Abstract

For a linearly polarized three-dimensional Gaussian beam in air that is normally incident upon a plane interface with a uniaxial crystal with optic axis in an arbitrary direction, we present integral representations for the transmitted field suitable for asymptotic analysis and efficient numerical evaluation and derive analytical expressions for transmitted nontruncated Gaussian beams for the cases in which the incident beam is polarized parallel to the plane containing the optic axis and the interface normal and transverse to it. The general solution for an arbitrary polarization state of an incident Gaussian beam follows by superposition of these two solutions.

© 2012 Optical Society of America

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    [CrossRef]
  47. G. Cincotti, A. Ciattoni, B. Crosignani, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680–1688 (2002).
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    [CrossRef]
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    [CrossRef]
  57. M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
    [CrossRef]
  58. J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
    [CrossRef]
  59. J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
    [CrossRef]
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2011 (2)

V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. II. Numerical results,” J. Eur. Opt. Soc. Rapid Pub. 6, 11036 (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]

2009 (2)

2006 (2)

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Effects of aperture size on focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 266, 438–447 (2006).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Comparisons of exact and paraxial intensities of electromagnetic waves focused into uniaxial crystals,” Pure Appl. Opt. 8, 709–719 (2006).
[CrossRef]

2005 (1)

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[CrossRef]

2004 (2)

S. Stallinga, “Light distribution close to focus in biaxially birefringent media,” J. Opt. Soc. Am. A 21, 1785–1798 (2004).
[CrossRef]

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79–92 (2004).
[CrossRef]

2003 (5)

A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. 224, 175–183 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20, 2163–2171 (2003).
[CrossRef]

G. Cincotti, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220, 33–40 (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]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

2002 (8)

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, B. Crosignani, and C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, B. Crosignani, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680–1688 (2002).
[CrossRef]

D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, F. Ravaccia, and C. Sapia, “Stokes parameters of a Gaussian beam in a calcite crystal,” Opt. Express 10, 699–706 (2002).

A. Ciattoni, G. Cincotti, and C. Palma, “Diffraction by elliptic and circular apertures in uniaxially anisotropic crystals: theory and experiment,” Pure Appl. Opt. 4, 424–432 (2002).
[CrossRef]

G. S. Sithambaranathan and J. J. Stamnes, “Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,” Opt. Commun. 209, 55–67 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, D. Provenziani, and P. De Porto, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Nonparaxial description of the reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1422–1432 (2002).
[CrossRef]

2001 (7)

2000 (2)

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

V. Dhayalan and J. J. Stamnes, “Comparison of exact asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. 39, 6332–6340 (2000).
[CrossRef]

1999 (3)

1998 (5)

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[CrossRef]

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

J. J. Stamnes and D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang and J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

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

1997 (1)

1996 (1)

1995 (3)

1993 (1)

1984 (1)

1983 (2)

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

J. J. Stamnes, B. Spelkjavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

1977 (2)

J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,” J. Opt. Soc. Am. A 68, 502–508 (1977).

J. J. Stamnes and 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]

1976 (2)

J. Gasper, G. C. Sherman, and J. J. Stamnes, “Reflection and refraction of an arbitrary wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 770–788 (1976).

1974 (1)

S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

1972 (1)

E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II.,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

1971 (1)

M. Lax and D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
[CrossRef]

1941 (1)

Bell, W. E.

W. E. Bell, Gas Laser Technology (Holt, Rinehart and Winston, 1969).

Bernacki, B. E.

Booker, G. R.

Ciattoni, A.

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79–92 (2004).
[CrossRef]

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20, 2163–2171 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. 224, 175–183 (2003).
[CrossRef]

G. Cincotti, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220, 33–40 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Diffraction by elliptic and circular apertures in uniaxially anisotropic crystals: theory and experiment,” Pure Appl. Opt. 4, 424–432 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, B. Crosignani, and C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

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, B. Crosignani, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680–1688 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Nonparaxial description of the reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1422–1432 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, D. Provenziani, and P. De Porto, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, F. Ravaccia, and C. Sapia, “Stokes parameters of a Gaussian beam in a calcite crystal,” Opt. Express 10, 699–706 (2002).

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

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

Cincotti, G.

G. Cincotti, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220, 33–40 (2003).
[CrossRef]

G. Cincotti, A. Ciattoni, B. Crosignani, and C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Diffraction by elliptic and circular apertures in uniaxially anisotropic crystals: theory and experiment,” Pure Appl. Opt. 4, 424–432 (2002).
[CrossRef]

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]

D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, F. Ravaccia, and C. Sapia, “Stokes parameters of a Gaussian beam in a calcite crystal,” Opt. Express 10, 699–706 (2002).

A. Ciattoni, G. Cincotti, D. Provenziani, and P. De Porto, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Nonparaxial description of the reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1422–1432 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, B. Crosignani, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680–1688 (2002).
[CrossRef]

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

Crosignani, B.

De Porto, P.

A. Ciattoni, G. Cincotti, D. Provenziani, and P. De Porto, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

Dhayalan, V.

V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. II. Numerical results,” J. Eur. Opt. Soc. Rapid Pub. 6, 11036 (2011).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Focal shifts on focusing through a plane interface,” Opt. Commun. 282, 2286–2291 (2009).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
[CrossRef]

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[CrossRef]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Comparison of exact asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. 39, 6332–6340 (2000).
[CrossRef]

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

Di Porto, P.

