Abstract

We investigate theoretical properties of beams of light with non-uniform polarization patterns. Specifically, we determine all possible configurations of cylindrically polarized modes (CPMs) of the electromagnetic field, calculate their total angular momentum and highlight the subtleties of their structure. Furthermore, a hybrid spatio-polarization description for such modes is introduced and developed. In particular, two independent Poincaré spheres have been introduced to represent simultaneously the polarization and spatial degree of freedom of CPMs. Possible mode-to-mode transformations accomplishable with the help of Bconventional polarization and spatial phase retarders are shown within this representation. Moreover, the importance of these CPMs in the quantum optics domain due to their classical features is highlighted.

© 2011 OSA

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  5. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000).
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  6. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [CrossRef] [PubMed]
  7. N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 273–276 (2001).
    [CrossRef] [PubMed]
  8. M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
    [CrossRef]
  9. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2006).
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  10. P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
    [CrossRef] [PubMed]
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  12. C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
    [CrossRef] [PubMed]
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  18. A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
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  19. J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons: The relativistic Field Theory of Charged Particles with Spin one-half (Addison-Wesley Publishing Company, 1955).
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    [CrossRef] [PubMed]
  22. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68(1), 012323 (2003).
    [CrossRef]
  23. M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79(4), 043825 (2009).
    [CrossRef]
  24. N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002).
    [CrossRef]
  25. J. N. Damask, Polarization Optics in Telecommunication (Springer, 2005).
  26. W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90(4), 043601 (2003).
    [CrossRef] [PubMed]
  27. M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2000).
  28. C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010).
    [CrossRef]

2011 (1)

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

2010 (3)

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105(3), 030407 (2010).
[CrossRef] [PubMed]

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
[CrossRef] [PubMed]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010).
[CrossRef]

2009 (4)

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102(16), 163602 (2009).
[CrossRef] [PubMed]

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009).
[CrossRef]

A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79(4), 043825 (2009).
[CrossRef]

2007 (2)

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
[CrossRef]

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

2006 (1)

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2006).
[CrossRef]

2003 (3)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68(1), 012323 (2003).
[CrossRef]

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90(4), 043601 (2003).
[CrossRef] [PubMed]

2002 (1)

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002).
[CrossRef]

2001 (1)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 273–276 (2001).
[CrossRef] [PubMed]

2000 (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000).
[CrossRef]

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335 (2000).
[CrossRef]

1999 (1)

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]

Aiello, A.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

Allen, L.

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]

Andersen, U. L.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102(16), 163602 (2009).
[CrossRef] [PubMed]

Azzam, R. M. A.

R. M. A. Azzam and N. Bashra, Ellipsometry and Polarized Light (North Holland Personal Library, 1988).

Banzer, P.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
[CrossRef] [PubMed]

Barreiro, J. T.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105(3), 030407 (2010).
[CrossRef] [PubMed]

Bashra, N.

R. M. A. Azzam and N. Bashra, Ellipsometry and Polarized Light (North Holland Personal Library, 1988).

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), 8185–8189 (1992).
[CrossRef] [PubMed]

Bernet, S.

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

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009).
[CrossRef]

Borges, C. V. S.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010).
[CrossRef]

Bowen, W. P.

M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79(4), 043825 (2009).
[CrossRef]

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90(4), 043601 (2003).
[CrossRef] [PubMed]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2000).

Courtial, J.

Damask, J. N.

J. N. Damask, Polarization Optics in Telecommunication (Springer, 2005).

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000).
[CrossRef]

Elser, D.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

Euser, T. G.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

Feurer, T.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2006).
[CrossRef]

Förtsch, M.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

Fürhapter, S.

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

Gabriel, C.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000).
[CrossRef]

Hell, S. W.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 273–276 (2001).
[CrossRef] [PubMed]

Hor-Meyll, M.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010).
[CrossRef]

Hsu, M. T. L.

