Abstract

The control of spatial and polarization modes of optical beams enables the production of topological singularities encoded on the polarization of the light. This allows the study of topological disclinations not easily found in the natural setting. In this article we report on the observation of new features in disclinations realized with singular optical beams. They were prepared using three spatial modes bearing optical vortices in non-separable superpositions with circular polarization states. The disclinations involve asymmetric rotational dislocations, whose optical counterparts in the optical far field are known as C-points, and which are classified as monstars. They have been known to have a singularity index that can be positive, or negative as reported by us recently. Here we report on monstars with an index of zero. Monstars are characterized by having sectors bound by radial lines that involve curved lines radiating from the singularity. We found that kinks in otherwise smooth line patterns of asymmetric disclinations are scars of a separate but related pattern of line-slope discontinuities, carried optically by C-lines in the far field. These scars are indicative of the underlying structure or symmetry of the pattern. We present a general formalism to understand and generate monstars, along with measurements: the experimental results are in excellent agreement with theoretical modelings.

© 2017 Optical Society of America

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References

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    [Crossref]
  29. V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15, 044027 (2013).
    [Crossref]
  30. M. Vasnetsov, M. Soskin, V. Pas’ko, and V. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18, 034003 (2016).
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  32. G. Milione, M. Lavery, H. Huang, Y. Ren, G. Xie, T. Nguyen, E. Karimi, L. Marrucci, D. Nolan, R. Alfano, and A. Wilner, “4 × 20  gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
    [Crossref]
  33. E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
    [Crossref]
  34. I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
    [Crossref]
  35. S. Vyas, Y. Kozawa, and S. Sato, “Polarization singularities in superposition of vector beams,” Opt. Express 21, 8972–8986 (2013).
    [Crossref]
  36. E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
    [Crossref]
  37. B. Khajavi and E. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
    [Crossref]
  38. S. Baumann, D. Kalb, L. MacMillan, and E. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
    [Crossref]
  39. B. Khajavi and E. J. Galvez, “Preparation of Poincaré beams with a same-path polarization/spatial-mode interferometer,” Opt. Eng. 54, 111305 (2015).
    [Crossref]
  40. I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
    [Crossref]
  41. P. Firby and C. Gardiner, Surface Topology (Ellis Horwood, 1982).
  42. I. Bendixon, “Sur les courbes définies par des équations différentielles,” Acta Math. 24, 1–88 (1901).
    [Crossref]
  43. I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss–Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
    [Crossref]
  44. M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
    [Crossref]
  45. M. R. Dennis, “Fermionic out-of-plane structure of polarization singularities,” Opt. Lett. 36, 3765–3767 (2011).
    [Crossref]

2016 (3)

M. Vasnetsov, M. Soskin, V. Pas’ko, and V. Vasil’ev, “A monstar portrait in the interior,” J. Opt. 18, 034003 (2016).
[Crossref]

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

B. Khajavi and E. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

2015 (3)

B. Khajavi and E. J. Galvez, “Preparation of Poincaré beams with a same-path polarization/spatial-mode interferometer,” Opt. Eng. 54, 111305 (2015).
[Crossref]

O. V. Manyuhina and M. J. Bowick, “Forming a cube from a sphere with tetratic order,” Phys. Rev. Lett. 114, 117801 (2015).
[Crossref]

G. Milione, M. Lavery, H. Huang, Y. Ren, G. Xie, T. Nguyen, E. Karimi, L. Marrucci, D. Nolan, R. Alfano, and A. Wilner, “4 × 20  gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
[Crossref]

2014 (2)

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

B. S. Murray, R. A. Pelcovits, and C. Rosenblatt, “Creating arbitrary arrays of two-dimensional topological defects,” Phys. Rev. E 90, 052501 (2014).
[Crossref]

2013 (3)

S. Gopalakrishnan, J. C. Y. Teo, and T. L. Hughes, “Disclination classes, fractional excitations, and the melting of quantum liquid crystals,” Phys. Rev. Lett. 111, 025304 (2013).
[Crossref]

