Abstract

Cascaded conical diffraction where optical elements modifying the local polarization state are intercalated between the aligned biaxial crystals is analyzed theoretically in the framework of paraxial diffraction theory. The obtained expressions are verified and confirmed experimentally for the case of a two-crystal cascade intercalated by a polarizer or a wave plate. The present approach can be used to realize a variety of vector beams with complex beam shapes composed of concentric rings with strongly modulated azimuthal intensity distribution. A potentially very fast switching of the overall beam shape is possible if the intercalated elements are electro-optically tunable retarders.

© 2017 Optical Society of America

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References

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  1. W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish Acad. 17, 1–144 (1837).
  2. H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxal crystals,” Philos. Mag. 2, 112–120 (1833).
  3. A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).
  4. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A: Pure Appl. Opt. 6, 289–300 (2004).
    [Crossref]
  5. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Progr. Opt. 50, 13–50 (2007).
    [Crossref]
  6. J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Opt. Commun. 105, 243–252 (1994).
    [Crossref]
  7. D. P. O’Dwyer, C. F. Phelan, K. E. Ballantine, Y. P. Rakovich, J. G. Lunney, and J. F. Donegan, “Conical diffraction of linearly polarised light controls the angular position of a microscopic object,” Opt. Express 18, 27319–27326 (2010).
    [Crossref]
  8. D. P. O’Dwyer, K. E. Ballantine, C. F. Phelan, J. G. Lunney, and J. F. Donegan, “Optical trapping using cascade conical refraction of light,” Opt. Express 20, 21119–21125 (2012).
    [Crossref]
  9. A. Turpin, V. Shvedov, C. Hnatovsky, Yu V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: A reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21, 26335–26340 (2013).
    [Crossref] [PubMed]
  10. A. Turpin, J. Polo, Yu. V. Loiko, J. Küber, F. Schmaltz, T. K. Kalkandjiev, V. Ahufinger, G. Birkl, and J. Mompart, “Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction,” Opt. Express 23, 1638–1650 (2015).
    [Crossref] [PubMed]
  11. A. Peinado, A. Turpin, A. Lizana, E. Fernandez, J. Mompart, and J. Campos, “Conical refraction as a tool for polarization metrology,” Opt. Lett. 38, 4100–4103 (2013).
    [Crossref] [PubMed]
  12. A. Peinado, A. Lizana, A. Turpin, C. Iemmi, T. K. Kalkandjiev, J. Mompart, and J. Campos, “Optimization, tolerance analysis and implementation of a Stokes polarimeter based on the conical refraction phenomenon,” Opt. Express 23, 5636–5652 (2015).
    [Crossref] [PubMed]
  13. S. D. Grant, S. Reynolds, and A. Abdolvand, “Optical sensing of polarization using conical diffraction phenomenon,” J. Opt. 18, 025609 (2016).
    [Crossref]
  14. A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Free-space optical polarization demultiplexing and multiplexing by means of conical refraction,” Opt. Lett. 37, 4197–4199 (2012).
    [Crossref] [PubMed]
  15. C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
    [Crossref]
  16. J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
    [Crossref]
  17. J. Hellström, H. Henricsson, V. Pasiskevicius, U. Bünting, and D. Haussmann, “Polarization-tunable Yb:KGW laser based on internal conical refraction,” Opt. Lett. 32, 2783–2785 (2007).
    [Crossref] [PubMed]
  18. A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rafailov, “Conical refraction Nd:KGd(WO4)2 laser,” Opt. Express 18, 2753–2759 (2010).
    [Crossref] [PubMed]
  19. V. Peet, “Improving directivity of laser beams by employing the effect of conical refraction in biaxial crystals,” Opt. Express 18, 19566–19573 (2010).
    [Crossref] [PubMed]
  20. A. Brenier, “Lasing with conical diffraction feature in the KGd(WO4)2:Nd biaxial crystal,” Appl. Phys. B 122, 237 (2016).
    [Crossref]
  21. R. Cattoor, I. Manek-Hönninger, D. Rytz, L. Canioni, and M. Eichhorn, “Laser action along and near the optic axis of a holmium-doped KY(WO4)2 crystal,” Opt. Lett. 39, 6407–6410 (2014).
    [Crossref] [PubMed]
  22. D. P. O’ Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010).
    [Crossref]
  23. 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. Express 19, 2580–2588 (2011).
    [Crossref]
  24. V. Peet, “Conical refraction in a degenerated two-crystal cascade,” J. Opt. 18, 015607 (2016).
    [Crossref]
  25. A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 10, 750–771 (2016).
    [Crossref]
  26. M. V. Berry, “Conical diffraction from an N-crystal cascade,” J. Opt. 12, 075704 (2010).
    [Crossref]
  27. A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express 21, 4503–4511 (2013).
    [Crossref] [PubMed]
  28. A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Multiple rings formation in cascaded conical refraction,” Opt. Lett. 38, 1455–1457 (2013).
    [Crossref] [PubMed]
  29. V. Peet, ‘Biaxial crystal as a versatile mode converter,” J. Opt. 12, 095706 (2010).
    [Crossref]
  30. A. Abdolvand, ‘Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys. B 103, 281–283 (2011).
    [Crossref]
  31. V. Peet, ‘Variable two-crystal cascade for conical refraction,” Opt. Lett. 40, 2405–2408 (2015).
    [Crossref] [PubMed]
  32. C. F. Phelan, K. E. Ballantine, P. R. Eastham, J. F. Donegan, and J. G. Lunney, ‘Conical diffraction of a Gaussian beam with a two crystal cascade,” Opt. Express 20, 13201–13207 (2012).
    [Crossref] [PubMed]
  33. S. D. Grant and A. Abdolvand, “Left- and right-circularly polarized light in cascade conical diffraction,” Opt. Lett. 37, 5226–5228 (2012).
    [Crossref] [PubMed]
  34. M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7, 839–854 (2013).
    [Crossref]
  35. W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
    [Crossref] [PubMed]
  36. M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
    [Crossref]
  37. V. Massidda, “Analytical calculation of a class of integrals containing exponential and trigonometric functions,” Math. Comp. 41, 555–557 (1983).
    [Crossref]
  38. A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

