Abstract

We report on the experimental and theoretical investigation of polarization conversion of linearly polarized Gaussian beam propagating in perpendicularly cut homogeneous uniaxial crystals. We derive analytical expressions, in good agreement with experimental data, for power transfer between components at normal incidence accompanied by the generation of a topological quadrupole. We extend the results to the oblique incidence case and confirm experimentally the optimal parameters for generation of a single charge on-axis optical vortex, including spectrally resolved measurements for the white-light beams.

© 2009 Optical Society of America

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  1. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London A 336, 165 (1974).
    [CrossRef]
  2. M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219 (Ed., E. Wolf, Elsevier, 2001).
    [CrossRef]
  3. M. R. Dennis, K. O’Holleran, and M. J. Padgett, "Singular Optics: Optical Vortices and Polarization Singularities," Prog. Opt. 52, 293 (Ed., E. Wolf, Elsevier, 2009).
    [CrossRef]
  4. V. Bazhenov, M. Vasnetsov, and M. Soskin, "Laser beams with screw dislocations in their wavefonts," JETP Lett. 52, 429 (1990).
  5. N. Heckenberg, R. McDuff, C. Smith, and A. White, "Generation of optical phase singularities by computergenerated holograms," Opt. Lett. 17, 221 (1992).
    [CrossRef] [PubMed]
  6. A. Ciattoni, G. Cincotti, and C. Palma, "Propagation of cylindrically symmetric fields in uniaxial crystals," J. Opt. Soc. Am. A 19, 792 (2002).
    [CrossRef]
  7. A. Ciattoni, G. Cancotti, and C. Palma, "Circularly polarized beams and vortex generation in uniaxial media," J. Opt. Soc. Am. A 20, 163 (2003).
    [CrossRef]
  8. A. Volyar and T. Fadeyeva, "Generation of singular beams in uniaxial crystals," Opt. Spectrosc. 94, 235 (2003).
    [CrossRef]
  9. M. Berry and M. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 459, 1261 (2003).
    [CrossRef]
  10. L. Marrucci, C. Manzo, and D. Paparo, "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media," Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  11. E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, "Optical vortices from liquid crystal droplets," Phys. Rev. Lett. 103, 103903 (2009).
    [CrossRef] [PubMed]
  12. A. V. Volyar and T. A. Fadeeva, "Optical vortices in crystals: Formation, annihilation, and decay of polarization ombilics," J. Technol. Phys. Lett. 29, 111-114 (2003).
    [CrossRef]
  13. A. V. Volyar and T. A. Fadeeva, "Decay and fusion of polarization umbilics in a singular beam passed through a crystal," Opt. Spectrosc. 95, 792-799 (2003).
    [CrossRef]
  14. 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 (2004).
  15. F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, "Polarization singularities from unfolding an optical vortex through a birefringent crystal," Phys. Rev. Lett. 95, 253901 (2005).
    [CrossRef] [PubMed]
  16. A. Volyar and T. Fadeyeva, "Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal," Opt. Spectrosc. 101, 450-457 (2006).
    [CrossRef]
  17. T. Fadeyeva, Yu. Egorov, A. Rubass, G. A. Swartzlander, Jr., and A. Volyar, "Indistinguishability limit for off-axis vortex beams in uniaxial crystals," Opt. Lett. 32, 3116 (2007).
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    [CrossRef]
  19. 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]
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    [CrossRef]
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  25. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, "Dynamics of optical spin-orbit coupling in uniaxial crystals," Opt. Lett. 34, 1021-1023 (2009).
    [CrossRef] [PubMed]
  26. A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krollikowski, and Yu. S. Kivshar, "Generation of single-charged optical vortices with a uniaxial crystal," Opt. Express 14, 3724 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3724.
    [CrossRef]
  27. M. Berry, "Coloured phase singularities," New J. Phys. 4, 66 (2002).
    [CrossRef]
  28. M. S. Soskin, P. V. Polyansky, and O. O. Arkheluyk, "Computer-synthesized hologram-based rainbow vortices," New J. Phys. 6, 196 (2004).
    [CrossRef]

2009

2008

2007

2006

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

A. Volyar and T. Fadeyeva, "Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal," Opt. Spectrosc. 101, 450-457 (2006).
[CrossRef]

A. Volyar, V. Shvedov, T. Fadeyeva, A. S. Desyatnikov, D. N. Neshev, W. Krollikowski, and Yu. S. Kivshar, "Generation of single-charged optical vortices with a uniaxial crystal," Opt. Express 14, 3724 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3724.
[CrossRef]

2005

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, "Polarization singularities from unfolding an optical vortex through a birefringent crystal," Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

2004

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 (2004).

