Abstract

The fringe pattern obtained when a divergent (or convergent) beam goes through a sample of birefringent crystal between two crossed polarizers contains information that is inherent to the crystalline sample under study. The formation of fringe patterns is analyzed from distinct approaches and with different degrees of approximation considering cones of light of large numerical aperture. We obtain analytic explicit formulas of the phase shift on the screen and compare them with the exact numerical solution. The results obtained are valid for arbitrary orientation of the optical axis and are not restricted either to low birefringence or to small angles of incidence. Moreover, they enable the extraction of the main features related to the characterization of uniaxial crystal slabs, such as the optical axis tilt angle and the principal refractive indices.

© 2012 Optical Society of America

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References

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  1. B. Van Horn and H. Winter, “Analysis of the conoscopic measurement for uniaxial liquid-crystal tilt angles,” Appl. Opt. 40, 2089–2094 (2001).
    [CrossRef]
  2. D. Su and C. Hsu, “Method for determining the optical axis and (ne, no) of a birefringent crystal,” Appl. Opt. 41, 3936–3940 (2002).
    [CrossRef]
  3. P. Lee, J. Pors, M. van Exter, and J. Woerdman, “Simple method for accurate characterization of birefringent crystals,” Appl. Opt. 44, 866–870 (2005).
    [CrossRef]
  4. F. E. Veiras, G. Pérez, M. T. Garea, and L. I. Perez, “Characterization of uniaxial crystals through the study of fringe patterns,” J. Phys.: Conf. Ser. 274, 012030 (2011).
    [CrossRef]
  5. F. E. Veiras, L. I. Perez, and M. T. Garea, “Phase shift formulas in uniaxial media: an application to waveplates,” Appl. Opt. 49, 2769–2777 (2010).
    [CrossRef]
  6. H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
    [CrossRef]
  7. D.-K. Yang and S.-T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).
  8. M. Françon, Encyclopedia of Physics. Fundamental of Optics, S. Flugge, ed.(Springer, 1956), vol. XXIV.
  9. M. C. Simon and M. T. Garea, “Plane parallel birefringent plates as polarization interferometers,” Optik 87, 95–102 (1991).
  10. F. E. Veiras, M. T. Garea, and L. I. Perez, “Fringe pattern analysis by means of wide angle conoscopic illumination of uniaxial crystals,” in Conf. Proc. VII International Conference of young scientists and specialists “Optics–2011” (St Petersburg, Russia, 17–21Oct.2011), pp. 55–57.
  11. J. W. GoodmanSpeckle Phenomena in Optics: Theory and Applications (Ben Roberts, 2007).
  12. J. W. Goodman, J W Statistical Optics (Wiley Classics Library, 2000).
  13. A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
    [CrossRef]
  14. D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
    [CrossRef]
  15. M. A. Alonso, “Exact description of free electromagnetic wave fields in terms of rays,” Opt. Express 11, 3128–3135 (2003).
    [CrossRef]
  16. M. Anwar and R. Small, “Geometrical-optics solution for self-focusing in nonlinear optics,” J. Opt. Soc. Am. 71, 124–126 (1981).
    [CrossRef]
  17. P. Berczynski, “Complex geometrical optics of nonlinear inhomogeneous fibres,” J. Opt. 13, 035707 (2011).
    [CrossRef]
  18. M. Sluijter, M. Xu, H. P. Urbach, and D. K. G. de Boer, “Applicability of geometrical optics to in-plane liquid-crystal configurations,” Opt. Lett. 35, 487–489 (2010).
    [CrossRef]
  19. L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).
  20. L. I. Perez, “Nonspecular transverse effects of polarized and unpolarized symmetric beams in isotropic-uniaxial interfaces,” J. Opt. Soc. Am. 20, 741–752 (2003).
    [CrossRef]
  21. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
    [CrossRef]
  22. M. C. Simon and R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
    [CrossRef]
  23. M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
    [CrossRef]
  24. M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008).
    [CrossRef]
  25. M. C. Simon, “Image formation through monoaxial plane-parallel plates,” Appl. Opt. 27, 4176–4182 (1988).
    [CrossRef]
  26. Note that the sign of A is related to the arbitrary way in which we compute the phase difference between waves: ordinary minus extraordinary.

