Abstract

We systematically investigate the generation of optical chirality patterns by applying the superposition of two waves in three scenarios, namely free-space plane waves, evanescent waves of totally reflected light at dielectric interface and propagating surface plasmon waves on a metallic surface. In each scenario, the general analytical solution of the optical chirality pattern is derived for different polarization states and propagating directions of the two waves. The analytical solutions are verified by numerical simulations. Spatially structured optical chirality patterns can be generated in all scenarios if the incident polarization states and propagation directions are correctly chosen. Optical chirality enhancement can be obtained from the constructive interference of free-space circularly polarized light or enhanced evanescent waves of totally reflected light. Surface plasmon waves do not provide enhanced optical chirality unless the near-field intensity enhancement is sufficiently high. The structured optical chirality patterns may find applications in chirality sorting, chiral imaging and circular dichroism spectroscopy.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (8)

H. Hu, Q. Gan, and Q. Zhan, “Generation of a nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

M. L. Solomon, J. Hu, M. Lawrence, A. García-Etxarri, and J. A. Dionne, “Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces,” ACS Photonics 6(1), 43–49 (2019).
[Crossref]

K. Yao and Y. Zheng, “Near-ultraviolet dielectric metasurfaces: from surface-enhanced circular dichroism spectroscopy to polarization-preserving mirrors,” J. Phys. Chem. C 123(18), 11814–11822 (2019).
[Crossref]

X. Zhao and B. M. Reinhard, “Switchable chiroptical hot-spots in silicon nanodisk dimers,” ACS Photonics 6(8), 1981–1989 (2019).
[Crossref]

M. L. Tseng, Z. H. Lin, H. Y. Kuo, T. T. Huang, Y. T. Huang, T. L. Chung, C. H. Chu, J. S. Huang, and D. P. Tsai, “Stress-induced 3D chiral fractal metasurface for enhanced and stabilized broadband near-field optical chirality,” Adv. Opt. Mater. 7(15), 1900617 (2019).
[Crossref]

Q. Zhang, J. Li, and X. Liu, “Optical lateral forces and torques induced by chiral surface-plasmon-polaritons and their potential applications in recognition and separation of chiral enantiomers,” Phys. Chem. Chem. Phys. 21(3), 1308–1314 (2019).
[Crossref]

M. Li, S. Yan, Y. Zhang, Y. Liang, P. Zhang, and B. Yao, “Optical sorting of small chiral particles by tightly focused vector beams,” Phys. Rev. A 99(3), 033825 (2019).
[Crossref]

O. Arteaga, Z. El-Hachemi, and R. Ossikovski, “Snapshot circular dichroism measurements,” Opt. Express 27(5), 6746–6756 (2019).
[Crossref]

2018 (4)

K. C. Van Kruining, R. P. Cameron, and J. Götte, “Superpositions of up to six plane waves without electric-field interference,” Optica 5(9), 1091–1098 (2018).
[Crossref]

E. Mohammadi, K. Tsakmakidis, A. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
[Crossref]

G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Surface-enhanced chiroptical spectroscopy with superchiral surface waves,” Chirality 30(7), 883–889 (2018).
[Crossref]

J. García-Guirado, M. Svedendahl, J. Puigdollers, and R. Quidant, “Enantiomer-selective molecular sensing using racemic nanoplasmonic arrays,” Nano Lett. 18(10), 6279–6285 (2018).
[Crossref]

2017 (7)

G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
[Crossref]

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Appl. Phys. Lett. 110(22), 221108 (2017).
[Crossref]

C. Kramer, M. Schäferling, T. Weiss, H. Giessen, and T. Brixner, “Analytic optimization of near-field optical chirality enhancement,” ACS Photonics 4(2), 396–406 (2017).
[Crossref]

Y. Luo, C. Chi, M. Jiang, R. Li, S. Zu, Y. Li, and Z. Fang, “Plasmonic chiral nanostructures: chiroptical effects and applications,” Adv. Opt. Mater. 5(16), 1700040 (2017).
[Crossref]

J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and chiroptical effects in metal nanostructures: fundamentals and current trends,” Adv. Opt. Mater. 5(16), 1700182 (2017).
[Crossref]

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3(5), e1602735 (2017).
[Crossref]

T. Zhang, M. R. C. Mahdy, Y. Liu, J. H. Teng, C. T. Lim, Z. Wang, and C.-W. Qiu, “All-optical chirality-sensitive sorting via reversible lateral forces in interference fields,” ACS Nano 11(4), 4292–4300 (2017).
[Crossref]

2016 (5)

H. Chen, C. Liang, S. Liu, and Z. Lin, “Chirality sorting using two-wave-interference-induced lateral optical force,” Phys. Rev. A 93(5), 053833 (2016).
[Crossref]

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6(1), 36884 (2016).
[Crossref]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 3(4), 578–583 (2016).
[Crossref]

