Abstract

We examine the geometric phase Doppler effect that appears when a structured light interacts with a rotating structured material. In our scheme the structured light possesses a vortex phase and the structured material works as an inhomogeneous anisotropic plate. We show that the Doppler effect manifests itself as a frequency shift which can be interpreted in terms of a dynamic evolution of Pancharatnam-Berry phase on the hybrid-order Poincaré sphere. The frequency shift induced by the change rate of Pancharatnam-Berry phase with time is derived from both the Jones matrix calculations and the theory of the hybrid-order Poincaré sphere. Unlike the conventional rotational Doppler effect, the frequency shift is proportional to the variation of total angular momentum of light beam, irrespective of the orbital angular momentum of input beams.

© 2017 Optical Society of America

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References

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2017 (1)

2016 (1)

2015 (7)

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

Y. Liu, Y. Ke, J. Zhou, H. Luo, and S. Wen, “Manipulating the spin-dependent splitting by geometric Doppler effect,” Opt. Express 23(13), 16682–16692 (2015).
[Crossref] [PubMed]

D. Hakobyan and E. Brasselet, “Optical torque reversal and spin-orbit rotational Doppler shift experiments,” Opt. Express 23(24), 31230–31239 (2015).
[Crossref] [PubMed]

M. Seghilani, M. Myara, I. Sagnes, B. Chomet, R. Bendoula, and A. Garnache, “Self-mixing in low-noise semi-conductor vortex laser: detection of a rotational Doppler shift in backscattered light,” Opt. Lett. 40(24), 5778–5781 (2015).
[Crossref] [PubMed]

D. Hakobyan, H. Magallanes, G. Seniutinas, S. Juodkazis, and E. Brasselet, “Tailoring orbital angular momentum of light in the visible domain with metallic metasurfaces,” Adv. Opt. Mater. 4(2), 306 (2015).
[Crossref]

M. I. Shalaev, J. Sun, A. Tsukernik, A. Pandey, K. Nikolskiy, and N. M. Litchinitser, “High-efficiency all-dielectric metasurfaces for ultracompact beam manipulation in transmission mode,” Nano Lett. 15(9), 6261–6266 (2015).
[Crossref] [PubMed]

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, and S. Wen, “hybrid-order Poincaré sphere,” Phys. Rev. A 91(2), 023801 (2015).
[Crossref]

2014 (4)

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3, e167 (2014).
[Crossref]

Y. Liu, X. Ling, X. Yi, X. Zhou, H. Luo, and S. Wen, “Realization of polarization evolution on higher-order Poincaré sphere with metasurface,” Appl. Phys. Lett. 104(19), 191110 (2014).
[Crossref]

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics 8(8), 610–614 (2014).
[Crossref]

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
[Crossref] [PubMed]

2013 (2)

O. Korech, U. Steinitz, R. J. Gordon, I. S. Averbukh, and Y. Prior, “Observing molecular spinning via the rotational Doppler effect,” Nat. Photonics 7(9), 711–714 (2013).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barbett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

2012 (2)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics,  6(7), 488–496 (2012).
[Crossref]

N. M. Litchinitser, “Structured light meets structured matter,” Science,  337(6098), 1054–1055 (2012).
[Crossref] [PubMed]

2011 (4)

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 383–385 (2011).
[Crossref]

T. D. Thomas, E. Kukk, K. Ueda, T. Ouchi, K. Sakai, T. X. Carroll, C. Nicolas, O. Travnikova, and C. Miron, “Experimental observation of rotational Doppler broadening in a molecular system,” Phys. Rev. Lett. 106(19), 193009 (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]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19(10), 9714–9736 (2011).
[Crossref] [PubMed]

2010 (1)

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric Doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

2009 (2)

2008 (4)

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett. 34(8), 1225–1227 (2008).
[Crossref]

A. Niv, Y. Gorodetski, V. Kleiner, and E. Hasman, “Topological spin-orbit interaction of light in anisotropic inhomogeneous subwavelength structures,” Opt. Lett. 33(24), 2910–2912 (2008).
[Crossref] [PubMed]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

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

2007 (1)

2006 (2)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

S. Barreiro, J.W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[Crossref] [PubMed]

2005 (2)

M. Michalski, W Hütner, and H. Schimming, “Experimental demonstration of the rotational frequency shift in a molecular system,” Phys. Rev. Lett. 95(20), 203005 (2005).
[Crossref] [PubMed]