2011

F. E. Veiras, G. Pérez, M. T. Garea, and L. I. Perez, “Characterization of uniaxial crystals through the study of fringe patterns,” J. Phys.: Conf. Ser. 274, 012030 (2011).
[CrossRef]

H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
[CrossRef]

P. Berczynski, “Complex geometrical optics of nonlinear inhomogeneous fibres,” J. Opt. 13, 035707 (2011).
[CrossRef]

2010

2009

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

2008

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008).
[CrossRef]

2007

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
[CrossRef]

2005

2003

L. I. Perez, “Nonspecular transverse effects of polarized and unpolarized symmetric beams in isotropic-uniaxial interfaces,” J. Opt. Soc. Am. 20, 741–752 (2003).
[CrossRef]

M. A. Alonso, “Exact description of free electromagnetic wave fields in terms of rays,” Opt. Express 11, 3128–3135 (2003).
[CrossRef]

2002

2001

2000

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

1991

M. C. Simon and M. T. Garea, “Plane parallel birefringent plates as polarization interferometers,” Optik 87, 95–102 (1991).

1988

1986

1983

1981

Alonso, M. A.

Anwar, M.

Berczynski, P.

P. Berczynski, “Complex geometrical optics of nonlinear inhomogeneous fibres,” J. Opt. 13, 035707 (2011).
[CrossRef]

de Boer, D. K. G.

Dong, X.

H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
[CrossRef]

Echarri, R. M.

Ferri, F.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

Françon, M.

M. Françon, Encyclopedia of Physics. Fundamental of Optics, S. Flugge, ed.(Springer, 1956), vol. XXIV.

Gao, X.

H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
[CrossRef]

Garea, M. T.

F. E. Veiras, G. Pérez, M. T. Garea, and L. I. Perez, “Characterization of uniaxial crystals through the study of fringe patterns,” J. Phys.: Conf. Ser. 274, 012030 (2011).
[CrossRef]

F. E. Veiras, L. I. Perez, and M. T. Garea, “Phase shift formulas in uniaxial media: an application to waveplates,” Appl. Opt. 49, 2769–2777 (2010).
[CrossRef]

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

M. C. Simon and M. T. Garea, “Plane parallel birefringent plates as polarization interferometers,” Optik 87, 95–102 (1991).

F. E. Veiras, M. T. Garea, and L. I. Perez, “Fringe pattern analysis by means of wide angle conoscopic illumination of uniaxial crystals,” in Conf. Proc. VII International Conference of young scientists and specialists “Optics–2011” (St Petersburg, Russia, 17–21Oct.2011), pp. 55–57.

Gatti, A.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

Goodman, J. W.

J. W. GoodmanSpeckle Phenomena in Optics: Theory and Applications (Ben Roberts, 2007).

J. W. Goodman, J W Statistical Optics (Wiley Classics Library, 2000).

Gottschalk, K. V.

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
[CrossRef]

Guo, H.

H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
[CrossRef]

Hsu, C.

Lee, P.

Magatti, D.

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

Perez, L. I.

F. E. Veiras, G. Pérez, M. T. Garea, and L. I. Perez, “Characterization of uniaxial crystals through the study of fringe patterns,” J. Phys.: Conf. Ser. 274, 012030 (2011).
[CrossRef]

F. E. Veiras, L. I. Perez, and M. T. Garea, “Phase shift formulas in uniaxial media: an application to waveplates,” Appl. Opt. 49, 2769–2777 (2010).
[CrossRef]

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008).
[CrossRef]

L. I. Perez, “Nonspecular transverse effects of polarized and unpolarized symmetric beams in isotropic-uniaxial interfaces,” J. Opt. Soc. Am. 20, 741–752 (2003).
[CrossRef]

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

F. E. Veiras, M. T. Garea, and L. I. Perez, “Fringe pattern analysis by means of wide angle conoscopic illumination of uniaxial crystals,” in Conf. Proc. VII International Conference of young scientists and specialists “Optics–2011” (St Petersburg, Russia, 17–21Oct.2011), pp. 55–57.