2016 (5)

S. D. Grant, S. Reynolds, and A. Abdolvand, “Optical sensing of polarization using conical diffraction phenomenon,” J. Opt. 18, 025609 (2016).
[Crossref]

A. Brenier, “Lasing with conical diffraction feature in the KGd(WO4)2:Nd biaxial crystal,” Appl. Phys. B 122, 237 (2016).
[Crossref]

V. Peet, “Conical refraction in a degenerated two-crystal cascade,” J. Opt. 18, 015607 (2016).
[Crossref]

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 10, 750–771 (2016).
[Crossref]

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (3)

R. Cattoor, I. Manek-Hönninger, D. Rytz, L. Canioni, and M. Eichhorn, “Laser action along and near the optic axis of a holmium-doped KY(WO4)2 crystal,” Opt. Lett. 39, 6407–6410 (2014).
[Crossref] [PubMed]

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

2013 (5)

2012 (4)

2011 (2)

2010 (6)

2007 (2)

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Progr. Opt. 50, 13–50 (2007).
[Crossref]

J. Hellström, H. Henricsson, V. Pasiskevicius, U. Bünting, and D. Haussmann, “Polarization-tunable Yb:KGW laser based on internal conical refraction,” Opt. Lett. 32, 2783–2785 (2007).
[Crossref] [PubMed]

2004 (1)

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A: Pure Appl. Opt. 6, 289–300 (2004).
[Crossref]

1999 (1)

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

1994 (1)

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

1983 (1)

V. Massidda, “Analytical calculation of a class of integrals containing exponential and trigonometric functions,” Math. Comp. 41, 555–557 (1983).
[Crossref]

1978 (2)

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).