M. S. Soskin, P. V. Polyansky, and O. O. Arkheluyk, "Computer-synthesized hologram-based rainbow vortices," New J. Phys. 6, 196 (2004).
[CrossRef]

2003

A. V. Volyar and T. A. Fadeeva, "Optical vortices in crystals: Formation, annihilation, and decay of polarization ombilics," J. Technol. Phys. Lett. 29, 111-114 (2003).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, "Decay and fusion of polarization umbilics in a singular beam passed through a crystal," Opt. Spectrosc. 95, 792-799 (2003).
[CrossRef]

A. Ciattoni, G. Cancotti, and C. Palma, "Circularly polarized beams and vortex generation in uniaxial media," J. Opt. Soc. Am. A 20, 163 (2003).
[CrossRef]

A. Volyar and T. Fadeyeva, "Generation of singular beams in uniaxial crystals," Opt. Spectrosc. 94, 235 (2003).
[CrossRef]

M. Berry and M. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 459, 1261 (2003).
[CrossRef]

2002

1998

M. S. Soskin and M. V. Vasnetsov, "Nonlinear singular optics," J. Opt. A:Pure Appl. Opt. 7, 301311 (1998).
[CrossRef]

1992

1990

V. Bazhenov, M. Vasnetsov, and M. Soskin, "Laser beams with screw dislocations in their wavefonts," JETP Lett. 52, 429 (1990).

1974

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London A 336, 165 (1974).
[CrossRef]

Arkheluyk, O. O.

M. S. Soskin, P. V. Polyansky, and O. O. Arkheluyk, "Computer-synthesized hologram-based rainbow vortices," New J. Phys. 6, 196 (2004).
[CrossRef]

Bazhenov, V.

V. Bazhenov, M. Vasnetsov, and M. Soskin, "Laser beams with screw dislocations in their wavefonts," JETP Lett. 52, 429 (1990).

Berry, M.

M. Berry and M. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 459, 1261 (2003).
[CrossRef]

M. Berry, "Coloured phase singularities," New J. Phys. 4, 66 (2002).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London A 336, 165 (1974).
[CrossRef]

Brasselet, E.

Cancotti, G.

Ciattoni, A.

Cincotti, G.

Dennis, M.

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, "Polarization singularities from unfolding an optical vortex through a birefringent crystal," Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

M. Berry and M. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 459, 1261 (2003).
[CrossRef]

Desyatnikov, A. S.

Dreischuh, A.

Egorov, Yu.

Fadeeva, T. A.

A. V. Volyar and T. A. Fadeeva, "Optical vortices in crystals: Formation, annihilation, and decay of polarization ombilics," J. Technol. Phys. Lett. 29, 111-114 (2003).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, "Decay and fusion of polarization umbilics in a singular beam passed through a crystal," Opt. Spectrosc. 95, 792-799 (2003).
[CrossRef]

Fadeyeva, T.

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]

Flossmann, F.

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, "Polarization singularities from unfolding an optical vortex through a birefringent crystal," Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Heckenberg, N.

Izdebskaya, Y.

Juodkazis, S.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, "Optical vortices from liquid crystal droplets," Phys. Rev. Lett. 103, 103903 (2009).
[CrossRef] [PubMed]

Kivshar, Yu. S

Kivshar, Yu. S.

Krolikowski, W.

Krolikowski, W. Z.

Krollikowski, W.

Maier, M.

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, "Polarization singularities from unfolding an optical vortex through a birefringent crystal," Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Manzo, C.

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

Marrucci, L.

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

McDuff, R.

Misawa, H.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, "Optical vortices from liquid crystal droplets," Phys. Rev. Lett. 103, 103903 (2009).
[CrossRef] [PubMed]

Murazawa, N.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, "Optical vortices from liquid crystal droplets," Phys. Rev. Lett. 103, 103903 (2009).
[CrossRef] [PubMed]

Neshev, D. N.

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London A 336, 165 (1974).
[CrossRef]

Palma, C.

Paparo, D.

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

Polyansky, P. V.

M. S. Soskin, P. V. Polyansky, and O. O. Arkheluyk, "Computer-synthesized hologram-based rainbow vortices," New J. Phys. 6, 196 (2004).
[CrossRef]

Rode, A. V.

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]

Schwarz, U.

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, "Polarization singularities from unfolding an optical vortex through a birefringent crystal," Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Shvedov, V.

Shvedov, V. G.

Smith, C.

Soskin, M.