M. Schäferling, N. Engheta, H. Giessen, and T. Weiss, “Reducing the complexity: Enantioselective chiral near-fields by diagonal slit and mirror configuration,” ACS Photonics 3(6), 1076–1084 (2016).
[Crossref]

S. Zu, Y. Bao, and Z. Fang, “Planar plasmonic chiral nanostructures,” Nanoscale 8(7), 3900–3905 (2016).
[Crossref]

2015 (3)

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5(1), 17534 (2015).
[Crossref]

M. H. Alizadeh and B. M. Reinhard, “Transverse chiral optical forces by chiral surface plasmon polaritons,” ACS Photonics 2(12), 1780–1788 (2015).
[Crossref]

A. Hayat, J. B. Mueller, and F. Capasso, “Lateral chirality-sorting optical forces,” Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015).
[Crossref]

2014 (7)

S. Wang and C. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014).
[Crossref]

G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5(1), 3577 (2014).
[Crossref]

A. Canaguier-Durand and C. Genet, “Chiral near fields generated from plasmonic optical lattices,” Phys. Rev. A 90(2), 023842 (2014).
[Crossref]

M. Schäferling, X. Yin, N. Engheta, and H. Giessen, “Helical plasmonic nanostructures as prototypical chiral near-field sources,” ACS Photonics 1(6), 530–537 (2014).
[Crossref]

S. Hashiyada, T. Narushima, and H. Okamoto, “Local optical activity in achiral two-dimensional gold nanostructures,” J. Phys. Chem. C 118(38), 22229–22233 (2014).
[Crossref]

D. Lin and J.-S. Huang, “Slant-gap plasmonic nanoantennas for optical chirality engineering and circular dichroism enhancement,” Opt. Express 22(7), 7434–7445 (2014).
[Crossref]

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity of interfering waves,” J. Mod. Opt. 61(1), 25–31 (2014).
[Crossref]

2013 (3)

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
[Crossref]

T. Davis and E. Hendry, “Superchiral electromagnetic fields created by surface plasmons in nonchiral metallic nanostructures,” Phys. Rev. B 87(8), 085405 (2013).
[Crossref]

A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013).
[Crossref]

2012 (3)

M. Schäferling, X. Yin, and H. Giessen, “Formation of chiral fields in a symmetric environment,” Opt. Express 20(24), 26326–26336 (2012).
[Crossref]

F. Eftekhari and T. Davis, “Strong chiral optical response from planar arrays of subwavelength metallic structures supporting surface plasmon resonances,” Phys. Rev. B 86(7), 075428 (2012).
[Crossref]

M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Phys. Rev. X 2(3), 031010 (2012).
[Crossref]

2011 (2)

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332(6027), 333–336 (2011).
[Crossref]

N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole Transitions,” J. Phys. Chem. B 115(18), 5304–5311 (2011).
[Crossref]

2010 (2)

E. Hendry, T. Carpy, J. Johnston, M. Popland, R. Mikhaylovskiy, A. Lapthorn, S. Kelly, L. Barron, N. Gadegaard, and M. Kadodwala, “Ultrasensitive detection and characterization of biomolecules using superchiral fields,” Nat. Nanotechnol. 5(11), 783–787 (2010).
[Crossref]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104(16), 163901 (2010).
[Crossref]

2008 (1)

1964 (1)

D. M. Lipkin, “Existence of a new conservation law in electromagnetic theory,” J. Math. Phys. 5(5), 696–700 (1964).
[Crossref]

Adamo, G.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Appl. Phys. Lett. 110(22), 221108 (2017).
[Crossref]

Alizadeh, M. H.

M. H. Alizadeh and B. M. Reinhard, “Transverse chiral optical forces by chiral surface plasmon polaritons,” ACS Photonics 2(12), 1780–1788 (2015).
[Crossref]

Altug, H.

E. Mohammadi, K. Tsakmakidis, A. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
[Crossref]

Andronaki, S. A.

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6(1), 36884 (2016).
[Crossref]

Arteaga, O.

Askarpour, A.

E. Mohammadi, K. Tsakmakidis, A. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
[Crossref]

Baimuratov, A. S.

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6(1), 36884 (2016).
[Crossref]

Bao, Y.

S. Zu, Y. Bao, and Z. Fang, “Planar plasmonic chiral nanostructures,” Nanoscale 8(7), 3900–3905 (2016).
[Crossref]

Baranov, A. V.

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6(1), 36884 (2016).
[Crossref]

Barnett, S. M.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity of interfering waves,” J. Mod. Opt. 61(1), 25–31 (2014).
[Crossref]

Barron, L.

E. Hendry, T. Carpy, J. Johnston, M. Popland, R. Mikhaylovskiy, A. Lapthorn, S. Kelly, L. Barron, N. Gadegaard, and M. Kadodwala, “Ultrasensitive detection and characterization of biomolecules using superchiral fields,” Nat. Nanotechnol. 5(11), 783–787 (2010).
[Crossref]

Baumberg, J. J.