Pérez, G.

F. E. Veiras, G. Pérez, M. T. Garea, and L. I. Perez, “Characterization of uniaxial crystals through the study of fringe patterns,” J. Phys.: Conf. Ser. 274, 012030 (2011).
[CrossRef]

Pors, J.

Simon, M. C.

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008).
[CrossRef]

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
[CrossRef]

M. C. Simon and M. T. Garea, “Plane parallel birefringent plates as polarization interferometers,” Optik 87, 95–102 (1991).

M. C. Simon, “Image formation through monoaxial plane-parallel plates,” Appl. Opt. 27, 4176–4182 (1988).
[CrossRef]

M. C. Simon and R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
[CrossRef]

M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
[CrossRef]

Sluijter, M.

Small, R.

Su, D.

Sui, G.

H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
[CrossRef]

Urbach, H. P.

van Exter, M.

Van Horn, B.

Veiras, F. E.

F. E. Veiras, G. Pérez, M. T. Garea, and L. I. Perez, “Characterization of uniaxial crystals through the study of fringe patterns,” J. Phys.: Conf. Ser. 274, 012030 (2011).
[CrossRef]

F. E. Veiras, L. I. Perez, and M. T. Garea, “Phase shift formulas in uniaxial media: an application to waveplates,” Appl. Opt. 49, 2769–2777 (2010).
[CrossRef]

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008).
[CrossRef]

F. E. Veiras, M. T. Garea, and L. I. Perez, “Fringe pattern analysis by means of wide angle conoscopic illumination of uniaxial crystals,” in Conf. Proc. VII International Conference of young scientists and specialists “Optics–2011” (St Petersburg, Russia, 17–21Oct.2011), pp. 55–57.

Weng, X.

H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
[CrossRef]

Winter, H.

Woerdman, J.

Wu, S.-T.

D.-K. Yang and S.-T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

Xu, M.

Yang, D.-K.

D.-K. Yang and S.-T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

Zhuang, S.

H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
[CrossRef]

AIP Conf. Proc.

M. C. Simon, L. I. Perez, and F. E. Veiras, “Parallel beams and fans of rays in uniaxial crystals,” AIP Conf. Proc. 992, 714–719 (2008).
[CrossRef]

Appl. Opt.

J. Opt.

P. Berczynski, “Complex geometrical optics of nonlinear inhomogeneous fibres,” J. Opt. 13, 035707 (2011).
[CrossRef]

J. Opt. Soc. Am.

L. I. Perez, “Nonspecular transverse effects of polarized and unpolarized symmetric beams in isotropic-uniaxial interfaces,” J. Opt. Soc. Am. 20, 741–752 (2003).
[CrossRef]

M. Anwar and R. Small, “Geometrical-optics solution for self-focusing in nonlinear optics,” J. Opt. Soc. Am. 71, 124–126 (1981).
[CrossRef]

J. Phys.: Conf. Ser.

F. E. Veiras, G. Pérez, M. T. Garea, and L. I. Perez, “Characterization of uniaxial crystals through the study of fringe patterns,” J. Phys.: Conf. Ser. 274, 012030 (2011).
[CrossRef]

Opt. Commun.

H. Guo, X. Weng, G. Sui, X. Dong, X. Gao, and S. Zhuang, “Propagation of an arbitrary incident light in a uniaxially planar slab,” Opt. Commun. 284, 5509–5512 (2011).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

L. I. Perez and M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

M. C. Simon and K. V. Gottschalk, “Waves and rays in uniaxial birefringent crystals,” Optik 118, 457–470 (2007).
[CrossRef]

M. C. Simon and M. T. Garea, “Plane parallel birefringent plates as polarization interferometers,” Optik 87, 95–102 (1991).

