Abstract

We study the transmission of a two-dimensional (2-D) TM Gaussian beam through a plane interface between an isotropic medium (e.g., air) and a uniaxially anisotropic crystal. The optic axis of the crystal is taken to be in the plane of incidence but is arbitrarily oriented relative to the interface normal. We show that, in the paraxial approximation, a nontruncated transmitted 2-D TM Gaussian beam inside a uniaxial crystal can be expressed in a form similar to that of a scalar Gaussian beam that propagates in a homogeneous medium. We also show that the transmitted beam corresponding to an incident 2-D TM Gaussian beam with its main propagation direction along the interface normal is tilted inside the crystal by the same angle as is the transmitted axial ray that corresponds to a normally incident ray.

© 2001 Optical Society of America

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References

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  1. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, IEEE Series on Electromagnetic Waves (Institute of Electrical and Electronics Engineers, New York, 1994).
  2. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 780–788 (1976).
    [CrossRef]
  3. E. Lalor, “The angular-spectrum representation of electromagnetic fields in crystals. II,” J. Math. Phys. 13, 443–449 (1972).
    [CrossRef]
  4. M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. 13, 443–449 (1971).
  5. J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
    [CrossRef]
  6. H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  7. 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]
  8. 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. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  9. P. Török, P. Varga, 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]
  10. T. D. Visser, S. H. Wiersma, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
    [CrossRef]
  11. V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
    [CrossRef]
  12. S. H. Wiersma, P. Török, T. D. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
    [CrossRef]
  13. D. Jiang, J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
    [CrossRef]
  14. J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
    [CrossRef]
  15. V. Dhayalan, J. J. Stamnes, “Comparison of exact and asymptotic results for the focusing of electromagnetic waves through a plane interface,” Appl. Opt. 39, 6332–6340 (2000).
    [CrossRef]
  16. 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]
  17. J. J. Stamnes, S. Glen have prepared a paper titled “Reflection and refraction of electromagnetic waves into a biaxial crystal” for submission to J. Opt. Soc. Am. A.
  18. J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
    [CrossRef]
  19. D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in uniaxial crystals,” Opt. Commun. 163, 55–71 (1999).
    [CrossRef]
  20. D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–344 (2000).
    [CrossRef]
  21. D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart & Winston, New York, 1969).
  22. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).
  23. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 12.5, pp. 331–350.
  24. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
    [CrossRef]
  25. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  26. C. J. R. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
    [CrossRef]
  27. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 4.2.
  28. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Chap. 9, Eqs. (9.7a)–(9.7g).
  29. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 7.2.
  30. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]

2000

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

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–344 (2000).
[CrossRef]

1999

1998

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

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

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

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

1997

1996

1995

1984

1983

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

1977

1976

1972

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

1971

M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. 13, 443–449 (1971).

Bell, W. E.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart & Winston, New York, 1969).

Booker, G. R.

Dhayalan, V.

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

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

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, IEEE Series on Electromagnetic Waves (Institute of Electrical and Electronics Engineers, New York, 1994).

Gasper, J.

Glen, S.

J. J. Stamnes, S. Glen have prepared a paper titled “Reflection and refraction of electromagnetic waves into a biaxial crystal” for submission to J. Opt. Soc. Am. A.

Jiang, D.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–344 (2000).
[CrossRef]

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

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

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

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, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. 13, 443–449 (1971).

Lee, S. W.

Ling, H.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, IEEE Series on Electromagnetic Waves (Institute of Electrical and Electronics Engineers, New York, 1994).

Nelson, D. F.

M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. 13, 443–449 (1971).

Nemeth, G.

Pedersen, H. M.

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

Saghafi, S.

Seshadri, S. R.

Sheppard, C. J. R.

Sherman, G. C.

Sinclair, D. C.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart & Winston, New York, 1969).

Spjelkavik, B.

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

Stamnes, J. J.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–344 (2000).
[CrossRef]

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

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

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

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

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

J. J. Stamnes, S. Glen have prepared a paper titled “Reflection and refraction of electromagnetic waves into a biaxial crystal” for submission to J. Opt. Soc. Am. A.

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 4.2.

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Chap. 9, Eqs. (9.7a)–(9.7g).

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 7.2.

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 12.5, pp. 331–350.

Török, P.

Varga, P.

Visser, T. D.

Wiersma, S. H.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).

Appl. Opt.