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]

Feit, M. D.

Felsen, L. B.

S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

L. B. Felsen, Radiation and Scattering of Waves (Prentice-Hall, 1973).

Fleck, J. A.

Frette, Ø.

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Effects of aperture size on focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 266, 438–447 (2006).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Comparisons of exact and paraxial intensities of electromagnetic waves focused into uniaxial crystals,” Pure Appl. Opt. 8, 709–719 (2006).
[CrossRef]

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]

Gasper, J.

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]

Jain, M.

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Comparisons of exact and paraxial intensities of electromagnetic waves focused into uniaxial crystals,” Pure Appl. Opt. 8, 709–719 (2006).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Effects of aperture size on focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 266, 438–447 (2006).
[CrossRef]

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[CrossRef]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

Jiang, D.

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
[CrossRef]

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[CrossRef]

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

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

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

J. J. Stamnes and D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang and J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

Jones, R. C.

Laczic, Z.

Lalor, E.

E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II.,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

Lax, M.

M. Lax and D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
[CrossRef]

Lee, S. W.

Ling, H.

Lotsberg, J. K.

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Comparisons of exact and paraxial intensities of electromagnetic waves focused into uniaxial crystals,” Pure Appl. Opt. 8, 709–719 (2006).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Effects of aperture size on focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 266, 438–447 (2006).
[CrossRef]

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[CrossRef]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

Mansuripur, M.

Nelson, D. F.

M. Lax and D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
[CrossRef]

Nemeth, G.

Palma, C.

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79–92 (2004).
[CrossRef]

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20, 2163–2171 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. 224, 175–183 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Nonparaxial description of the reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1422–1432 (2002).
[CrossRef]

D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, F. Ravaccia, and C. Sapia, “Stokes parameters of a Gaussian beam in a calcite crystal,” Opt. Express 10, 699–706 (2002).

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, B. Crosignani, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680–1688 (2002).
[CrossRef]

G. Cincotti, A. Ciattoni, B. Crosignani, and C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Diffraction by elliptic and circular apertures in uniaxially anisotropic crystals: theory and experiment,” Pure Appl. Opt. 4, 424–432 (2002).
[CrossRef]

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

Pedersen, H. M.

J. J. Stamnes, B. Spelkjavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Provenziani, D.

D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, F. Ravaccia, and C. Sapia, “Stokes parameters of a Gaussian beam in a calcite crystal,” Opt. Express 10, 699–706 (2002).

A. Ciattoni, G. Cincotti, D. Provenziani, and P. De Porto, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

Ravaccia, F.

Saghafi, S.

Sapia, C.

G. Cincotti, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220, 33–40 (2003).
[CrossRef]

D. Provenziani, A. Ciattoni, G. Cincotti, C. Palma, F. Ravaccia, and C. Sapia, “Stokes parameters of a Gaussian beam in a calcite crystal,” Opt. Express 10, 699–706 (2002).

Seshadri, S. R.

Shephard, C. J. R.

Sherman, G. C.

J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,” J. Opt. Soc. Am. A 68, 502–508 (1977).

J. J. Stamnes and 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]

J. Gasper, G. C. Sherman, and J. J. Stamnes, “Reflection and refraction of an arbitrary wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 770–788 (1976).

Shin, S. Y.

S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

Sithambaranathan, G.

Sithambaranathan, G. S.

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[CrossRef]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

G. S. Sithambaranathan and J. J. Stamnes, “Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,” Opt. Commun. 209, 55–67 (2002).
[CrossRef]

G. S. Sithambaranathan and J. J. Stamnes, “Transmission of a Gaussian beam into a biaxial crystal,” J. Opt. Soc. Am. A 18, 1670–1677 (2001).
[CrossRef]

Spelkjavik, B.

J. J. Stamnes, B. Spelkjavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Stallinga, S.

Stamnes, J. J.

V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. II. Numerical results,” J. Eur. Opt. Soc. Rapid Pub. 6, 11036 (2011).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Focal shifts on focusing through a plane interface,” Opt. Commun. 282, 2286–2291 (2009).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Comparisons of exact and paraxial intensities of electromagnetic waves focused into uniaxial crystals,” Pure Appl. Opt. 8, 709–719 (2006).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Effects of aperture size on focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 266, 438–447 (2006).
[CrossRef]

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[CrossRef]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

G. S. Sithambaranathan and J. J. Stamnes, “Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,” Opt. Commun. 209, 55–67 (2002).
[CrossRef]

G. S. Sithambaranathan and J. J. Stamnes, “Transmission of a Gaussian beam into a biaxial crystal,” J. Opt. Soc. Am. A 18, 1670–1677 (2001).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
[CrossRef]

J. J. Stamnes and G. Sithambaranathan, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,” J. Opt. Soc. Am. A 18, 3119–3129 (2001).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Comparison of exact asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. 39, 6332–6340 (2000).
[CrossRef]

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

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

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

D. Jiang and J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

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

J. J. Stamnes and D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

J. J. Stamnes, B. Spelkjavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

J. J. Stamnes and 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]

J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,” J. Opt. Soc. Am. A 68, 502–508 (1977).

J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 770–788 (1976).

J. Gasper, G. C. Sherman, and J. J. Stamnes, “Reflection and refraction of an arbitrary wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

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]

Török, P.

