Abstract

The dynamics of the spin-orbit coupling in elegant and standard Hermite–Gaussian (HG), Laguerre–Gaussian (LG), and Bessel–Gaussian (BG) beams propagating through a uniaxial crystal are analyzed. We consider the structure of the electric fields of the paraxial beams and show that the extreme values of the spin and orbital angular momenta are inherent in the elegant HG and LG of high orders. The spin-orbit coupling in the BG beam of the lowest order can result in nearly 100% energy transport from a vortex-free beam to the vortex-bearing beam at a relatively small crystal length. The extreme spin-orbit coupling does not manifest itself in standard HG and LG beams.

© 2010 Optical Society of America

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  1. A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel'dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204-8208 (1992).
    [CrossRef] [PubMed]
  2. V. S. Liberman and B. Ya. Zel'dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199-5207 (1992).
    [CrossRef] [PubMed]
  3. M. Soskin and M. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (North-Holand, 2001), Vol. 42, pp. 219-276.
    [CrossRef]
  4. C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Fiber optical vortices,” in Lasers, Optics and Electro-Optics Research Trends, L.I.Chen, ed. (Nova Science Publishers, 2007), pp. 131-223.
  5. K. Yu. Bliokh and Yu. P. Bliokh, “Polarization, transverse shifts and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
    [CrossRef]
  6. V. Fedoseev, “Conservation laws and angular transverse shift of reflected and transmitted light beams,” Opt. Commun. 282, 1247-1251 (2009).
    [CrossRef]
  7. K. Bliokh, “Geometrodynamics of polarized light: Berry phase, spin Hall effect in gradient-index medium,” J. Opt. A, Pure Appl. Opt. 11, 094009-094023 (2009).
    [CrossRef]
  8. T. Fadeyeva, A. Rubass, Yu. Egorov, A. Volyar, and G. Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1634-1641 (2008).
    [CrossRef]
  9. T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “Transverse shift of a high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 053815 (2009).
    [CrossRef]
  10. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Lasers Photonics Rev. 2, 299-313 (2008).
    [CrossRef]
  11. K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31-35 (2002).
  12. L. Allen, S. Barnett, and M. Padgett, Angular Momentum (IOP, 2003).
    [CrossRef]
  13. A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
    [CrossRef]
  14. J. A. Fleck and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920-928 (1983).
    [CrossRef]
  15. S. R. Seshadri, “Basic elliptical Gaussian wave and beam in a uniaxial crystal,” J. Opt. Soc. Am. A 20, 1818-1826 (2003).
    [CrossRef]
  16. A. Ciattoni, G. Cincotti, and C. Palma, “Circular polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20, 163-171 (2003).
    [CrossRef]
  17. G. Cincotti, A. Ciattoni, and G. Palma, “Hermite-Gaussian beams in uniaxilly anisotropic crystals,” IEEE J. Quantum Electron. 37, 517-524 (2001).
    [CrossRef]
  18. G. Cincotti, A. Ciatoni, and C. Palma, “Laguerre-Gaussian and Bessel-Gaussian beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680-1688 (2002).
    [CrossRef]
  19. A. Ciatoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
    [CrossRef]
  20. A. Ciatoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
    [CrossRef]
  21. A. Volyar and T. Fadeyeva, “Laguerre-Gaussian beams with complex and real arguments in uniaxial crystals,” Opt. Spectrosc. 101, 297-304 (2006).
    [CrossRef]
  22. A. Volyar and T. Fadeyeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 264-274 (2003).
    [CrossRef]
  23. Yu. Egorov, T. Fadeyeva, and A. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A, Pure Appl. Opt. 6, S217-S228 (2004).
    [CrossRef]
  24. F. Flossman, 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]
  25. F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
    [CrossRef]
  26. T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “The matrix model of the vortex-beam quadrefringence in a uniaxial crystal,” Ukr. J. Phys. Opt. 10, 109-123 (2009), http://www.ifo.lviv.ua/journal/2009/2009_3_10_01.html.
    [CrossRef]
  27. E. Brasselet, Ya. Izdebskaya, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Yu. Kivshar, “Dynamics of optical spin-orbit coupling in crystals,” Opt. Lett. 34, 1021-1023 (2009).
    [CrossRef] [PubMed]
  28. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  29. A. E. Siegman, Lasers (University Science Books, 1986).
  30. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3, 465-469 (1986).
    [CrossRef]
  31. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093-1094 (1973).
    [CrossRef]
  32. S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699-700 (1977).
    [CrossRef]
  33. D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259, 409-414 (2006).
    [CrossRef]
  34. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
    [CrossRef]
  35. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).
  36. M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6-11 (1998).
    [CrossRef]
  37. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated hologram,” J. Mod. Opt. 45, 1231-1237 (1998).
    [CrossRef]
  38. S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
    [CrossRef]
  39. M. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order Laguerre-Gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25, 1642-1651 (2008).
    [CrossRef]
  40. D. McGoloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
    [CrossRef]
  41. F. Gori, G. Gauttari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  42. A. P. Kiselev, “The localized light waves: paraxial and exact solution to the wave equation,” Opt. Spectrosc. 102, 697-717 (2007).
    [CrossRef]
  43. M. Parras, R. Barghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18, 177-184 (2001).
    [CrossRef]
  44. T. A. Fadeyeva, A. F. Rubass, B. V. Sokolenko, and A. V. Volyar, “The vortex-beam 'precession' in a rotating uniaxial crystal,” J. Opt. A, Pure Appl. Opt. 11, 094008 (2009).
    [CrossRef]