Phys. Rev. A

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[CrossRef]

D. Magatti, A. Gatti, and F. Ferri, “Three-dimensional coherence of light speckles: experiment,” Phys. Rev. A 79, 053831 (2009).
[CrossRef]

Other

D.-K. Yang and S.-T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

M. Françon, Encyclopedia of Physics. Fundamental of Optics, S. Flugge, ed.(Springer, 1956), vol. XXIV.

F. E. Veiras, M. T. Garea, and L. I. Perez, “Fringe pattern analysis by means of wide angle conoscopic illumination of uniaxial crystals,” in Conf. Proc. VII International Conference of young scientists and specialists “Optics–2011” (St Petersburg, Russia, 17–21Oct.2011), pp. 55–57.

J. W. GoodmanSpeckle Phenomena in Optics: Theory and Applications (Ben Roberts, 2007).

J. W. Goodman, J W Statistical Optics (Wiley Classics Library, 2000).

Note that the sign of A is related to the arbitrary way in which we compute the phase difference between waves: ordinary minus extraordinary.

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Figures (12)

Fig. 1.
Fig. 1.

Scheme of the experience. Light propagates in the direction of the x axis, while the screen is placed perpendicular to it. θ is the angle between the optical axis and the interface, (x,y,z), is the lab frame fixed to the first interface of the crystal where z is the projection of the optical axis on the interface of the plate. The diffuser is located at xP and the screen at xD.

Fig. 2.
Fig. 2.

A ray impinging on the first interface (x=0) of a plane-parallel uniaxial plate. (x,σ,t) is the coordinate system associated to the incident wave. x, t is the plane of incidence. As in Fig. 1(b), (x,y,z) is the lab frame. θ is the angle between the optical axis and the interface.

Fig. 3.
Fig. 3.

Calculated three-dimensional spot diagram. The blue circles correspond to the points of intersection of ordinary rays (along the x axis). The red diamonds correspond to the extraordinary points XeZ(α,δ) (on the plane z=Δz), and the green squares correspond to the extraordinary points XeY(α,δ) (on the plane y=0) for α=0°;5°;10°;15° and δ=0°;10°;;360°. λ=632.8nm, no=1.5426, ne=1.5516, xP=0, H=3mm. (a) θ=0°. (b) θ=45°.

Fig. 4.
Fig. 4.

X coordinate of the ordinary and extraordinary intersection points. XeZ (red solid line). XeY (green squares). Xo(α) (blue circles). 0°<α<=30° and 0°<δ=360°; λ=632.8nm, no=1.5426, ne=1.5516, xP=0, H=3mm, D=10cm. (a) θ=0°. (b) θ=45°

Fig. 5.
Fig. 5.

XeZXo (red solid line), XeYXo (green squares), and XeYXeZ (blue circles). 0°<α<=30° and 0°<δ=360°; λ=632.8nm, no=1.5426, ne=1.5516, xP=0, H=3mm, D=10cm. (a) θ=0°. (b) θ=45°.

Fig. 6.
Fig. 6.

Images and wavefronts. For the ordinary case, there is a point image that originates spherical wavefronts, and for the extraordinary case, there is an astigmatic image that originates cylindrical wavefronts. Once the images are located, it is possible to construct the fringe patterns by superposing the wavefronts that emerge from them. (a) θ=0°. (b) θ=45°.

Fig. 7.
Fig. 7.

Interference patterns generated with the exact phase shift formulas. λ=632.8nm, no=1.5426, ne=1.5516, xP=0, H=3mm, xD=10cm. (a) θ=45°. (b) θ=85°.

Fig. 8.
Fig. 8.

Interference patterns generated with different phase shift formulas Δϕ(xD,y,z). λ=632.8nm, no=1.5426, ne=1.5516, θ=5°, xP=0, H=3mm, xD=10cm. (a) Δϕexact(xD,y,z). (b) Δϕimage(xD,y,z).

Fig. 9.
Fig. 9.

Subtraction (absolute value) of fringe patterns generated with different approximations (exact-approximated). λ=632.8nm, no=1.5426, ne=1.5516, θ=5°, xP=0, H=3mm, xD=10cm. (a) ΔϕQNP(xD,y,z), MN%=0.3%. (b) Δϕsec(xD,y,z), MN%=6.43%. (c) Δϕimage(xD,y,z), MN%=22.4%.