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
[Crossref]

Biagioni, P.

G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Surface-enhanced chiroptical spectroscopy with superchiral surface waves,” Chirality 30(7), 883–889 (2018).
[Crossref]

G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Brasselet, E.

G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5(1), 3577 (2014).
[Crossref]

Brixner, T.

C. Kramer, M. Schäferling, T. Weiss, H. Giessen, and T. Brixner, “Analytic optimization of near-field optical chirality enhancement,” ACS Photonics 4(2), 396–406 (2017).
[Crossref]

Cameron, R. P.

K. C. Van Kruining, R. P. Cameron, and J. Götte, “Superpositions of up to six plane waves without electric-field interference,” Optica 5(9), 1091–1098 (2018).
[Crossref]

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity of interfering waves,” J. Mod. Opt. 61(1), 25–31 (2014).
[Crossref]

Canaguier-Durand, A.

A. Canaguier-Durand and C. Genet, “Chiral near fields generated from plasmonic optical lattices,” Phys. Rev. A 90(2), 023842 (2014).
[Crossref]

A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013).
[Crossref]

Capasso, F.

A. Hayat, J. B. Mueller, and F. Capasso, “Lateral chirality-sorting optical forces,” Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015).
[Crossref]

Carpy, T.

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[Crossref]

Rukhlenko, I. D.

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6(1), 36884 (2016).
[Crossref]

Schäferling, M.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3(5), e1602735 (2017).
[Crossref]

C. Kramer, M. Schäferling, T. Weiss, H. Giessen, and T. Brixner, “Analytic optimization of near-field optical chirality enhancement,” ACS Photonics 4(2), 396–406 (2017).
[Crossref]

M. Schäferling, N. Engheta, H. Giessen, and T. Weiss, “Reducing the complexity: Enantioselective chiral near-fields by diagonal slit and mirror configuration,” ACS Photonics 3(6), 1076–1084 (2016).
[Crossref]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 3(4), 578–583 (2016).
[Crossref]

M. Schäferling, X. Yin, N. Engheta, and H. Giessen, “Helical plasmonic nanostructures as prototypical chiral near-field sources,” ACS Photonics 1(6), 530–537 (2014).
[Crossref]

M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Phys. Rev. X 2(3), 031010 (2012).
[Crossref]

M. Schäferling, X. Yin, and H. Giessen, “Formation of chiral fields in a symmetric environment,” Opt. Express 20(24), 26326–26336 (2012).
[Crossref]

Sibilia, C.

J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and chiroptical effects in metal nanostructures: fundamentals and current trends,” Adv. Opt. Mater. 5(16), 1700182 (2017).
[Crossref]

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
[Crossref]

Solomon, M. L.

M. L. Solomon, J. Hu, M. Lawrence, A. García-Etxarri, and J. A. Dionne, “Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces,” ACS Photonics 6(1), 43–49 (2019).
[Crossref]

Sun, H.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Appl. Phys. Lett. 110(22), 221108 (2017).
[Crossref]

Sun, M.

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5(1), 17534 (2015).
[Crossref]

Svedendahl, M.

J. García-Guirado, M. Svedendahl, J. Puigdollers, and R. Quidant, “Enantiomer-selective molecular sensing using racemic nanoplasmonic arrays,” Nano Lett. 18(10), 6279–6285 (2018).
[Crossref]

Tang, Y.

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332(6027), 333–336 (2011).
[Crossref]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104(16), 163901 (2010).
[Crossref]

Tavakoli, A.

E. Mohammadi, K. Tsakmakidis, A. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
[Crossref]

Teh, B. H.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Appl. Phys. Lett. 110(22), 221108 (2017).
[Crossref]

Teng, J.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Appl. Phys. Lett. 110(22), 221108 (2017).
[Crossref]

Teng, J. H.

T. Zhang, M. R. C. Mahdy, Y. Liu, J. H. Teng, C. T. Lim, Z. Wang, and C.-W. Qiu, “All-optical chirality-sensitive sorting via reversible lateral forces in interference fields,” ACS Nano 11(4), 4292–4300 (2017).
[Crossref]

Tepliakov, N. V.

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6(1), 36884 (2016).
[Crossref]

Tian, X.

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5(1), 17534 (2015).
[Crossref]

Tkachenko, G.

G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5(1), 3577 (2014).
[Crossref]

Tsai, D. P.

M. L. Tseng, Z. H. Lin, H. Y. Kuo, T. T. Huang, Y. T. Huang, T. L. Chung, C. H. Chu, J. S. Huang, and D. P. Tsai, “Stress-induced 3D chiral fractal metasurface for enhanced and stabilized broadband near-field optical chirality,” Adv. Opt. Mater. 7(15), 1900617 (2019).
[Crossref]

Tsakmakidis, K.

