Abstract

Optical forces acting on particles - controlled by the intensity, polarization and direction of optical beams - have become an important tool in manipulation, sorting and analysis of nano/micro-particles. The nature of these forces has been well understood in reciprocal structures exhibiting time-reversal symmetries. Here, we investigate the nature of optical forces in non-reciprocal structures with non-degenerate counter-propagating modes. We consider the specific case of non-reciprocity induced via translational motion and show that the two counter-propagating modes in a moving slab-waveguide are not degenerate which results in a non-zero lateral and longitudinal force on a nanoparticle. We prove that these anomalous forces are fundamentally connected to near-field photonic spin in optical waveguides and explain their directionality using universal spin-momentum locking of evanescent waves. The presented results show that the interplay of photon spin and non-reciprocity can lead to unique avenues of controlling nanoscale optical forces on-chip.

© 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 (4)

L. Liu, A. Di Donato, V. Ginis, S. Kheifets, A. Amirzhan, and F. Capasso, “Three-dimensional measurement of the helicity-dependent forces on a mie particle,” Phys. Rev. Lett. 120, 223901 (2018).
[Crossref] [PubMed]

F. Le Kien, S. S. S. Hejazi, V. G. Truong, S. Nic Chormaic, and T. Busch, “Chiral force of guided light on an atom,” Phys. Rev. A 97, 063849 (2018).
[Crossref]

R. Luo, Y. Li, S. Deng, C. Peng, and Z. Li, “Effective suppression of residual coherent phase error in a dual-polarization fiber optic gyroscope,” Opt. Lett. 43, 815–818 (2018).
[Crossref] [PubMed]

R. R. O. Weernink, P. Barcellona, and S. Y. Buhmann, “Lateral casimir-polder forces by breaking time-reversal symmetry,” Phys. Rev. A 97, 032507 (2018).
[Crossref]

2017 (6)

S. Pendharker, Y. Guo, F. Khosravi, and Z. Jacob, “PT-symmetric spectral singularity and negative-frequency resonance,” Phys. Rev. A 95, 033817 (2017).
[Crossref]

R. Luo, Y. Li, S. Deng, D. He, C. Peng, and Z. Li, “Compensation of thermal strain induced polarization nonreciprocity in dual-polarization fiber optic gyroscope,” Opt. Express 25, 26747–26759 (2017).
[Crossref] [PubMed]

A. Espinosa-Soria, F. J. Rodriguez-Fortuño, A. Griol, and A. Martínez, “On-chip optimal stokes nanopolarimetry based on spin-orbit interaction of light,” Nano Lett. 17, 3139–3144 (2017). PMID: .
[Crossref] [PubMed]

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473 (2017).
[Crossref] [PubMed]

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, 4292–4300 (2017).
[Crossref] [PubMed]

2016 (8)

F. Kalhor, T. Thundat, and Z. Jacob, “Universal spin-momentum locked optical forces,” Appl. Phys. Lett. 108, 061102 (2016).
[Crossref]

T. Van Mechelen and Z. Jacob, “Universal spin-momentum locking of evanescent waves,” Optica 3, 118–126 (2016).
[Crossref]

W. Ye, F. Zeuner, X. Li, B. Reineke, S. He, C.-W. Qiu, J. Liu, Y. Wang, S. Zhang, and T. Zentgraf, “Spin and wavelength multiplexed nonlinear metasurface holography,” Nat. Commun. 7, 11930 (2016).
[Crossref] [PubMed]

A. Espinosa-Soria and A. Martínez, “Transverse spin and spin-orbit coupling in silicon waveguides,” IEEE Photonics Technol. Lett. 28, 1561–1564 (2016).
[Crossref]

D. F. Kornovan, A. S. Sheremet, and M. I. Petrov, “Collective polaritonic modes in an array of two-level quantum emitters coupled to an optical nanofiber,” Phys. Rev. B 94, 245416 (2016).
[Crossref]

M. Antognozzi, C. Bermingham, R. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. Dennis, A. Bekshaev, K. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12, 731 (2016).
[Crossref]

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

2015 (7)

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon–emitter coupling in chiral photonic circuits,” Nat. nanotechnology 10, 775–778 (2015).
[Crossref]

B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6, 6695 (2015).
[Crossref] [PubMed]

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

S. Scheel, S. Y. Buhmann, C. Clausen, and P. Schneeweiss, “Directional spontaneous emission and lateral casimir-polder force on an atom close to a nanofiber,” Phys. Rev. A 92, 043819 (2015).
[Crossref]

F. J. Rodríguez-fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
[Crossref] [PubMed]

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

S. Sukhov, V. Kajorndejnukul, R. R. Naraghi, and A. Dogariu, “Dynamic consequences of optical spin–orbit interaction,” Nat. Photonics 9, 809 (2015).
[Crossref]