V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15, 044027 (2013).
[Crossref]

S. Vyas, Y. Kozawa, and S. Sato, “Polarization singularities in superposition of vector beams,” Opt. Express 21, 8972–8986 (2013).
[Crossref]

2012 (4)

2011 (3)

S. C. Čopar and S. Z. Žumer, “Nematic braids: Topological invariants and rewiring of disclinations,” Phys. Rev. Lett. 106, 177801 (2011).
[Crossref]

I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss–Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
[Crossref]

M. R. Dennis, “Fermionic out-of-plane structure of polarization singularities,” Opt. Lett. 36, 3765–3767 (2011).
[Crossref]

2010 (2)

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010).
[Crossref]

G. M. Grason, “Topological defects in twisted bundles of two-dimensionally ordered filaments,” Phys. Rev. Lett. 105, 045502 (2010).
[Crossref]

2009 (3)

S. Baumann, D. Kalb, L. MacMillan, and E. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1, 1–57 (2009).
[Crossref]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref]

2008 (1)

2006 (1)

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

2005 (2)

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref]

A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Rotating vectorial vortices produced by space-variant sub wavelength gratings,” Opt. Lett. 30, 2933–2935 (2005).
[Crossref]

2003 (1)

2002 (3)

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
[Crossref]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

2001 (1)

I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
[Crossref]

1997 (1)

M. Kamionkowski, A. Kosowsky, and A. Stebbins, “Statistics of cosmic microwave background polarization,” Phys. Rev. D 55, 7368–7388 (1997).
[Crossref]

1987 (1)

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. observation on the electric field,” Proc. R. Soc. London A 414, 447–468 (1987).
[Crossref]

1983 (1)

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
[Crossref]

1979 (1)

N. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51, 591–648 (1979).
[Crossref]

1977 (1)

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[Crossref]

1901 (1)

I. Bendixon, “Sur les courbes définies par des équations différentielles,” Acta Math. 24, 1–88 (1901).
[Crossref]

1882 (1)

H. Poincaré, “Memoire sur les courbes definies par une equation differentielle (ii),” J. Math. Pures Appl. 8, 251–296 (1882).

1881 (1)

H. Poincaré, “Memoire sur les courbes definies par une equation differentielle,” J. Math. Pures Appl. 7, 375–422 (1881).

Alfano, R.

Alonso, M. A.

Alpmann, C.

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

Baumann, S.

Beckley, A. M.

Bendixon, I.

I. Bendixon, “Sur les courbes définies par des équations différentielles,” Acta Math. 24, 1–88 (1901).
[Crossref]

Berry, M. V.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[Crossref]

Beuman, T.

T. Beuman, A. Turner, and V. Vitelli, “Stochastic geometry and topology of non-Gaussian fields,” Proc. Natl. Acad. Sci. U.S.A. 109, 19943–19948 (2012).
[Crossref]

Biener, G.

Bowick, M. J.

O. V. Manyuhina and M. J. Bowick, “Forming a cube from a sphere with tetratic order,” Phys. Rev. Lett. 114, 117801 (2015).
[Crossref]

Brasselet, E.

E. Brasselet, “Tunable optical vortex arrays from a single nematic topological defect,” Phys. Rev. Lett. 108, 087801 (2012).
[Crossref]

Brown, T. G.

Cardano, F.

Copar, S. C.

S. C. Čopar and S. Z. Žumer, “Nematic braids: Topological invariants and rewiring of disclinations,” Phys. Rev. Lett. 106, 177801 (2011).
[Crossref]

de Lisio, C.

De Martini, F.

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref]

Denisenko, V.

Dennis, M. R.

M. R. Dennis, “Fermionic out-of-plane structure of polarization singularities,” Opt. Lett. 36, 3765–3767 (2011).
[Crossref]

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[Crossref]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref]

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213, 201–221 (2002).
[Crossref]

Denz, C.

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

Firby, P.

