Abstract

We derive the analytical expression of the Stokes parameters corresponding to a Gaussian beam propagating along the optical axis of a uniaxial crystal, pointing the simultaneous effects of anisotropy and diffraction out. The theoretical results are compared with experimental measurements at the output of a calcite crystal, showing a good agreement.

© 2002 Optical Society of America

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References

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  1. R. M. A. Azzam, B. E. Merrill, and N. M. Bashara, “Trajectories describing the evolution of polarized light in homogeneous anisotropic media and liquid crystals,” Appl. Opt. 12, 4, 764–771 (1973).
    [Crossref] [PubMed]
  2. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 23, 1948–1950 (1996).
    [Crossref] [PubMed]
  3. J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, “A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,” IEEE J. Lightwave Tech. 16, 2, 200–206 (1998).
    [Crossref]
  4. W. M. Shute, C. S. Brown, and J. Jarzynski, “Polarization model for a helically wound optical fiber,” J. Opt. Soc. Am. A 14, 12, 3251–3261 (1997).
    [Crossref]
  5. Z. K. Ioannidis, R. Kadiwar, and I. Giles, “Anisotropic polarization maintaining optical fiber ring resonators,” IEEE J. Lightwave Tech. 14, 3, 377–384 (1996).
    [Crossref]
  6. M. J. Bloemer and J. W. Haus, “Broadband waveguide polarizers based on the anisotropic optical constants of nanocomposite films,” IEEE J. Lightwave Tech. 14, 6, 1534–1540 (1996).
    [Crossref]
  7. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  8. W. A. Shurcliff, Polarized light (Harvard Univ. Press, Cambridge, MA, 1962).
  9. M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford1993).
  10. E. Collett, Polarized light (Marcel Dekker, New York, 1992).
  11. R. M. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, Amsterdam, 1977).
  12. R. C. Jones, “New calculus for the treatment of optical systems” J. Opt. Soc. Am. A 31, 488–450 (1941).
    [Crossref]
  13. C. Brosseau, Fundamentals of polarized light (Wiley, New York, 1998).
  14. L. Dettwiller, “General expression of light intensity emerging from a linear anisotropic device using Stokes parameters,” J. Mod. Opt. 42, 4, 841–848 (1995).
    [Crossref]
  15. C. Brosseau, “Evolution of the Stokes parameters in optically anisotropic media” Opt. Lett. 20, 11, 1221–1223 (1995).
    [Crossref] [PubMed]
  16. J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
    [Crossref]
  17. J.J. Stamnes and G.C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. A 66, 780–788 (1976).
    [Crossref]
  18. J.A. Fleck and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. A 73, 920–926 (1983).
    [Crossref]
  19. J. M. Liu and L. Gomelsky, “Vectorial beam propagation method,” J. Opt. Soc. Am. A 9, 9, 1574–1585 (1992).
    [Crossref]
  20. S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
    [Crossref]
  21. R. Martínez-Herrero, J. M. Movilla, and P. M. Mejías, “Radiation of electromagnetic fields in uniaxially anisotropic media”, J. Opt. Soc. Am. A 18, 8, 2009–2014 (2001).
    [Crossref]
  22. G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001).
    [Crossref]
  23. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A ,  19, 792–796 (2002).
    [Crossref]

2002 (1)

2001 (2)

R. Martínez-Herrero, J. M. Movilla, and P. M. Mejías, “Radiation of electromagnetic fields in uniaxially anisotropic media”, J. Opt. Soc. Am. A 18, 8, 2009–2014 (2001).
[Crossref]

G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001).
[Crossref]

2000 (2)

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[Crossref]

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
[Crossref]

1998 (1)

J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, “A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,” IEEE J. Lightwave Tech. 16, 2, 200–206 (1998).
[Crossref]

1997 (1)

1996 (3)

Z. K. Ioannidis, R. Kadiwar, and I. Giles, “Anisotropic polarization maintaining optical fiber ring resonators,” IEEE J. Lightwave Tech. 14, 3, 377–384 (1996).
[Crossref]