2009 (6)

V. Fedoseev, “Conservation laws and angular transverse shift of reflected and transmitted light beams,” Opt. Commun. 282, 1247-1251 (2009).
[CrossRef]

K. Bliokh, “Geometrodynamics of polarized light: Berry phase, spin Hall effect in gradient-index medium,” J. Opt. A, Pure Appl. Opt. 11, 094009-094023 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “Transverse shift of a high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 053815 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “The matrix model of the vortex-beam quadrefringence in a uniaxial crystal,” Ukr. J. Phys. Opt. 10, 109-123 (2009), http://www.ifo.lviv.ua/journal/2009/2009_3_10_01.html.
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, B. V. Sokolenko, and A. V. Volyar, “The vortex-beam 'precession' in a rotating uniaxial crystal,” J. Opt. A, Pure Appl. Opt. 11, 094008 (2009).
[CrossRef]

E. Brasselet, Ya. Izdebskaya, V. Shvedov, A. Desyatnikov, W. Krolikowski, and Yu. Kivshar, “Dynamics of optical spin-orbit coupling in crystals,” Opt. Lett. 34, 1021-1023 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (2)

K. Yu. Bliokh and Yu. P. Bliokh, “Polarization, transverse shifts and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
[CrossRef]

A. P. Kiselev, “The localized light waves: paraxial and exact solution to the wave equation,” Opt. Spectrosc. 102, 697-717 (2007).
[CrossRef]

2006 (3)

A. Volyar and T. Fadeyeva, “Laguerre-Gaussian beams with complex and real arguments in uniaxial crystals,” Opt. Spectrosc. 101, 297-304 (2006).
[CrossRef]

D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259, 409-414 (2006).
[CrossRef]

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef]

2005 (2)

D. McGoloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

F. Flossman, 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]

2004 (3)

Yu. Egorov, T. Fadeyeva, and A. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A, Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

A. Ciatoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
[CrossRef]

2003 (4)

A. Volyar and T. Fadeyeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 264-274 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Circular polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20, 163-171 (2003).
[CrossRef]

S. R. Seshadri, “Basic elliptical Gaussian wave and beam in a uniaxial crystal,” J. Opt. Soc. Am. A 20, 1818-1826 (2003).
[CrossRef]

2002 (3)

G. Cincotti, A. Ciatoni, and C. Palma, “Laguerre-Gaussian and Bessel-Gaussian beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680-1688 (2002).
[CrossRef]

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31-35 (2002).

