin θ ) e i k z cos θ d θ H ρ A Z α 1 α 2 cos 1 2 θ sin 2 θ J 1 ( k ρ sin θ ) e i k z cos θ d θ , H z 2 i A Z α 1 α 2 cos 1 2 θ sin 2 θ J 0 ( k ρ sin θ ) e i k z cos θ d θ