Abstract

One difficulty of angular spectrum representation method in studying optical propagation inside anisotropic crystals is to calculate the double integrals containing highly oscillating functions. In this paper, we introduce an accuracy and numerically cheap method based on asymptotic expansion theory to overcome this difficulty, which therefore allows to compute optical fields with arbitrary incident beam and is not restricted to the paraxial limit. This numerical method is benchmarked against the analytical solutions in uniaxial crystals and excellent agreements between them are obtained. As an application, we adopt it to investigate the propagation of a Gaussian vortex-beam in a biaxial crystal. The general features of anisotropic dynamics and power conversion between field components are revealed. The numerical results is interpreted by making appropriate analytical approximation to the wave equations.

© 2013 OSA

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2013 (1)

N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun.286, 1–5 (2013).
[CrossRef]

2012 (3)

2011 (5)

D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “The creation and annihilation of optical vortices using cascade conical diffraction,” Opt. Express19, 2580–2588 (2011).
[CrossRef]

T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt.12, 62–82 (2011).
[CrossRef]

T. Fadeyeva, C. Alexeyev, A. Rubass, A. Zinov’ev, V. Konovalenko, and A. Volyar, “Subwave spikes of the orbital angular momentum of the vortex beams in a uniaxial crystal,” Opt. Lett.36, 4215–4217 (2011).
[CrossRef] [PubMed]

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun.284, 3183–3191 (2011).
[CrossRef]

2010 (3)

2009 (3)

2008 (2)

2006 (3)

S. Olver, “Moment-free numerical integration of highly oscillatory functions,” IMA J. Numer. Anal.26, 213–227 (2006).
[CrossRef]

D. Huybrechs and S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation,” SIAM J. Numer. Anal.44, 1026–1048 (2006).
[CrossRef]

A. Volyar and T. Fadeeva, “Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal,” Opt. Spectros.101, 450–457 (2006).
[CrossRef]

2005 (3)

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] [PubMed]

A. Iserles and S. P. Nørsett, “Efficient quadrature of highly oscillatory integrals using derivatives,” Proc. R. Soc. A461, 1383–1399 (2005).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A: Pure Appl. Opt.7, 685 (2005).
[CrossRef]

2004 (2)

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt.6, 289 (2004).
[CrossRef]

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

2003 (4)

A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Optics and Spectroscopy94, 235–244 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A20, 163–171 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A20, 2163–2171 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E67, 036618 (2003).
[CrossRef]

2002 (3)

2001 (2)

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A18, 1656–1661 (2001).
[CrossRef]

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001).
[CrossRef]

1999 (3)

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron.29, 1020 (1999).
[CrossRef]

A. Belsky and M. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun.167, 1–5 (1999).
[CrossRef]

M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A: Pure Appl. Opt.1, 601 (1999).
[CrossRef]

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A56, 4064–4075 (1997).
[CrossRef]

1996 (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. A45, 8185–8189 (1992).
[CrossRef] [PubMed]

1983 (1)

1976 (1)

Alexeyev, C.

T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt.12, 62–82 (2011).
[CrossRef]

T. Fadeyeva, C. Alexeyev, A. Rubass, A. Zinov’ev, V. Konovalenko, and A. Volyar, “Subwave spikes of the orbital angular momentum of the vortex beams in a uniaxial crystal,” Opt. Lett.36, 4215–4217 (2011).
[CrossRef] [PubMed]

T. Fadeyeva, C. Alexeyev, P. Anischenko, and A. Volyar, “The fine structure of the vortex-beams in the biaxial and biaxially-induced birefringent media caused by the conical diffraction,” arXiv:1107.5775 (2011).

Alexeyev, C. N.

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. A45, 8185–8189 (1992).
[CrossRef] [PubMed]

Anischenko, P.

T. Fadeyeva, C. Alexeyev, P. Anischenko, and A. Volyar, “The fine structure of the vortex-beams in the biaxial and biaxially-induced birefringent media caused by the conical diffraction,” arXiv:1107.5775 (2011).