V. Bazhenov, M. Vasnetsov, and M. Soskin, "Laser beams with screw dislocations in their wavefonts," JETP Lett. 52, 429 (1990).

Soskin, M. S.

M. S. Soskin, P. V. Polyansky, and O. O. Arkheluyk, "Computer-synthesized hologram-based rainbow vortices," New J. Phys. 6, 196 (2004).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Nonlinear singular optics," J. Opt. A:Pure Appl. Opt. 7, 301311 (1998).
[CrossRef]

Swartzlander, G.

Swartzlander, G. A.

Vasnetsov, M.

V. Bazhenov, M. Vasnetsov, and M. Soskin, "Laser beams with screw dislocations in their wavefonts," JETP Lett. 52, 429 (1990).

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, "Nonlinear singular optics," J. Opt. A:Pure Appl. Opt. 7, 301311 (1998).
[CrossRef]

Volyar, A.

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]

A. V. Volyar and T. A. Fadeeva, "Decay and fusion of polarization umbilics in a singular beam passed through a crystal," Opt. Spectrosc. 95, 792-799 (2003).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, "Optical vortices in crystals: Formation, annihilation, and decay of polarization ombilics," J. Technol. Phys. Lett. 29, 111-114 (2003).
[CrossRef]

White, A.

J. Opt. A: Pure Appl. Opt.

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 (2004).

J. Opt. Soc. Am. A

J. Technol. Phys. Lett.

A. V. Volyar and T. A. Fadeeva, "Optical vortices in crystals: Formation, annihilation, and decay of polarization ombilics," J. Technol. Phys. Lett. 29, 111-114 (2003).
[CrossRef]

JETP Lett.

V. Bazhenov, M. Vasnetsov, and M. Soskin, "Laser beams with screw dislocations in their wavefonts," JETP Lett. 52, 429 (1990).

New J. Phys.

M. Berry, "Coloured phase singularities," New J. Phys. 4, 66 (2002).
[CrossRef]

M. S. Soskin, P. V. Polyansky, and O. O. Arkheluyk, "Computer-synthesized hologram-based rainbow vortices," New J. Phys. 6, 196 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

A. Volyar and T. Fadeyeva, "Laguerre-Gaussian beams with complex and real arguments in a uniaxial crystal," Opt. Spectrosc. 101, 450-457 (2006).
[CrossRef]

A. V. Volyar and T. A. Fadeeva, "Decay and fusion of polarization umbilics in a singular beam passed through a crystal," Opt. Spectrosc. 95, 792-799 (2003).
[CrossRef]

A. Volyar and T. Fadeyeva, "Generation of singular beams in uniaxial crystals," Opt. Spectrosc. 94, 235 (2003).
[CrossRef]

Phys. Rev. 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]

Phys. Rev. Lett.

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

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, "Optical vortices from liquid crystal droplets," Phys. Rev. Lett. 103, 103903 (2009).
[CrossRef] [PubMed]

F. Flossmann, U. Schwarz, M. Maier, and M. Dennis, "Polarization singularities from unfolding an optical vortex through a birefringent crystal," Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A

M. Berry and M. Dennis, "The optical singularities of birefringent dichroic chiral crystals," Proc. R. Soc. Lond. A 459, 1261 (2003).
[CrossRef]

Proc. R. Soc. London A

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London A 336, 165 (1974).
[CrossRef]

Pure Appl. Opt.

M. S. Soskin and M. V. Vasnetsov, "Nonlinear singular optics," J. Opt. A:Pure Appl. Opt. 7, 301311 (1998).
[CrossRef]

Other

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," Prog. Opt. 47, 291-391 (Ed. E. Wolf, Elsevier, 2005).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219 (Ed., E. Wolf, Elsevier, 2001).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, "Singular Optics: Optical Vortices and Polarization Singularities," Prog. Opt. 52, 293 (Ed., E. Wolf, Elsevier, 2009).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Top row: intensity distributions I‖=|E‖|2 of the field polarized parallel to the incident Gaussian beam (φ0=0) with the waist w=4.6µm in calcite crystal (no =1.656 and ne =1.485 at λ=633 nm). Bottom row: cross-sections along x (red dashed curves) and y (black solid curves).

Fig. 2.
Fig. 2.

Experimental setup. L1,2: lenses or microobjectives; PBS: polarization beamsplitter; CCD: charge coupled device cameras; o.a.: optical axis, whose orientation is defined by the two angles α and ψ as shown in the inset. Intensities I =|E |2 and I =|E |2 for (a) normal and (b) oblique (see text for details) incidences, at the propagation length z=6 mm for input beam waist w=4.6µm in (a) and w=11µm in (b); in both cases λ=633 nm. Intensity distribution in panels (a,b) are false colored.