2014 (9)

M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs, “Polarization tailored light driven directional optical nanobeacon,” Nano Lett. 14, 2546–2551 (2014).
[Crossref] [PubMed]

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

A. S. Urban, S. Carretero-Palacios, A. A. Lutich, T. Lohmüller, J. Feldmann, and F. Jäckel, “Optical trapping and manipulation of plasmonic nanoparticles: fundamentals, applications, and perspectives,” Nanoscale 6, 4458–4474 (2014).
[Crossref] [PubMed]

S. Wang and C. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 4307 (2014).

J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346, 67–71 (2014).
[Crossref] [PubMed]

D. O’Connor, P. Ginzburg, F. Rodríguez-Fortuño, G. Wurtz, and A. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
[Crossref]

R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Quantum state-controlled directional spontaneous emission of photons into a nanophotonic waveguide,” Nat. Commun. 5, 5713 (2014).
[Crossref] [PubMed]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923 (2014).
[Crossref]

Y. Shoji and T. Mizumoto, “Magneto-optical non-reciprocal devices in silicon photonics,” Sci. Technol. Adv. Mater. 15, 014602 (2014).
[Crossref] [PubMed]

2013 (3)

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref] [PubMed]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light. Sci. & Appl. 2, e70 (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, 123037 (2013).
[Crossref]

2011 (1)

J. Maciejko, T. L. Hughes, and S.-C. Zhang, “The quantum spin hall effect,” Annu. Rev. Condens. Matter Phys. 2, 31–53 (2011).
[Crossref]

2007 (1)

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. Lukaszew, J. Skuza, D. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76, 153402 (2007).
[Crossref]

Albrecht, B.

R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Quantum state-controlled directional spontaneous emission of photons into a nanophotonic waveguide,” Nat. Commun. 5, 5713 (2014).
[Crossref] [PubMed]

Alizadeh, M. H.

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

Alù, A.

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref] [PubMed]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923 (2014).
[Crossref]

Amirzhan, A.

L. Liu, A. Di Donato, V. Ginis, S. Kheifets, A. Amirzhan, and F. Capasso, “Three-dimensional measurement of the helicity-dependent forces on a mie particle,” Phys. Rev. Lett. 120, 223901 (2018).
[Crossref] [PubMed]

Antognozzi, M.

M. Antognozzi, C. Bermingham, R. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. Dennis, A. Bekshaev, K. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12, 731 (2016).
[Crossref]

Antoniou, N.

J. Lin, J. P. B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan, and F. Capasso, “Polarization-controlled tunable directional coupling of surface plasmon polaritons,” Science 340, 331–334 (2013).
[Crossref] [PubMed]

Armelles, G.

J. B. González-Díaz, A. García-Martín, G. Armelles, J. M. García-Martín, C. Clavero, A. Cebollada, R. Lukaszew, J. Skuza, D. Kumah, and R. Clarke, “Surface-magnetoplasmon nonreciprocity effects in noble-metal/ferromagnetic heterostructures,” Phys. Rev. B 76, 153402 (2007).
[Crossref]

Bai, B.

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light. Sci. & Appl. 2, e70 (2013).
[Crossref]

Banzer, P.

M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs, “Polarization tailored light driven directional optical nanobeacon,” Nano Lett. 14, 2546–2551 (2014).
[Crossref] [PubMed]

Barcellona, P.

R. R. O. Weernink, P. Barcellona, and S. Y. Buhmann, “Lateral casimir-polder forces by breaking time-reversal symmetry,” Phys. Rev. A 97, 032507 (2018).
[Crossref]

Barnett, S. M.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

Bauer, T.

M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs, “Polarization tailored light driven directional optical nanobeacon,” Nano Lett. 14, 2546–2551 (2014).
[Crossref] [PubMed]

Bekshaev, A.

M. Antognozzi, C. Bermingham, R. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. Dennis, A. Bekshaev, K. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12, 731 (2016).
[Crossref]

Bermingham, C.

M. Antognozzi, C. Bermingham, R. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. Dennis, A. Bekshaev, K. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12, 731 (2016).
[Crossref]

Bliokh, K.

M. Antognozzi, C. Bermingham, R. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. Dennis, A. Bekshaev, K. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12, 731 (2016).
[Crossref]

Bliokh, K. Y.

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

Buhmann, S. Y.

R. R. O. Weernink, P. Barcellona, and S. Y. Buhmann, “Lateral casimir-polder forces by breaking time-reversal symmetry,” Phys. Rev. A 97, 032507 (2018).
[Crossref]

S. Scheel, S. Y. Buhmann, C. Clausen, and P. Schneeweiss, “Directional spontaneous emission and lateral casimir-polder force on an atom close to a nanofiber,” Phys. Rev. A 92, 043819 (2015).
[Crossref]

Busch, T.

