Abstract

Optical intrinsic angular momentum can be regarded as derivation from spatial superposition of optical vector fields embodied by spinning or/and spiraling the electric-field vector. We employ vectorial formulation derivation to comprehensively study all angular momentum contents of optical vector fields in arbitrary superposition states, including the longitudinal and transverse, spin and orbital (SAM and OAM) components. As for the orthogonal superposition fields, there inherently exists spin-orbit shift from longitudinal SAM to OAM, and the whole local spin flow manifests local multiple-fold helical trajectories. Especially, both the spin-orbit shift and transverse SAM could become considerable in the non-paraxial condition. Our studies here provide an explicit insight into the derivation and evolution, intrinsic correlations and salient features of various types of angular momentum components.

© 2017 Optical Society of America

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2017 (1)

L. Fang and J. Wang, “Intrinsic transverse spin angular momentum of fiber eigenmodes,” Phys. Rev. A 95(5), 053827 (2017).
[Crossref]

2016 (5)

L. Fang and J. Wang, “Mode conversion and orbital angular momentum transfer among multiple modes by helical gratings,” IEEE J. Quantum Electron. 52(8), 6600306 (2016).
[Crossref]

T. Van Mechelen and Z. Jacob, “Universal spin-momentum locking of evanescent waves,” Optica 3(2), 118–126 (2016).
[Crossref]

J. Wang, “Advances in communications using optical vortices,” Photon. Res. 4(5), B14–B28 (2016).
[Crossref]

F. Kalhor, T. Thundat, and Z. Jacob, “Universal spin-momentum locked optical forces,” Appl. Phys. Lett. 108(6), 061102 (2016).
[Crossref]

N. Radwell, R. D. Hawley, J. B. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref] [PubMed]

2015 (6)

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(6242), 1448–1451 (2015).
[Crossref] [PubMed]

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 4, 592 (2015).

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2(10), 864–868 (2015).
[Crossref]

Z. Zhang, J. Gan, X. Heng, Y. Wu, Q. Li, Q. Qian, D. Chen, and Z. Yang, “Optical fiber design with orbital angular momentum light purity higher than 99.9%,” Opt. Express 23(23), 29331–29341 (2015).
[Crossref] [PubMed]

2014 (7)

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3(3), e153 (2014).
[Crossref]

J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346(6205), 67–71 (2014).
[Crossref] [PubMed]

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

S. Golowich, “Asymptotic theory of strong spin-orbit coupling in optical fiber,” Opt. Lett. 39(1), 92–95 (2014).
[Crossref] [PubMed]

S. Chen, X. Zhou, Y. Liu, X. Ling, H. Luo, and S. Wen, “Generation of arbitrary cylindrical vector beams on the higher order Poincaré sphere,” Opt. Lett. 39(18), 5274–5276 (2014).
[Crossref] [PubMed]

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

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

2013 (2)

2012 (2)

2011 (4)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

A. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19(27), 26132–26149 (2011).
[Crossref] [PubMed]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

2010 (3)

K. Y. Bliokh, M. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 13289 (2010).
[Crossref]

H.-T. Wang, X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18(10), 10786–10795 (2010).
[Crossref] [PubMed]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref] [PubMed]

2009 (2)

2008 (3)

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008).
[Crossref] [PubMed]

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref] [PubMed]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2(4), 299–313 (2008).
[Crossref]

2007 (2)

R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express 15(23), 15214–15227 (2007).
[Crossref] [PubMed]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

2006 (3)

F. L. Kien, V. I. Balykin, and K. Hakuta, “Angular momentum of light in an optical nanofiber,” Phys. Rev. A 73(5), 053823 (2006).
[Crossref]

A. F. Abouraddy and K. C. Toussaint., “Three-Dimensional Polarization Control in Microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[Crossref] [PubMed]

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[Crossref] [PubMed]

2005 (1)

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005).
[Crossref] [PubMed]

2004 (4)

S. S. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92(21), 217901 (2004).
[Crossref] [PubMed]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6(2), 259–268 (2004).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

M. J. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

2003 (2)

M. Harwit, “Photon orbital angular momentum in astrophysics,” Astrophys. J. 597(2), 1266–1270 (2003).
[Crossref]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

2002 (1)

A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002).
[Crossref]

2001 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

1996 (1)

1995 (1)

W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser Acceleration of Relativistic Electrons Using the Inverse Cherenkov Effect,” Phys. Rev. Lett. 74(4), 546–549 (1995).
[Crossref] [PubMed]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1988 (1)

V. G. Fedoseyev, “Conservation laws and transverse motion of energy on reflection and transmission of electromagnetic waves,” J. Phys. Math. Gen. 21(9), 2045–2059 (1988).
[Crossref]

1987 (1)

M. A. Player, “Angular momentum balance and transverse shift on reflection of light,” J. Phys. Math. Gen. 20(12), 3667–3678 (1987).
[Crossref]

’t Hooft, G. W.