E. Mohammadi, K. Tsakmakidis, A. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
[Crossref]

Tseng, M. L.

M. L. Tseng, Z. H. Lin, H. Y. Kuo, T. T. Huang, Y. T. Huang, T. L. Chung, C. H. Chu, J. S. Huang, and D. P. Tsai, “Stress-induced 3D chiral fractal metasurface for enhanced and stabilized broadband near-field optical chirality,” Adv. Opt. Mater. 7(15), 1900617 (2019).
[Crossref]

Valev, V. K.

J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and chiroptical effects in metal nanostructures: fundamentals and current trends,” Adv. Opt. Mater. 5(16), 1700182 (2017).
[Crossref]

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
[Crossref]

Van Kruining, K. C.

Verbiest, T.

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
[Crossref]

Wang, S.

S. Wang and C. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014).
[Crossref]

Wang, Y.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Appl. Phys. Lett. 110(22), 221108 (2017).
[Crossref]

Wang, Z.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Appl. Phys. Lett. 110(22), 221108 (2017).
[Crossref]

T. Zhang, M. R. C. Mahdy, Y. Liu, J. H. Teng, C. T. Lim, Z. Wang, and C.-W. Qiu, “All-optical chirality-sensitive sorting via reversible lateral forces in interference fields,” ACS Nano 11(4), 4292–4300 (2017).
[Crossref]

Weiss, T.

C. Kramer, M. Schäferling, T. Weiss, H. Giessen, and T. Brixner, “Analytic optimization of near-field optical chirality enhancement,” ACS Photonics 4(2), 396–406 (2017).
[Crossref]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 3(4), 578–583 (2016).
[Crossref]

M. Schäferling, N. Engheta, H. Giessen, and T. Weiss, “Reducing the complexity: Enantioselective chiral near-fields by diagonal slit and mirror configuration,” ACS Photonics 3(6), 1076–1084 (2016).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

Yan, S.

M. Li, S. Yan, Y. Zhang, Y. Liang, P. Zhang, and B. Yao, “Optical sorting of small chiral particles by tightly focused vector beams,” Phys. Rev. A 99(3), 033825 (2019).
[Crossref]

Yang, N.

N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole Transitions,” J. Phys. Chem. B 115(18), 5304–5311 (2011).
[Crossref]

Yao, A. M.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity of interfering waves,” J. Mod. Opt. 61(1), 25–31 (2014).
[Crossref]

Yao, B.

M. Li, S. Yan, Y. Zhang, Y. Liang, P. Zhang, and B. Yao, “Optical sorting of small chiral particles by tightly focused vector beams,” Phys. Rev. A 99(3), 033825 (2019).
[Crossref]

Yao, K.

K. Yao and Y. Zheng, “Near-ultraviolet dielectric metasurfaces: from surface-enhanced circular dichroism spectroscopy to polarization-preserving mirrors,” J. Phys. Chem. C 123(18), 11814–11822 (2019).
[Crossref]

Yin, X.

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 3(4), 578–583 (2016).
[Crossref]

M. Schäferling, X. Yin, N. Engheta, and H. Giessen, “Helical plasmonic nanostructures as prototypical chiral near-field sources,” ACS Photonics 1(6), 530–537 (2014).
[Crossref]

M. Schäferling, X. Yin, and H. Giessen, “Formation of chiral fields in a symmetric environment,” Opt. Express 20(24), 26326–26336 (2012).
[Crossref]

Zhan, Q.

H. Hu, Q. Gan, and Q. Zhan, “Generation of a nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

Zhang, J.

S.-Y. Huang, J. Zhang, C. Karras, R. Förster, R. Heintzmann, and J.-S. Huang, “Chiral Structured Illumination Microscopy,” arXiv: 1908.09391 (2019).

Zhang, P.

M. Li, S. Yan, Y. Zhang, Y. Liang, P. Zhang, and B. Yao, “Optical sorting of small chiral particles by tightly focused vector beams,” Phys. Rev. A 99(3), 033825 (2019).
[Crossref]

Zhang, Q.

Q. Zhang, J. Li, and X. Liu, “Optical lateral forces and torques induced by chiral surface-plasmon-polaritons and their potential applications in recognition and separation of chiral enantiomers,” Phys. Chem. Chem. Phys. 21(3), 1308–1314 (2019).
[Crossref]

Zhang, T.

T. Zhang, M. R. C. Mahdy, Y. Liu, J. H. Teng, C. T. Lim, Z. Wang, and C.-W. Qiu, “All-optical chirality-sensitive sorting via reversible lateral forces in interference fields,” ACS Nano 11(4), 4292–4300 (2017).
[Crossref]

Zhang, Y.