F. Le Kien, S. S. S. Hejazi, V. G. Truong, S. Nic Chormaic, and T. Busch, “Chiral force of guided light on an atom,” Phys. Rev. A 97, 063849 (2018).
[Crossref]

Cai, L.

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
[Crossref]

Cameron, R. P.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

Canaguier-Durand, A.

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

Capasso, F.

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P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473 (2017).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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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, 4292–4300 (2017).
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ACS Photonics (1)

M. H. Alizadeh and B. M. Reinhard, “Transverse chiral optical forces by chiral surface plasmon polaritons,” ACS Photonics 2, 1780–1788 (2015).
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Annu. Rev. Condens. Matter Phys. (1)

J. Maciejko, T. L. Hughes, and S.-C. Zhang, “The quantum spin hall effect,” Annu. Rev. Condens. Matter Phys. 2, 31–53 (2011).
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Appl. Phys. Lett. (1)

F. Kalhor, T. Thundat, and Z. Jacob, “Universal spin-momentum locked optical forces,” Appl. Phys. Lett. 108, 061102 (2016).
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IEEE Photonics Technol. Lett. (1)

A. Espinosa-Soria and A. Martínez, “Transverse spin and spin-orbit coupling in silicon waveguides,” IEEE Photonics Technol. Lett. 28, 1561–1564 (2016).
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Light. Sci. & Appl. (1)

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light. Sci. & Appl. 2, e70 (2013).
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Nano Lett. (2)

A. Espinosa-Soria, F. J. Rodriguez-Fortuño, A. Griol, and A. Martínez, “On-chip optimal stokes nanopolarimetry based on spin-orbit interaction of light,” Nano Lett. 17, 3139–3144 (2017). PMID: .
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Nanoscale (1)

A. S. Urban, S. Carretero-Palacios, A. A. Lutich, T. Lohmüller, J. Feldmann, and F. Jäckel, “Optical trapping and manipulation of plasmonic nanoparticles: fundamentals, applications, and perspectives,” Nanoscale 6, 4458–4474 (2014).
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Nat. Commun. (7)

S. Wang and C. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 4307 (2014).

F. J. Rodríguez-fortuño, N. Engheta, A. Martínez, and A. V. Zayats, “Lateral forces on circularly polarizable particles near a surface,” Nat. Commun. 6, 8799 (2015).
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F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref] [PubMed]

D. O’Connor, P. Ginzburg, F. Rodríguez-Fortuño, G. Wurtz, and A. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
[Crossref]

R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Quantum state-controlled directional spontaneous emission of photons into a nanophotonic waveguide,” Nat. Commun. 5, 5713 (2014).
[Crossref] [PubMed]

W. Ye, F. Zeuner, X. Li, B. Reineke, S. He, C.-W. Qiu, J. Liu, Y. Wang, S. Zhang, and T. Zentgraf, “Spin and wavelength multiplexed nonlinear metasurface holography,” Nat. Commun. 7, 11930 (2016).
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B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6, 6695 (2015).
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Nat. nanotechnology (1)

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon–emitter coupling in chiral photonic circuits,” Nat. nanotechnology 10, 775–778 (2015).
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Nat. Photonics (2)

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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S. Sukhov, V. Kajorndejnukul, R. R. Naraghi, and A. Dogariu, “Dynamic consequences of optical spin–orbit interaction,” Nat. Photonics 9, 809 (2015).
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Nat. Phys. (2)

M. Antognozzi, C. Bermingham, R. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. Dennis, A. Bekshaev, K. Bliokh, and F. Nori, “Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever,” Nat. Phys. 12, 731 (2016).
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N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923 (2014).
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Nature (1)

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541, 473 (2017).
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R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
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Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (5)

L. Cai, M. Liu, S. Chen, Y. Liu, W. Shu, H. Luo, and S. Wen, “Quantized photonic spin hall effect in graphene,” Phys. Rev. A 95, 013809 (2017).
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Figures (4)