1837 (1)

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish Acad. 17, 1–144 (1837).

1833 (1)

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxal crystals,” Philos. Mag. 2, 112–120 (1833).

Abdolvand, A.

S. D. Grant, S. Reynolds, and A. Abdolvand, “Optical sensing of polarization using conical diffraction phenomenon,” J. Opt. 18, 025609 (2016).
[Crossref]

S. D. Grant and A. Abdolvand, “Left- and right-circularly polarized light in cascade conical diffraction,” Opt. Lett. 37, 5226–5228 (2012).
[Crossref] [PubMed]

A. Abdolvand, ‘Conical diffraction from a multi-crystal cascade: experimental observations,” Appl. Phys. B 103, 281–283 (2011).
[Crossref]

A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rafailov, “Conical refraction Nd:KGd(WO4)2 laser,” Opt. Express 18, 2753–2759 (2010).
[Crossref] [PubMed]

Aguiló, M.

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Ahufinger, V.

Alpmann, C.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7, 839–854 (2013).
[Crossref]

Ballantine, K. E.

Belskii, A. M.

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Berry, M. V.

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

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Progr. Opt. 50, 13–50 (2007).
[Crossref]

M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A: Pure Appl. Opt. 6, 289–300 (2004).
[Crossref]

Birkl, G.

Boulanger, B.

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

Braitbart, P. O.

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

Brenier, A.

A. Brenier, “Lasing with conical diffraction feature in the KGd(WO4)2:Nd biaxial crystal,” Appl. Phys. B 122, 237 (2016).
[Crossref]

Bünting, U.

Campos, J.

Canioni, L.

Caron, J.

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

Cattoor, R.

Chen, P.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

Chigrinov, V.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

Denz, C.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7, 839–854 (2013).
[Crossref]

Diaz, F.

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Donegan, J. F.

Eastham, P. R.

Eichhorn, M.

Esseling, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7, 839–854 (2013).
[Crossref]

Fallet, C.

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

Fernandez, E.

Fève, J. P.

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

Grant, S. D.

S. D. Grant, S. Reynolds, and A. Abdolvand, “Optical sensing of polarization using conical diffraction phenomenon,” J. Opt. 18, 025609 (2016).
[Crossref]

S. D. Grant and A. Abdolvand, “Left- and right-circularly polarized light in cascade conical diffraction,” Opt. Lett. 37, 5226–5228 (2012).
[Crossref] [PubMed]

Hamilton, W. R.

W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Trans. Royal Irish Acad. 17, 1–144 (1837).

Haussmann, D.

Hellström, J.

Henricsson, H.

Hnatovsky, C.

Hu, W.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

Iemmi, C.

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Progr. Opt. 50, 13–50 (2007).
[Crossref]

Ji, W.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

Kalkandjiev, T. K.

Khapalyuk, A. P.

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 436–439 (1978).

A. M. Belskii and A. P. Khapalyuk, “Propagation of confined light beams along the beam axes (axes of single ray velocity) of biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Krolikowski, W.

Küber, J.

Lee, C. H.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

Lin, T. H.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

Lizana, A.

Lloyd, H.

H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxal crystals,” Philos. Mag. 2, 112–120 (1833).

Loiko, Yu V.

Loiko, Yu. V.

Lu, Y. Q.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

Lunney, J. G.

Manek-Hönninger, I.

Marnier, G.

J. P. Fève, B. Boulanger, and G. Marnier, “Experimental study of internal and external conical refractions in KTP,” Opt. Commun. 105, 243–252 (1994).
[Crossref]

Massidda, V.

V. Massidda, “Analytical calculation of a class of integrals containing exponential and trigonometric functions,” Math. Comp. 41, 555–557 (1983).
[Crossref]

Ming, Y.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

Moisan, L.

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

Mompart, J.

