Abstract

The splitting of a single optical vortex into four separate ones in a singular beam is theoretically and experimentally described for the propagation of obliquely incident light in a uniaxial crystal. We also find the condition under which the generated vortices in each of the four individual beams propagate independently without changing their structure and have different locations in all beams for any crystal lengths.

© 2008 Optical Society of America

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References

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  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, E.Wolf, ed. (North-Holland, 2001), Vol. 42, pp. 219-276.
    [CrossRef]
  2. 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]
  3. A. Ciattoni, G. Cancotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20, 163-171 (2003).
    [CrossRef]
  4. A. Ciattoni, G. Cancotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in uniaxial crystal,” Phys. Rev. E 67, 036618 (2003).
    [CrossRef]
  5. N. S. Kasak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020-1024 (1999).
    [CrossRef]
  6. M. Berry and M. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
    [CrossRef]
  7. M. Berry, M. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7, 685-690 (2005).
    [CrossRef]
  8. M. Berry, “The optical singularities of bianisotropic crystals,” Proc. R. Soc. London, Ser. A 461, 2071-2098 (2005).
    [CrossRef]
  9. A. Volyar and T. Fadeyeva, “Generation of singular beams in uniaxial crystals,” Opt. Spectrosc. 94, 264-274 (2003).
    [CrossRef]
  10. Yu. Egorov, T. Fadeyeva, and A. Volyar, “The fine structure of singular beams in crystals: Colours and polarization,” J. Opt. A, Pure Appl. Opt. 6, S217-S228 (2004).
    [CrossRef]
  11. A. Volyar and T. Fadeyeva, “Laguerre-Gaussian beams with complex and real arguments in uniaxial crystals,” Opt. Spectrosc. 101, 297-304 (2006).
    [CrossRef]
  12. A. Volyar, V. Shvedov, T. Fadeyeva, A. Desyatnikov, D. Neshev, W. Krollikowski, and Yu. Kivshar, “Generation of single-charged optical vortices with a uniaxial crystal,” Opt. Express 14, 3724-3729 (2006).
    [CrossRef]
  13. F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402-11411 (2006).
    [CrossRef]
  14. F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
    [CrossRef]
  15. S. R. Seshadri, “Basic elliptical Gaussian wave and beam in a uniaxial crystal,” J. Opt. Soc. Am. A 20, 1818-1826 (2003).
    [CrossRef]
  16. S. Y. Chin, and L. B. Felson, “Gaussian beam in anisotropic media,” Appl. Phys. 5, 239-225 (1974).
    [CrossRef]
  17. N. N. Rozanov, “Propagation of laser radiation in anisotropic media,” Opt. Spectrosc. 93, 808-813 (2002).
    [CrossRef]
  18. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics, 1999).
  19. I. D. Maleev and G. A. Swartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169-1176 (2003).
    [CrossRef]
  20. 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, 3116-3118 (2007).
    [CrossRef] [PubMed]
  21. T. A. Fadeyeva and A. V. Volyar, “Vector singularities analysis by the computer differential polarimeter,” Proc. SPIE 5257, 231-235 (2003).
    [CrossRef]

2007

2006

2005

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

M. Berry, M. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7, 685-690 (2005).
[CrossRef]

M. Berry, “The optical singularities of bianisotropic crystals,” Proc. R. Soc. London, Ser. A 461, 2071-2098 (2005).
[CrossRef]

2004

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]

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

2003

I. D. Maleev and G. A. Swartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169-1176 (2003).
[CrossRef]

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

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

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

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

M. Berry and M. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

T. A. Fadeyeva and A. V. Volyar, “Vector singularities analysis by the computer differential polarimeter,” Proc. SPIE 5257, 231-235 (2003).
[CrossRef]

2002

N. N. Rozanov, “Propagation of laser radiation in anisotropic media,” Opt. Spectrosc. 93, 808-813 (2002).
[CrossRef]

2001

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

1999

N. S. Kasak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020-1024 (1999).
[CrossRef]

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

1974

S. Y. Chin, and L. B. Felson, “Gaussian beam in anisotropic media,” Appl. Phys. 5, 239-225 (1974).
[CrossRef]

Berry, M.

M. Berry, M. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7, 685-690 (2005).
[CrossRef]

M. Berry, “The optical singularities of bianisotropic crystals,” Proc. R. Soc. London, Ser. A 461, 2071-2098 (2005).
[CrossRef]

M. Berry and M. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

Berry, M. V.

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]

Cancotti, G.

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

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

Chin, S. Y.

S. Y. Chin, and L. B. Felson, “Gaussian beam in anisotropic media,” Appl. Phys. 5, 239-225 (1974).
[CrossRef]

Ciattoni, A.

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

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

Dennis, M.