J. Math. Phys.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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

H. Ling, 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, 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. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, 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, 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. 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. Sheppard, S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
[CrossRef]

Opt. Acta

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

Opt. Commun.

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

D. Jiang, 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, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves into a uniaxial crystal,” Opt. Commun. 174, 321–344 (2000).
[CrossRef]

Phys. Rev.

M. Lax, D. F. Nelson, “Linear and nonlinear electrodynamics in elastic anisotropic dielectrics,” Phys. Rev. 13, 443–449 (1971).

Pure Appl. Opt.

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

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

Other

J. J. Stamnes, S. Glen have prepared a paper titled “Reflection and refraction of electromagnetic waves into a biaxial crystal” for submission to J. Opt. Soc. Am. A.

D. C. Sinclair, W. E. Bell, Gas Laser Technology (Holt, Rinehart & Winston, New York, 1969).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, New York, 1976).

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 12.5, pp. 331–350.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, IEEE Series on Electromagnetic Waves (Institute of Electrical and Electronics Engineers, New York, 1994).

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 4.2.

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Chap. 9, Eqs. (9.7a)–(9.7g).

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), Sec. 7.2.

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Figures (9)

Fig. 1
Fig. 1

Two-dimensional Gaussian beam diffracted through an aperture of width 2a in the plane z=0 and subsequently transmitted through a plane interface into a uniaxial crystal with the principal permittivities o and e and with the optic axis along the unit vector sˆ in the plane of incidence (the xz plane).

Fig. 2
Fig. 2

Axial intensities in LiNbO3 with its optic axis making an angle θ with the interface normal of θ=0 (solid curve), θ=π/4 (dashed curve), and θ=π/2 (dashed–dotted curve).

Fig. 3
Fig. 3

Axial variation of the beam width σ in LiNbO3 for the cases in which the angle θ between the optic axis and the interface normal is θ=0 (solid curve), θ=π/4 (dashed curve), and θ=π/2 (dashed–dotted curve).

Fig. 4
Fig. 4

Axial variation of the radius of curvature R of the transmitted beam in LiNbO3 for the cases in which the angle θ between the optic axis and the interface normal is θ=0 (solid curve), θ=π/4 (dashed curve), and θ=π/2 (dashed–dotted curve).

Fig. 5
Fig. 5

Axial variation of the beam width σ in LiNbO3 with its optic axis making an angle of θ=π/4 with the interface normal by use of Eq. (34) (solid curve) and expression (49) (dashed curve).

Fig. 6
Fig. 6

Axial variation of the radius of curvature R of the transmitted beam in LiNbO3 with its optic axis making an angle of θ=π/4 with the interface normal by use of Eq. (33) (solid curve) and Eq. (51) (dashed curve).

Fig. 7
Fig. 7

Copolarized axial intensities in LiNbO3 with its optic axis making an angle of θ=π/4 with the interface normal by use of Eq. (32) (solid curve) and Eq. (52) (dashed curve).

Fig. 8
Fig. 8

Axial variation of the relative width of the transmitted beam Rσ in Eqs. (53) in LiNbO3 with its optic axis making angles of θ=π/6 (solid curve), θ=π/4 (dashed curve), θ=π/3 (dashed–dotted curve), and θ=π/2 (dotted curve).

Fig. 9
Fig. 9

Variation of Rf in Eqs. (53) in LiNbO3 with the angle θ between the optic axis of the uniaxial crystal and the interface normal.

Equations (64)