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 10, 750–771 (2016).
[Crossref]

A. Peinado, A. Lizana, A. Turpin, C. Iemmi, T. K. Kalkandjiev, J. Mompart, and J. Campos, “Optimization, tolerance analysis and implementation of a Stokes polarimeter based on the conical refraction phenomenon,” Opt. Express 23, 5636–5652 (2015).
[Crossref] [PubMed]

A. Turpin, J. Polo, Yu. V. Loiko, J. Küber, F. Schmaltz, T. K. Kalkandjiev, V. Ahufinger, G. Birkl, and J. Mompart, “Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction,” Opt. Express 23, 1638–1650 (2015).
[Crossref] [PubMed]

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express 21, 4503–4511 (2013).
[Crossref] [PubMed]

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Multiple rings formation in cascaded conical refraction,” Opt. Lett. 38, 1455–1457 (2013).
[Crossref] [PubMed]

A. Turpin, V. Shvedov, C. Hnatovsky, Yu V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: A reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21, 26335–26340 (2013).
[Crossref] [PubMed]

A. Peinado, A. Turpin, A. Lizana, E. Fernandez, J. Mompart, and J. Campos, “Conical refraction as a tool for polarization metrology,” Opt. Lett. 38, 4100–4103 (2013).
[Crossref] [PubMed]

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Free-space optical polarization demultiplexing and multiplexing by means of conical refraction,” Opt. Lett. 37, 4197–4199 (2012).
[Crossref] [PubMed]

Nikolov, V.

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

O’ Dwyer, D. P.

O’Dwyer, D. P.

Oddos, S.

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

Pasiskevicius, V.

Peet, V.

Peinado, A.

Phelan, C. F.

Polo, J.

Pujol, M.C.

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Rafailov, E. U.

Rakovich, Y. P.

Reynolds, S.

S. D. Grant, S. Reynolds, and A. Abdolvand, “Optical sensing of polarization using conical diffraction phenomenon,” J. Opt. 18, 025609 (2016).
[Crossref]

Rico, M.

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Rytz, D.

Schmaltz, F.

Shorte, S. L.

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

Shvedov, V.

Sirat, G. Y.

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

Solans, X.

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Solé, R.

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Tinevez, J.Y.

C. Fallet, J. Caron, S. Oddos, J.Y. Tinevez, L. Moisan, G. Y. Sirat, P. O. Braitbart, and S. L. Shorte, “Conical diffraction as a versatile building block to implement new imaging modalities for superresolution in fluorescence microscopy,” Proc. SPIE 9169, 916905 (2014).
[Crossref]

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

Tomizawa, H.

Turpin, A.

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Conical refraction: fundamentals and applications,” Laser Photonics Rev. 10, 750–771 (2016).
[Crossref]

A. Turpin, J. Polo, Yu. V. Loiko, J. Küber, F. Schmaltz, T. K. Kalkandjiev, V. Ahufinger, G. Birkl, and J. Mompart, “Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction,” Opt. Express 23, 1638–1650 (2015).
[Crossref] [PubMed]

A. Peinado, A. Lizana, A. Turpin, C. Iemmi, T. K. Kalkandjiev, J. Mompart, and J. Campos, “Optimization, tolerance analysis and implementation of a Stokes polarimeter based on the conical refraction phenomenon,” Opt. Express 23, 5636–5652 (2015).
[Crossref] [PubMed]

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express 21, 4503–4511 (2013).
[Crossref] [PubMed]

A. Turpin, V. Shvedov, C. Hnatovsky, Yu V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: A reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21, 26335–26340 (2013).
[Crossref] [PubMed]

A. Peinado, A. Turpin, A. Lizana, E. Fernandez, J. Mompart, and J. Campos, “Conical refraction as a tool for polarization metrology,” Opt. Lett. 38, 4100–4103 (2013).
[Crossref] [PubMed]

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Multiple rings formation in cascaded conical refraction,” Opt. Lett. 38, 1455–1457 (2013).
[Crossref] [PubMed]

A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, and J. Mompart, “Free-space optical polarization demultiplexing and multiplexing by means of conical refraction,” Opt. Lett. 37, 4197–4199 (2012).
[Crossref] [PubMed]

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Woerdemann, M.