2001 (3)

A. Ciatoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
[CrossRef]

G. Cincotti, A. Ciattoni, and G. Palma, “Hermite-Gaussian beams in uniaxilly anisotropic crystals,” IEEE J. Quantum Electron. 37, 517-524 (2001).
[CrossRef]

M. Parras, R. Barghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18, 177-184 (2001).
[CrossRef]

1998 (2)

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6-11 (1998).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated hologram,” J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

1992 (2)

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel'dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204-8208 (1992).
[CrossRef] [PubMed]

V. S. Liberman and B. Ya. Zel'dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199-5207 (1992).
[CrossRef] [PubMed]

1987 (1)

F. Gori, G. Gauttari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

1986 (1)

1983 (1)

1977 (1)

1973 (1)

Abraham, E.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

Alexeyev, C. N.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Fiber optical vortices,” in Lasers, Optics and Electro-Optics Research Trends, L.I.Chen, ed. (Nova Science Publishers, 2007), pp. 131-223.

Allen, L.

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

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated hologram,” J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

L. Allen, S. Barnett, and M. Padgett, Angular Momentum (IOP, 2003).
[CrossRef]

Ando, T.

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated hologram,” J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

Barghi, R.

Barnett, S.

L. Allen, S. Barnett, and M. Padgett, Angular Momentum (IOP, 2003).
[CrossRef]

Berry, M. V.

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6-11 (1998).
[CrossRef]

Bliokh, K.

K. Bliokh, “Geometrodynamics of polarized light: Berry phase, spin Hall effect in gradient-index medium,” J. Opt. A, Pure Appl. Opt. 11, 094009-094023 (2009).
[CrossRef]

Bliokh, K. Yu.

K. Yu. Bliokh and Yu. P. Bliokh, “Polarization, transverse shifts and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
[CrossRef]

Bliokh, Yu. P.

K. Yu. Bliokh and Yu. P. Bliokh, “Polarization, transverse shifts and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Brasselet, E.

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Ciatoni, A.

A. Ciatoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
[CrossRef]

G. Cincotti, A. Ciatoni, and C. Palma, “Laguerre-Gaussian and Bessel-Gaussian beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680-1688 (2002).
[CrossRef]

A. Ciatoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
[CrossRef]

Ciattoni, A.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Circular polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20, 163-171 (2003).
[CrossRef]

G. Cincotti, A. Ciattoni, and G. Palma, “Hermite-Gaussian beams in uniaxilly anisotropic crystals,” IEEE J. Quantum Electron. 37, 517-524 (2001).
[CrossRef]

Cincotti, G.

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Circular polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20, 163-171 (2003).
[CrossRef]

G. Cincotti, A. Ciatoni, and C. Palma, “Laguerre-Gaussian and Bessel-Gaussian beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680-1688 (2002).
[CrossRef]

A. Ciatoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
[CrossRef]

G. Cincotti, A. Ciattoni, and G. Palma, “Hermite-Gaussian beams in uniaxilly anisotropic crystals,” IEEE J. Quantum Electron. 37, 517-524 (2001).
[CrossRef]

Deng, D.

D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259, 409-414 (2006).
[CrossRef]

Dennis, M. R.

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef]

F. Flossman, 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]

Desyatnikov, A.

Dholakia, K.

D. McGoloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31-35 (2002).

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated hologram,” J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

Dooghin, A. V.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel'dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204-8208 (1992).
[CrossRef] [PubMed]

Egorov, Yu.

T. Fadeyeva, A. Rubass, Yu. Egorov, A. Volyar, and G. Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1634-1641 (2008).
[CrossRef]

Yu. Egorov, T. Fadeyeva, and A. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A, Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

Fadeyeva, T.