Varga, P.

Visser, T. D.

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]

Wiersma, S. H.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1976).

Zhao, X.

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
[CrossRef]

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[CrossRef]

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]

Appl. Opt. (2)

Appl. Phys. (1)

S. Y. Shin and L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Cincotti, A. Ciattoni, B. Crosignani, and C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. II. Numerical results,” J. Eur. Opt. Soc. Rapid Pub. 6, 11036 (2011).
[CrossRef]

J. Math. Phys. (1)

E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II.,” J. Math. Phys. 13, 443–449 (1972).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (23)

S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001).
[CrossRef]

S. Stallinga, “Light distribution close to focus in biaxially birefringent media,” J. Opt. Soc. Am. A 21, 1785–1798 (2004).
[CrossRef]

M. Jain, J. K. Lotsberg, J. J. Stamnes, Ø. Frette, V. Dhayalan, D. Jiang, and X. Zhao, “Numerical and experimental results for focusing of three-dimensional electromagnetic waves into uniaxial crystals,” J. Opt. Soc. Am. A 26, 691–698 (2009).
[CrossRef]

G. Cincotti, A. Ciattoni, B. Crosignani, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680–1688 (2002).
[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. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20, 2163–2171 (2003).
[CrossRef]

J. J. Stamnes and G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,” J. Opt. Soc. Am. A 68, 502–508 (1977).

J. J. Stamnes and G. Sithambaranathan, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface separating an isotropic and a biaxial medium,” J. Opt. Soc. Am. A 18, 3119–3129 (2001).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
[CrossRef]

G. S. Sithambaranathan and J. J. Stamnes, “Transmission of a Gaussian beam into a biaxial crystal,” J. Opt. Soc. Am. A 18, 1670–1677 (2001).
[CrossRef]

S. R. Seshadri, “Paraxial wave equation for the extraordinary-mode beam in a uniaxial crystal,” J. Opt. Soc. Am. A 18, 2628–2629 (2001).
[CrossRef]

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

A. Ciattoni, G. Cincotti, and C. Palma, “Nonparaxial description of the reflection and transmission at the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1422–1432 (2002).
[CrossRef]

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

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

P. Török, P. Varga, and G. Nemeth, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
[CrossRef]

T. D. Visser and S. H. Wiersma, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
[CrossRef]

S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, “Comparison of different theories for focusing through a plane dielectric interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[CrossRef]

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[CrossRef]

S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
[CrossRef]

C. J. R. Shephard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
[CrossRef]

Opt. Acta (1)

J. J. Stamnes, B. Spelkjavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Opt. Commun. (13)

G. Cincotti, A. Ciattoni, and C. Sapia, “Radially and azimuthally polarized vortices in uniaxial crystals,” Opt. Commun. 220, 33–40 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79–92 (2004).
[CrossRef]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, and V. Dhayalan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[CrossRef]

J. K. Lotsberg, X. Zhao, M. Jain, V. Dhayalan, G. S. Sithambaranathan, J. J. Stamnes, and D. Jiang, “Focusing of electromagnetic waves into a biaxial crystal, experimental results,” Opt. Commun. 250, 231–240 (2005).
[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]

M. Jain, J. K. Lotsberg, J. J. Stamnes, and Ø. Frette, “Effects of aperture size on focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 266, 438–447 (2006).
[CrossRef]

A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. 224, 175–183 (2003).
[CrossRef]

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

G. S. Sithambaranathan and J. J. Stamnes, “Analytical approach to the transmission of a Gaussian beam into a biaxial crystal,” Opt. Commun. 209, 55–67 (2002).
[CrossRef]

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

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

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

J. J. Stamnes and V. Dhayalan, “Focal shifts on focusing through a plane interface,” Opt. Commun. 282, 2286–2291 (2009).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (1)

M. Lax and D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. B 4, 3694–3731 (1971).
[CrossRef]

Phys. Rev. E (1)

A. Ciattoni, G. Cincotti, D. Provenziani, and P. De Porto, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

Pure Appl. Opt. (5)

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

J. J. Stamnes and D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang and J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Diffraction by elliptic and circular apertures in uniaxially anisotropic crystals: theory and experiment,” Pure Appl. Opt. 4, 424–432 (2002).
[CrossRef]

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L. B. Felsen, Radiation and Scattering of Waves (Prentice-Hall, 1973).

W. E. Bell, Gas Laser Technology (Holt, Rinehart and Winston, 1969).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1976).

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986).

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Equations (164)