M. Li, S. Yan, Y. Zhang, Y. Liang, P. Zhang, and B. Yao, “Optical sorting of small chiral particles by tightly focused vector beams,” Phys. Rev. A 99(3), 033825 (2019).
[Crossref]

Zhao, X.

X. Zhao and B. M. Reinhard, “Switchable chiroptical hot-spots in silicon nanodisk dimers,” ACS Photonics 6(8), 1981–1989 (2019).
[Crossref]

Zheng, Y.

K. Yao and Y. Zheng, “Near-ultraviolet dielectric metasurfaces: from surface-enhanced circular dichroism spectroscopy to polarization-preserving mirrors,” J. Phys. Chem. C 123(18), 11814–11822 (2019).
[Crossref]

Zu, S.

Y. Luo, C. Chi, M. Jiang, R. Li, S. Zu, Y. Li, and Z. Fang, “Plasmonic chiral nanostructures: chiroptical effects and applications,” Adv. Opt. Mater. 5(16), 1700040 (2017).
[Crossref]

S. Zu, Y. Bao, and Z. Fang, “Planar plasmonic chiral nanostructures,” Nanoscale 8(7), 3900–3905 (2016).
[Crossref]

ACS Nano (1)

T. Zhang, M. R. C. Mahdy, Y. Liu, J. H. Teng, C. T. Lim, Z. Wang, and C.-W. Qiu, “All-optical chirality-sensitive sorting via reversible lateral forces in interference fields,” ACS Nano 11(4), 4292–4300 (2017).
[Crossref]

ACS Photonics (8)

M. H. Alizadeh and B. M. Reinhard, “Transverse chiral optical forces by chiral surface plasmon polaritons,” ACS Photonics 2(12), 1780–1788 (2015).
[Crossref]

C. Kramer, M. Schäferling, T. Weiss, H. Giessen, and T. Brixner, “Analytic optimization of near-field optical chirality enhancement,” ACS Photonics 4(2), 396–406 (2017).
[Crossref]

M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 3(4), 578–583 (2016).
[Crossref]

M. Schäferling, N. Engheta, H. Giessen, and T. Weiss, “Reducing the complexity: Enantioselective chiral near-fields by diagonal slit and mirror configuration,” ACS Photonics 3(6), 1076–1084 (2016).
[Crossref]

E. Mohammadi, K. Tsakmakidis, A. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
[Crossref]

M. L. Solomon, J. Hu, M. Lawrence, A. García-Etxarri, and J. A. Dionne, “Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces,” ACS Photonics 6(1), 43–49 (2019).
[Crossref]

X. Zhao and B. M. Reinhard, “Switchable chiroptical hot-spots in silicon nanodisk dimers,” ACS Photonics 6(8), 1981–1989 (2019).
[Crossref]

M. Schäferling, X. Yin, N. Engheta, and H. Giessen, “Helical plasmonic nanostructures as prototypical chiral near-field sources,” ACS Photonics 1(6), 530–537 (2014).
[Crossref]

Adv. Mater. (1)

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
[Crossref]

Adv. Opt. Mater. (3)

M. L. Tseng, Z. H. Lin, H. Y. Kuo, T. T. Huang, Y. T. Huang, T. L. Chung, C. H. Chu, J. S. Huang, and D. P. Tsai, “Stress-induced 3D chiral fractal metasurface for enhanced and stabilized broadband near-field optical chirality,” Adv. Opt. Mater. 7(15), 1900617 (2019).
[Crossref]

Y. Luo, C. Chi, M. Jiang, R. Li, S. Zu, Y. Li, and Z. Fang, “Plasmonic chiral nanostructures: chiroptical effects and applications,” Adv. Opt. Mater. 5(16), 1700040 (2017).
[Crossref]

J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and chiroptical effects in metal nanostructures: fundamentals and current trends,” Adv. Opt. Mater. 5(16), 1700182 (2017).
[Crossref]

Appl. Phys. Lett. (1)

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Appl. Phys. Lett. 110(22), 221108 (2017).
[Crossref]

Chirality (1)

G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Surface-enhanced chiroptical spectroscopy with superchiral surface waves,” Chirality 30(7), 883–889 (2018).
[Crossref]

J. Math. Phys. (1)

D. M. Lipkin, “Existence of a new conservation law in electromagnetic theory,” J. Math. Phys. 5(5), 696–700 (1964).
[Crossref]

J. Mod. Opt. (1)

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Optical helicity of interfering waves,” J. Mod. Opt. 61(1), 25–31 (2014).
[Crossref]

J. Phys. Chem. B (1)

N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole Transitions,” J. Phys. Chem. B 115(18), 5304–5311 (2011).
[Crossref]

J. Phys. Chem. C (2)

S. Hashiyada, T. Narushima, and H. Okamoto, “Local optical activity in achiral two-dimensional gold nanostructures,” J. Phys. Chem. C 118(38), 22229–22233 (2014).
[Crossref]