T. Fadeyeva, A. Rubass, Yu. Egorov, A. Volyar, and G. Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1634-1641 (2008).
[CrossRef]

A. Volyar and T. Fadeyeva, “Laguerre-Gaussian beams with complex and real arguments in uniaxial crystals,” Opt. Spectrosc. 101, 297-304 (2006).
[CrossRef]

Yu. Egorov, T. Fadeyeva, and A. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A, Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

A. Volyar and T. Fadeyeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 264-274 (2003).
[CrossRef]

Fadeyeva, T. A.

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “Transverse shift of a high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 053815 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, B. V. Sokolenko, and A. V. Volyar, “The vortex-beam 'precession' in a rotating uniaxial crystal,” J. Opt. A, Pure Appl. Opt. 11, 094008 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “The matrix model of the vortex-beam quadrefringence in a uniaxial crystal,” Ukr. J. Phys. Opt. 10, 109-123 (2009), http://www.ifo.lviv.ua/journal/2009/2009_3_10_01.html.
[CrossRef]

Fedoseev, V.

V. Fedoseev, “Conservation laws and angular transverse shift of reflected and transmitted light beams,” Opt. Commun. 282, 1247-1251 (2009).
[CrossRef]

Feit, M. D.

Felsen, L. B.

Fleck, J. A.

Flossman, F.

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef]

F. Flossman, 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]

Franke-Arnold, S.

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

Fukuchi, N.

Gauttari, G.

F. Gori, G. Gauttari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Gori, F.

F. Gori, G. Gauttari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Hara, T.

Inoue, T.

Izdebskaya, Ya.

Kennedy, S. A.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Kiselev, A. P.

A. P. Kiselev, “The localized light waves: paraxial and exact solution to the wave equation,” Opt. Spectrosc. 102, 697-717 (2007).
[CrossRef]

Kivshar, Yu.

Krolikowski, W.

Kundikova, N. D.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel'dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204-8208 (1992).
[CrossRef] [PubMed]

Liberman, V. S.

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel'dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204-8208 (1992).
[CrossRef] [PubMed]

V. S. Liberman and B. Ya. Zel'dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199-5207 (1992).
[CrossRef] [PubMed]

MacDonald, M.

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31-35 (2002).

Maier, M.

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef]

F. Flossman, 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]

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Matsumoto, M.

McGoloin, D.

D. McGoloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Ohtake, Y.

Padgett, M.

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

L. Allen, S. Barnett, and M. Padgett, Angular Momentum (IOP, 2003).
[CrossRef]

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated hologram,” J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

Padovani, C.

F. Gori, G. Gauttari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Palma, C.

A. Ciatoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Circular polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20, 163-171 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[CrossRef]

G. Cincotti, A. Ciatoni, and C. Palma, “Laguerre-Gaussian and Bessel-Gaussian beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680-1688 (2002).
[CrossRef]

A. Ciatoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
[CrossRef]

Palma, G.

G. Cincotti, A. Ciattoni, and G. Palma, “Hermite-Gaussian beams in uniaxilly anisotropic crystals,” IEEE J. Quantum Electron. 37, 517-524 (2001).
[CrossRef]

Parras, M.

Porterfield, J. Z.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

Rubass, A.

Rubass, A. F.

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “Transverse shift of a high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 053815 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, B. V. Sokolenko, and A. V. Volyar, “The vortex-beam 'precession' in a rotating uniaxial crystal,” J. Opt. A, Pure Appl. Opt. 11, 094008 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “The matrix model of the vortex-beam quadrefringence in a uniaxial crystal,” Ukr. J. Phys. Opt. 10, 109-123 (2009), http://www.ifo.lviv.ua/journal/2009/2009_3_10_01.html.
[CrossRef]

Santarsiero, M.

Schwarz, U. T.

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
[CrossRef]

F. Flossman, 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]

Seshadri, S. R.

Shin, S. Y.

Shvedov, V.