M. J. Bloemer and J. W. Haus, “Broadband waveguide polarizers based on the anisotropic optical constants of nanocomposite films,” IEEE J. Lightwave Tech. 14, 6, 1534–1540 (1996).
[Crossref]

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 23, 1948–1950 (1996).
[Crossref] [PubMed]

1995 (2)

L. Dettwiller, “General expression of light intensity emerging from a linear anisotropic device using Stokes parameters,” J. Mod. Opt. 42, 4, 841–848 (1995).
[Crossref]

C. Brosseau, “Evolution of the Stokes parameters in optically anisotropic media” Opt. Lett. 20, 11, 1221–1223 (1995).
[Crossref] [PubMed]

1992 (1)

1983 (1)

J.A. Fleck and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. A 73, 920–926 (1983).
[Crossref]

1976 (1)

J.J. Stamnes and G.C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. A 66, 780–788 (1976).
[Crossref]

1973 (1)

1941 (1)

R. C. Jones, “New calculus for the treatment of optical systems” J. Opt. Soc. Am. A 31, 488–450 (1941).
[Crossref]

Azzam, R. M.

R. M. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, Amsterdam, 1977).

Azzam, R. M. A.

Barbosa-García, O.

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
[Crossref]

Bashara, N. M.

Bloemer, M. J.

M. J. Bloemer and J. W. Haus, “Broadband waveguide polarizers based on the anisotropic optical constants of nanocomposite films,” IEEE J. Lightwave Tech. 14, 6, 1534–1540 (1996).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford1993).

Brosseau, C.

Brown, C. S.

Ciattoni, A.

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A ,  19, 792–796 (2002).
[Crossref]

G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001).
[Crossref]

Cincotti, G.

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A ,  19, 792–796 (2002).
[Crossref]

G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001).
[Crossref]

Collett, E.

E. Collett, Polarized light (Marcel Dekker, New York, 1992).

Dettwiller, L.

L. Dettwiller, “General expression of light intensity emerging from a linear anisotropic device using Stokes parameters,” J. Mod. Opt. 42, 4, 841–848 (1995).
[Crossref]

Díaz-Torres, L. A.

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
[Crossref]

Digonnet, J. J. F.

J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, “A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,” IEEE J. Lightwave Tech. 16, 2, 200–206 (1998).
[Crossref]

Falquier, D. G.

J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, “A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,” IEEE J. Lightwave Tech. 16, 2, 200–206 (1998).
[Crossref]

Feit, M. D.

J.A. Fleck and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. A 73, 920–926 (1983).
[Crossref]

Fleck, J.A.

J.A. Fleck and M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. A 73, 920–926 (1983).
[Crossref]

Giles, I.

Z. K. Ioannidis, R. Kadiwar, and I. Giles, “Anisotropic polarization maintaining optical fiber ring resonators,” IEEE J. Lightwave Tech. 14, 3, 377–384 (1996).
[Crossref]

Gomelsky, L.

Haus, J. W.

M. J. Bloemer and J. W. Haus, “Broadband waveguide polarizers based on the anisotropic optical constants of nanocomposite films,” IEEE J. Lightwave Tech. 14, 6, 1534–1540 (1996).
[Crossref]

Ioannidis, Z. K.

Z. K. Ioannidis, R. Kadiwar, and I. Giles, “Anisotropic polarization maintaining optical fiber ring resonators,” IEEE J. Lightwave Tech. 14, 3, 377–384 (1996).
[Crossref]

Jarzynski, J.

Jones, R. C.

R. C. Jones, “New calculus for the treatment of optical systems” J. Opt. Soc. Am. A 31, 488–450 (1941).
[Crossref]

Kadiwar, R.

Z. K. Ioannidis, R. Kadiwar, and I. Giles, “Anisotropic polarization maintaining optical fiber ring resonators,” IEEE J. Lightwave Tech. 14, 3, 377–384 (1996).
[Crossref]

Liu, J. M.