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005).
[Crossref] [PubMed]

Abouraddy, A. F.

A. F. Abouraddy and K. C. Toussaint., “Three-Dimensional Polarization Control in Microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[Crossref] [PubMed]

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Aiello, A.

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2(10), 864–868 (2015).
[Crossref]

K. Y. Bliokh, M. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 13289 (2010).
[Crossref]

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005).
[Crossref] [PubMed]

S. S. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92(21), 217901 (2004).
[Crossref] [PubMed]

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Alhassen, F.

P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
[Crossref] [PubMed]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2(4), 299–313 (2008).
[Crossref]

M. J. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

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W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser Acceleration of Relativistic Electrons Using the Inverse Cherenkov Effect,” Phys. Rev. Lett. 74(4), 546–549 (1995).
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Steinhauer, L.

W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser Acceleration of Relativistic Electrons Using the Inverse Cherenkov Effect,” Phys. Rev. Lett. 74(4), 546–549 (1995).
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G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
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G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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S. S. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92(21), 217901 (2004).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Light Sci. Appl. (1)

L. Chen, J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light Sci. Appl. 3(3), e153 (2014).
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S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2(5-6), 455–477 (2013).
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N. Radwell, R. D. Hawley, J. B. Götte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
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K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. 5, 3300 (2014).
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S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
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D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, “Spin-orbit coupling in surface plasmon scattering by nanostructures,” Nat. Commun. 5, 5327 (2014).
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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. Photonics (1)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
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J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
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Photon. Res. (1)

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K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 4, 592 (2015).

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L. Fang and J. Wang, “Intrinsic transverse spin angular momentum of fiber eigenmodes,” Phys. Rev. A 95(5), 053827 (2017).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85(6), 1577–1581 (2012).
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K. Y. Bliokh, M. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82(6), 13289 (2010).
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Phys. Rev. Lett. (8)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
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P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of Orbital Angular Momentum Transfer between Acoustic and Optical Vortices in Optical Fiber,” Phys. Rev. Lett. 96(4), 043604 (2006).
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W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laser Acceleration of Relativistic Electrons Using the Inverse Cherenkov Effect,” Phys. Rev. Lett. 74(4), 546–549 (1995).
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G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005).
[Crossref] [PubMed]

S. S. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92(21), 217901 (2004).
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Science (3)

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[Crossref] [PubMed]

J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346(6205), 67–71 (2014).
[Crossref] [PubMed]

K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348(6242), 1448–1451 (2015).
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Z. Wang, B. Chen, R. Wang, S. Shi, Q. Qiu, and X. Li, “New investigations on the transverse spin of structured optical fields,” https://arxiv.org/abs/1605.03103 (2016).

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Figures (5)

Fig. 1
Fig. 1 Arbitrary polarization states and OAM evolution from arbitrary superposition states ψ 2 1 ( Δ φ , γ ) on the basis of HE21 mode with the amount of spin-orbit shift ξ = 0.12 . Positions A and B correspond to superposition states ψ 2 1 ( π / 6 , π / 4 ) and ψ 2 1 ( π / 2 , π / 6 ) , respectively.

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Fig. 2
Fig. 2 Mean longitudinal angular momentum per photon versus γ and Δ φ on the basis of vector modes HE21 (a) and EH11 (b), both with ξ = 0.12 .

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Fig. 3
Fig. 3 Local spin flow trajectories of (a) TM01 mode and (b) TE01 mode.

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Fig. 4
Fig. 4 Contrast between OAM and SAM. Spatial phase distribution and local spin flow trajectories of superposed fields for (a) ψ 11 ( + π / 2 , π / 4 ) and (d) ψ 11 ( − π / 2 , π / 4 ) .

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Fig. 5
Fig. 5 Contrast between OAM and SAM. Spatial phase distribution and local spin flow trajectories of superposed fields for (a) ψ 21 ( + π / 2 , π / 4 ) and (d) ψ 21 ( − π / 2 , π / 4 ) .