G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30(10), 1207–1209 (2005).
[Crossref] [PubMed]

2003 (1)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

1999 (1)

1998 (2)

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[Crossref]

1996 (2)

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 (1992).
[Crossref] [PubMed]

1988 (2)

T. H. Chyba, L. J. Wang, L. Mandel, and R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13(7), 562–564 (1988).
[Crossref] [PubMed]

R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61(1), 19–22 (1988).
[Crossref] [PubMed]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392(1802), 45–57 (1984).
[Crossref]

1979 (1)

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating halfwave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
[Crossref]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115 (1936).
[Crossref]

Ahmed, N.

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

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics,  6(7), 488–496 (2012).
[Crossref]

Aiello, A.

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]

Allen, L.

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

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
[Crossref]

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 (1992).
[Crossref] [PubMed]

Arnold, S.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating halfwave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
[Crossref]

Ashrafi, N.

Ashrafi, S.

Averbukh, I. S.

O. Korech, U. Steinitz, R. J. Gordon, I. S. Averbukh, and Y. Prior, “Observing molecular spinning via the rotational Doppler effect,” Nat. Photonics 7(9), 711–714 (2013).
[Crossref]

Bao, C.

Barbett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barbett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Barreiro, S.

S. Barreiro, J.W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[Crossref] [PubMed]

Beijersbergen, M. W.

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 (1992).
[Crossref] [PubMed]

Bendoula, R.

Benseny, A.

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 383–385 (2011).
[Crossref]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392(1802), 45–57 (1984).
[Crossref]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115 (1936).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

Bomzon, Z.

Boyd, R. W.

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3, e167 (2014).
[Crossref]

Brasselet, E.

D. Hakobyan, H. Magallanes, G. Seniutinas, S. Juodkazis, and E. Brasselet, “Tailoring orbital angular momentum of light in the visible domain with metallic metasurfaces,” Adv. Opt. Mater. 4(2), 306 (2015).
[Crossref]

D. Hakobyan and E. Brasselet, “Optical torque reversal and spin-orbit rotational Doppler shift experiments,” Opt. Express 23(24), 31230–31239 (2015).
[Crossref] [PubMed]

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics 8(8), 610–614 (2014).
[Crossref]

Calvo, G. F.

Cao, Y.

Carroll, T. X.

T. D. Thomas, E. Kukk, K. Ueda, T. Ouchi, K. Sakai, T. X. Carroll, C. Nicolas, O. Travnikova, and C. Miron, “Experimental observation of rotational Doppler broadening in a molecular system,” Phys. Rev. Lett. 106(19), 193009 (2011).
[Crossref] [PubMed]

Chen, S.

Chomet, B.

Chyba, T. H.

Courtial, J.

M. J. Padgett and J. Courtial, “Poincaré sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999).
[Crossref]

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J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
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M. I. Shalaev, J. Sun, A. Tsukernik, A. Pandey, K. Nikolskiy, and N. M. Litchinitser, “High-efficiency all-dielectric metasurfaces for ultracompact beam manipulation in transmission mode,” Nano Lett. 15(9), 6261–6266 (2015).
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L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
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Picón, A.

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E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003).
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E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3, e167 (2014).
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J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
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T. D. Thomas, E. Kukk, K. Ueda, T. Ouchi, K. Sakai, T. X. Carroll, C. Nicolas, O. Travnikova, and C. Miron, “Experimental observation of rotational Doppler broadening in a molecular system,” Phys. Rev. Lett. 106(19), 193009 (2011).
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Schadt, M.

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E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3, e167 (2014).
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M. I. Shalaev, J. Sun, A. Tsukernik, A. Pandey, K. Nikolskiy, and N. M. Litchinitser, “High-efficiency all-dielectric metasurfaces for ultracompact beam manipulation in transmission mode,” Nano Lett. 15(9), 6261–6266 (2015).
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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]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003).
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S. Barreiro, J.W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[Crossref] [PubMed]

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T. D. Thomas, E. Kukk, K. Ueda, T. Ouchi, K. Sakai, T. X. Carroll, C. Nicolas, O. Travnikova, and C. Miron, “Experimental observation of rotational Doppler broadening in a molecular system,” Phys. Rev. Lett. 106(19), 193009 (2011).
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[Crossref] [PubMed]

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[Crossref]