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7, 839–854 (2013).
[Crossref]

Zaldo, C.

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Zhang, L.

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
[Crossref] [PubMed]

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[Crossref]

M.C. Pujol, M. Rico, C. Zaldo, R. Solé, V. Nikolov, X. Solans, M. Aguiló, and F. Diaz, “Crystalline structure and optical spectroscopy of Er3+-doped KGd(WO4)2 single crystals,” Appl. Phys. B 68, 187–197 (1999).
[Crossref]

Cell Adhes. Migr. (1)

J. Caron, C. Fallet, J.Y. Tinevez, L. Moisan, P. O. Braitbart, G. Y. Sirat, and S. L. Shorte, “Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging,” Cell Adhes. Migr. 8, 430–439 (2014).
[Crossref]

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[Crossref]

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[Crossref]

V. Peet, ‘Biaxial crystal as a versatile mode converter,” J. Opt. 12, 095706 (2010).
[Crossref]

S. D. Grant, S. Reynolds, and A. Abdolvand, “Optical sensing of polarization using conical diffraction phenomenon,” J. Opt. 18, 025609 (2016).
[Crossref]

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[Crossref]

M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7, 839–854 (2013).
[Crossref]

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A. Turpin, Yu. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express 21, 4503–4511 (2013).
[Crossref] [PubMed]

A. Turpin, V. Shvedov, C. Hnatovsky, Yu V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: A reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21, 26335–26340 (2013).
[Crossref] [PubMed]

A. Turpin, J. Polo, Yu. V. Loiko, J. Küber, F. Schmaltz, T. K. Kalkandjiev, V. Ahufinger, G. Birkl, and J. Mompart, “Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction,” Opt. Express 23, 1638–1650 (2015).
[Crossref] [PubMed]

A. Peinado, A. Lizana, A. Turpin, C. Iemmi, T. K. Kalkandjiev, J. Mompart, and J. Campos, “Optimization, tolerance analysis and implementation of a Stokes polarimeter based on the conical refraction phenomenon,” Opt. Express 23, 5636–5652 (2015).
[Crossref] [PubMed]

A. Abdolvand, K. G. Wilcox, T. K. Kalkandjiev, and E. U. Rafailov, “Conical refraction Nd:KGd(WO4)2 laser,” Opt. Express 18, 2753–2759 (2010).
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D. P. O’ Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010).
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V. Peet, “Improving directivity of laser beams by employing the effect of conical refraction in biaxial crystals,” Opt. Express 18, 19566–19573 (2010).
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D. P. O’Dwyer, C. F. Phelan, K. E. Ballantine, Y. P. Rakovich, J. G. Lunney, and J. F. Donegan, “Conical diffraction of linearly polarised light controls the angular position of a microscopic object,” Opt. Express 18, 27319–27326 (2010).
[Crossref]

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. Express 19, 2580–2588 (2011).
[Crossref]

C. F. Phelan, K. E. Ballantine, P. R. Eastham, J. F. Donegan, and J. G. Lunney, ‘Conical diffraction of a Gaussian beam with a two crystal cascade,” Opt. Express 20, 13201–13207 (2012).
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Proc. SPIE (1)

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[Crossref]

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W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6, 25528 (2016).
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Figures (7)