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Equations (23)

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L z = − i ε 0 2 ω ( ψ r ∗ ∂ ∂ ϕ ψ r + ψ ϕ ∗ ∂ ∂ ϕ ψ ϕ + ψ z ∗ ∂ ∂ ϕ ψ z − ψ r ∗ ψ ϕ + ψ ϕ ∗ ψ r ) e ⇀ z ,
L ϕ = − i ε 0 r 2 ω ( ψ r ∗ ∂ ∂ z ψ r + ψ ϕ ∗ ∂ ∂ z ψ ϕ + ψ z ∗ ∂ ∂ z ψ z ) e ⇀ ϕ ,
S = − i ε 0 2 ω [ ( ψ ϕ ∗ ψ z − ψ z ∗ ψ ϕ ) e ⇀ r + ( ψ z ∗ ψ r − ψ r ∗ ψ z ) e ⇀ ϕ + ( ψ r ∗ ψ ϕ − ψ ϕ ∗ ψ r ) e ⇀ z ] .
W = 1 2 ε ( | ψ r | 2 + | ψ ϕ | 2 + | ψ z | 2 ) ,
ψ m n ( φ 1 , φ 2 , γ ) = [ cos γ ⋅ V m n ( φ 1 ) + i sin γ ⋅ V m n ( φ 2 ) ] exp [ i ( ω t + β z ) ]
V m n ( φ k ) = E r cos ( m ϕ − φ k ) ⋅ e ⇀ r + E ϕ sin ( m ϕ − φ k ) ⋅ e ⇀ ϕ + i E z cos ( m ϕ − φ k ) ⋅ e ⇀ z ( k = 1 , 2 ) ,
L z = ε 0 sin 2 γ ⋅ sin Δ φ 4 ω [ m ( E r 2 + E ϕ 2 + E z 2 ) +2 E r E ϕ ] ,
S z = − ε 0 2 ω sin 2 γ ⋅ sin Δ φ ⋅ E r E ϕ .
S ϕ = − ε 0 ω E r E z [ cos 2 γ cos 2 ( m ϕ − φ 1 ) + sin 2 γ cos 2 ( m ϕ − φ 2 ) ] ,
S r = ε 0 2 ω E ϕ E z [ cos 2 γ sin 2 ( m ϕ − φ 1 ) + sin 2 γ sin 2 ( m ϕ − φ 2 ) ] .
n i 2 〈 S z 〉 〈 W 〉 = n i 2 ∬ S z r d r d ϕ ∬ W r d r d ϕ = − ς ω sin 2 γ ⋅ sin Δ φ ⋅ ( 1 − ξ ) ,
n i 2 〈 L z 〉 〈 W 〉 = n i 2 ∬ L z r d r d ϕ ∬ W r d r d ϕ = 1 ω sin 2 γ ⋅ sin Δ φ ⋅ [ m + ς ( 1 − ξ ) ] ,
ξ = 1 − ς 2 ∫ E r E ϕ r d r ∫ ( | E r | 2 + | E ϕ | 2 + | E z | 2 ) r d r .
n i 2 ( 〈 S z 〉 + 〈 L z 〉 ) 〈 W 〉 = 1 ω m sin 2 γ ⋅ sin Δ φ .
n i 2 〈 L ϕ 〉 〈 W 〉 = r β ω ,
n i 2 〈 S ϕ 〉 〈 W 〉 = − π ω ∫ E r E z r d r ∫ ( | E r | 2 + | E ϕ | 2 + | E z | 2 ) r d r ,
n i 2 〈 S r 〉 〈 W 〉 = 0 ,
ψ 21 1 = cos γ ⋅ cos Δ φ ⋅ ψ 21 ,
ψ 21 2 = cos γ ⋅ sin Δ φ ⋅ ψ 21 ,
ψ 2 1 3 = sin γ ⋅ ψ 21 ,
| ψ 21 1 | 2 + | ψ 21 2 | 2 + | ψ 21 3 | 2 = | ψ 21 | 2 ,
ψ m n ( φ 1 , φ 2 , γ ) = ( 1 + η ) 2 1 + η 2 ( cos γ e − i φ 1 + i sin γ e − i φ 2 ) e i ( m + 1 ) ϕ σ − + ( 1 − η ) 2 1 + η 2 ( cos γ e − i φ 1 + i sin γ e − i φ 2 ) e i ( m − 1 ) ϕ σ + + ( 1 − η ) 2 1 + η 2 ( cos γ e i φ 1 + i sin γ e i φ 2 ) e − i ( m − 1 ) ϕ σ − , + ( 1 + η ) 2 1 + η 2 ( cos γ e i φ 1 + i sin γ e i φ 2 ) e − i ( m + 1 ) ϕ σ +
η ≃ ± i ∫ 0 ∞ ∫ 0 2 π E ϕ ⋅ H r r d r d ϕ ∫ 0 ∞ ∫ 0 2 π E r ⋅ H ϕ r d r d ϕ ,

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