M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79(4), 043825 (2009).
[CrossRef]

Huguenin, J. A. O.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010).
[CrossRef]

Huse, N.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 273–276 (2001).
[CrossRef] [PubMed]

Jauch, J. M.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons: The relativistic Field Theory of Charged Particles with Spin one-half (Addison-Wesley Publishing Company, 1955).

Jesacher, A.

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

Joly, N. Y.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

Khoury, A. Z.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010).
[CrossRef]

Konermann, H.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
[CrossRef]

Korolkova, N.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002).
[CrossRef]

Kwiat, P. G.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105(3), 030407 (2010).
[CrossRef] [PubMed]

Lam, P. K.

M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79(4), 043825 (2009).
[CrossRef]

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90(4), 043601 (2003).
[CrossRef] [PubMed]

Lassen, M.

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102(16), 163602 (2009).
[CrossRef] [PubMed]

Leuchs, G.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
[CrossRef] [PubMed]

A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102(16), 163602 (2009).
[CrossRef] [PubMed]

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000).
[CrossRef]

Lindlein, N.

A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
[CrossRef]

Loudon, R.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002).
[CrossRef]

R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).

Maiwald, R.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
[CrossRef]

Marquardt, Ch.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Maurer, C.

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

Meier, M.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2006).
[CrossRef]

Mejías, P. M.

R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2000).

Okamoto, M.

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68(1), 012323 (2003).
[CrossRef]

Padgett, M. J.

Peres, A.

A. Peres, Quantum Theory: Concept and Methods (Kluwer Academic, Boston, 1995).

Peschel, U.

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
[CrossRef] [PubMed]

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
[CrossRef]

Piquero, G.

R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

Quabis, S.

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000).
[CrossRef]

Ralph, T. C.

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90(4), 043601 (2003).
[CrossRef] [PubMed]

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002).
[CrossRef]

Ritsch-Marte, M.

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

Rohrlich, F.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons: The relativistic Field Theory of Charged Particles with Spin one-half (Addison-Wesley Publishing Company, 1955).

Romano, V.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2006).
[CrossRef]

Russell, P. St. J.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

Sasada, H.

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68(1), 012323 (2003).
[CrossRef]

Schnabel, R.

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90(4), 043601 (2003).
[CrossRef] [PubMed]

Schönle, A.

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 273–276 (2001).
[CrossRef] [PubMed]

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Silberhorn, Ch.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002).
[CrossRef]

Sondermann, M.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
[CrossRef]

Spreeuw, R. J. C.

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]

Wei, T.-C.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105(3), 030407 (2010).
[CrossRef] [PubMed]

Woerdman, J. P.

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]

Zhong, W.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

Appl. Phys. A (1)

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2006).
[CrossRef]

Appl. Phys. B (1)

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–492 (2007).
[CrossRef]

J. Biomed. Opt. (1)

N. Huse, A. Schönle, and S. W. Hell, “Z-polarized confocal microscopy,” J. Biomed. Opt. 6, 273–276 (2001).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt. (1)

M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009).
[CrossRef]

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

N. J. Phys. (1)

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

Opt. Comm. (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Comm. 179, 1 – 7 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (5)

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]

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68(1), 012323 (2003).
[CrossRef]

M. T. L. Hsu, W. P. Bowen, and P. K. Lam, “Spatial-state stokes-operator squeezing and entanglement for optical beams,” Phys. Rev. A 79(4), 043825 (2009).
[CrossRef]

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and Ch. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65(5), 052306 (2002).
[CrossRef]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010).
[CrossRef]

Phys. Rev. Lett. (6)

W. P. Bowen, R. Schnabel, P. K. Lam, and T. C. Ralph, “Experimental investigation of criteria for continuous variable entanglement,” Phys. Rev. Lett. 90(4), 043601 (2003).
[CrossRef] [PubMed]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous variable entanglement and squeezing of orbital angular momentum states,” Phys. Rev. Lett. 102(16), 163602 (2009).
[CrossRef] [PubMed]

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106(6), 060502 (2011).
[CrossRef] [PubMed]

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105(3), 030407 (2010).
[CrossRef] [PubMed]

A. Aiello, N. Lindlein, Ch. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Other (8)

A. E. Siegman, Lasers (University Science Books, 1986).

R. Martinez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009).
[CrossRef]

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons: The relativistic Field Theory of Charged Particles with Spin one-half (Addison-Wesley Publishing Company, 1955).