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s ^ = s x e ^ x + s y e ^ y + s z e ^ z ,
E t ( r , t ) = { [ E o t ( r ) + E e t ( r ) ] e i ω t } ,
E p t ( r ) = A [ E x i ( r t , 0 ) F E p , x ( r t r t , z 0 , z ) + E y i ( r t , 0 ) F E p , y ( r t r t , z 0 , z ) ] d x d y .
F E p , j ( r t r t , z 0 , z ) = ( 1 2 π ) 2 f E p , j ( k t ) e i h p ( k t · ( r t r t ) , z 0 , z ) d k x d k y ,
h p ( k t · ( r t r t ) , z 0 , z ) = k t · ( r t r t ) + k z 1 z 0 + k z p ( z z 0 ) ; k t = k x e ^ x + k y e ^ y ,
f E o , j ( k t ) = ( k o ) 2 k 1 2 [ k o × s ^ ] [ T α T E , o k t ( k t × e ^ z k t ) + c k 1 2 ω k z 1 T α T M , o k t k t k t ] · e ^ j ,
f E e , j ( k t ) = ω c k 1 2 [ ( k o ) 2 s ^ ( k e · s ^ ) k e ] [ T α T E , e k t ( k t × e ^ z k t ) + c k 1 2 ω k z 1 T α T M , e k t k t k t ] · e ^ j ,
k z 1 = ( k 1 2 k t 2 ) 1 2 ; k t 2 = k x 2 + k y 2 ; k 1 2 = ω 2 c 2 ϵ 1 μ 1 ; k p = k t + k z p e ^ z ,
k z o = [ ( k o ) 2 k t 2 ] 1 2 ; k z e = α + β ,
α = χ s z 1 + χ s z 2 k t · s ^ ; χ = ( k e ) 2 ( k o ) 2 1 ; ( k p ) 2 = ω 2 c 2 μ ϵ p ,
β = ( k e ) 2 k t 2 1 + χ s z 2 α 2 χ s z 2 = ( k e ) 2 1 + χ s z 2 ( 1 + χ s z 2 ) k t 2 + χ ( k t · s ^ ) 2 ( 1 + χ s z 2 ) 2 .
E p t ( r ) = E x s F E p , x ( r t r s t , z 0 , z ) ; r s t = x s e ^ x + y s e ^ y ,
E p t ( r ) = E y s F E p , x ( r t r s t , z 0 , z ) ,
T α p , o = α ˜ o t α ˜ p i = E ˜ o t E ˜ p i | e p i | | e o t | = T p , o | e p i | | e o t | ; T α p , e = α ˜ e t α ˜ p i = E ˜ e t E ˜ p i | e p i | | e e t | = T p , e | e p i | | e e t | ,
e T E i = k 1 2 c ω k t × e ^ z ; | e T E i | = k 1 2 c ω k t ; e T M i = ( k t + k z 1 e ^ z ) × ( k t × e ^ z ) ; | e T M i | = k 1 k t ,
e o t = ( k o ) 2 c ω k o × s ^ ; e e t = ( k o ) 2 s ^ ( k e · s ^ ) k e .
T α T E , o = det Ω 3 T E det Ω ; T α T M , o = det Ω 3 T M det Ω ; T α T E , e = det Ω 4 T E det Ω ; T α T M , e = det Ω 4 T M det Ω ,
( A 1 0 C 1 D 1 0 B 2 C 2 D 2 0 B 3 C 3 D 3 A 4 0 C 4 D 4 ) = ( k 1 2 c ω k t 0 ( k o ) 2 c ω A o ( k o ) 2 b 0 k t k z 1 ( k o ) 2 c ω b k z o B e 0 k 1 2 c ω μ 1 k t ( k o ) 4 c 2 ω 2 μ b ( k o ) 2 c ω μ A e k 1 2 c 2 ω 2 μ 1 k t k z 1 0 ( k o ) 2 c 2 ω 2 μ B o ( k o ) 2 c ω μ b k z e ) ,
( E 1 0 0 E 4 ) = ( k 1 2 c ω k t 0 0 k 1 2 c 2 ω 2 μ 1 k t k z 1 ) = ( A 1 0 0 A 4 ) ,
( 0 F 2 F 3 0 ) = ( 0 k t k z 1 k 1 2 c ω μ 1 k t 0 ) = ( 0 B 2 B 3 0 ) .
A p = s z k t 1 k t ( k t · s ^ ) ( k p · e ^ z ) ,
B p = ( k o ) 2 k t ( k t · s ^ ) k t ( k p · s ^ ) ,
b = 1 k t ( k t × s ^ ) · e ^ z .
det Ω = ( A 1 D 4 A 4 D 1 ) ( B 2 C 3 B 3 C 2 ) ( A 1 C 4 A 4 C 1 ) ( B 2 D 3 B 3 D 2 ) ,
det Ω 3 T E = 2 A 1 A 4 ( B 2 D 3 B 3 D 2 ) ,
det Ω 3 T M = 2 B 2 B 3 ( A 1 D 4 A 4 D 1 ) ,