P. Firby and C. Gardiner, Surface Topology (Ellis Horwood, 1982).

Flossmann, F.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref]

Freund, I.

I. Freund, “Möbius strips and twisted ribbons in intersecting Gauss–Laguerre beams,” Opt. Commun. 284, 3816–3845 (2011).
[Crossref]

M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. 28, 1475–1477 (2003).
[Crossref]

I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).
[Crossref]

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

I. Freund, “Polarization flowers,” Opt. Commun. 199, 47–63 (2001).
[Crossref]

Galvez, E.

B. Khajavi and E. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

S. Baumann, D. Kalb, L. MacMillan, and E. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
[Crossref]

Galvez, E. J.

B. Khajavi and E. J. Galvez, “Preparation of Poincaré beams with a same-path polarization/spatial-mode interferometer,” Opt. Eng. 54, 111305 (2015).
[Crossref]

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by non-separable superpositions of Laguerre–Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012).
[Crossref]

Gardiner, C.

P. Firby and C. Gardiner, Surface Topology (Ellis Horwood, 1982).

Gopalakrishnan, S.

S. Gopalakrishnan, J. C. Y. Teo, and T. L. Hughes, “Disclination classes, fractional excitations, and the melting of quantum liquid crystals,” Phys. Rev. Lett. 111, 025304 (2013).
[Crossref]

Grason, G. M.

G. M. Grason, “Topological defects in twisted bundles of two-dimensionally ordered filaments,” Phys. Rev. Lett. 105, 045502 (2010).
[Crossref]

Guilfoyle, B.

B. Guilfoyle and W. Klingenberg, “Proof of the Caratheodory conjecture,” arXiv:0808.0851v3 (2013).

Hagen, H.

X. Tricoche, G. Scheuermann, and H. Hagen, “A topology simplification method for 2d vector fields,” in Proceedings of Visualization (2000), pp. 359–366.

Hajnal, J. V.

J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. observation on the electric field,” Proc. R. Soc. London A 414, 447–468 (1987).
[Crossref]

Hannay, J. H.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A 10, 1809–1821 (1977).
[Crossref]

Hasman, E.

Huang, H.

Hughes, T. L.

S. Gopalakrishnan, J. C. Y. Teo, and T. L. Hughes, “Disclination classes, fractional excitations, and the melting of quantum liquid crystals,” Phys. Rev. Lett. 111, 025304 (2013).
[Crossref]

Kalb, D.

Kamionkowski, M.

M. Kamionkowski, A. Kosowsky, and A. Stebbins, “Statistics of cosmic microwave background polarization,” Phys. Rev. D 55, 7368–7388 (1997).
[Crossref]

Karimi, E.

Khadka, S.

Khajavi, B.

B. Khajavi and E. Galvez, “High-order disclinations in space-variant polarization,” J. Opt. 18, 084003 (2016).
[Crossref]

B. Khajavi and E. J. Galvez, “Preparation of Poincaré beams with a same-path polarization/spatial-mode interferometer,” Opt. Eng. 54, 111305 (2015).
[Crossref]

Kleiner, V.

Klingenberg, W.

B. Guilfoyle and W. Klingenberg, “Proof of the Caratheodory conjecture,” arXiv:0808.0851v3 (2013).

Kosowsky, A.

M. Kamionkowski, A. Kosowsky, and A. Stebbins, “Statistics of cosmic microwave background polarization,” Phys. Rev. D 55, 7368–7388 (1997).
[Crossref]

Kozawa, Y.

Kumar, V.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

V. Kumar, G. M. Philip, and N. K. Viswanathan, “Formation and morphological transformation of polarization singularities: hunting the monstar,” J. Opt. 15, 044027 (2013).
[Crossref]

Lavery, M.

MacMillan, L.

Maier, M.

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[Crossref]

Manyuhina, O. V.

O. V. Manyuhina and M. J. Bowick, “Forming a cube from a sphere with tetratic order,” Phys. Rev. Lett. 114, 117801 (2015).
[Crossref]

Manzo, C.