K. Yao and Y. Zheng, “Near-ultraviolet dielectric metasurfaces: from surface-enhanced circular dichroism spectroscopy to polarization-preserving mirrors,” J. Phys. Chem. C 123(18), 11814–11822 (2019).
[Crossref]

Nano Lett. (1)

J. García-Guirado, M. Svedendahl, J. Puigdollers, and R. Quidant, “Enantiomer-selective molecular sensing using racemic nanoplasmonic arrays,” Nano Lett. 18(10), 6279–6285 (2018).
[Crossref]

Nanoscale (1)

S. Zu, Y. Bao, and Z. Fang, “Planar plasmonic chiral nanostructures,” Nanoscale 8(7), 3900–3905 (2016).
[Crossref]

Nat. Commun. (2)

S. Wang and C. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014).
[Crossref]

G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5(1), 3577 (2014).
[Crossref]

Nat. Nanotechnol. (1)

E. Hendry, T. Carpy, J. Johnston, M. Popland, R. Mikhaylovskiy, A. Lapthorn, S. Kelly, L. Barron, N. Gadegaard, and M. Kadodwala, “Ultrasensitive detection and characterization of biomolecules using superchiral fields,” Nat. Nanotechnol. 5(11), 783–787 (2010).
[Crossref]

New J. Phys. (1)

A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013).
[Crossref]

Opt. Express (4)

Optica (1)

Phys. Chem. Chem. Phys. (1)

Q. Zhang, J. Li, and X. Liu, “Optical lateral forces and torques induced by chiral surface-plasmon-polaritons and their potential applications in recognition and separation of chiral enantiomers,” Phys. Chem. Chem. Phys. 21(3), 1308–1314 (2019).
[Crossref]

Phys. Rev. A (3)

M. Li, S. Yan, Y. Zhang, Y. Liang, P. Zhang, and B. Yao, “Optical sorting of small chiral particles by tightly focused vector beams,” Phys. Rev. A 99(3), 033825 (2019).
[Crossref]

H. Chen, C. Liang, S. Liu, and Z. Lin, “Chirality sorting using two-wave-interference-induced lateral optical force,” Phys. Rev. A 93(5), 053833 (2016).
[Crossref]

A. Canaguier-Durand and C. Genet, “Chiral near fields generated from plasmonic optical lattices,” Phys. Rev. A 90(2), 023842 (2014).
[Crossref]

Phys. Rev. B (3)

T. Davis and E. Hendry, “Superchiral electromagnetic fields created by surface plasmons in nonchiral metallic nanostructures,” Phys. Rev. B 87(8), 085405 (2013).
[Crossref]

F. Eftekhari and T. Davis, “Strong chiral optical response from planar arrays of subwavelength metallic structures supporting surface plasmon resonances,” Phys. Rev. B 86(7), 075428 (2012).
[Crossref]

G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
[Crossref]

Phys. Rev. Lett. (2)

H. Hu, Q. Gan, and Q. Zhan, “Generation of a nondiffracting superchiral optical needle for circular dichroism imaging of sparse subdiffraction objects,” Phys. Rev. Lett. 122(22), 223901 (2019).
[Crossref]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104(16), 163901 (2010).
[Crossref]

Phys. Rev. X (1)

M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Phys. Rev. X 2(3), 031010 (2012).
[Crossref]

Proc. Natl. Acad. Sci. U. S. A. (1)

A. Hayat, J. B. Mueller, and F. Capasso, “Lateral chirality-sorting optical forces,” Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015).
[Crossref]

Sci. Adv. (1)

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Sci. Adv. 3(5), e1602735 (2017).
[Crossref]

Sci. Rep. (2)

X. Tian, Y. Fang, and M. Sun, “Formation of enhanced uniform chiral fields in symmetric dimer nanostructures,” Sci. Rep. 5(1), 17534 (2015).
[Crossref]

I. D. Rukhlenko, N. V. Tepliakov, A. S. Baimuratov, S. A. Andronaki, Y. K. Gun’ko, A. V. Baranov, and A. V. Fedorov, “Completely chiral optical force for enantioseparation,” Sci. Rep. 6(1), 36884 (2016).
[Crossref]

Science (1)

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332(6027), 333–336 (2011).
[Crossref]

Other (3)

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).

H. Raether, Surface plasmons on smooth surfaces. In: Surface Plasmons on Smooth and Rough Surfaces and on Gratings111 (Springer, 1988).

S.-Y. Huang, J. Zhang, C. Karras, R. Förster, R. Heintzmann, and J.-S. Huang, “Chiral Structured Illumination Microscopy,” arXiv: 1908.09391 (2019).