R. M. A. Azzam and N. Bashra, Ellipsometry and Polarized Light (North Holland Personal Library, 1988).

R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2000).

A. Peres, Quantum Theory: Concept and Methods (Kluwer Academic, Boston, 1995).

J. N. Damask, Polarization Optics in Telecommunication (Springer, 2005).

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Figures (9)

Fig. 1
Fig. 1

Illustrating the two sets of pairs of co- and counter-rotating unit vectors { + (θ), θ ^ + (θ)} and { (θ), θ ^ (θ)}, respectively. The vector (θ) is the mirror-image with respect to the vertical axis y of +(θ), while θ ^ (θ) is the mirror-image with respect to the horizontal axis x of θ ^ + (θ).

Fig. 2
Fig. 2

Complex polarization patterns of (a) u R + , (b) u A + , (c) u R , (d) u A , underlayed with the doughnut shaped intensity distribution.

Fig. 3
Fig. 3

Poincaré sphere representation for an arbitrary two-dimensional system. Here, the radius r of the sphere is fixed to one, and spherical angles {ϑ,ϕ} are related to the Stokes parameters via the relation: {cos ϑ,sin ϑ cos ϕ,sin ϑ sin ϕ} = {S 1/S 0, S 2/S 0, S 3/S 0}, where S 0 denotes the total intensity of the beam.

Fig. 4
Fig. 4

Fundamental idea behind the new Poincaré sphere representation: Combining the SMPS (a) and the PPS (b) to the hybrid Poincaré spheres (HPSs) (c) and (d).

Fig. 5
Fig. 5

Polarization states on the HPSs, represented on the sphere of the “+” modes (a) and the “−” modes (b). The Stokes parameters on both the “+” and “−” hybrid Poincaré sphere are explicitly marked.

Fig. 6
Fig. 6

(a) Experimental setup to measure hybrid Stokes parameters on the “+” HPS: Measurement of S 1 + : (A) = (B) = (C) = (D) no optical element. Measurement of S 2 + : (A) = (C) = (D) HWP 22.5°, (B) HWP 112.5°. Measurement of S 3 + : (A) = (C) HWP 22. 5°, QWP 0°, (B) HWP 112.5°, QWP 90°, (D) HWP 22.5°, QWP 90°. S 0 + can be measured in every case if one uses the sum signal between both ports. In (b), the asymmetric Mach Zehnder interferometer is displayed which acts as a spatial mode splitter (SMS) and has been used three times is the Stokes measurement setup.

Fig. 7
Fig. 7

Manipulation of states on the HPSs, represented on the sphere of the “+” modes (a) and the “−” modes (b). This special case can be carried out by both waveplates and pairs of cylindrical lenses. The distance between the (c) two cylindrical lenses is 2f, where f denotes their focal length.

Fig. 8
Fig. 8

Transformations on the “+” HPS: Without altering the rotational symmetry of the state, these transformations are possible with the help of conventional phase retarders.

Fig. 9
Fig. 9

By a mirror image symmetry operation one can pass from one sphere to the respective other sphere. This action is practically performed by means of a HWP.

