Abstract

Angular spectra of reflected and transmitted fields, induced by an arbitrary electromagnetic beam passing through the planar interface between a homogeneous medium and a uniaxially anisotropic medium, are derived and related to the incident medium. By using these formulas, we obtain the expressions for paraxial and slightly nonparaxial fields. The reflected paraxial field is related to the incident one by means of Fresnel relations; the transmitted paraxial field is the superposition of an ordinary and an extraordinary beam, multiplied by the Fresnel coefficient. We find that the nonparaxial corrections, owing to the medium discontinuity, are larger than their free-propagation counterparts and that they are very simply related to the paraxial solutions of the incident beam. The case of two homogeneous media with different refractive indices is also discussed. The general expressions obtained are applied to the case of a nonparaxial Gaussian beam.

© 2002 Optical Society of America

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References

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  1. J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
    [CrossRef]
  2. J. J. Stamnes, D. Jiang, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
    [CrossRef]
  3. D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
    [CrossRef]
  4. H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  5. P. Török, P. Varga, Z. Laczic, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  6. P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  7. D. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab,” Pure Appl. Opt. 7, 33–52 (1998).
    [CrossRef]
  8. J. J. Stamnes, G. C. Sherman, “Reflection and refraction of an arbitrary wave at a plane interface separating two uniaxial crystals,” J. Opt. Soc. Am. 67, 683–695 (1977).
    [CrossRef]
  9. T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
    [CrossRef]
  10. T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1989).
    [CrossRef]
  11. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  12. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  13. A. Yu. Savchencko, B. Ya. Zel’dovich, “Wave propagation in a guiding structure: one step beyond the paraxial approximation,” J. Opt. Soc. Am. B 13, 273–281 (1996).
    [CrossRef]
  14. H. C. Kim, Y. H. Lee, “Higher-order corrections to the electric field vector of a Gaussian beam,” J. Opt. Soc. Am. A 16, 2232–2238 (1999).
    [CrossRef]
  15. A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809–819 (2000).
    [CrossRef]
  16. A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
    [CrossRef]
  17. Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
    [CrossRef]
  18. Q. Cao, “Corrections to the paraxial approximation solutions in transversely nonuniform refractive-index media,” J. Opt. Soc. Am. A 16, 2494–2499 (1999).
    [CrossRef]
  19. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  20. A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
    [CrossRef]
  21. A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beam characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
    [CrossRef]
  22. A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
    [CrossRef]
  23. G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2001).
    [CrossRef]
  24. R. M. Herrero, J. M. Movilla, P. M. Mejias, “Beam propagation through uniaxial anisotropic media: global changes in the spatial profile,” J. Opt. Soc. Am. A 18, 2009–2014 (2001).
    [CrossRef]
  25. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).
  26. H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

2002

2001

A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
[CrossRef]

R. M. Herrero, J. M. Movilla, P. M. Mejias, “Beam propagation through uniaxial anisotropic media: global changes in the spatial profile,” J. Opt. Soc. Am. A 18, 2009–2014 (2001).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beam characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

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

2000

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809–819 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

1999

1998

Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

D. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

1996

1995

1989

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1989).
[CrossRef]

1984

1983

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

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

1977

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Booker, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

Cao, Q.

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

Ciattoni, A.

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

A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beam characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

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

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B 17, 809–819 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Cincotti, G.

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

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beam characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

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

Crosignani, B.

Deng, X.

Dhayalan, D.

D. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

Di Porto, P.

Feit, M. D.

Fleck, J. A.

Herrero, R. M.

Jiang, D.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

J. J. Stamnes, D. Jiang, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

Kim, H. C.

Kozaki, S.

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1989).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
[CrossRef]

Laczic, Z.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lee, S. W.

Lee, Y. H.

Ling, H.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mejias, P. M.

Movilla, J. M.

Palma, C.

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

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beam characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

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

Savchencko, A. Yu.

Sherman, G. C.

Sonoda, T.

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1989).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
[CrossRef]

Stamnes, J. J.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

J. J. Stamnes, D. Jiang, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

D. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

J. J. Stamnes, G. C. Sherman, “Reflection and refraction of an arbitrary wave at a plane interface separating two uniaxial crystals,” J. Opt. Soc. Am. 67, 683–695 (1977).
[CrossRef]

Török, P.

Varga, P.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

Yariv, A.

Zel’dovich, B. Ya.

Electron. Commun. Jpn.