Siegman, A. E.

Sokolenko, B. V.

T. A. Fadeyeva, A. F. Rubass, B. V. Sokolenko, and A. V. Volyar, “The vortex-beam 'precession' in a rotating uniaxial crystal,” J. Opt. A, Pure Appl. Opt. 11, 094008 (2009).
[CrossRef]

Soskin, M.

M. Soskin and M. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (North-Holand, 2001), Vol. 42, pp. 219-276.
[CrossRef]

Spalding, G.

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31-35 (2002).

Swartzlander, G.

Szabo, M. J.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Teslow, H.

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Vasnetsov, M.

M. Soskin and M. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (North-Holand, 2001), Vol. 42, pp. 219-276.
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

Volyar, A.

T. Fadeyeva, A. Rubass, Yu. Egorov, A. Volyar, and G. Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1634-1641 (2008).
[CrossRef]

A. Volyar and T. Fadeyeva, “Laguerre-Gaussian beams with complex and real arguments in uniaxial crystals,” Opt. Spectrosc. 101, 297-304 (2006).
[CrossRef]

Yu. Egorov, T. Fadeyeva, and A. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A, Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

A. Volyar and T. Fadeyeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 264-274 (2003).
[CrossRef]

Volyar, A. V.

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “Transverse shift of a high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 053815 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, B. V. Sokolenko, and A. V. Volyar, “The vortex-beam 'precession' in a rotating uniaxial crystal,” J. Opt. A, Pure Appl. Opt. 11, 094008 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “The matrix model of the vortex-beam quadrefringence in a uniaxial crystal,” Ukr. J. Phys. Opt. 10, 109-123 (2009), http://www.ifo.lviv.ua/journal/2009/2009_3_10_01.html.
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Fiber optical vortices,” in Lasers, Optics and Electro-Optics Research Trends, L.I.Chen, ed. (Nova Science Publishers, 2007), pp. 131-223.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Yavorsky, M. A.

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Fiber optical vortices,” in Lasers, Optics and Electro-Optics Research Trends, L.I.Chen, ed. (Nova Science Publishers, 2007), pp. 131-223.

Zauderer, E.

Zel'dovich, B. Ya.

V. S. Liberman and B. Ya. Zel'dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199-5207 (1992).
[CrossRef] [PubMed]

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel'dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204-8208 (1992).
[CrossRef] [PubMed]

Contemp. Phys. (1)

D. McGoloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Cincotti, A. Ciattoni, and G. Palma, “Hermite-Gaussian beams in uniaxilly anisotropic crystals,” IEEE J. Quantum Electron. 37, 517-524 (2001).
[CrossRef]

J. Mod. Opt. (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre-Gaussian modes by computer-generated hologram,” J. Mod. Opt. 45, 1231-1237 (1998).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (4)

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

Yu. Egorov, T. Fadeyeva, and A. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. A, Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

K. Bliokh, “Geometrodynamics of polarized light: Berry phase, spin Hall effect in gradient-index medium,” J. Opt. A, Pure Appl. Opt. 11, 094009-094023 (2009).
[CrossRef]

T. A. Fadeyeva, A. F. Rubass, B. V. Sokolenko, and A. V. Volyar, “The vortex-beam 'precession' in a rotating uniaxial crystal,” J. Opt. A, Pure Appl. Opt. 11, 094008 (2009).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (7)

Lasers Photonics Rev. (1)

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

Opt. Commun. (5)

A. Ciatoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
[CrossRef]

A. Ciatoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
[CrossRef]

V. Fedoseev, “Conservation laws and angular transverse shift of reflected and transmitted light beams,” Opt. Commun. 282, 1247-1251 (2009).
[CrossRef]

D. Deng, “Propagation of elegant Hermite cosine Gaussian laser beams,” Opt. Commun. 259, 409-414 (2006).
[CrossRef]

F. Gori, G. Gauttari, and C. Padovani, “Bessel-Gaussian beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Spectrosc. (3)