Fig. 1
Fig. 1 Arrangement for cascaded conical diffraction of N crystals with strength parameter ρn intercalated by N − 1 polarization transforming elements with Jones matrices Jm. The inset on the right shows the situation for the first crystal. The Poynting vector directions S⃗ associated to the common wave-vector k⃗ parallel to the optical axis lie on a cone containing the vector k⃗. The direction γ⃗1 of displacement of the conical diffraction cone points towards the Poynting vector S⃗* that has a maximum walk-off angle with k⃗. The inset on top left shows the orientation of the projection of the index ellipsoid (indicatrix) on the xz-plane for the first crystal.
Fig. 2
Fig. 2 Cascaded conical diffraction for two crossed crystals (γ2 = π/2) and circular polarized input as obtained in the focal image plane ζ = 0. Panel (a) shows the calculated circular symmetric intensity distribution in absence of any intercalated polarization transforming element. ρx and ρy are the projections of the normalized radius ρ into the x and y axis, respectively. Panel (b) shows the experimental observation for the case where a λ/4-plate under the angle θ = 0 is introduced between the KGW crystals and panel (c) shows the corresponding theoretical intensity distribution. Panel (d) is the same as panel (c) but for the modulus of the D⃗ vector instead of the intensity. Panel (e) gives the theoretical angular dependence of the intensity distribution along the internal ring (blue solid line) and the external ring (red dotted line), and panel (f) gives the corresponding experimental angular dependence. The value φ = 0 correspond to the points on the horizontal axis in (b) and (c).
Fig. 3
Fig. 3 Cascaded conical diffraction for two crossed crystals as in Fig. 2 but for an intercalated polarizer under the angle θ = π/2. (a) Theoretical distribution of the modulus of the D⃗ vector in the plane ζ = 0; (b) Experimental intensity distribution; (c) Theoretical and experimental angular dependence of the intensity distribution along the internal ring (blue solid line) and the external ring (red dotted line). The inset in the lower graph in (c) gives a zoom for the weak signal in the external ring for φ between 0 and 90 degrees.
Fig. 4
Fig. 4 Cascaded conical diffraction for two cascaded parallel crystals (γ2 = 0) intercalated by a λ/4-plate under the angle θ = π/4. (a) Theoretical distribution of the modulus of the D⃗ vector in the plane ζ = 0; (b) Experimental intensity distribution; (c) Theoretical and experimental angular dependence of the intensity distribution along the internal ring (blue solid line) and the external ring (red dotted line).
Fig. 5
Fig. 5 Cascaded conical diffraction for two cascaded parallel crystals (γ2 = 0) intercalated by a polarizer under the angle θ = π/2. (a) Theoretical distribution of the modulus of the D⃗ vector in the plane ζ = 0; (b) Experimental intensity distribution; (c) Theoretical and experimental angular dependence of the intensity distribution along the internal ring (blue solid line) and the external ring (red dotted line).
Fig. 6
Fig. 6 Expected intensity distribution in the plane ζ = 0 for the cascaded conical diffraction of more than two crystals and intercalation of polarizing elements. (a) Three crystals with normalized strength parameters ρ1 = 40, ρ2 = 15, ρ3 = 5 and orientations γ1 = 0 and γ2 = γ3 = 90 deg. A λ/4-plate is placed between the first and the second crystal under the angle θ1 = 45 deg, and a λ/2-plate under the angle θ2 = 90 deg is placed between the second and the third crystal. (b) Four crystals with ρ1 = 40, ρ2 = 15, ρ3 = 5, ρ4 = 70 and orientations γ1 = 0, γ2 = γ3 = 90 deg, and γ4 = 135 deg. The polarization transforming elements are a polarizer followed by a λ/2-plate and another polarizer, their orientations are θ1 = 0 deg, θ2 = 45 deg and θ3 = 135 deg.
Fig. 7
Fig. 7 Modified Belskii-Khapalyuk integrals B0, B1 and B2 in (21) as a function of the normalized radius ρ for the case ζ = 0 and a(κ) = exp(−κ2/2). The panels in the left column give the real part and the panels in the right column give the imaginary part of the integrals. The top panels are for B0(ρ̃+) (solid lines) and B0(ρ̃) (dotted lines). The corresponding functions for B1(ρ̃±) and B2(ρ̃±) are in the middle panels and bottom panels, respectively. Here ρ̃+ρ1 + ρ2 and ρ̃ρ1ρ2, with ρ1 = 98.6 and ρ2 = 76.8. The vertical lines correspond to the conditions ρ = ρ̃ and ρ = ρ̃+.