Anischenko, P. M.

Band, Y. B.

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. A45, 8185–8189 (1992).
[CrossRef] [PubMed]

Belsky, A.

A. Belsky and M. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun.167, 1–5 (1999).
[CrossRef]

Belyi, V. N.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001).
[CrossRef]

Berry, M. V.

M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt.12, 075704 (2010).
[CrossRef]

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A: Pure Appl. Opt.7, 685 (2005).
[CrossRef]

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt.6, 289 (2004).
[CrossRef]

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2007), pp. 13–50.
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).

Brasselet, E.

Chen, F.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

Chen, L.

Chen, Y.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Chen, Z.

Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun.284, 3183–3191 (2011).
[CrossRef]

Ciattoni, A.

Cincotti, G.

Crosignani, B.

deDoncker Kapenga, E.

R. Piessens, E. deDoncker Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Dennis, M. R.

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] [PubMed]

Desyatnikov, A.

Desyatnikov, A. S.

Donegan, J. F.

Dreger, M. A.

M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A: Pure Appl. Opt.1, 601 (1999).
[CrossRef]

Eastham, P. R.

Egorov, Y.

Fadeeva, T.

A. Volyar and T. Fadeeva, “Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal,” Opt. Spectros.101, 450–457 (2006).
[CrossRef]

A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Optics and Spectroscopy94, 235–244 (2003).
[CrossRef]

Fadeyeva, T.

T. Fadeyeva, C. Alexeyev, A. Rubass, A. Zinov’ev, V. Konovalenko, and A. Volyar, “Subwave spikes of the orbital angular momentum of the vortex beams in a uniaxial crystal,” Opt. Lett.36, 4215–4217 (2011).
[CrossRef] [PubMed]

T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt.12, 62–82 (2011).
[CrossRef]

T. Fadeyeva, A. Rubass, Y. Egorov, A. Volyar, and J. Grover Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A25, 1634–1641 (2008).
[CrossRef]

T. Fadeyeva, C. Alexeyev, P. Anischenko, and A. Volyar, “The fine structure of the vortex-beams in the biaxial and biaxially-induced birefringent media caused by the conical diffraction,” arXiv:1107.5775 (2011).

Fadeyeva, T. A.

Feit, M. D.

Fleck, J. J. A.

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] [PubMed]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A56, 4064–4075 (1997).
[CrossRef]

Grover Swartzlander, J.

Guo, Q.

Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun.284, 3183–3191 (2011).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A56, 4064–4075 (1997).
[CrossRef]

Huybrechs, D.

D. Huybrechs and S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation,” SIAM J. Numer. Anal.44, 1026–1048 (2006).
[CrossRef]

D. Huybrechs and S. Olver, “Highly oscillatory quadrature,” in Highly Oscillatory Problems (Cambridge University, 2009), pp. 25–50.
[CrossRef]

Iserles, A.

A. Iserles and S. P. Nørsett, “Efficient quadrature of highly oscillatory integrals using derivatives,” Proc. R. Soc. A461, 1383–1399 (2005).
[CrossRef]

Izdebskaya, Y.

Izdebskaya, Y. V.

Jeffrey, M. R.

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A: Pure Appl. Opt.7, 685 (2005).
[CrossRef]

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2007), pp. 13–50.
[CrossRef]

Kahaner, D.

R. Piessens, E. deDoncker Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Katranji, E. G.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001).
[CrossRef]

Kazak, N. S.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001).
[CrossRef]

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron.29, 1020 (1999).
[CrossRef]

Khilo, N. A.

N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun.286, 1–5 (2013).
[CrossRef]

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001).
[CrossRef]

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron.29, 1020 (1999).
[CrossRef]

King, T. A.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001).
[CrossRef]

Kivshar, Y.

Kivshar, Y. S.

Konovalenko, V.

Krolikowski, W.

Kudryavtseva, M.

T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt.12, 62–82 (2011).
[CrossRef]

Li, J.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Lu, X.