A. Volyar and T. Fadeyeva, “Laguerre-Gaussian beams with complex and real arguments in uniaxial crystals,” Opt. Spectrosc. 101, 297-304 (2006).
[CrossRef]

A. Volyar and T. Fadeyeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 264-274 (2003).
[CrossRef]

A. P. Kiselev, “The localized light waves: paraxial and exact solution to the wave equation,” Opt. Spectrosc. 102, 697-717 (2007).
[CrossRef]

Phys. Rev. A (4)

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “Transverse shift of a high-order paraxial vortex-beam induced by a homogeneous anisotropic medium,” Phys. Rev. A 79, 053815 (2009).
[CrossRef]

A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Ya. Zel'dovich, “Optical Magnus effect,” Phys. Rev. A 45, 8204-8208 (1992).
[CrossRef] [PubMed]

V. S. Liberman and B. Ya. Zel'dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199-5207 (1992).
[CrossRef] [PubMed]

S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. Abraham, “Creation of Laguerre-Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002).
[CrossRef]

Phys. Rev. E (2)

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
[CrossRef]

K. Yu. Bliokh and Yu. P. Bliokh, “Polarization, transverse shifts and angular momentum conservation laws in partial reflection and refraction of an electromagnetic wave packet,” Phys. Rev. E 75, 066609 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

F. Flossman, 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]

Phys. World (1)

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31-35 (2002).

Proc. SPIE (1)

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6-11 (1998).
[CrossRef]

Ukr. J. Phys. Opt. (1)

T. A. Fadeyeva, A. F. Rubass, and A. V. Volyar, “The matrix model of the vortex-beam quadrefringence in a uniaxial crystal,” Ukr. J. Phys. Opt. 10, 109-123 (2009), http://www.ifo.lviv.ua/journal/2009/2009_3_10_01.html.
[CrossRef]

Other (6)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

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

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Nauka, 1983).

L. Allen, S. Barnett, and M. Padgett, Angular Momentum (IOP, 2003).
[CrossRef]

M. Soskin and M. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (North-Holand, 2001), Vol. 42, pp. 219-276.
[CrossRef]

C. N. Alexeyev, A. V. Volyar, and M. A. Yavorsky, “Fiber optical vortices,” in Lasers, Optics and Electro-Optics Research Trends, L.I.Chen, ed. (Nova Science Publishers, 2007), pp. 131-223.

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Figures (8)

Fig. 1
Fig. 1

Sketch of the beam propagation through the crystal and polarization filters: Pol, a polarizer; BP, a quarter-wave retarder; A, a transmittance axis of the polarizer; C, a unit vector of the crystal optical axis; e and o, crystallographic axes of the birefringent phase retarder. (a), (b), (c) Intensity distributions of the circularly polarized components of the elegant HG beam with m = n = 4 , w 0 = 8 μ m , and z = 4.7 mm .

Fig. 2
Fig. 2

Transformation of the SAM S z ( m , n ) ( z ) and L z ( m , n ) ( z ) in the elegant HG beam along the Li Nb O 3 crystal length z for n o = 2.3 , n 3 = 2.2 , w 0 = 10 μ m , and λ = 0.6328 μ m

Fig. 3
Fig. 3

Extreme values of the energy efficiency η max as a function of a sum of indices m + n in the elegant HG beam with w 0 = 15 μ m .

Fig. 4
Fig. 4

SAM S z ( p , l ) as a function of the crystal length z for different kinds of elegant LG mode beams.

Fig. 5
Fig. 5

(a) S z ( 0 ) in the BG beam as a function of the crystal length z: K = i 1.5 10 3 , w 0 = 10 5 m . (b) S z as a function of the beam parameter K o : w 0 = 30 μ m , z = 2 cm

Fig. 6
Fig. 6

Evolution of the SAM S z along the crystal length for standard LG beams with w 0 = 10 μ m .