det Ω 4 T E = 2 A 1 A 4 ( B 2 C 3 B 3 C 2 ) ,
det Ω 4 T M = 2 B 2 B 3 ( A 1 C 4 A 4 C 1 ) .
| e o t | = | e T E t | = k 2 c ω k t ; | e e t | = | e T M t | = k k t ,
T T E T T E , T E = k 2 k 1 2 T α T E , T E ; T T M T T M , T M = k k 1 T α T M , T M .
s ^ = s x e ^ x + s z e ^ z .
k z e = α + β ; α = χ s z 1 + χ s z 2 s x k x ; β = ( k o ) 2 ( 1 + χ ) 1 + χ s z 2 1 + χ ( 1 + χ s z 2 ) 2 [ k x 2 + k y 2 1 + χ s z 2 1 + χ ] .
F E p , j ( r t ) = 1 2 π η p | H p | 1 2 f E p , j ( k t s ) e i h p ( k t s ) ( j = x , y ) ,
h p ( k t s ) k t s · ( r t r t ) + k z 1 s z 0 + k z p , s ( z z 0 ) ; k z 1 s = k z 1 ( k x s , k y s ) ; k z p , s = k z p ( k x s , k y s ) ,
H p = [ 2 h p k x 2 2 h p k y 2 ( 2 h p k x k y ) 2 ] k x s , k y s ,
η p = { 1 if H < 0 i if H > 0 and 2 h p k x 2 | k x s , k y s > 0 i if H > 0 and 2 h p k x 2 | k x s , k y s < 0 .
h p k x | k x s , k y s = x x + z 0 k z 1 k x | k x s , k y s + ( z z 0 ) k z p k x | k x s , k y s = 0 ,
h p k x | k x s , k y s = y y + z 0 k z 1 k y | k x s , k y s + ( z z 0 ) k z p k y | k x s , k y s = 0 .
E p t ( r ) = A [ P p , x ( r t , r , z 0 ) + P p , y ( r t , r , z 0 ) ] e i h p ( k t s ) d x d y ,
P p , j ( r t , r , z 0 ) = η p f E p , j ( k t s ) 2 π | H p | 1 2 E j i ( r t , 0 ) .
E y i ( r t , 0 ) = { exp { x 2 + y 2 2 σ 0 2 } for ( x , y ) inside A 0 for ( x , y ) outside A ,
k z 1 k 1 k x 2 + k y 2 2 k 1 ; k z o k o k x 2 + k y 2 2 k o ,
h o ( k x , k y ) k 1 ( z 0 + k o k 1 ( z z 0 ) ) + k x ( x x ) + k y ( y y ) k x 2 + k y 2 2 k 1 ( z 0 + k 1 k o ( z z 0 ) ) .
k x s = k 1 Z ˜ [ x x ] ; k y s = k 1 Z ˜ [ y y ] ,
Z ˜ = z 0 + k 1 k o ( z z 0 ) .
h o ( k t s ) = k 1 [ Z + ( x x ) 2 + ( y y ) 2 2 Z ˜ ] ,
Z = z 0 + k o k 1 ( z z 0 ) .
2 h o k x 2 | k x = k y = 0 = 2 h o k y 2 | k x = k y = 0 = Z k 1 ; 2 h o k x k y | k x = k y = 0 = 0 ,
| H o | 1 2 = Z ˜ k 1 = λ 1 2 π Z ˜ ; η o = i .
E o t ( x , y , z ) = f E o , y ( 0 , 0 ) i λ 1 Z exp { i k 1 [ Z + x 2 + y 2 2 Z ] } e [ ϕ ( x ) + ϕ ( y ) ] d x d y ,
ϕ ( x ) = a 1 x 2 + b 1 x ; a 1 = 1 2 [ 1 σ 0 2 i k 1 Z ˜ ] ; b 1 = i k 1 x Z ˜ ,
ϕ ( y ) = a 1 y 2 + b 2 y ; b 2 = i k 1 y Z .
exp [ ϕ ( x ) ] d x = π a 1 exp ( b 1 2 / 4 a 1 ) ,
a 1 = | a 1 | e i arctan ( k 1 σ 0 2 / Z ˜ ) ,
1 | a 1 | = 2 σ 0 Z ˜ k 1 σ ( Z ˜ ) ; σ 2 ( Z ˜ ) = σ 0 2 + ( Z ˜ / k 1 σ 0 ) 2 ,
{ b 1 2 / 4 a 1 } = x 2 2 σ 2 ( Z ˜ ) ; { b 1 2 / 4 a 1 } = k 1 x 2 2 Z ˜ σ 0 2 σ 2 ( Z ˜ ) .
E o t ( x , y , z ) = f E o , y ( 0 , 0 ) σ 0 i σ ( Z ˜ ) exp [ x 2 + y 2 2 σ 2 ( Z ˜ ) ] exp { i k 1 [ Z + x 2 + y 2 2 R ( Z ˜ ) ] } e i arctan ( k 1 σ 0 2 / Z ) ,
1 R ( Z ˜ ) = 1 Z ˜ ( 1 σ 0 2 σ 2 ( Z ˜ ) ) ,
R ( Z ˜ ) = Z ˜ 1 σ 0 2 / σ 2 ( Z ˜ ) = Z ˜ + k 1 2 σ 0 4 Z ˜ ,
f E o , y ( 0 , 0 ) = 2 μ k 1 μ k 1 + μ 1 k o e ^ y .