Equations (135)

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E ( r , t ) = 1 2 ( u ( r ) e i χ + u * ( r ) e i χ ) ,
u ( r ) = f 1 ( r ) x ^ + f 2 ( r ) y ^ .
2 f 1 , 2 x 2 + 2 f 1 , 2 y 2 + 2 i k f 1 , 2 z = 0.
x = x x ^ + y y ^ .
r ^ ( θ ) = cos θ x ^ + sin θ y ^ ,
θ ^ ( θ ) = sin θ x ^ + cos θ y ^ .
x ^ = ( 1 0 ) , y ^ = ( 0 1 ) ,
r ^ ( θ ) = R ( θ ) x ^ , θ ^ ( θ ) = R ( θ ) y ^ ,
R ( θ ) = ( cos θ sin θ sin θ cos θ ) .
R ( φ ) r ^ ( θ ) = R ( θ + φ ) x ^ = r ^ ( θ + φ ) ,
R ( φ ) θ ^ ( θ ) = R ( θ + φ ) y ^ = θ ^ ( θ + φ ) .
r ^ ± ( θ ) = ± R ( ± θ ) x ^ , θ ^ ± ( θ ) = ± R ( ± θ ) y ^ ,
R ( φ ) r ^ ± ( θ ) = r ^ ± ( θ ± φ ) , R ( φ ) θ ^ ± ( θ ) = θ ^ ± ( θ ± φ ) .
u ± ( r , θ ) = r ^ ± ( θ ) a ( r ) + θ ^ ± ( θ ) b ( r ) ,
u ± ( r , θ ± φ ) = R ( φ ) u ± ( r , θ ) ,
R ( φ ) u ± ( r , θ ) = ( cos φ sin φ sin φ cos φ ) ( x ^ · u ± y ^ · u ± ) .
x ( r , θ ± φ ) = r cos ( θ ± φ ) = r ( cos θ cos φ sin θ sin φ ) ,
y ( r , θ ± φ ) = r sin ( θ ± φ ) = r ( sin θ cos φ ± cos θ sin φ ) .
x ( r , θ ± φ ) = R ( ± φ ) x ( r , θ ) .
u ± ( R ( ± φ ) x ) = R ( φ ) u ± ( x ) ,
u + ( x ) = R ( φ ) u + ( R ( φ ) x ) ,
u ( x ) = R ( φ ) u ( R ( φ ) x ) .
W ( x ) = R ( φ ) V ( R ( φ ) x ) .
R ( 2 φ ) u ( x ) = R ( φ ) u ( R ( φ ) x ) .
u ( r , θ ) = r ^ ( cos θ f 1 + sin θ f 2 ) + θ ^ ( sin θ f 1 + cos θ f 2 ) .
f 1 ( r ) = a ( r ) cos θ b ( r ) sin θ ,
f 2 ( r ) = a ( r ) sin θ + b ( r ) cos θ .
f 1 ( r ) = x α ( r ) y β ( r ) ,
f 2 ( r ) = y α ( r ) + x β ( r ) ,
ψ 10 ( x ) = x e ( x 2 + y 2 ) / w 0 2 / w 0 ,
ψ 01 ( x ) = y e ( x 2 + y 2 ) / w 0 2 / w 0 ,
f 1 ( r ) = 1 2 ( A ψ 10 B ψ 01 ) ,
f 2 ( r ) = 1 2 ( B ψ 10 + A ψ 01 ) ,
u = 1 2 ( A ψ 10 B ψ 01 ) x ^ + 1 2 ( B ψ 10 + A ψ 01 ) y ^ .
u ^ R + = 1 2 ( ψ 10 x ^ + ψ 01 y ^ ) ,
u ^ A + = 1 2 ( ψ 01 x ^ + ψ 10 y ^ ) .
{ ψ 10 , ψ 01 } { x ^ , y ^ } = { ψ 10 x ^ , ψ 10 y ^ , ψ 01 x ^ , ψ 01 y ^ } .
u ^ R = 1 2 ( ψ 10 x ^ + ψ 01 y ^ ) ,
u ^ A = 1 2 ( ψ 01 x ^ + ψ 10 y ^ ) .