Fig. 7
Fig. 7

Intensity distributions of the RHP and LHP field components along the Bessel-Gaussian beam of the lowest order, l = 0 , in a uniaxial crystal.

Fig. 8
Fig. 8

Evolution of the RHP and LHP components in the standard HG H ( 10 , 10 ) and LG L ( 5 , 10 ) mode beams along the Li Nb O 3 crystal length: w 0 = 10 μ m .

Equations (35)

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I total ( z ) = I + ( z ) + I ( z ) = S [ | Ψ + | 2 + | Ψ | 2 ] d S = I 0 ( z = 0 ) ,
I + = S | Ψ 0 + Ψ e | 2 d S = S [ | Ψ o | 2 + | Ψ e | 2 + 2 Re ( Ψ o Ψ e * ) ] d S .
I o ( z ) = I e ( z ) = S | Ψ o , e | 2 d S = I o ( z = 0 ) = I e ( z = 0 ) = 1 4 I 0 ( z = 0 ) .
I + = 1 2 I 0 + 2 Re S Ψ o Ψ e * d S ,
I = 1 2 I 0 2 Re S Ψ o Ψ e * d S .
S z = I + I I + + I = 4 I 0 Re S Ψ o Ψ e * d S = 1 I o Re S Ψ o Ψ e * d S ,
P = 1 I o | S Ψ o Ψ e * d S | .
η = I I + + I .
η = 1 S z 2 ,
E ̃ + = eH + ( m , n ) = [ eH m , n ( o ) + eH m , n ( e ) ] 2 ,
E ̃ = eH ( m , n ) = ( w 0 ) m + n 2 m + n x m y m G ,
eH m , n ( o , e ) ( x , y , z ) = Ψ 0 ( o , e ) ( σ o , e ) m + n H m ( X σ o , e ) H n ( Y σ o , e ) ,
I x = H 2 m + δ ( b x ) H 2 n + δ ( c x ) exp { p x 2 } d x = ( 2 m + δ ) ! ( 2 n + δ ) ! π p k = 0 min ( m , n ) ( b 2 p p ) m k ( c 2 p p ) n k ( 2 b c p ) 2 k + δ ( m k ) ! ( n k ) ! ( 2 k + δ ) ! , δ = 0 , 1 ,
S z ( m , n ) = Re [ ( 2 σ o + σ e * ) 2 ( m + n ) + 2 δ + 1 ] .
L z ( m , n ) = i eH m , n | x y y x | eH m , n I 0 = σ S z ( m , n ) ,
E p , l ( ± ) = ( E + , p , l ( ± ) E , p , l ( ± ) ) = e L p , l ( ± ) = 1 2 l ̂ p , l ( ± ) ( Ψ 0 ( o ) + Ψ 0 ( e ) G ) ,
l ̂ p , l ( ± ) = z p [ e ± i φ ( r ± i r φ ) ] ( l )
e L p , l ( ) = 1 2 ( r e i φ w 0 ) l { ( 1 σ o p + l L p ( l ) ( R o 2 ) w 0 2 ( p + 1 ) e i 2 φ r 2 σ o p + l 1 L p + 1 ( l 2 ) ( R o 2 ) ) Ψ 0 ( o ) + ( 1 σ e p + l L p ( l ) ( R e 2 ) w 0 2 ( p + 1 ) e i 2 φ r 2 σ e p + l 1 L p + 1 ( l 2 ) ( R e 2 ) ) Ψ 0 ( e ) } ,
0 x λ e p x L m λ ( b x ) L n λ ( c x ) d x = Γ ( m + n + λ + 1 ) m ! n ! p m + n + λ + 1 ( p b ) m ( p c ) n × F 1 2 ( m , n ; m n λ ; p ( p b c ) ( p b ) ( p c ) ) .