E x i ( r t , 0 ) = { exp { x 2 + y 2 2 σ 0 2 } for ( x , y ) inside A 0 for ( x , y ) outside A ,
k z e = δ k x + β ,
δ = χ s x s z 1 + χ s z 2 .
β k e ( 1 + χ s z 2 ) 1 / 2 k x 2 1 + χ 2 k e ( 1 + χ s z 2 ) 3 / 2 k y 2 1 2 k e ( 1 + χ s z 2 ) 1 / 2 .
h e ( k x , k y ) k 1 Z e + k x [ X ˜ x ] + k y ( y y ) k x 2 2 k 1 Z x k y 2 2 k 1 Z y ,
X ˜ = x δ ( z z 0 ) ; Z e = z 0 + k e k 1 ( 1 + χ s z 2 ) 1 / 2 ( z z 0 ) ,
Z x = z 0 + k 1 ( 1 + χ ) k e ( 1 + χ s z 2 ) 3 / 2 ( z z 0 ) ; Z y = z 0 + k 1 k e ( 1 + χ s z 2 ) 1 / 2 ( z z 0 ) .
k x s = k 1 Z x [ X ˜ x ] ; k y s = k 1 Z y [ y y ] ,
h e ( k x s , k y s ) k 1 Z e + k 1 2 Z x [ X ˜ x ] 2 + k 1 2 Z y ( y y ) 2 .
2 h e k x 2 | k x = k y = 0 = Z x k 1 ; 2 h e k y 2 | k x = k y = 0 = Z y k 1 ; 2 h e k x k y | k x = k y = 0 = 0 ,
| H e | 1 2 = ( Z x Z y ) 1 / 2 k 1 ; η e = i .
E e t ( x , y , z ) = k 1 f E e , x ( 0 , 0 ) i 2 π ( Z x Z y ) 1 / 2 exp { i k 1 [ Z e + X ˜ 2 2 Z x + y 2 2 Z y ] } e [ ϕ ( x ) + ϕ ( y ) ] d x d y ,
ϕ ( x ) = a 1 x 2 + b 1 x ; a 1 = 1 2 [ 1 σ 0 2 i k 1 Z x ] ; b 1 = i k 1 X ˜ Z x ,
ϕ ( y ) = a 2 y 2 + b 2 y a 2 = a 1 = 1 2 [ 1 σ 0 2 i k 1 Z y ] ; b 2 = i k 1 y Z y .
exp [ ϕ ( x ) ] d x = π a 1 exp ( b 1 2 / 4 a 1 ) ,
a 1 = | a 1 | e i arctan ( k 1 σ 0 2 / Z ˜ ) ,
1 | a 1 | = 2 σ 0 Z x k 1 σ ( Z x ) ; σ 2 ( Z x ) = σ 0 2 + ( Z x / k 1 σ 0 ) 2 ,
{ b 1 2 / 4 a 1 } = X ˜ 2 2 σ 2 ( Z x ) ; { b 1 2 / 4 a 1 } = k 1 X ˜ 2 2 Z x σ 0 2 σ 2 ( Z x ) .
E e t ( x , y , z ) = f E e , x ( 0 , 0 ) i σ 0 σ ( Z x ) σ 0 σ ( Z y ) exp [ X ˜ 2 2 σ 2 ( Z x ) y 2 2 σ 2 ( Z y ) ] exp { i k 1 [ Z e + X ˜ 2 2 R ( Z x ) + y 2 2 R ( Z y ) ] } exp { i 1 2 [ arctan ( k 1 σ 0 2 / Z x ) + arctan ( k 1 σ 0 2 / Z y ) ] } ,
σ 2 ( Z j ) = σ 0 2 + ( Z j / k 1 σ 0 ) 2 ,
1 R ( Z j ) = 1 Z j ( 1 σ 0 2 σ 2 ( Z j ) ) ,
R ( Z j ) = Z j 1 σ 0 2 / σ 2 ( Z j ) = Z j + k 1 2 σ 0 4 Z j .
f E e , x ( 0 , 0 ) = 2 μ k 1 μ k 1 + μ 1 k z e ( 0 , 0 ) { e ^ x δ e ^ z } ; ( k z e ( 0 , 0 ) k o ) 2 = 1 + χ 1 + χ s z 2 ,
X ˜ = x ; Z e = z 0 + k o k 1 ( z z 0 ) = Z ,
Z x = Z y = z 0 + k 1 k o ( 1 + χ ) ( z z 0 ) ,
f E e , x ( 0 , 0 ) = 2 μ k 1 μ k 1 + μ 1 k o e ^ x .
X ˜ = x ; Z e = z 0 + k e k 1 ( z z 0 ) ,
Z x = z 0 + k 1 ( 1 + χ ) k e ( z z 0 ) ; Z y = z 0 + k 1 k e ( z z 0 ) .
f E e , x ( 0 , 0 ) = 2 μ k 1 μ k 1 + μ 1 k e e ^ x .
S e = k e + χ ( k e · s ^ ) s ^ | k e + χ ( k e · s ^ ) s ^ | .
tan γ = S e · e ^ x S e · e ^ z = χ s x s z 1 + χ s z 2 ,
| tan γ max | = | χ | 2 ( 1 + χ ) ,
A 1 C 4 A 4 C 1 = k 1 2 ( k o ) 2 c 3 ω 3 μ 1 μ [ μ 1 ( k t B o ) μ k z 1 ( k t A o ) ] ,
B 2 D 3 B 3 D 2 = c ω μ 1 μ [ μ k 1 2 ( k t B e ) μ 1 ( k o ) 2 k z 1 ( k t A e ) ] ,
A 1 D 4 A 4 D 1 = k 1 2 ( k o ) 2 c 2 ω 2 μ 1 μ [ μ k z 1 + μ 1 k z e ] ( k t b ) ,