Martínez-Herrero, R.

Mejías, P. M.

Meneses-Nava, M. A.

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
[Crossref]

Merrill, B. E.

Mosiño, J. F.

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
[Crossref]

Movilla, J. M.

Palma, C.

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A ,  19, 792–796 (2002).
[Crossref]

G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001).
[Crossref]

Schadt, M.

Selleri, S.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[Crossref]

Shaw, H. J.

J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, “A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,” IEEE J. Lightwave Tech. 16, 2, 200–206 (1998).
[Crossref]

Sherman, G.C.

J.J. Stamnes and G.C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. A 66, 780–788 (1976).
[Crossref]

Shurcliff, W. A.

W. A. Shurcliff, Polarized light (Harvard Univ. Press, Cambridge, MA, 1962).

Shute, W. M.

Stalder, M.

Stamnes, J.J.

J.J. Stamnes and G.C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. A 66, 780–788 (1976).
[Crossref]

Starodumov, A.

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
[Crossref]

Vega-Durán, J. T.

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
[Crossref]

Vincetti, L.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[Crossref]

Wagener, J. L.

J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, “A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,” IEEE J. Lightwave Tech. 16, 2, 200–206 (1998).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford1993).

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Zoboli, M.

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[Crossref]

Appl. Opt. (1)

IEEE J. Lightwave Tech. (3)

Z. K. Ioannidis, R. Kadiwar, and I. Giles, “Anisotropic polarization maintaining optical fiber ring resonators,” IEEE J. Lightwave Tech. 14, 3, 377–384 (1996).
[Crossref]

M. J. Bloemer and J. W. Haus, “Broadband waveguide polarizers based on the anisotropic optical constants of nanocomposite films,” IEEE J. Lightwave Tech. 14, 6, 1534–1540 (1996).
[Crossref]

J. L. Wagener, D. G. Falquier, J. J. F. Digonnet, and H. J. Shaw, “A Mueller matrix formalism for modelling polarization effects in erbium-doped fiber,” IEEE J. Lightwave Tech. 16, 2, 200–206 (1998).
[Crossref]

IEEE J. Quantum Electron. (2)

S. Selleri, L. Vincetti, and M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[Crossref]

G. Cincotti, A. Ciattoni, and C. Palma, “Hermite-Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 12, 1517–1524 (2001).
[Crossref]

J. Mod. Opt. (1)

L. Dettwiller, “General expression of light intensity emerging from a linear anisotropic device using Stokes parameters,” J. Mod. Opt. 42, 4, 841–848 (1995).
[Crossref]

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

Opt. Commun. (1)

J. F. Mosiño, O. Barbosa-García, A. Starodumov, L. A. Díaz-Torres, M. A. Meneses-Nava, and J. T. Vega-Durán, “Evolution of partially polarized light through non-depolarizing anisotropic media the Stokes parameters in optically anisotropic media” Opt. Commun. 173, 57–71 (2000).
[Crossref]

Opt. Lett. (2)

Other (6)

C. Brosseau, Fundamentals of polarized light (Wiley, New York, 1998).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

W. A. Shurcliff, Polarized light (Harvard Univ. Press, Cambridge, MA, 1962).

M. Born and E. Wolf, Principles of optics (Pergamon Press, Oxford1993).

E. Collett, Polarized light (Marcel Dekker, New York, 1992).

R. M. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland, Amsterdam, 1977).

Supplementary Material (4)

» Media 1: MPG (321 KB)     
» Media 2: MPG (319 KB)     
» Media 3: MPG (307 KB)     
» Media 4: MPG (308 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Animations of the evolution of the Stokes parameters associated with a Gaussian beam propagating along the optical axis of a calcite crystal, from the z = 0 to the z = 4zRo planes; the transverse coordinates are normalized with respect to the input spot size w 0. [Media 1] [Media 2] [Media 3] [Media 4]

Fig. 2.
Fig. 2.