Lunney, J. G.

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] [PubMed]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A56, 4064–4075 (1997).
[CrossRef]

Mansuripur, M.

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A: Pure Appl. Opt.7, 685 (2005).
[CrossRef]

Neshev, D. N.

Nørsett, S. P.

A. Iserles and S. P. Nørsett, “Efficient quadrature of highly oscillatory integrals using derivatives,” Proc. R. Soc. A461, 1383–1399 (2005).
[CrossRef]

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, Bristol, 1999).

O’Dwyer, D. P.

Olver, S.

S. Olver, Numerical Approximation of Highly Oscillatory Integrals (PhD thesis, 2008).

S. Olver, “Moment-free numerical integration of highly oscillatory functions,” IMA J. Numer. Anal.26, 213–227 (2006).
[CrossRef]

D. Huybrechs and S. Olver, “Highly oscillatory quadrature,” in Highly Oscillatory Problems (Cambridge University, 2009), pp. 25–50.
[CrossRef]

Palma, C.

Phelan, C. F.

Piessens, R.

R. Piessens, E. deDoncker Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Porto, P. D.

Provenziani, D.

A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E66, 036614 (2002).
[CrossRef]

Rakovich, Y. P.

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. A79, 053815 (2009).
[CrossRef]

Ryzhevich, A. A.

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001).
[CrossRef]

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron.29, 1020 (1999).
[CrossRef]

Schwarz, U. T.

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] [PubMed]

Sherman, G. C.

Shvedov, V.

Shvedov, V. G.

Sokolenko, B.

T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt.12, 62–82 (2011).
[CrossRef]

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A56, 4064–4075 (1997).
[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. A45, 8185–8189 (1992).
[CrossRef] [PubMed]

Stamnes, J. J.

Stepanov, M.

A. Belsky and M. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun.167, 1–5 (1999).
[CrossRef]

Trippenbach, M.

Uberhuber, C.

R. Piessens, E. deDoncker Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Vandewalle, S.

D. Huybrechs and S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation,” SIAM J. Numer. Anal.44, 1026–1048 (2006).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A56, 4064–4075 (1997).
[CrossRef]

Volyar, A.

T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt.12, 62–82 (2011).
[CrossRef]

T. Fadeyeva, C. Alexeyev, A. Rubass, A. Zinov’ev, V. Konovalenko, and A. Volyar, “Subwave spikes of the orbital angular momentum of the vortex beams in a uniaxial crystal,” Opt. Lett.36, 4215–4217 (2011).
[CrossRef] [PubMed]

T. Fadeyeva, A. Rubass, Y. Egorov, A. Volyar, and J. Grover Swartzlander, “Quadrefringence of optical vortices in a uniaxial crystal,” J. Opt. Soc. Am. A25, 1634–1641 (2008).
[CrossRef]

A. Volyar and T. Fadeeva, “Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal,” Opt. Spectros.101, 450–457 (2006).
[CrossRef]

A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Optics and Spectroscopy94, 235–244 (2003).
[CrossRef]

T. Fadeyeva, C. Alexeyev, P. Anischenko, and A. Volyar, “The fine structure of the vortex-beams in the biaxial and biaxially-induced birefringent media caused by the conical diffraction,” arXiv:1107.5775 (2011).

Volyar, A. V.

Wang, Y.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Weber, H.

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. A45, 8185–8189 (1992).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).

Wong, R.

R. Wong, Asymptotic Approximations of Integrals (SIAM, Philadelphia, 2001).
[CrossRef]

Xin, Y.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Xu, S.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Zhao, Q.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Zhou, M.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

Zinov’ev, A.