B 2 C 3 B 3 C 2 = ( k o ) 2 c 2 ω 2 μ 1 μ [ μ 1 ( k o ) 2 k z 1 + μ k 1 2 k z o ] ( k t b ) ,
2 A 1 A 4 = 2 k 1 4 c 3 ω 3 μ 1 k t 2 k z 1 ,
2 B 2 B 3 = 2 k 1 2 c ω μ 1 k t 2 k z 1 .
k t A p = s z k t 2 k x s x k z p ,
k t B p = ( k o ) 2 k x s x k t 2 ( k x s x + k z p s z ) ,
k t b = k y s x ,
A 1 C 4 A 4 C 1 = k 1 2 ( k o ) 2 c 3 ω 3 μ 1 μ { μ 1 [ ( k o ) 2 k x s x k t 2 ( k x s x + k z o s z ) μ k z 1 ( s z k t 2 k x s x k z o ) ] } ,
B 2 D 3 B 3 D 2 = c ω μ 1 μ { μ k 1 2 [ ( k o ) 2 k x s x k t 2 ( k x s x + k z e s z ) ] μ 1 ( k o ) 2 k z 1 [ s z k t 2 k x s x k z e ] } ,
A 1 D 4 A 4 D 1 = k 1 2 ( k o ) 2 c 2 ω 2 μ 1 μ [ μ k z 1 + μ 1 k z e ] ( k y s x ) ,
B 2 C 3 B 3 C 2 = ( k o ) 2 c 2 ω 2 μ 1 μ [ μ 1 ( k o ) 2 k z 1 + μ k 1 2 k z o ] ( k y s x ) ,
2 A 1 A 4 = 2 k 1 4 c 3 ω 3 μ 1 k t 2 k z 1 ,
2 B 2 B 3 = 2 k 1 2 c ω μ 1 k t 2 k z 1 .
A 1 C 4 A 4 C 1 = k 1 2 ( k o ) 2 c 3 k x ω 3 μ 1 μ { μ 1 s x k x 2 + s z ( μ k z 1 + μ 1 k z o ) k x s x ( μ k z 1 k z o + μ 1 ( k o ) 2 ) } ,
B 2 D 3 B 3 D 2 = c k x ω μ 1 μ { μ k 1 2 s x k x 2 + ( μ k 1 2 k z e + μ 1 ( k o ) 2 k z 1 ) s z k x ( k o ) 2 s x ( μ k 1 2 + μ 1 k z 1 k z e ) } ,
A 1 D 4 A 4 D 1 = 0 ,
B 2 C 3 B 3 C 2 = 0 ,
2 A 1 A 4 = 2 k 1 4 c 3 ω 3 μ 1 k x 2 k z 1 ,
2 B 2 B 3 = 2 k 1 2 c ω μ 1 k x 2 k z 1 .
det Ω = ( A 1 C 4 A 4 C 1 ) ( B 2 D 3 B 3 D 2 ) ,
det Ω 3 T E = 2 A 1 A 4 ( B 2 D 3 B 3 D 2 ) ,
det Ω 3 T M = 2 B 2 B 3 ( A 1 D 4 A 4 D 1 ) = 0 ,
det Ω 4 T E = 2 A 1 A 4 ( B 2 C 3 B 3 C 2 ) = 0 ,
det Ω 4 T M = 2 B 2 B 3 ( A 4 C 1 A 1 C 4 ) ,
T α T E , o = det Ω 3 T E det Ω = 2 A 1 A 4 A 1 C 4 A 4 C 1 ,
T α T M , o = det Ω 3 T M det Ω = 0 ,
T α T E , e = det Ω 4 T E det Ω = 0 ,
T α T M , e = det Ω 4 T M det Ω = 2 B 2 B 3 B 2 D 3 B 3 D 2 .
T α T E , o = ( k 1 k o ) 2 2 μ k z 1 k x μ 1 s x k x 2 + s z ( μ k z 1 + μ 1 k z o ) k x s x ( μ k z 1 k z o + μ 1 ( k o ) 2 ) .
T α T M , e = 2 μ k 1 2 k z 1 k x μ k 1 2 s x k x 2 + ( μ k 1 2 k z e + μ 1 ( k o ) 2 k z 1 ) s z k x ( k o ) 2 s x ( μ k 1 2 + μ 1 k z 1 k z e ) .
f E o , x ( k x , k y = 0 ) = 0 ,
f E o , y ( k x , k y = 0 ) = ( k o ) 2 k 1 2 [ k o × s ^ ] T α T E , o k x .
lim k x 0 = k o s x e ^ y .
lim k x 0 f E o , y ( k x , k y = 0 ) = 2 μ k 1 μ k 1 + μ 1 k o e ^ y .
f E e , y ( k x , k y = 0 ) = 0 ,
f E e , x ( k x , k y = 0 ) = [ ( k o ) 2 s ^ ( k e · s ^ ) k e ] T α T M , e k z 1 k x .
( k o ) 2 s ^ ( k e · s ^ ) k e = ( k o ) 2 s x { e ^ x + s z s x [ 1 ( k z e k o ) 2 ] e ^ x }
( k z e ( 0 , 0 ) k o ) 2 = 1 + χ 1 + χ s z 2
lim k x 0 [ ( k o ) 2 s ^ ( k e · s ^ ) k e ] = ( k o ) 2 s x { e ^ x δ e ^ z } ,
δ = χ s x s z 1 + χ s z 2 .
lim k x 0 f E e , x ( k x , k y = 0 ) = 2 μ k 1 μ k 1 + μ 1 k z e ( 0 , 0 ) { e ^ x δ e ^ z } .