Equations (23)

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D ( ρ , φ , ζ ) = 1 2 π 0 2 π 0 e i κ ρ cos ( ϕ φ ) e i κ 2 2 ζ U tot D 0 ( κ , ϕ ) κ d κ d ϕ .
D 0 ( κ , ϕ ) = a ( κ ) d 0 ,
U tot ( κ , ϕ ) = U N ( κ , ϕ , γ N ) U N 1 ( κ , ϕ , γ N 1 ) U 2 ( κ , ϕ , γ 2 ) U 1 ( κ , ϕ , 0 ) ,
U n ( κ , ϕ , γ n ) = exp [ i ρ n κ ( cos ( ϕ γ n ) sin ( ϕ γ n ) sin ( ϕ γ n ) cos ( ϕ γ n ) ) ] = ( cos ( κ ρ n ) i sin ( κ ρ n ) cos ( ϕ γ n ) i sin ( κ ρ n ) sin ( ϕ γ n ) i sin ( κ ρ n ) sin ( ϕ γ n ) cos ( κ ρ n ) + i sin ( κ ρ n ) cos ( ϕ γ n ) ) .
U tot ( κ , ϕ ) = U N ( κ , ϕ , γ N ) J N 1 ( θ N 1 ) J 2 ( θ 2 ) U 2 ( κ , ϕ , γ 2 ) J 1 ( θ 1 ) U 1 ( κ , ϕ , 0 ) ,
I ( ρ , φ , ζ ) = D D * = | D | 2 .
J λ / 4 ( θ ) = 1 2 ( cos ( 2 θ ) + i sin ( 2 θ ) sin ( 2 θ ) cos ( 2 θ ) + i )
D x ( ρ , φ , ζ ) = [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ ) + B 0 ( ρ ˜ ) + B 0 ( ρ ˜ ) ] ( e i 2 θ + i ) + [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ + ) B 0 ( ρ ˜ ) B 0 ( ρ ˜ ) ] i e i γ 2 [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) B 1 ( ρ ˜ ) + B 1 ( ρ ˜ ) ] ( e i ( γ 2 φ + 2 θ ) + i e i ( φ γ 2 ) ) [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) + B 1 ( ρ ˜ ) B 1 ( ρ ˜ ) ] ( e i ( φ 2 θ ) + i e i φ ) + [ B 2 ( ρ ˜ + ) + B 2 ( ρ ˜ + ) B 2 ( ρ ˜ ) B 2 ( ρ ˜ ) ] e i ( 2 φ 2 θ γ 2 ) ,
D y ( ρ , φ , ζ ) = [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ ) + B 0 ( ρ ˜ ) ] ( i e i 2 θ + 1 ) [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ + ) B 0 ( ρ ˜ ) B 0 ( ρ ˜ ) ] e i γ 2 [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) B 1 ( ρ ˜ ) + B 1 ( ρ ˜ ) ] ( i e i ( γ 2 φ + 2 θ ) + e i ( φ γ 2 ) ) [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) + B 1 ( ρ ˜ ) B 1 ( ρ ˜ ) ] ( i e i ( φ 2 θ ) + i e i φ ) [ B 2 ( ρ ˜ + ) + B 2 ( ρ ˜ + ) B 2 ( ρ ˜ ) B 2 ( ρ ˜ ) ] i e i ( 2 φ 2 θ γ 2 ) ,
J λ / 2 ( θ ) = ( cos ( 2 θ ) sin ( 2 θ ) sin ( 2 θ ) cos ( 2 θ ) )
D x ( ρ , φ , ζ ) = [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ + ) B 0 ( ρ ˜ ) + B 0 ( ρ ˜ ) ] e i 2 θ [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) B 1 ( ρ ˜ ) + B 1 ( ρ ˜ ) ] e i ( γ 2 φ + 2 θ ) [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) + B 1 ( ρ ˜ ) B 1 ( ρ ˜ ) ] e i ( φ 2 θ ) + [ B 2 ( ρ ˜ + ) + B 2 ( ρ ˜ + ) B 2 ( ρ ˜ ) B 2 ( ρ ˜ ) ] e i ( 2 φ 2 θ γ 2 ) ,