( u ( x ) , v ( x ) ) = i = 1 2 u i * ( x ) v i ( x ) d x d y ,
u ( r , θ α ) = R ( α ) u ( r , θ ) .
e i * · e j = δ i j , i , j { 1 , 2 } ,
v 1 = ψ 10 ( x , y , z ) , v 2 = ψ 01 ( x , y , z ) ,
v i * v j d x d y = δ i j , i , j { 1 , 2 } .
u ^ R + = i = 1 2 ( λ i ) 1 / 2 e ^ i v i ,
K = 1 / i = 1 2 λ i 2 = 2 .
v 1 = ψ 01 ( x , y , z ) , v 2 = ψ 10 ( x , y , z ) ,
u ^ AB + = 1 2 [ ψ 10 ( A x ^ + B y ^ ) + ψ 01 ( B x ^ + A y ^ ) ] 1 2 f 1 ψ 10 + 1 2 f 2 ψ 01 ,
f 1 * · f 2 = AB * A * B = 0 ,
u ^ AB + = A 2 ( x ^ i y ^ ) ( ψ 10 ± i ψ 01 ) ,
p ( r ) = Im [ u * · ( ) u ] + 1 2 Im [ × ( u * × u ) ] p orb ( r ) + p sp ( r ) ,
p x ( r ) = 1 k Im [ f 1 * ( x f 1 ) + f 2 * ( x f 2 ) ] + 1 k Im [ y ( f 1 * f 2 ) ] ,
p y ( r ) = 1 k Im [ f 1 * ( y f 1 ) + f 1 * ( y f 2 ) ] 1 k Im [ x f 1 * f 2 ] ,
p z ( r ) = | f 1 | 2 + | f 2 | 2 ,
P = p ( r ) d x d y P orb + P sp ,
j ( r ) = r × p ( r ) = r × [ p orb ( r ) + p sp ( r ) ] l ( r ) + s ( r ) ,
l x ( r ) = y ( | f 1 | 2 + | f 2 | 2 ) z k Im ( f 1 * y f 1 + f 2 * y f 2 ) ,
l y ( r ) = x ( | f 1 | 2 + | f 2 | 2 ) + z k Im ( f 1 * x f 1 + f 2 * x f 2 ) ,
l z ( r ) = Im [ x ( f 1 * y f 1 + f 2 * y f 2 ) y ( f 1 * x f 1 + f 2 * x f 2 ) ] ,
s x ( r ) = z k Im [ x ( f 1 * f 2 ) ] ,
s y ( r ) = z k Im [ y ( f 1 * f 2 ) ] ,
s z ( r ) = 1 k Im [ x x ( f 1 * f 2 ) + y y ( f 1 * f 2 ) ] .
J = j ( r ) d x d y L + S ,
[ x x ( f 1 * f 2 ) + y y ( f 1 * f 2 ) ] d x d y = 2 f 1 * f 2 d x d y .
S z = i k ( f 1 * f 2 f 1 f 2 * ) d x d y .
L x = y ( | f 1 | 2 + | f 2 | 2 ) d x d y ,
L y = x ( | f 1 | 2 + | f 2 | 2 ) d x d y ,
L z = 0.
{ ψ 10 , ψ 01 } { x ^ , y ^ } = { ψ 10 x ^ , ψ 10 y ^ , ψ 01 x ^ , ψ 01 y ^ } .
{ ψ 10 , ψ 01 } { x ^ , y ^ } = { u R + , u A + } { u R , u A } .
S 0 ± = f R ± * f R ± + f A ± * f A ± ,
S 1 ± = f R ± * f R ± f A ± * f A ± ,
S 2 ± = f R ± * f A ± + f A ± * f R ± ,
S 3 ± = i ( f R ± * f A ± f A ± * f R ± ) ,
E = f A ± u A ± + f R ± u R ± .
f A ± = cos ( θ / 2 ) ,
f R ± = exp ( i ϕ ) sin ( θ / 2 ) .