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1989).
[CrossRef]

IEEE J. Quantum Electron.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
[CrossRef]

R. M. Herrero, J. M. Movilla, P. M. Mejias, “Beam propagation through uniaxial anisotropic media: global changes in the spatial profile,” J. Opt. Soc. Am. A 18, 2009–2014 (2001).
[CrossRef]

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

H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
[CrossRef]

H. C. Kim, Y. H. Lee, “Higher-order corrections to the electric field vector of a Gaussian beam,” J. Opt. Soc. Am. A 16, 2232–2238 (1999).
[CrossRef]

Q. Cao, “Corrections to the paraxial approximation solutions in transversely nonuniform refractive-index media,” J. Opt. Soc. Am. A 16, 2494–2499 (1999).
[CrossRef]

Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

P. Török, P. Varga, Z. Laczic, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beam characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

Phys. Rev. A

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Pure Appl. Opt.

D. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

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Figures (9)

Fig. 1
Fig. 1

Plot of the x-component of the paraxial reflected beam |E1(R)|/E¯ versus the normalized variable x/w0 for three values of z/zR, where zR=k0nw0/2 is the Rayleigh distance in the homogeneous medium with refractive index n. We assume n=1, n0=2.616, and ne=2.903 (focusing into a rutile crystal from air) and w0=λ. All the plots are evaluated in the x=y plane.

Fig. 2
Fig. 2

Plot of the x component of the nonparaxial reflected beam |E2(R)|/E¯.

Fig. 3
Fig. 3

(a) Plot of the x component of the nonparaxial reflected beam |E3(R)|/E¯, (b) plot of the y component of the nonparaxial reflected beam |E3(R)|/E¯.

Fig. 4
Fig. 4

Plot of the longitudinal component of the reflected beam |Ez(R)|/E¯.

Fig. 5
Fig. 5

(a) Plot of the x-component of the ordinary paraxial transmitted beam |Eo1(T)|/E¯ versus the normalized variable x/w0, for three values of z/zRo, where zR=k0now0/2 is the Rayleigh distance in a homogeneous medium of refraction index no. (b) Plot of the y component of the ordinary paraxial transmitted beam |Eo1(T)|/E¯.

Fig. 6
Fig. 6

(a) Plot of the x component of the ordinary nonparaxial transmitted beam |Eo2(T)|/E¯, (b) plot of the y component of the ordinary nonparaxial transmitted beam |Eo2(T)|/E¯.

Fig. 7
Fig. 7

(a) Plot of the x component of the extraordinary paraxial transmitted beam |Ee1(T)|/E¯, (b) plot of the y component of the extraordinary paraxial transmitted beam |Ee1(T)|/E¯.

Fig. 8
Fig. 8

(a) Plot of the x-component of the extraordinary nonparaxial transmitted beam |Ee2(T)|/E¯, (b) plot of the y component of the extraordinary nonparaxial transmitted beam |Ee2(T)|/E¯.

Fig. 9
Fig. 9

Plot of the longitudinal component of the transmitted beam |Ez(T)|/E¯.

Equations (66)