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T. D. Thomas, E. Kukk, K. Ueda, T. Ouchi, K. Sakai, T. X. Carroll, C. Nicolas, O. Travnikova, and C. Miron, “Experimental observation of rotational Doppler broadening in a molecular system,” Phys. Rev. Lett. 106(19), 193009 (2011).
[Crossref] [PubMed]

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E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3, e167 (2014).
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Figures (5)

Fig. 1
Fig. 1

Schematic illustration of the propagation of circular polarized light beams passing through two half-wave q plates, wherein the first one is fixed while the second one rotates with a angular velocity. The red and blue arrows represent the left- and right-handed circularly polarized waves, respectively. And the relative displacements of arrows between the input and the output are resulting from the induced geometric phase.

Fig. 2
Fig. 2

Schematic illustration of the evolution of optical field in the rotational Doppler effect. The q plate rotating uniformly with an angular velocity ω′ is illuminated by a circularly polarized vortex wave with an angular frequency ω. Here, σ = −1 and σ = +1 represent the left- and right-handed circular polarization, respectively. And the output beams in (a) and (b) possess two opposite helical wave fronts with different angular frequencies, ω + Δω and ω − Δω.

Fig. 3
Fig. 3

Schematic illustration showing realization of the evolution along different longitude lines on the hybrid-order Poincaré sphere. The sphere is assumed with state σ = +1 and l = 0 in the north pole, and the state σ = −1 and m = +1 in the south pole. Insets (t0)–(t3) show the rotating q plate (q = 1/2) with different initial angle α0. Here the sense of the positive rotating angle is chosen as anticlockwise which means ω′ < 0, α0 < 0 corresponding to the longitude line’s moving direction: t0t1t2t3 and polarization states moving from north pole to south pole.

Fig. 4
Fig. 4

Schematic illustration showing realization of the evolution along different longitude lines on the hybrid-order Poincaré sphere. The state on the pole is assumed as the same as Fig. 3. Insets (t0)–(t3) show the rotating q plate (q = 1/2) with different initial angle α0. Here, ω′ > 0, α0 > 0 corresponding to the longitude line’s moving direction: t′0t′1t′2t′3 and polarization states moving from north pole to south pole.

Fig. 5
Fig. 5

3D spatial structure of output beam screws with pitch zp after the rotating q plate. The rotational Doppler effect induces rotation of intensity pattern with the rotational angular velocity ϖ = σω′(q − 1)/q. Here, q = 1/2, σ = +1. (a) the positive rotation (ω′ > 0, ϖ < 0), (b) the opposite rotation (ω′ < 0, ϖ > 0).

Equations (23)

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T ( r , φ ) = M ( α ) J ( δ ) M 1 ( α ) .
J = [ exp ( i δ / 2 ) 0 0 exp ( i δ / 2 ) ] .
M ( α ) = ( cos α sin α sin α cos α ) .
T ( r , φ ) = cos δ 2 ( 1 0 0 1 ) + i sin δ 2 ( cos 2 α sin 2 α sin 2 α cos 2 α ) .
α ( r , φ ) = q φ + α 0 ,
| ψ I = 2 2 ( e ^ x i σ e ^ y ) .
| ψ II = 2 2 ( e ^ x + i σ e ^ y ) exp [ i ( l φ 2 σ α 0 ) ] ,
α ( r , φ ) = q ( φ β ) + ( α 0 + β ) ,
| ψ III = 2 2 ( e ^ x i σ e ^ y ) exp [ i ( m φ 2 σ q β + 2 σ β ) ] ,
γ = 2 σ ( q 1 ) β .
Δ ω = d γ d t ,
Δ ω = 2 σ ω ( q 1 ) ,
| ψ ( θ , Φ ) = cos θ 2 | N l + sin θ 2 | S m exp ( i σ Φ ) ,
γ ( C ) = C d S V ( R ) ,
V ( R ) = R × A .
γ ( C ) = m l 2 σ 4 Ω .
γ ( C ) = σ ( q 1 ) 2 Ω .
Δ ω = 2 σ ω ( q 1 ) .
k + z + k + r 2 R + ( z ) + m + φ + Φ + = const ,
k z + k r 2 R ( z ) + m φ + Φ = const ,
( k + k ) z + φ ( m + m ) = 0 .
z p = π ( m + m ) c | Δ ω | ,
t p = 2 π | q | | ( 1 q ) ω | .

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