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Figures (6)

Fig. 1.
Fig. 1. Schematic of two free-space plane waves at the same wavelength propagating along the $z$ axis at incident angle ${\alpha _{1,2}}$ and orientation angle ${\phi _{1,2}}$ respect to the $+ x$ axis.
Fig. 2.
Fig. 2. Numerically simulated normalized OC patterns generated by the superposition of (a,f) a s- and a p-pol. light beam; (b,g) two p- or s-pol. light beams; (c,h) two same-handed CPL beams (e.g., left-handed); (d,i) two opposite-handed CPL beams and (e,j) a p-pol. and a left-handed CPL beam. The two beams are (a-e) cross-propagating and (f-j) counter-propagating at an incident angle of ${\alpha _1} = {\alpha _2} = 41^\circ$ . The insets indicate the polarization states of the two beams and the projection of the incident directions on the $xy$ plane. Scale bars: $2{\pi /}{k_\textrm{0}}$ .
Fig. 3.
Fig. 3. Schematic of two incident plane waves totally reflected at the prism-air interface. The incident angle is ${\alpha _{1,2}}$ and the orientation angle is ${\phi _{1,2}}$ respect to the $+ x$ axis.
Fig. 4.
Fig. 4. Numerically simulated normalized OC patterns generated by the superposition of two TIR-based EWs excited by (a,f) a s- and a p-pol. light beam; (b,g) two p- or s-pol. light beams; (c,h) two same-handed CPL beams (e.g., left-handed); (d,i) two opposite-handed CPL beams and (e,j) a p-pol. and a left-handed CPL beam. The two incident beams are (a-e) cross-propagating and (f-j) counter-propagating at an incident angle of ${\alpha _1} = {\alpha _2} = 41^\circ$ . The insets indicate the polarization states of the two incident beams and the projection of the incident directions on the $xy$ plane. Scale bars: $2{\pi /}{k_\textrm{0}}$ .
Fig. 5.
Fig. 5. Generation of OC patterns by the superposition of two SPWs. (a) Schematic of two SPWs propagating in arbitrary in-plane directions ${\phi _{1,2}}$ respect to the $+ x$ axis. Numerically simulated normalized OC patterns generated by (b) two cross-propagating SPWs and (c) two counter-propagating SPWs. The insets in (b) and (c) show the top views of the SPWs.
Fig. 6.
Fig. 6. Transmittance of the s- ( $|{{t_s}(\alpha )} |$ ) and p-pol. component ( $|{{t_p}(\alpha )} |$ ) of the incident light, and $T(\alpha )$ change with the incident angle $\alpha $ .

Tables (3)

Tables Icon

Table 1. Analytical solutions of the normalized OC generated by the superposition of two free-space plane waves cross-propagating (second column) and counter-propagating (third column) at incident angles of α .

Tables Icon

Table 2. Analytical solutions of the normalized OC generated by the superposition of two TIR-based EWs excited by two incident plane waves cross-propagating (second column) and counter-propagating (third column) at incident angles of α .

Tables Icon

Table 3. Calculation parameters for the SPW

Equations (17)