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

Marrucci, L.

G. Milione, M. Lavery, H. Huang, Y. Ren, G. Xie, T. Nguyen, E. Karimi, L. Marrucci, D. Nolan, R. Alfano, and A. Wilner, “4 × 20  gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
[Crossref]

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
[Crossref]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref]

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

Mermin, N.

N. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51, 591–648 (1979).
[Crossref]

Milione, G.

Mokhun, A. I.

I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208, 223–253 (2002).
[Crossref]

Murray, B. S.

B. S. Murray, R. A. Pelcovits, and C. Rosenblatt, “Creating arbitrary arrays of two-dimensional topological defects,” Phys. Rev. E 90, 052501 (2014).
[Crossref]

Nabarro, F.

F. Nabarro, Theory of Crystals Dislocations (Oxford University, 1967).

Nagali, E.

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref]

Nguyen, T.

Niv, A.

Nolan, D.

Nomoto, S.

Nye, J. F.

J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).
[Crossref]

Oswald, P.

P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals (Taylor & Francis, 2005).

Otte, E.

E. Otte, C. Alpmann, and C. Denz, “Higher-order polarization singularities in tailored vector beams,” J. Opt. 18, 074012 (2016).
[Crossref]

Paparo, D.

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

Pas’ko, V.

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F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the apparatus. Optical elements include single-mode fiber (SMF), lenses (L), spatial light modulators (SLMs), half-wave plate (H), quarter-wave plate (Q), polarizer (P), filter (F) digital camera (DC), and collimator (C).

Fig. 2.
Fig. 2.

Plots of the radial orientation angle for different values of parameters in the case (2, 3): (a) β=10°, γ=180°; (b) β=23.33°, γ=180°; (c) β=30°, γ=180°; (d) β=30°, γ=0°.

Fig. 3.
Fig. 3.

Disclination lines for the case (1,2)=(2,3), γ=π, δ=0 for different values of β.

Fig. 4.
Fig. 4.

Monstar for (a) (1, 1) and (b) (2, 2), both with β=40° and γ=180°, for which cases IC=0. The second and third rows show, respectively, the measured and modeled patterns in the space-variant polarization of the light. False color indicates orientation relative to the radial direction. Saturation encodes light intensity.

Fig. 5.
Fig. 5.

Case 1,2=(2,3), γ=180° for two values of β: 40° in column (a) and 90° in column (b). The first row shows the calculated disclination patterns; solid red lines are radial lines and dashed red lines are discontinuities in the β=45° pattern. The second and third rows show, respectively, the measured and modeled patterns in the space-variant polarization of the light. False color indicates orientation relative to the radial direction. Saturation encodes light intensity.

Fig. 6.
Fig. 6.

Monstar for 1,2=(1,4) for two values of β: β=50° in column (a) and γ=180° in column (b). The format and color coding is the same one described in the caption of Fig. 5.

Tables (2)

Tables Icon

Table 1. Types of Disclination Transformations That Can Be Investigated with Set Values of Topological Charges 1 and 2 by Varying β in Eq. (1)

Tables Icon

Table 2. Definition of the Type of Disclination Lines That Are Contained in the Angular Sectors of a Pattern

Equations (15)

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

E=E0[eiα+δe^R+ei2ϕe^L]er2/w2iωt,
α=arg(cosβei1ϕ+sinβei1ϕeiγ)
IC=12πdθ.
θr=θϕ.
IC=1+12πdθr.
IC=1+e2h2,
N=1π|dθr|,
N=|2(IC1)|.
cosβsinΨ+sinβsinϕ=0,
Ψ=(122)ϕδ+2nπ,
ϕ=(122)ϕ+γδ+2nπ,
ϕR=cos1(sign[cos(1ϕγ/2)]).
ϕn=(2n1)π+γ21.
φR=tan1(tan(1ϕγ/2)1tanβ1+tanβ),
dθrdϕ=0.

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