Fig. 3.
Fig. 3.

Polarization conversion with respect to the normalized propagation distance in the case of normal incidence for λ=630 nm and ε=0.109. Solid red lines: numerical integration of exact solution (2, 3), and dashed blue lines: approximate formulas (7).

Fig. 4.
Fig. 4.

(a) Ratio P /P as a function of the crystal thickness for three different waists for λ=0.6328µm. Solid lines: theory; markers: experiment; black: w=1.8µm; red: w=4.4µm; blue: w=11.2µm. (b) Power ratio P /P as a function of z/w 2 for the data of panel (a), the markers and the solid curve being respectively experimental data and theoretical prediction. The panels (1,2) correspond to the intensity distribution I ‖,⊥, respectively, and refer the set of parameters of the data indicated by the arrows in panel (a). Similarly, the panels (3,4) correspond to another set of parameters, see panel (a).

Fig. 5.
Fig. 5.

Trajectories of single charge vortices which are the closest to the optical axis in reduced coordinates. (a–c) Distance from the optical axis ρ 0. (d–f) Polar angle ψ 0 in the (x, y) plane for φ0=0. Blue curves: analytical expressions Eqs. (11, 12); dashed curves: asymptotes Eqs. (13, 14), and red curves: exact results Eqs. (2, 3).

Fig. 6.
Fig. 6.

(a) White light multipole at normal incidence for different propagation distance z inside the crystal (z=2, 6 and 10 mm). (b) Multipole spectral components, red (630 nm), green (550 nm) and blue (440 nm), for a crystal slab with fixed thickness 6 mm. (c) Angle defining the optical vortices trajectories as a function of the propagation distance for three different wavelengths; symbols: experimental data, solid curves: results from Eq. (15).

Fig. 7.
Fig. 7.

Illustration of the titled geometry for the uniaxial crystal in the (x, z) plane.

Fig. 8.
Fig. 8.

Experimental (top row) and numerical (bottom row) results for generation of single-charge vortex beam, λ=632.8 nm and w=11 µm. Output intensity patterns I for (a) three propagation lengths z for an internal angle α=0.81°, and (b) different internal angles α with crystal length z=6 mm. The experimental angles in (b, top) also take into account the refraction at the input interface of the crystal. Numerical results in panels (a,b) are false colored.

Fig. 9.
Fig. 9.

Output intensity profiles I (upper line) and I (bottom line) for (a) experimental data and (b) patterns calculated using Eq. (1). Note the relative shift of spectral components visible in the contour plots in panels (b), which explains the rainbow coloring of intensities in panels (a). Experimental spectrally resolved power measurements in (c) and (d), where solid (dashed) curves refer to P (P ), are compared with numerical results in (e) and (f) for an input Gaussian beam with w=6µm. In panels (c,d,f) the color labeling is red, green and blue for λ=630, 550 and 440 nm, respectively.

Equations (17)

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

(2+2iknoz)E=γ (·E) ,
Ex=E02{G(e)+G(o)+cos2φ[G(e)G(o)+ir2(ZeβeG(e)ZoβoG(o))]},
Ey=E02sin2φ[G(e)G(o)+ir2(ZeβeG(e)ZoβoG(o))],
(ExEy) E0 G M̂ (cosφ0sinφ0),
M̂=(cosδisinδcos2φisinδsin2φisinδsin2φcosδ+isinδcos2φ),
(EE) E0 G (cosδisinδcos2(φφ0)isinδsin2(φφ0)),
P(ζ)=P0P(ζ),P(ζ)=P04ε2ζ2(1+ζ2)2ε2ζ6+(1+ζ2)2,
P(ζ1) P04 ε2 ζ2 , P (ζ1) P04 ε2ζ21+ε2ζ2 ,
cos A (coshB+cos2(φφ0)sinhB)=0,
sin A (sinhB+cos2(φφ0)coshB)=0,
ρ0 = π2ε ζ2+1ζζ21 ,
ψ0 = sign (m) {(1m)π+12arccos[tanh(πζξ21)]},
ρ0 π2εζ for ζ 1 and ρ0 πζ2ε for ζ 1 ,
ψ0 {3π4,π4,π43π4} for ζ 1 or ζ 1 .
α0 λ2z(none).
z = z cos α + r cos (φψ)sinα,
r2 = r2 [1cos2(φψ)sin2α]rzsin2α+z2sin2α.

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