M. Berry and M. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

Dennis, M. R.

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

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Desyatnikov, A.

Egorov, Yu.

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, 3116-3118 (2007).
[CrossRef] [PubMed]

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

Fadeyeva, T.

T. Fadeyeva, Yu. Egorov, A. Rubass, G. A. Swartzlander, Jr., and A. Volyar, “Indistinguishability limit for off-axis vortex beams in uniaxial crystals,” Opt. Lett. 32, 3116-3118 (2007).
[CrossRef] [PubMed]

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

A. Volyar, V. Shvedov, T. Fadeyeva, A. Desyatnikov, D. Neshev, W. Krollikowski, and Yu. Kivshar, “Generation of single-charged optical vortices with a uniaxial crystal,” Opt. Express 14, 3724-3729 (2006).
[CrossRef]

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

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

Fadeyeva, T. A.

T. A. Fadeyeva and A. V. Volyar, “Vector singularities analysis by the computer differential polarimeter,” Proc. SPIE 5257, 231-235 (2003).
[CrossRef]

Felson, L. B.

S. Y. Chin, and L. B. Felson, “Gaussian beam in anisotropic media,” Appl. Phys. 5, 239-225 (1974).
[CrossRef]

Flossman, F.

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

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Jeffrey, M.

M. Berry, M. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7, 685-690 (2005).
[CrossRef]

Kasak, N. S.

N. S. Kasak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020-1024 (1999).
[CrossRef]

Khio, N. A.

N. S. Kasak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020-1024 (1999).
[CrossRef]

Kivshar, Yu.

Krollikowski, W.

Maier, M.

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

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Maleev, I. D.

Mansuripur, M.

M. Berry, M. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7, 685-690 (2005).
[CrossRef]

Neshev, D.

Nye, J. F.

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

Palma, C.

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

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

Rozanov, N. N.

N. N. Rozanov, “Propagation of laser radiation in anisotropic media,” Opt. Spectrosc. 93, 808-813 (2002).
[CrossRef]

Rubass, A.

Ryzhevich, A. A.

N. S. Kasak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020-1024 (1999).
[CrossRef]

Schwarz, U. T.

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

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Seshadri, S. R.

Shvedov, V.

Soskin, M. S.

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

Swartzlander, G. A.

Vasnetsov, M. V.

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

Volyar, A.

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, 3116-3118 (2007).
[CrossRef] [PubMed]

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

A. Volyar, V. Shvedov, T. Fadeyeva, A. Desyatnikov, D. Neshev, W. Krollikowski, and Yu. Kivshar, “Generation of single-charged optical vortices with a uniaxial crystal,” Opt. Express 14, 3724-3729 (2006).
[CrossRef]

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

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

Volyar, A. V.

T. A. Fadeyeva and A. V. Volyar, “Vector singularities analysis by the computer differential polarimeter,” Proc. SPIE 5257, 231-235 (2003).
[CrossRef]

Appl. Phys.

S. Y. Chin, and L. B. Felson, “Gaussian beam in anisotropic media,” Appl. Phys. 5, 239-225 (1974).
[CrossRef]

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-S228 (2004).
[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]

M. Berry, M. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7, 685-690 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Opt. Spectrosc.

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

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

N. N. Rozanov, “Propagation of laser radiation in anisotropic media,” Opt. Spectrosc. 93, 808-813 (2002).
[CrossRef]

Phys. Rev. E

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

Phys. Rev. Lett.

F. Flossman, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901 (2005).
[CrossRef]

Proc. R. Soc. London, Ser. A

M. Berry, “The optical singularities of bianisotropic crystals,” Proc. R. Soc. London, Ser. A 461, 2071-2098 (2005).
[CrossRef]

M. Berry and M. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. R. Soc. London, Ser. A 459, 1261-1292 (2003).
[CrossRef]

Proc. SPIE

T. A. Fadeyeva and A. V. Volyar, “Vector singularities analysis by the computer differential polarimeter,” Proc. SPIE 5257, 231-235 (2003).
[CrossRef]

Quantum Electron.

N. S. Kasak, N. A. Khio, and A. A. Ryzhevich, “Generation of Bessel light beams under the condition of internal conical refraction,” Quantum Electron. 29, 1020-1024 (1999).
[CrossRef]

Other

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

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

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Figures (8)

Fig. 1
Fig. 1

Sketched representation of the off-axis individual paraxial beams in the uniaxial anisotropic medium.

Fig. 2
Fig. 2

Variations of the total beam intensities I + and I for E + and E components of a singular beam in a LiNbO 3 crystal with n o = ε = 2.3 , and n o = ε 3 = 2.2 .