Appl. Opt. (1)

Eur. Phys. J. Appl. Phys. (1)

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys.53, 20701 (2011).
[CrossRef]

IMA J. Numer. Anal. (1)

S. Olver, “Moment-free numerical integration of highly oscillatory functions,” IMA J. Numer. Anal.26, 213–227 (2006).
[CrossRef]

J. Opt. (1)

M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt.12, 075704 (2010).
[CrossRef]

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

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt.6, 289 (2004).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A: Pure Appl. Opt.7, 685 (2005).
[CrossRef]

M. A. Dreger, “Optical beam propagation in biaxial crystals,” J. Opt. A: Pure Appl. Opt.1, 601 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

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

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

Numerical Approximation of Highly Oscillatory Integrals (1)

S. Olver, Numerical Approximation of Highly Oscillatory Integrals (PhD thesis, 2008).

Opt. Commun. (4)

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

A. Belsky and M. Stepanov, “Internal conical refraction of coherent light beams,” Opt. Commun.167, 1–5 (1999).
[CrossRef]

Z. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun.284, 3183–3191 (2011).
[CrossRef]

N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun.286, 1–5 (2013).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Opt. Spectros. (1)

A. Volyar and T. Fadeeva, “Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal,” Opt. Spectros.101, 450–457 (2006).
[CrossRef]

Optics and Spectroscopy (1)

A. Volyar and T. Fadeeva, “Generation of singular beams in uniaxial crystals,” Optics and Spectroscopy94, 235–244 (2003).
[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. A79, 053815 (2009).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45, 8185–8189 (1992).
[CrossRef] [PubMed]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A56, 4064–4075 (1997).
[CrossRef]

C. N. Alexeyev, “Circular array of anisotropic fibers: A discrete analog of a q plate,” Phys. Rev. A86, 063830 (2012).
[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. E67, 036618 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E66, 036614 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

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] [PubMed]

Proc. R. Soc. A (1)

A. Iserles and S. P. Nørsett, “Efficient quadrature of highly oscillatory integrals using derivatives,” Proc. R. Soc. A461, 1383–1399 (2005).
[CrossRef]

Proc. SPIE (1)

V. N. Belyi, T. A. King, N. S. Kazak, N. A. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Methods of formation and nonlinear conversion of Bessel optical vortices,” Proc. SPIE4403, 229–240 (2001).
[CrossRef]

Quantum Electron. (1)

N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Generation of Bessel light beams under the conditions of internal conical refraction,” Quantum Electron.29, 1020 (1999).
[CrossRef]

SIAM J. Numer. Anal. (1)

D. Huybrechs and S. Vandewalle, “On the evaluation of highly oscillatory integrals by analytic continuation,” SIAM J. Numer. Anal.44, 1026–1048 (2006).
[CrossRef]

Ukr. J. Phys. Opt. (1)

T. Fadeyeva, C. Alexeyev, B. Sokolenko, M. Kudryavtseva, and A. Volyar, “Non-canonical propagation of high-order elliptic vortex beams in a uniaxially anisotropic medium,” Ukr. J. Phys. Opt.12, 62–82 (2011).
[CrossRef]

Other (8)

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, Bristol, 1999).

T. Fadeyeva, C. Alexeyev, P. Anischenko, and A. Volyar, “The fine structure of the vortex-beams in the biaxial and biaxially-induced birefringent media caused by the conical diffraction,” arXiv:1107.5775 (2011).

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1999).

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

R. Piessens, E. deDoncker Kapenga, C. Uberhuber, and D. Kahaner, Quadpack: A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

D. Huybrechs and S. Olver, “Highly oscillatory quadrature,” in Highly Oscillatory Problems (Cambridge University, 2009), pp. 25–50.
[CrossRef]

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2007), pp. 13–50.
[CrossRef]

R. Wong, Asymptotic Approximations of Integrals (SIAM, Philadelphia, 2001).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

A comparison of asymptotic expansion and analytical solutions. (a) and (b) show the moduli of E+ and E, the circularly polarized components of Gaussian beam propagating in uniaxial crystal, as a function of transverse radius r. (c) and (d) show the absolute errors of asymptotic expansion for |E+| and |E| with respect to the analytical results. The inset inside (d) is the errors of |E| in the region of small radius r. The values of parameters used in calculations: z = 100000μm, no = 1.656, ne = 1.458, λ = 0.633μm, and s = 4.59μm.