Fig. 10.
Fig. 10.

Percentage difference between the interference patterns generated with different phase shift formulas Δϕ(xD,y,z) and the exact formulas. ΔϕQNP(xD,y,z) (red solid line), Δϕsec(xD,y,z) (green squares), and Δϕimage(xD,y,z) (blue circles). λ=632.8nm, no=1.5426, ne=1.5516, θ=5°, xP=0, H=3mm, xD=10cm. (a) 10cm×10cm (NA0.58). (b) 5cm×5cm (NA0.33).

Fig. 11.
Fig. 11.

Coefficient analysis. (a) Phase shift for normal incidence A. (b) Linear coefficient B/|ρ|. (c) CY (red solid line) and CZ (green squares). λ=632.8nm, no=1.5426, ne=1.5516, xP=0, H=3mm, xD=10cm.

Fig. 12.
Fig. 12.

(a) CZ/CY. (b) |ZC| (logarithmic scale). λ=632.8nm, no=1.5426, ne=1.5516, xP=0, xD=10cm, H=3mm.

Equations (28)

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

I=Io+Ie+2(IoIe)12cosΔϕ,
Xo(α)=xP+H[1ncosα(no2n2sin2α)12]
Δz=H(ne2no2)sinθcosθno2cos2θ+ne2sin2θ.
XeZ(α,δ)=xP+H{1nne2nocosαhx[ne2hx(ne2cos2δ+hxsin2δ)n2sin2α]12},
XeY(α,δ)=xP+H{1nnocosα[ne2hx(ne2cos2δ+hxsin2δ)n2sin2α]12}.
θΔzmax=±arccos[ne(ne2+no2)12].
xord=xP+H[1nno],
xeZ=xP+H[1nneno(no2cos2θ+ne2sin2θ)32],
xeY=xP+H[1nnone(no2cos2θ+ne2sin2θ)12].
Δϕaberra=2πnλv[y2xoxeY2(xDxo)(xDxeY)+z2xoxeZ2(xDxo)(xDxeZ)+zΔzxDxeZ].
Δϕimage(y,z)=Δϕaberra(y,z)+Δϕα=0,
Δϕα=0=2πλvH(nonα=0)
nα=0=none(no2cos2θ+ne2sin2θ)12
Δϕimage(xD,y,z)2πHλv(A+nB|ρ|z+n2CYρ2y2+n2CZρ2z2),
A=nonone(no2cos2θ+ne2sin2θ)12,
B=(no2ne2)sinθcosθ(no2cos2θ+ne2sin2θ),
CY=no2ne(no2cos2θ+ne2sin2θ)122none(no2cos2θ+ne2sin2θ)12,
CZ=no2ne(no2cos2θ+ne2sin2θ)322no(no2cos2θ+ne2sin2θ)32.
Δϕexact(α,δ)=2πHλv((no2n2sin2α)12+n(no2ne2)sinθcosθcosδsinαne2sin2θ+no2cos2θ+no{ne2(ne2sin2θ+no2cos2θ)[ne2(ne2no2)cos2θsin2δ]n2sin2α}12ne2sin2θ+no2cos2θ).
(y2+z2)12=(xDHxP)tanα+Hnsinα(no2n2sin2α)12.
cosδ=z(y2+z2)12,sin2δ=y2y2+z2.
sinα={y2+z2[xDXo(α)]2+y2+z2}12.
sinα{y2+z2[xDxo]2+y2+z2}12.
Δϕsec(α,δ)=2πHλv[A+Bcosδnsinα+(CYsin2δ+CZcos2δ)n2sin2α],
MN%=i=1Aj=1B|mijnij|AB100,
θCZ=0=arcsin([(no2ne)23no2ne2no2]12).
|CZCY|12=|no2ne2nehx3/2no2hxnehx3/2|1/2
zC=no(no2ne2)hxsinθcosθ[xDxPH(1nno)]n(no2nehx32).

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