u ^ R + x ^ ψ 10 + y ^ ψ 01 fixed spatial mode HWP , HWP u ^ A + y ^ ψ 10 + x ^ ψ 01 , u ^ R + x ^ ψ 10 + y ^ ψ 01 fixed polarization two pairs of cyl . lenses u ^ A + x ^ ψ 01 + y ^ ψ 10 .
{ θ , ϕ } { θ = π θ , ϕ } ϕ { 0 , π 2 , π , 3 π 2 } ,
{ θ , ϕ } { θ = π θ , ϕ = π + ϕ } ϕ { 0 , π 2 , π , 3 π 2 } ,
{ θ , ϕ } { θ = θ , ϕ = π + ϕ } ϕ [ 0 , 2 π [ ,
E ^ ( r , t ) = E ^ + ( r , t ) + E ^ ( r , t ) ,
E ^ + ( r , t ) = 1 2 λ = 1 2 n , m e ^ λ a ^ λ n m ψ n m e i χ ,
[ a ^ λ n m , a ^ λ n m ] = δ λ λ δ n n δ m m .
u AB ( r ) = 1 2 [ x ^ ( A ψ 10 B ψ 01 ) + y ^ ( B ψ 10 + A ψ 01 ) ] = A u R + B u A ,
( u AB ( r ) , u A B ( r ) ) = A * A + B * B .
𝒮 = 1 2 [ α u AB ( r ) e i χ + α * u AB * ( r ) e i χ ] .
0 | E ^ + | 1 = 1 2 u AB ( r ) e i χ .
a ^ AB = A ( a ^ x 10 + a ^ y 01 2 ) + B ( a ^ x 01 + a ^ y 10 2 ) ,
[ a ^ A B , a ^ AB ] = A * A + B * B ,
D ^ i ( β ) = exp ( β a ^ i β * a ^ i ) ,
a ^ 1 = a ^ x 10 , a ^ 2 = a ^ y 01 , a ^ 3 = a ^ x 01 , a ^ 4 = a ^ y 10 .
v i ( r , t ) e ^ λ ψ n m exp ( i χ ) ,
E ^ + = 1 2 i = 1 v i a ^ i .
E ^ + | β i = 1 2 j = 1 v j a ^ j | β i = 1 2 v i β | β i ,
E ^ + | α 1 , α 2 , α 3 , α 4 = 1 2 i = 1 v i a ^ i | α 1 , α 2 , α 3 , α 4 = 1 2 i = 1 4 v i α i | α 1 , α 2 , α 3 , α 4 ,
𝒮 = α 1 , α 2 , α 3 , α 4 | E ^ + + E ^ | α 1 , α 2 , α 3 , α 4 = 1 2 i = 1 4 ( v i α i + v i * α i * ) = 1 2 [ x ^ ( α 1 ψ 10 + α 3 ψ 01 ) + y ^ ( α 4 ψ 10 + α 2 ψ 01 ) ] e i χ + c . c .
α 1 = α 2 = A * α 2 , α 3 = α 4 = B * α 2 ,
D ^ 4 ( α 4 ) D ^ 3 ( α 3 ) D ^ 2 ( α 2 ) D ^ 1 ( α 1 ) = exp i = 1 4 ( α i a ^ i + α i * a ^ i ) = exp ( α a ^ AB α * a ^ AB ) ,
A ^ ( ξ ) = i = 1 4 ξ i * a ^ i , with i = 1 4 | ξ i | 2 = 1 ,
[ A ^ ( ξ ) , A ^ ( ξ ) ] = 1.
0 | E ^ + | 1 = 1 2 0 | i = 1 v i a ^ i j = 1 4 ξ j a ^ j | 0 = 1 2 i , j ξ j v i 0 | a ^ i a ^ j | 0 = 1 2 i = 1 4 v i ξ i .
ξ 1 = ξ 2 = A * 1 2 , ξ 3 = ξ 4 = B * 1 2 ,
A ^ ( ξ ) = A ( a ^ x 10 + a ^ y 01 2 ) + B ( a ^ x 01 + a ^ y 10 2 ) = a ^ AB ,