Fig. 2
Fig. 2

The real parts of functions u+ = c1ic3 (a) and u = c1 + ic3 (b) in momentum space. The initial field is Gaussian beam of Eq. (17).

Fig. 3
Fig. 3

The absolute errors of |E+| (a) and |E| (b) as a function of the width dk of the integration domain for various transverse radii r. The inset in subfigure (b) is a sketch of the square integration domain, in which S denotes the position of the stationary point and the concentric circles are the oscillating periods of the integrand. The values of parameters used in the calculations are the same as those in Fig. 1.

Fig. 4
Fig. 4

The absolute errors of |E+| (a) and |E| (b) as a function of the width dk of the integration domain for various transverse radii r. A coarse-grained average is performed between two square domains of integration; see the inset in subfigure (b) for a sketch. The values of parameters used in the calculations are the same as those in Fig. 1.

Fig. 5
Fig. 5

The profiles of |Ex| (first row), |Ey| (second row), and intensity I = |Ex|2 + |Ey|2 (third row) of a circularly polarized Gaussian vortex beam propagating in biaxial crystals at a distance z = 5000μm. The deviations for each subfigures are: (a), (f), and (k): Δn = 0, η = 0; (b), (g), and (l): Δn = 0.0001, η = 0.0005; (c), (h), and (m): Δn = 0.001, η = 0.0051; (d), (i), and (n): Δn = 0.05, η = 0.2525; (e), (j), and (o): Δn = 0.198, η = 1. The x and y coordinates are in unit of μm. The colorbars for each rows are shown on the right sides.

Fig. 6
Fig. 6

The profiles of |Ex| (first row), |Ey| (second row), and intensity I = |Ex|2 + |Ey|2 (third row) of a linearly polarized Gaussian vortex beam propagating in biaxial crystals at a distance z = 5000μm. The deviations for each subfigures are: (a), (f), and (k): Δn = 0, η = 0; (b), (g), and (l): Δn = 0.0001, η = 0.0005; (c), (h), and (m): Δn = 0.001, η = 0.0051; (d), (i), and (n): Δn = 0.05, η = 0.2525; (e), (j), and (o): Δn = 0.198, η = 1. The x and y coordinates are in unit of μm. The colorbars for the first and third rows are shown on the right sides of them. In the second row, the colorbar of subfigures (f), (g), and (h) is the same as that of the first row, and the colorbar on the right side is only for subfigures (i) and (j).

Equations (37)