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

=no2000no2000ne2,
E(I)(r, z)=dkexp(ikr+iWz)E˜(I)(k),
Ez(I)(r, z)=-dkexp(ikr+iWz)×1WkE˜(I)(k),
E(I)=Ex(I)e^x+Ey(I)e^y,r=xe^x+ye^y,
k=kxe^x+kye^y,dk=dkxdky,
E˜(I)(k)=1(2π)2drexp(-ikr)E(I)(r, 0).
E(R)(r, z)=dkexp(ikr-iWz)E˜(R)(k),
Ez(R)(r, z)=dkexp(ikr-iWz) 1Wk
E˜(R)(k),
E(T)(r, z)=dkexp(ikr)exp(iWoz)Po+expi none WezPeE˜(T)(k),
Ez(T)(r, z)=-nonedkexp(ikr)×expi none Wez1WekE˜(T)(k),
Po=1k2ky2-kxky-kxkykx2,
Pe=1k2kx2kxkykxkyky2.
E(I)(r, 0)+E(R)(r, 0)=E(T)(r, 0),
H(I)(r, 0)+H(R)(r, 0)=H(T)(r, 0),
H=1iωμ0 ×E,
E˜(I)+E˜(R)=E˜(T),
WPo+k02n2WPe(E˜(I)-E˜(R))
=WoPo+k02noneWePeE˜(T)
E˜(R)(k)=R(k)E˜(I)(k),
E˜(T)(k)=T(k)E˜(I)(k),
R(k)=W-WoW+WoPo+n2We-noneWn2We+noneWPe,
T(k)=2WW+WoPo+2n2Wen2We+noneWPe.
E(R)(r, z)=dkexp(ikr-iWz)×R(k)E˜(I)(k),
Ez(R)(r, z)=dkexp(ikr-iWz)×1WkR(k)E˜(I)(k),
E(T)(r, z)=dkexp(ikr)exp(iWoz)Po+expi none WezPeT(k)E˜(I)(k),
Ez(T)(r, z)=-nonedkexp(ikr)×expi none Wez1WekT(k)×E˜(I)(k).
E(I)(r, z)=exp(ik0nz)A(I)(r, z),
E(R)(r, z)=exp(-ik0nz) n-non+noA(I)(r, -z),
E(T)(r, z)=exp(ik0noz) 2nn+no [Ao(T)(r, z)+Ae(T)(r, z)],
A(I)(r, z)=dkexpikr-ik22k0n zE˜(I)(k),
Ao(T)(r, z)=dkexpikr-ik22k0no zPo×E˜(I)(k),
Ae(T)(r, z)=dkexpikr-inok22k0ne2 zPe×E˜(I)(k),
R=n-non+no1001,T=2nn+no1001,
E(T)(r, z)=exp(ik0nz) 2nn+nA(I)r, nn z,
E(I)(r, z)=exp(ik0nz)A(I)(r, z),
Ez(I)(r, z)=exp(ik0nz) ik0n A(I)(r, z),
E(R)(r, z)=exp(-ik0nz)n-non+no-hok02 2-he-hok02D^eA(I)(r, -z),
Ez(R)(r, z)=-n-non+no Ez(I)(r, -z),
E(T)(r, z)=exp(ik0noz)2nn+noAo(T)(r, z)-hok02D^oA(I)r, nno z+exp(ik0noz)×2nn+noAe(T)(r, z)-hek02D^eA(I)r, nnone2 z,
Ez(T)(r, z)=expik0no1-n2ne2z2non+non2ne2×Ez(I)r, nnone2 z,
ho=(n2-no2)/[nno(n+no)2],
he=[no(ne2-n2)]/[nne2(n+no)2],
=xe^x+ye^y,
D^o=y2-xy-xyx2,D^e=x2xyxyy2,
R(k)=n-non+no1001+hok2k021001+(he-ho) k2k02Pe,
T(k)=2nn+no1001+hok2k021001+(he-ho) k2k02Pe,
E(R)(r, z)=exp(-ik0nz)n-nn+n+hk02Dˆ×A(I)(r, -z),
Ez(R)(r, z)=-n-nn+n Ez(I)(r, -z),
E(T)(r, z)=exp(ik0noz)2nn+n+hk02Dˆ×A(I)r, nn z,
Ez(T)(r, z)=expik0n1-n2n2z2n+nnn×Ez(I)r, nn z,
Dˆ=x2-y22xy2xyy2-x2.
E(I)(r, 0)=E¯ exp-r2w0210.
E(R)(r, z)=E1(R)(r, z)+E2(R)(r, z)+E3(R)(r, z),
E1(R)(r, z)=exp(-ik0nz) n-non+no E¯ w02q(-z)×exp-r2q(-z)10,
E2(R)(r, z)=exp(-ik0nz)E¯ 4w02q2(-z)hok021-r2q(-z)×exp-r2q(-z)10,
E3(R)(r, z)=exp(-ik0nz)E¯ 2w02q2(-z)ho-hek02×exp-r2q(-z)2x2q(-z)-12xyq(-z),
Ez(R)(r, z)=exp(-ik0nz) n-non+no E¯ 2w02q2(-z)ixk0n×exp-r2q(-z),
E(T)(r, z)=Eo1(T)(r, z)+Eo2(T)(r, z)+Ee1(T)(r, z)+Ee2(T)(r, z),
Eo1(T)(r, z)=exp(ik0noz) 2nn+no E¯ w02qo(z)1r2×exp-r2qo(z)y2-xy+qo(z)2r2×1-exp-r2qo(z)x2-y22xy,
Eo2(T)(r, z)=-exp(ik0noz)E¯2w02qo2(z)hok02exp-r2qo(z)×2y2qo(z)-1-2xyqo(z),
Ee1(T)(r, z)=exp(ik0noz)E¯2nn+now02qe(z)1r2×exp-r2qe(z)x2xy-qe(z)2r2×1-exp-r2qe(z)x2-y22xy,
Ee2(T)(r, z)=-exp(ik0noz)E¯ 2w02qe2(z)hek02exp-r2qe(z)×2x2qe(z)-12xyqe(z),
Ez(T)(r, z)=-exp(ik0noz) 2non+no E¯ 2w02qe2(z)inxk0ne2×exp-r2qe(z).
q(z)=w02+i 2zk0n,qo(z)=w02+i 2zk0no,
qe(z)=w02+i 2znok0ne2.

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