Laboratory setup.

Fig. 3.
Fig. 3.

Numerical (left hand) and experimental (right hand) results for Stokes parameters of a Gaussian beam with input spot size w 0 = 10μm propagated for a distance z = 20zRo along the optical axis of a calcite crystal.

Fig. 4.
Fig. 4.

The same as Fig. 3 for w 0 = 15μm and z = 8.5zR0 .

Fig. 5.
Fig. 5.

Comparison between numerical and experimental results for the Stokes parameters of Fig. 3 evaluated for x = y or x = 6w 0. Solid line refers to experimental values and dashed line to numerical values.

Fig. 6.
Fig. 6.

Comparison between numerical and experimental results for the Stokes parameters of Fig. 4 evaluated for x = y or x = 6w 0. Solid line refers to experimental values and dashed line to numerical values.

Equations (34)

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s 0 ( x , y , z ) = E x ( x , y , z ) 2 + E y ( x , y , z ) 2
s 1 ( x , y , z ) = E x ( x , y , z ) 2 E y ( x , y , z ) 2
s 2 ( x , y , z ) = E x ( x , y , z ) E y * ( x , y , z ) + E y ( x , y , z ) E x * ( x , y , z )
s 3 ( x , y , z ) = i [ E y ( x , y , z ) E x * ( x , y , z ) E x ( x , y , z ) E y * ( x , y , z ) ] ,
E x ( x , y , z ) = E ̅ w 0 2 x 2 + y 2 exp ( i k 0 n o z ) [ ( y 2 Q o ( z ) + y 2 x 2 2 ( x 2 + y 2 ) ) exp ( x 2 + y 2 Q o ( z ) )
+ ( x 2 Q e ( z ) + x 2 y 2 2 ( x 2 + y 2 ) ) exp ( x 2 + y 2 Q e ( z ) ) ]
E y ( x , y , z ) = E ̅ w 0 2 x 2 + y 2 e i k 0 n o z [ ( x y Q o ( z ) + x y x 2 + y 2 ) exp ( x 2 + y 2 Q o ( z ) )
+ ( x y Q e ( z ) + x y x 2 + y 2 ) exp ( x 2 + y 2 Q e ( z ) ) ] ,
s n ( x , y , z ) = E ̅ 2 w 0 4 ( x 2 + y 2 ) 2 { f n ( o ) ( x , y , z ) exp ( 2 w 0 2 ( x 2 + y 2 ) Q o ( z ) 2 )
+ f n ( e ) ( x , y , z ) exp ( 2 w 0 2 ( x 2 + y 2 ) Q e ( z ) 2 )
+ f n ( o e ) ( x , y , z ) exp [ w 0 2 ( x 2 + y 2 ) ( 1 Q o ( z ) 2 + 1 Q e ( z ) 2 ) ] }
( n = 0,1,2,3 ) ,
f n ( o e ) ( x , y , z ) = f n ( s ) ( x , y , z ) sin [ 2 z k 0 n o ( 1 Q o ( z ) 2 n o 2 n e 2 1 Q e ( z ) 2 ) ( x 2 + y 2 ) ]
f n ( c ) ( x , y , z ) cos [ 2 z k 0 n o ( 1 Q o ( z ) 2 n o 2 n e 2 1 Q e ( z ) 2 ) ( x 2 + y 2 ) ] ,