Fig. 3
Fig. 3

Sketch of the experimental setup: P 1 , P 2 , polarizers; λ 4 , quarter-wave plate; L 1 L 5 , lenses; W, optical wedge; D 1 D 3 , diaphragms; SM, semitransparent mirrors; M, mirrors; CCD, CCD camera; (a) beam axis trajectories in LiNbO 3 crystal; c ̂ , a unit vector of the crystal optical axis.

Fig. 4
Fig. 4

Theoretically predicted and experimentally measured intensity distributions and interferential patterns of the E component of the off-axis beam via different incident angles α i n .

Fig. 5
Fig. 5

Trajectory of optical vortices imprinted in the E component: (a) Region of the vortex trajectory in the X Y coordinates for the LiNbO 3 crystal inside the range of angles Δ α o ( 0 , 1 ° ) and (b) vortex trajectory in the X Y α cordinates, with the crystal length z = 2 cm .

Fig. 6
Fig. 6

Curves α o = f ( ρ , z ) outlining the indistinguishability range for two singular beams.

Fig. 7
Fig. 7

C-lines for the singular beam with w 0 = 50 μ m and z = 2 cm .

Fig. 8
Fig. 8

Maps of polarization states and integral curves for directions of the azimuthal angle of the polarization ellipses for the angles α o = 5.3 ° ( w 0 = 50 μ m , z = 2 cm ) positioned against a background of the intensity distributions: (a), (b), theory; (c), (d), experiment.

Equations (23)

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

E ( x , y , z = 0 , α o ) = A ( α o ) [ ( x i y ) w 0 ] exp ( r 2 w 0 2 )
ε ̂ = ( ε 1 0 0 0 ε 2 0 0 0 ε 3 ) ,
( 2 + k 2 ε ̂ ) E = ( E ) ,
E ( x , y , z , α ) = E ̃ ( x , y , z , α ) exp ( i k o z ) ,
( 2 + 2 i k o z ) E ̃ = Δ ε ε 3 ( E ̃ ) ,
E ̃ ( o ) = ( x ̂ y y ̂ x ) Ψ o
E ̃ ( e ) = Ψ e ,
( 2 + 2 i k o z ) Ψ o = 0 ,
( 2 + 2 i k e z ) Ψ e = 0 ,
Ψ ̃ o = σ o 1 exp { ( x 2 + y o 2 ) w 0 2 σ o } exp ( α o 2 k o z o 2 ) ,
Ψ ̃ e = σ e 1 exp { ( x 2 + y e 2 ) w 0 2 σ e } exp ( α e 2 k e z e 2 ) ,
E ( o ) = E ̃ ( o ) exp ( i k o z ) = { e ̂ + x i y o σ o ( z , k o ) e ̂ x + i y o σ o ( z , k o ) } Ψ ̃ o w 0 exp ( i k o z ) ,
E ( e ) = E ̃ ( e ) exp ( i k o z ) = { e ̂ + x i y o σ e ( z , k e ) + e ̂ x + i y o σ e ( z , k e ) } Ψ ̃ e w 0 exp ( i k o z ) .
E 1 = E ̃ ( o ) + E ̃ ( e ) = e ̂ + x i y o w 0 ( Ψ ̃ o σ o + Ψ ̃ e σ e ) e ̂ x + i y o σ o ( Ψ ̃ o σ o Ψ ̃ e σ e ) .
G ̃ = e ̂ + ( Ψ ̃ o + Ψ ̃ e ) e ̂ ( x + i y o r o ) 2 [ w 0 2 r o 2 ( σ o Ψ ̃ o σ e Ψ ̃ e ) + ( Ψ ̃ o Ψ ̃ e ) ] ,
E ̃ + = { x i ( y α o z ) w 0 σ o Ψ ̃ o + x i ( y α e z ) w 0 σ e Ψ ̃ e } ,
E ̃ = x + i y o w 0 [ Ψ ̃ o σ o Ψ ̃ e σ e ] + α ¯ ( x + i y o r o ) 2 [ w 0 2 r o 2 ( σ o Ψ ̃ o σ e Ψ ̃ e ) + ( Ψ ̃ o Ψ ̃ e ) ] .
P z ( E + 2 + E 2 ) d x d y = I + + I .
Re [ E ± ( x , y , z , α o ) ] = 0 , Im [ E ± ( x , y , z , α o ) ] = 0 .
z 2 ( α o 2 w 0 2 Δ ε 2 2 ε 3 2 1 z o 2 ) 1 .
E ̃ + X i Y o Z o Ψ ̃ o + X i Y e Z e Ψ ̃ e ,
E ̃ ( X + 1 α ¯ ) i Y o Z o Ψ ̃ o ( X + 1 α ¯ ) i Y e Z e Ψ ̃ e ,
Δ x = w 0 α ¯ = λ π n o α o = λ π n e α e ,

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