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

C ε 0 2 E × E + 1 2 μ 0 B × B = ε 0 ω 2 Im ( E ~ B ~ ) ,
E ~ 1 s , 2 s = E 1 s , 2 s e i k 1 , 2 r + i ( ω t + φ 1 , 2 ) e i Δ θ 1 , 2 ( sin ϕ 1 , 2 x ^ + cos ϕ 1 , 2 y ^ ) ,
B ~ 1 s , 2 s = k 0 ω E 1 s , 2 s e i k 1 , 2 r + i ( ω t + φ 1 , 2 ) e i Δ θ 1 , 2 ( cos α 1 , 2 cos ϕ 1 , 2 x ^ cos α 1 , 2 sin ϕ 1 , 2 y ^ + sin α 1 , 2 z ^ ) .
B ~ 1 p , 2 p = B 1 p , 2 p e i k 1 , 2 r + i ( ω t + φ 1 , 2 ) ( sin ϕ 1 , 2 x ^ + cos ϕ 1 , 2 y ^ ) ,
E ~ 1 p , 2 p = c 2 k 0 ω B 1 p , 2 p e i k 1 , 2 r + i ( ω t + φ 1 , 2 ) ( cos α 1 , 2 cos ϕ 1 , 2 x ^ cos α 1 , 2 sin ϕ 1 , 2 y ^ + sin α 1 , 2 z ^ ) ,
C = ε 0 k 0 2 Im { E 1 s E 1 p e i Δ θ 1 E 1 s E 1 p e i Δ θ 1 + E 2 s E 2 p e i Δ θ 2 E 2 s E 2 p e i Δ θ 2 + e i Φ [ cos ( ϕ 1 ϕ 2 ) ( E 1 s E 2 p e i Δ θ 1 cos α 1 cos α 2 E 1 p E 2 s e i Δ θ 2 ) + sin ( ϕ 1 ϕ 2 ) ( cos α 2 E 1 s E 2 s e i ( Δ θ 1 Δ θ 2 ) + cos α 1 E 1 p E 2 p ) sin α 1 sin α 2 E 1 p E 2 s e i Δ θ 2 ] , + e i Φ [ cos ( ϕ 1 ϕ 2 ) ( E 1 p E 2 s e i Δ θ 2 cos α 1 cos α 2 E 1 s E 2 p e i Δ θ 1 ) sin ( ϕ 1 ϕ 2 ) ( cos α 1 E 1 s E 2 s e i ( Δ θ 1 Δ θ 2 ) + cos α 2 E 1 p E 2 p ) sin α 1 sin α 2 E 1 s E 2 p e i Δ θ 1 ] }
Φ = k 0 [ x ( sin α 2 cos ϕ 2 sin α 1 cos ϕ 1 ) + y ( sin α 2 sin ϕ 2 sin α 1 sin ϕ 1 ) + z ( cos α 2 cos α 1 ) ] + φ 2 φ 1 .
E ~ 1 s , 2 s = t s ( α ) E 1 s , 2 s e κ ( α ) z e i n k 0 ( x sin α cos ϕ 1 , 2 + y sin α sin ϕ 1 , 2 ) + i ( ω t + φ 1 , 2 ) e i Δ θ 1 , 2 ( sin ϕ 1 , 2 x ^ + cos ϕ 1 , 2 y ^ ) ,
B ~ 1 s , 2 s = t s ( α ) i ω E 1 s , 2 s e κ ( α ) z e i n k 0 ( x sin α cos ϕ 1 , 2 + y sin α sin ϕ 1 , 2 ) + i ( ω t + φ 1 , 2 ) e i Δ θ 1 , 2 [ κ ( α ) cos ϕ 1 , 2 x ^ + κ ( α ) sin ϕ 1 , 2 y ^ + i n k 0 sin α z ^ ] ;
B ~ 1 p , 2 p = t p ( α ) B 1 p , 2 p e κ ( α ) z e i n k 0 ( x sin α cos ϕ 1 , 2 + y sin α sin ϕ 1 , 2 ) + i ( ω t + φ 1 , 2 ) ( sin ϕ 1 , 2 x ^ + cos ϕ 1 , 2 y ^ ) ,
E ~ 1 p , 2 p = c 2 t p ( α ) i ω B 1 p , 2 p e κ ( α ) z e i n k 0 ( x sin α cos ϕ 1 , 2 + y sin α sin ϕ 1 , 2 ) + i ( ω t + φ 1 , 2 ) [ κ ( α ) cos ϕ 1 , 2 x ^ + κ ( α ) sin ϕ 1 , 2 y ^ + i n k 0 sin α z ^ ] .
C = ε 0 k 0 e 2 κ ( α ) z 2 Im { t ( α ) ( 2 n 2 sin 2 α 1 ) ( E 1 s E 1 p e i Δ θ 1 + E 2 s E 2 p e i Δ θ 2 ) + t ( α ) ( E 1 s E 1 p e i Δ θ 1 + E 2 s E 2 p e i Δ θ 2 ) + t ( α ) cos ( ϕ 1 ϕ 2 ) ( E 1 s E 2 p e i Δ θ 1 e i Φ + E 1 p E 2 s e i Δ θ 2 e i Φ ) t ( α ) [ ( n 2 sin 2 α 1 ) cos ( ϕ 1 ϕ 2 ) + n 2 sin 2 α ] ( E 1 p E 2 s e i Δ θ 2 e i Φ + E 1 s E 2 p e i Δ θ 1 e i Φ ) } ,
Φ = n k 0 sin α [ x ( cos ϕ 2 cos ϕ 1 ) + y ( sin ϕ 2 sin ϕ 1 ) ] + φ 2 φ 1 .
B ~ 1 , 2 = B SP e i q z e i k SP ( x cos ϕ 1 , 2 + y sin ϕ 1 , 2 ) + i ( ω t + φ 1 , 2 ) ( sin ϕ 1 , 2 x ^ + cos ϕ 1 , 2 y ^ ) ,
E ~ 1 , 2 = c 2 i ω B SP e i q z e i k SP ( x cos ϕ 1 , 2 + y sin ϕ 1 , 2 ) + i ( ω t + φ 1 , 2 ) ( i q cos ϕ 1 , 2 x ^ i q sin ϕ 1 , 2 y ^ + i k SP z ^ ) ,
C = ε 0 q E SP 2 e 2 q z e k SP [ x ( cos ϕ 1 + cos ϕ 2 ) + y ( sin ϕ 1 + sin ϕ 2 ) ] sin ( ϕ 1 ϕ 2 ) sin Φ ,
Φ = k SP [ x ( cos ϕ 2 cos ϕ 1 ) + y ( sin ϕ 2 sin ϕ 1 ) ] + φ 2 φ 1 ,

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