f 0 ( o ) ( x , y , z ) = 1 4 + ( w 0 2 + x 2 + y 2 ) y 2 Q o ( z ) 2
f 0 ( e ) ( x , y , z ) = 1 4 + ( w 0 2 + x 2 + y 2 ) x 2 Q e ( z ) 2
f 0 ( s ) ( x , y , z ) = 2 z k 0 n o ( y 2 Q o ( z ) 2 n o 2 n e 2 x 2 Q e ( z ) 2 )
f 0 ( c ) ( x , y , z ) = ( 1 2 + x 2 w 0 2 Q e ( z ) 2 + y 2 w 0 2 Q o ( z ) 2 )
f 1 ( o ) ( x , y , z ) = x 4 + y 4 6 x 2 y 2 4 ( x 2 + y 2 ) 2 + y 2 ( y 4 x 4 + ( y 2 3 x 2 ) w 0 2 ) ( x 2 + y 2 ) Q o ( z ) 2
f 1 ( e ) ( x , y , z ) = x 4 + y 4 6 x 2 y 2 4 ( x 2 + y 2 ) 2 + x 2 ( x 4 y 4 + ( x 2 3 y 2 ) w 0 2 ) ( x 2 + y 2 ) Q e ( z ) 2
f 1 ( s ) ( x , y , z ) = 2 z k 0 n o ( x 2 ( x 2 3 y 2 ) x 2 + y 2 n o 2 n e 2 1 Q e ( z ) 2 y 2 ( y 2 3 x 2 ) x 2 + y 2 1 Q o ( z ) 2
+ 4 x 2 y 2 n e 2 n o 2 n e 2 w 0 2 Q o ( z ) 2 Q e ( z ) 2 )
f 1 ( c ) ( x , y , z ) = 1 2 + 4 x 2 y 2 ( 1 ( x 2 + y 2 ) 2 + 4 z 2 k 0 2 n e 2 + w 0 4 Q o ( z ) 2 Q e ( z ) 2 ) + y 2 ( y 2 3 x 2 ) x 2 + y 2 w 0 2 Q o ( z ) 2
+ x 2 ( x 2 3 y 2 ) x 2 + y 2 w 0 2 Q e ( z ) 2
f 2 ( o ) ( x , y , z ) = xy ( x 2 y 2 ( x 2 + y 2 ) 2 2 y 2 Q o ( z ) 2 3 y 2 x 2 x 2 + y 2 w 0 2 Q o ( z ) 2 )
f 2 ( e ) ( x , y , z ) = xy ( x 2 y 2 ( x 2 + y 2 ) 2 + 2 x 2 Q e ( z ) 2 + 3 x 2 y 2 x 2 + y 2 w 0 2 Q e ( z ) 2 )
f 2 ( s ) ( x , y , z ) = 2 zxy k 0 n 0 ( 3 y 2 x 2 x 2 + y 2 1 Q o ( z ) 2 + 3 x 2 y 2 x 2 + y 2 n o 2 n e 2 1 Q e ( z ) 2
+ 2 ( x 2 y 2 ) n o 2 n e 2 n e 2 w 0 2 Q o ( z ) 2 Q e ( z ) 2 )
f 2 ( c ) ( x , y , z ) = xy [ 3 y 2 x 2 x 2 + y 2 w 0 2 Q o ( z ) 2 3 x 2 y 2 x 2 + y 2 w 0 2 Q e ( z ) 2
+ ( y 2 x 2 ) ( 1 ( x 2 + y 2 ) 2 + w 0 4 + 4 z 2 k 0 2 n e 2 Q o ( z ) 2 Q e ( z ) 2 ) ]
f 3 ( o ) ( x , y , z ) = xy 2 z k 0 n 0 1 Q o ( z ) 2
f 3 ( e ) ( x , y , z ) = xy 2 z k 0 n 0 n o 2 n e 2 1 Q e ( z ) 2
f 3 ( s ) ( x , y , z ) = xy ( w 0 2 Q o ( z ) 2 + w 0 2 Q e ( z ) 2 + 2 ( x 2 + y 2 ) w 0 4 4 z 2 k 0 2 n e 2 Q o ( z ) 2 Q e ( z ) 2 )
f 3 ( c ) ( x , y , z ) = xy 2 z k 0 n o ( n o 2 n e 2 1 Q e ( z ) 2 1 Q o ( z ) 2 + 2 ( x 2 + y 2 ) w 0 2 Q o ( z ) 2 Q e ( z ) 2 )

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