D y ( ρ , φ , ζ ) = [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ ) + B 0 ( ρ ˜ ) ] i e i 2 θ [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) B 1 ( ρ ˜ ) + B 1 ( ρ ˜ ) ] i e i ( γ 2 φ + 2 θ ) [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) + B 1 ( ρ ˜ ) B 1 ( ρ ˜ ) ] i e i ( φ 2 θ ) [ B 2 ( ρ ˜ + ) + B 2 ( ρ ˜ + ) B 2 ( ρ ˜ ) B 2 ( ρ ˜ ) ] i e i ( 2 φ 2 θ γ 2 ) .
J pol ( θ ) = ( cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ ) .
D x ( ρ , φ , ζ ) = [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ ) + B 0 ( ρ ˜ ) + B 0 ( ρ ˜ ) ] ( e i 2 θ + 1 ) + [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ + ) B 0 ( ρ ˜ ) B 0 ( ρ ˜ ) ] e i γ 2 [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) B 1 ( ρ ˜ ) + B 1 ( ρ ˜ ) ] ( e i ( γ 2 φ + 2 θ ) + e i ( φ γ 2 ) ) [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) + B 1 ( ρ ˜ ) B 1 ( ρ ˜ ) ] ( e i ( φ 2 θ ) + e i φ ) + [ B 2 ( ρ ˜ + ) + B 2 ( ρ ˜ + ) B 2 ( ρ ˜ ) B 2 ( ρ ˜ ) ] e i ( 2 φ 2 θ γ 2 ) ,
D y ( ρ , φ , ζ ) = [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ ) + B 0 ( ρ ˜ ) ] ( i e i 2 θ + i ) [ B 0 ( ρ ˜ + ) + B 0 ( ρ ˜ + ) B 0 ( ρ ˜ ) B 0 ( ρ ˜ ) ] i e i γ 2 [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) B 1 ( ρ ˜ ) + B 1 ( ρ ˜ ) ] ( i e i ( γ 2 φ + 2 θ ) e i ( φ γ 2 ) ) [ B 1 ( ρ ˜ + ) B 1 ( ρ ˜ + ) + B 1 ( ρ ˜ ) B 1 ( ρ ˜ ) ] ( i e i ( φ 2 θ ) i e i φ ) [ B 2 ( ρ ˜ + ) + B 2 ( ρ ˜ + ) B 2 ( ρ ˜ ) B 2 ( ρ ˜ ) ] i e i ( 2 φ 2 θ γ 2 ) .
e ± im ϕ e i β e i κ ρ ˜ ,
p e i κ ρ ˜ e i β 1 2 π 0 2 π e i κ ρ cos ( ϕ φ ) e ± i m ϕ d ϕ .
1 2 π 0 2 π e im φ e a cos ( ϕ α 1 ) e 2 b cos 2 ( ϕ α 2 ) d ϕ = e b e i m α 1 k e 2 i k ( α 1 α 2 ) I 2 k + m ( a ) I m ( b ) ,
p = e i κ ρ ˜ e i β e ± im φ ( i ) m J m ( κ ρ ) .
e ± im ϕ e i β e i κ ρ ˜ q = e i β e ± im φ B m ( ρ , ρ ˜ , ζ ) .
B m ( ρ , ρ ˜ , ζ ) ( i ) m 0 e i κ 2 2 ζ e i κ ρ ˜ J m ( κ ρ ) a ( κ ) κ d κ .
Re [ B m ( ρ , ρ ˜ , 0 ) ] = ( 1 ) m Re [ B m ( ρ , ρ ˜ , 0 ) ] ,
Im [ B m ( ρ , ρ ˜ , 0 ) ] = ( 1 ) m + 1 Im [ B m ( ρ , ρ ˜ , 0 ) ] .

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