A 4 C 1 A 1 C 4 = k 1 2 ( k o ) 2 c 3 ω 3 μ 1 μ s z k y 2 [ μ k z 1 + μ 1 k z o ] ,
B 2 D 3 B 3 D 2 = c ω μ 1 μ k y 2 s z [ μ k 1 2 k z e + μ 1 ( k o ) 2 k z 1 ] ,
A 1 D 4 A 4 D 1 = k 1 2 ( k o ) 2 c 2 ω 2 μ 1 μ [ μ k z 1 + μ 1 k z e ] ( k y s x ) ,
B 2 C 3 B 3 C 2 = ( k o ) 2 c 2 ω 2 μ 1 μ [ μ 1 ( k o ) 2 k z 1 + μ k 1 2 k z o ] ( k y s x ) ,
2 A 1 A 4 = 2 k 1 4 c 3 ω 3 μ 1 k y 2 k z 1 ,
2 B 2 B 3 = 2 k 1 2 c ω μ 1 k y 2 k z 1 .
det Ω = k 1 2 ( k o ) 2 c 4 ω 4 μ 1 2 μ 2 k y 2 ( s z 2 R 1 k y 2 + s x 2 R 2 ) ,
R 1 = ( μ k z 1 + μ 1 k z o ) ( μ k 1 2 k z e + μ 1 ( k o ) 2 k z 1 ) ,
R 2 = ( k o ) 2 ( μ k z 1 + μ 1 k z e ) ( μ 1 ( k o ) 2 k z 1 + μ k 1 2 k z o ) ,
det Ω 3 T E = 2 k 1 4 c 4 ω 4 μ 1 2 μ k z 1 s z k y 4 [ μ k 1 2 k z e + μ 1 ( k o ) 2 k z 1 ] ,
det Ω 3 T M = 2 k 1 4 ( k o ) 2 c 3 ω 3 μ 1 2 μ k z 1 s x k y 3 [ μ k z 1 + μ 1 k z e ] ,
det Ω 4 T E = 2 k 1 4 ( k o ) 2 c 5 ω 5 μ 1 2 μ k z 1 s x k y 3 [ μ 1 ( k o ) 2 k z 1 + μ k 1 2 k z o ] ,
det Ω 4 T M = 2 k 1 4 ( k o ) 2 c 4 ω 4 μ 1 2 μ k z 1 s z k y 4 [ μ k z 1 + μ 1 k z o ] .
T α T E , o = det Ω 3 T E det Ω = 2 k 1 2 μ k z 1 s z ( k o ) 2 μ k 1 2 k z e + μ 1 ( k o ) 2 k z 1 s z 2 R 1 k y 2 + s x 2 R 2 k y 2 ,
T α T M , o = det Ω 3 T M det Ω = 2 k 1 2 ω c μ k z 1 s x μ k z 1 + μ 1 k z e s z 2 R 1 k y 2 + s x 2 R 2 k y ,
T α T E , e = det Ω 4 T E det Ω = 2 k 1 2 c ω μ k z 1 s x μ 1 ( k o ) 2 k z 1 + μ k 1 2 k z o s z 2 R 1 k y 2 + s x 2 R 2 k y ,
T α T M , e = det Ω 4 T M det Ω = 2 k 1 2 μ k z 1 s z μ k z 1 + μ 1 k z o s z 2 R 1 k y 2 + s x 2 R 2 k y 2 ,
f E o , x ( k x = 0 , k y ) = ( k o ) 2 k 1 2 [ k o × s ^ ] T α T E , o k y 2 k y ,
f E o , y ( k x = 0 , k y ) = ( k o ) 2 k z 1 [ k o × s ^ ] c ω T α T M , o k y ,
f E e , x ( k x = 0 , k y ) = 1 k 1 2 [ ( k o ) 2 s ^ ( k e · s ^ ) k e ] ω c T α T E , e k y ,
f E e , y ( k x = 0 , k y ) = 1 k z 1 [ ( k o ) 2 s ^ ( k e · s ^ ) k e ] T α T M , e k y .
lim k y 0 f E o , x ( k x = 0 , k y ) = 0 ,
lim k y 0 [ k o × s ^ ] = k o s x e ^ y ,
lim k y 0 f E o , y ( k x = 0 , k y ) = 2 μ k 1 μ 1 k o + μ k 1 e ^ y .
lim k y 0 [ ( k o ) 2 s ^ ( k e · s ^ ) k e ] = ( k o ) 2 s x { e ^ x + s z s x [ 1 ( k z e ( 0 , 0 ) k o ) 2 ] e ^ z } ,
s z s x [ 1 ( k z e ( 0 , 0 ) k o ) 2 ] = s z s x [ 1 χ + 1 1 + χ s z 2 ] = s z s x χ ( s z 2 1 ) 1 + χ s z 2 = χ s x s z 1 + χ s z 2 ,
lim k x , k y 0 [ ( k o ) 2 s ^ ( k e · s ^ ) k e ] = ( k o ) 2 s x { e ^ x δ e ^ z } ; δ = χ s x s z 1 + χ s z 2 .
lim k y 0 f E e , y ( k x = 0 , k y ) = 0 ,
lim k y 0 f E e , x ( k x = 0 , k y ) = 2 μ k 1 μ k 1 + μ 1 k z e ( 0 , 0 ) { e ^ x δ e ^ z } ; k z e ( 0 , 0 ) = k o 1 + χ 1 + χ s z 2 ,

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