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u(x, z)=1(iλ)1/2-aaui(x, 0) zRexp(ikR)Rdx,
ui(x, 0)=exp(-x2/2σ02),
u(x, z)=exp(ikz)I(x, z, λ, σ0, a),
I(x, z, λ, σ0, a)=exp(ikx2/2z)(iλz)1/2-aaexp[-ϕ(x)]dx
ϕ(x)=a1x2+b1x,
a1=121σ02-ikz,
b1=ikxz.
u(x, z)=exp(ikz)I(x, z, λ, σ0, )
=σ0iσ(z)1/2exp-x22σ2(z)×expikz+x22R(z)×expi 12arctan(kσ02/z),
R(z)=z+k2σ04z,σ2(z)=σ02+(z/kσ0)2.
sˆ=sxe^x+sze^z=e^xsin θ+e^zcos θ.
Et(x, z)=Eet(x, z)=-Exi(x, 0)FEe,x(x)dx,
FEe,x(x)=12π-fEe,x[kx(x)]exp{ihe[kx(x)]}dkx,
he(kx)=kx(x-x)+kz1z0+kze(z-z0),
fEe,x(kx)=-Tαeekxkz1 [(ko)2(sˆ)-(ket  sˆ)ket],
kz1=(k12-kx2)1/2,k12=ω2c2 1μ1,
ket=e^xkx+e^zkze,
kze=-δkx+β,δ=(ko)2ke2(θ) χ cos θ sin θ,
β=(koke)2ke4(θ) [ke2(θ)-kx2],
ke2(θ)=(ko)2(1+χ cos2 θ),
ke(0)=ke,ke(π/2)=ko,
χ=(ke)2(ko)2-1,(kp)2=ω2c2 μp,(p=o, e).
Exi(x, 0)=E0exp(-x2/2σ02)if|x|<a0if|x|>a,
Tαee=2μk12kxkz1μ1(ko)2kz1Aet-μk12Bet,
Aet=kxcos θ-kzesin θ,
Bet=(ko)2sin θ-kx(kxsin θ+kzecos θ).
FEe,x(x)=12π2π|he(kxs)|1/2fEe,x(kxs)×expihe(kxs)+i π4sgn[he(kxs)],
Eet(x, z)-aaG(x)exp[iH(x)]dx,
G(x)=E0fEe,x[kxs(x)]{2π|he[kxs(x)]|}1/2exp-x22σ02,
H(x)=he[kxs(x)]+π4sgn[he(kxs)].
he(kx)=k1Z+kx[x-x-δ(z-z0)]-kx22k1 Z˜,
Z=z0+kokek1ke(θ) (z-z0),
Z˜=z0+k1koke[ke(θ)]3 (z-z0).
kxs=k1Z˜ [x-x-δ(z-z0)],
he(kxs)=k1Z+[x-x-δ(z-z0)]22Z˜.
he=-λ12π Z˜,
Eet(x, z)t0fEe,x(kx=0)exp(ik1Z)I(X˜, Z˜, λ1, σ0, a),
X˜=x-δ(z-z0)
Eet(x, z)=E0fEe,x(kx=0)σ0iσ(Z˜)1/2×exp-[x-δ(z-z0)]22σ2(Z˜)×expik1Z+[x-δ(z-z0)]22R(Z˜)×exp[i12arctan(k1σ02/Z˜)],
R(Z˜)=Z˜+k12σ04Z˜,
σ2(Z˜)=σ02×(Z˜/k1σ0)2.
fEe,x(kx=0)=21+μ1μkokek1ke(θ) (e^x-δe^z).
β=(ko)2-(ko)2(ke)2 kx2=(kze)2.
fEe,x(kx=0)=21+μ1koμk1e^x.
β=(ke)2-(ke)2(ko)2 kx2=(kze)2.
fEe,x(kx=0)=21+μ1keμk1e^x.
S^et=ket+χ(ket  sˆ)sˆ|ket+χ(ket  sˆ)sˆ|.
tan γ=S^xetS^zet=χ sin θ cos θ1+χ cos2 θ.
|tan γmax| =|χ|2(χ+1)1/2.
Et(x, z)=E0fEx(kx=0)σ0iσ(Z˜)1/2exp-x22σ2(Z˜)
×expik1Z+x22R(Z˜)
×exp[i12arctan(k1σ02/Z˜)],
fEx=21+μ1kμk1e^x.
Z˜(θ)=Z˜0+k1koke[ke(θ)]31-ke(θ)ke3(z-z0),
Z˜0=Z˜(θ=0)=z0+k1ko(ke)2 (z-z0).
Z˜(θ)=Z˜0+ΔZ˜,
ΔZ˜=32k1ko χ(z-z0)sin2 θ.
σ2[Z˜(θ)]=σ02+Z˜0+ΔZ˜k1σ02σ02+2 ΔZ˜Z˜0Z˜0k1σ02.
Δσ2=σ2[Z˜(θ)]-σ2[Z˜(θ=0)]=3Z˜0(z-z0) k1ko χ 1(k1σ0)2sin2 θ.
R[Z˜(θ)]=R(Z˜0)+ΔZ˜1-k12σ04Z˜02.
Exet(x=0, z, θ)Exet(x=0, z, θ=0)2=Rf Rσ,
Rf=fExe,x(kx=0, θ)fExe,x(kx=0, θ=0)2=11+Cχ sin2 θ2,
Rσ=σ(Z˜0)σ[Z˜(θ)],
C=142μ1koμk1+μ1ko.

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