Fig. 1
Fig. 1 Spin-momentum locking in a ridge waveguide. (a) Schematic of a ridge waveguide and the EM triplet introduced by Mechelen et. al. [16], for the fundamental TE-like mode. (b) z ^ component of electric spin-density S e in y-z plane for the fundamental mode. (c) y ^ component of magnetic spin-density S m in y-z plane for the fundamental mode.
Fig. 2
Fig. 2 (a) Schematic of a chiral particle moving near a slab-waveguide. It can be equivalently analyzed by transforming to the frame of the particle where the waveguide is moving instead. (b) The dispersion curves of positive (blue) and negative (red) momentum modes in the full ωk plane when the slab is stationary (solid line) and moving (dashed line). The thickness of the waveguide is d = 1.5µm and the dielectric constant is ϵr = 9. The frequency axis is normalized to k0 = ω0/c, where ω0 = 2π1014rad/s. (c) Shows the field plots in the slab-waveguide at points n1, p1, n2 and p2. The color plot represents the y ^ component of the third Stokes parameter S 3 = S 3 y ^ and the electric field is shown by arrows. (d) The electric-spin density S e = ϵ 0 S 3 / ( 2 ω ) of the positive- and negative-momentum modes as a function of slab velocity at dz = 0.1d, where d is the thickness of the slab.
Fig. 3
Fig. 3 Velocity of motion greater than Cherenkov velocity. (a) Dispersion curves in the moving slab waveguide when the velocity of motion is large enough to transform the negative frequency modes to the positive frequency region. Above the Cherenkov velocity the negative-momentum mode is also transformed to the positive-momentum. (b) Shows the spin of the modes as latitude on the Poincare sphere. It can be seen that the spin switches abruptly from LHC to RHC at the Cherenkov velocity.
Fig. 4
Fig. 4 (a) Two degenerate and counter-propagating modes in a stationary waveguide are symmetric, resulting in a balance of radiation pressure force and scatttering recoil force on a particle in its vicinity. However, when the slab is moving, the two modes transform asymmetrically resulting in an unbalanced force. (b) ΔFy represents the net optical force on a chiral particle due to the two modes in the lateral direction y ^ as function of velocity of motion. Fyf and Fyb is the force on the particle in y ^ direction due to the positive (forward) momentum mode and the negative (backward) momentum mode, respectively.(c) ΔFx represents the net force in the x ^ direction as function of velocity of motion. Fxf and Fxb represent the force in the x ^ direction due to the positive-momentum mode and the negative-momentum mode, respectively. All forces are computed at a distance 0.1d from the interface, where d = 1.5µm is the thickness of the slab waveguide. For computation of optical force, electric field of the form E x = 1 e i ( k x x ω t ) e i k z 2 z V / m, is assumed in the region z > d/2. This corresponds to the approximate Poynting vector of 17 µW/m2 at the center of the stationary waveguide.

Equations (22)

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s ^ = k ^ × η ^ ,
[ c D H ] = [ c ϵ 0 ϵ χ η μ 1 c μ 0 ] [ E c B ]
ϵ = γ 2 [ ϵ r γ 2 0 0 0 ϵ r β x 2 0 0 0 ϵ r β x 2 ]
χ = γ 2 ϵ 0 μ 0 [ 0 0 0 0 0 β x ( ϵ r 1 ) 0 β x ( ϵ r 1 ) 0 ]
η = γ 2 ϵ 0 μ 0 [ 0 0 0 0 0 β x ( ϵ r 1 ) 0 β x ( ϵ r 1 ) 0 ]
μ 1 = γ 2 [ 1 γ 2 0 0 0 β x 2 ϵ r + 1 0 0 0 β x 2 ϵ r + 1 ]
k E = ω B
k H = ω D
k = [ 0 k z k y k z 0 k x k y k x 0 ]
H = [ η + μ 1 μ 0 ω k ] E
E = [ ω ϵ 0 ϵ + χ k ] k H
( k μ 1 k ω μ 0 + ω ϵ 0 ϵ + k η + χ k ) E = 0
det ( k μ 1 k ω μ 0 + ω ϵ 0 ϵ + k η + χ k ) = 0
H y = C e i ( k x x ω t ) e i k z 1 z + D e i ( k x x ω t ) e i k z 1 z
H y = A e i ( k x x ω t ) e i k z 2 z
H y = B e i ( k x x ω t ) e i k z 3 z
F = F g r + F o p + F s r ,
F g r = Δ U ,
F o p = k 0 c ( I m [ α e e ] ϵ 0 + I m [ α m m ] μ 0 ) N I m [ α e m ] × N c k 0 2 ( I m [ α e e ] ϵ 0 × s e + I m [ α m m ] μ 0 × s m ) + ω 2 I m [ α e m ] ( s e + s m ) ,
F s r = c k 0 4 6 π { ( R e [ α e e α m m * ] + | α e m | 2 ) N + μ 0 ϵ 0 R e [ α e e α m m * ] S 3 e + ϵ 0 μ 0 R e [ α m m α e m * ] S 3 m 1 2 I m [ α e e α m m * ] I m [ E × H * ] } ,
T i j = ϵ 0 ( E i E j 1 2 δ i j E 2 ) + 1 μ 0 ( B i B j 1 2 δ i j B 2 ) ,
F = v T a .

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