S z ( p , l , + ) = S z ( p , l , ) = Re [ ( 2 σ o + σ e * ) 2 p + l + 1 ] ,
F 1 2 ( m , n ; m n λ ; p ( p b c ) ( p b ) ( p c ) ) = F 1 2 ( m , n ; m n λ ; 0 ) = 1 .
B G o , e ( l ) = 1 σ o , e J l ( i K o , e z o , e σ o , e r ) e r 2 w 0 2 ± i l φ K o , e 2 ( 2 k o , e z o , e σ o , e ) ,
E + ( l ) = 1 2 { B G o ( l ) + B G e ( l ) } ,
e t ( t x ) α 2 J α ( 2 t x ) = k = 0 t k k ! Γ ( k + α + 1 ) L k ( α ) ( x ) ,
E + ( l ) = 1 2 ( r e i φ w 0 ) l { Ψ 0 ( o ) σ o l j = 0 ( K o 2 2 k o ) j j ! ( j + l ) ! ( 1 σ o ) j L j ( l ) ( R o 2 ) + Ψ 0 ( e ) σ e l j = 0 ( K o 2 2 k o ) j j ! ( j + l ) ! ( 1 σ e ) j L j ( l ) ( R e 2 ) } ,
E ( l ) = 1 2 ( r e i φ w 0 ) l w 0 2 e i 2 φ r 2 j = 0 ( j + 1 ) j ! ( j + l ) ! ( K o 2 2 k o ) j { Ψ 0 ( o ) σ o j + l 1 L j + 1 ( l 2 ) ( R o 2 ) Ψ 0 ( o ) σ e j + l 1 L j + 1 ( l 2 ) ( R e 2 ) } .
S z ( l ) = Re { exp ( K o 2 k o z o [ 1 ( σ o + σ e * ) 1 2 ] ) I l ( K o 2 k o z o ( σ o + σ e * ) ) 2 ( σ o + σ e * ) } I l ( K o 2 2 k o z o ) ,
0 r exp ( p r 2 ) J v ( b r ) J v ( c r ) d r = 1 2 p exp ( b 2 + c 2 4 p ) I v ( b c 2 p ) .
E p , l , ( + ) ( ) = L p , l , ( + ) ( ) = ( σ o * σ o ) ( l + 2 p ) 2 ( r e i φ w 0 | σ o | ) l L p ( l ) ( 2 r 2 w 0 2 | σ o | 2 ) Ψ 0 ( o ) + ( σ e * σ e ) ( l + 2 p ) 2 ( r e i φ w 0 | σ e | ) l L p ( l ) ( 2 r 2 w 0 2 | σ e | 2 ) Ψ 0 ( e ) .
L n ( α ) ( μ x ) = j = 0 n ( n + α j ) μ n j ( 1 μ ) j L n j ( α ) ( x )
L p , l , ( + ) ( ) = Ψ 0 ( o ) ( r e i φ w 0 σ o ) l j = 0 p ( p + l j ) ( 2 σ o ) p j L p j ( l ) ( R o 2 ) + Ψ 0 ( e ) ( r e i φ w 0 σ e ) l j = 0 p ( p + l j ) ( 2 σ e ) p j L p j ( l ) ( R e 2 ) .
L p , l , ( ) ( ) = ( r w 0 ) l 2 e i ( l 2 ) φ j = 0 p ( p + l j ) ( j + 1 ) { ( 2 σ o ) p j Ψ 0 ( o ) σ o p + l j 1 L p j + 1 ( l ) ( R o 2 ) ( 2 σ e ) p j Ψ 0 ( e ) σ e p + l j 1 L p j + 1 ( l ) ( R e 2 ) } .
S z = ( 2 p + l ) ! p ! ( p + l ) ! Re { ( 2 σ o + σ e * ) 2 p + l + 1 ( σ o σ e * σ o * σ e ) 2 p + l F 1 2 ( p , p l ; 2 p l ; | σ 0 + σ e * 2 | 2 ) } ,
S z = ( 1 + Z 2 ) N cos N Γ ,
tan [ N arctan Z ] = Z .

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