S ^ A ( ζ ) = exp [ 1 2 ζ * ( a ^ A ) 2 1 2 ζ ( a ^ A ) 2 ] ,
S ^ A ( ζ ) = S ^ 3 ( ζ / 2 ) S ^ 4 ( ζ / 2 ) S ^ 34 ( ζ / 2 ) ,
S ^ i ( ζ ) = exp [ 1 2 ζ * ( a ^ i ) 2 1 2 ζ ( a ^ i ) 2 ] ,
S ^ i , j ( ζ ) = exp ( ζ * a ^ i a ^ j ζ a ^ i a ^ j ) .
| ζ A = S ^ 3 ( ζ / 2 ) S ^ 4 ( ζ / 2 ) S ^ 34 ( ζ ) | 0 .
S ^ 34 ( ζ ) | 0 = 1 cosh s n = 0 ( e i ϑ tanh s ) n | n 3 | n 4 ,
x ^ | e 1 , y ^ | e 2 ,
ψ 10 ( x ) | ψ 1 ( x ) , ψ 01 ( x ) | ψ 2 ( x ) ,
u R ± | u R ± = 1 2 ( ± | ψ 1 | e 1 + | ψ 2 | e 2 ) ,
u A ± | u A ± = 1 2 ( | ψ 2 | e 1 + | ψ 1 | e 2 ) .
{ u R + , u A + , u R , u A } { | Φ + , | Ψ , | Φ , | Ψ + } ,
| u ± ( x ) = G ^ ± ( φ ) | u ± ( x ) ,
G ^ ± ( φ ) | ψ i ( x ) | e j = | ψ i ( R ( φ ) x ) R ^ ( φ ) | e j ,
R ^ ( φ ) | e j = k = 1 2 R k j ( φ ) | e k ,
ψ 10 ( R ( φ ) x ) = ψ 10 ( x ) cos φ ± ψ 01 ( x ) sin φ ,
ψ 01 ( R ( φ ) x ) = ψ 10 ( x ) sin φ + ψ 01 ( x ) cos φ .
| ψ i ( R ( φ ) x ) = R ^ ( φ ) | ψ i ( x ) = l = 1 2 R i l ( φ ) | ψ l ( x ) .
G ^ ± ( φ ) | ψ i ( x ) | e j = R ^ ( φ ) | ψ i ( x ) R ^ ( φ ) | e j = k , l R i l ( φ ) | ψ l ( x ) R k j ( φ ) | e k , = k , l R i l ( φ ) R j k T ( φ ) | ψ l ( x ) | e k , = k , l [ R ( φ ) R ( φ ) ] i j , l k | ψ l ( x ) | e k , = R ^ ( φ ) R ^ ( φ ) | ψ i ( x ) | e j ,
G ^ ± ( φ ) = R ^ ( φ ) R ^ ( φ ) .
G ^ ± ( φ ) T ^ | u ± ( x ) = T ^ | u ± ( x ) .
T ^ | u ± ( x ) = T ^ G ^ ± ( φ ) | u ± ( x ) .
G ^ ± ( φ ) T ^ | u ± ( x ) = T ^ G ^ ± ( φ ) | u ± ( x ) .
[ G ^ ± ( φ ) T ^ T ^ G ^ ± ( φ ) ] | u ± ( x ) = 0.
[ G ^ ± ( φ ) , T ^ ] | u ± ( x ) = 0.
[ R ^ ( φ ) R ^ ( φ ) ] [ M ^ P ^ ] = [ M ^ P ^ ] [ R ^ ( φ ) R ^ ( φ ) ] ,
R ^ ( φ ) M ^ [ R ^ ( φ ) ] 1 R ^ ( φ ) P ^ [ R ^ ( φ ) ] 1 = M ^ P ^ ,
[ R ^ ( φ ) , M ^ ] = 0 , [ R ^ ( φ ) , P ^ ] = 0 ,
M = ( m 1 m 2 m 2 m 1 ) and P = ( p 1 p 2 p 2 p 1 ) ,
M = ( m 1 m 2 m 2 m 1 ) and P = ( p 1 p 2 p 2 p 1 ) ,
P = 1 2 ( 1 ± i i 1 ) ,
P = ( cos ( 2 α ) sin ( 2 α ) sin ( 2 α ) cos ( 2 α ) ) ,

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