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2 E ( E ) + k 0 2 ε E = 0
ε = [ n x 2 0 0 0 n y 2 0 0 0 n z 2 ]
E ( r , z ) = d 2 k exp ( i k r ) E ˜ ( k , z ) ,
E ˜ ( k , z ) = [ c 1 c 3 c 5 ] exp ( λ 1 z ) + [ c 2 c 4 c 6 ] exp ( λ 2 z ) .
λ 1 , λ 2 = 1 2 ( K ± L )
[ c 1 c 2 ] = P E ˜ ( 0 ) , [ c 3 c 4 ] = Q E ˜ ( 0 )
[ c 5 c 6 ] = 1 q [ λ 1 ( k x c 1 + k y c 3 ) λ 2 ( k x c 1 + k y c 3 ) ]
P = 1 L [ ( M L ) 2 ( 1 β γ ) k x k y ( M + L ) 2 ( 1 β γ ) k x k y ]
Q = 1 L [ ( 1 α γ ) k x k y ( M + L ) 2 ( 1 α γ ) k x k y ( M L ) 2 ]
E ˜ ( 0 ) = 1 ( 2 π ) 2 d 2 r exp ( i k r ) E ( r , 0 ) .
α = k 0 2 n x 2 , β = k 0 2 n y 2 , γ = k 0 2 n z 2 , q = γ k 2
K = ( α + β ) ( 1 + α γ ) k x 2 ( 1 + β γ ) k y 2
L = ( α β ) 2 + ( 1 α γ ) 2 k x 4 + ( 1 β γ ) 2 k y 4 + 2 ( α β ) ( 1 α γ ) k x 2 + 2 ( β α ) ( 1 β γ ) k y 2 + 2 ( 1 α γ ) ( 1 β γ ) k x 2 k y 2
M = ( α β ) + ( 1 α γ ) k x 2 ( 1 β γ ) k y 2
λ 1 = ( k 0 2 n o 2 k 2 ) , λ 2 = [ k 0 2 n o 2 ( n o n e ) 2 k 2 ]
P = 1 k 2 [ k y 2 k x k y k x 2 k x k y ] , Q = 1 k 2 [ k x k y k x 2 k x k y k y 2 ]
E ( r , 0 ) = exp ( r 2 2 s 2 ) e ^ + ,
[ E + E ] = 1 2 [ 1 i 1 i ] [ E x E y ] .
E ˜ ( 0 ) = s 2 2 π exp ( k 2 s 2 2 ) e ^ + .
I ( λ ) = D g ( x , y ) exp [ i λ f ( x , y ) ] d x d y
E x = c 1 ( k x , k y ) exp [ i z f 1 ( k x , k y ) ] d k x d k y + c 2 ( k x , k y ) exp [ i z f 2 ( k x , k y ) ] d k x d k y
f 1 ( k x , k y ) = λ 1 + x z k x + y z k y f 2 ( k x , k y ) = λ 2 + x z k x + y z k y
k x o = k 0 n o x r o , k y o = k 0 n o y r o
k x e = k 0 n e x r e , k y e = k 0 n e y r e
f 1 ( k x , k y ) = f 1 ( k x o , k y o ) + 1 2 a 1 ( k x k x o ) 2 + 1 2 b 1 ( k y k y o ) 2 + c 1 ( k x k x o ) ( k y k y o ) +
E x 2 π i | a 1 b 1 c 1 2 | c 1 ( k x o , k y o ) exp [ i z f 1 ( k x o , k y o ) ] z 2 π i | a 2 b 2 c 2 2 | c 2 ( k x e , k y e ) exp [ i z f 2 ( k x e , k y e ) ] z
E ( r , z ) 2 π i k 0 n o z [ c 1 c 3 c 5 ] ( k x o , k y o ) exp ( i k 0 n o r o ) r o 2 + [ c 2 c 4 c 6 ] ( k x e , k y e ) exp ( i k 0 n e r e ) r e 2
E ( r , 0 ) = ( x + i y ) exp ( r 2 s 2 ) d ^ in ,
E ˜ ( 0 ) = s 4 8 π ( k y i k x ) exp ( k 2 s 2 4 ) d ^ in .
α β 2 n x ( n x n z ) η k 0 2 k 2 .
L = [ ( α β ) + ( 1 α γ ) k x 2 ( 1 β γ ) k y 2 ] 2
λ 1 = β + k x 2 + β γ k y 2 , λ 2 = α + α γ k x 2 + k y 2
P = [ 0 ( 1 β γ ) k x k y L 1 ( 1 β γ ) k x k y L ] , Q = [ ( 1 α γ ) k x k y L 1 ( 1 α γ ) k x k y L 0 ]
P = [ 0 0 1 0 ] , Q = [ k x k y γ k x 2 1 k x k y γ k x 2 0 ]
E x ( r , z ) = exp ( i k 0 n x z ) d k exp ( i k r ) exp ( i z n x 2 k x 2 + n z 2 k y 2 2 k 0 n x n z 2 ) E ˜ x ( 0 )
E y ( r , z ) = exp ( i k 0 n y z ) d k exp ( i k r ) exp ( i z n z 2 k x 2 + n y 2 k y 2 2 k 0 n y n z 2 ) E ˜ y ( 0 )
T = [ sin ϕ cos ϕ 0 cos θ cos ϕ cos θ sin ϕ sin θ sin θ cos θ sin θ sin ϕ cos θ ] .

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