Abstract

We show that, for a TM (or p-state) Gaussian beam incident onto an absorbing medium at and around Brewster’s dip, the reflected beam always remains Gaussian and undergoes a Goos–Hänchen-like (GH) shift, an angular shift, a focal shift, and a beam-waist modification, provided that the beam is sufficiently collimated that the third-order change of the (logarithmic) reflection coefficient can be ignored in the angular range of beam divergence. For weak absorption, not only are a large negative GH shift and an odd-functioned-like focal shift with greater magnitude found but also the angular shift, though small by itself, is shown to give an even larger lateral net shift at a distance beyond the Rayleigh range.

© 2006 Optical Society of America

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References

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  1. H. M. Lai and S. W. Chan, "Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media," Opt. Lett. 27, 680-682 (2002).
    [CrossRef]
  2. F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 1, 333-334 (1947).
    [CrossRef]
  3. F. Goos and H. Hänchen, "Neumessung des Strahlversetzungseffeketes bei Totalreflexion," Ann. Phys. 5, 251-252 (1949).
    [CrossRef]
  4. K. Artmann, "Berechnung der Seitenversetzung des totalreflektierten Strahles," Ann. Phys. 2, 87-102 (1948).
    [CrossRef]
  5. C.-F. Li, "Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects," Phys. Rev. Lett. 91, 133903 (2003).
    [CrossRef] [PubMed]
  6. C.-F. Li and Q. Wang, "Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration," Phys. Rev. E 69, 055601(R) (2004).
    [CrossRef]
  7. L.-G. Wang and S.-Y. Zhu, "Large and negative Goos-Hänchen shifts from a weakly absorbing left-handed slab," J. Appl. Phys. 98, 043522 (2005).
    [CrossRef]
  8. L.-G. Wang and S.-Y. Zhu, "Large negative lateral shifts from Kretschmann-Raether configuration with left-handed materials," Appl. Phys. Lett. 87, 221102 (2005).
    [CrossRef]
  9. J. W. Ra, H. L. Bertoni, and L. B. Felsen, "Reflection and transmission of beams at a dielectric interface," SIAM J. Appl. Math. 24, 396-413 (1973).
    [CrossRef]
  10. Y. M. Antar and W. M. Boerner, "Gaussian beam interaction with a planar dielectric interface," Can. J. Phys. 52, 962-972 (1974).
  11. M. McGuirk and C. K. Carniglia, "An angular spectrum representation approach to the Goos-Hänchen shift," J. Opt. Soc. Am. 67, 103-107 (1977).
    [CrossRef]
  12. C. K. Carniglia and K. R. Brownstein, "Focal shift and ray model for total internal reflection," J. Opt. Soc. Am. 67, 121-123 (1977).
    [CrossRef]
  13. T. Tamir, "Nonspecular phenomena in beam fields reflected by multilayered media," J. Opt. Soc. Am. A 3, 558-564 (1986).
    [CrossRef]
  14. C. C. Chan and T. Tamir, "Angular shift of a Gaussian beam reflected near the Brewster angle," Opt. Lett. 10, 378-380 (1985).
    [CrossRef] [PubMed]
  15. J.-J. Greffet and C. Baylard, "Nonspecular reflection from a lossy dielectric," Opt. Lett. 18, 1129-1131 (1993).
    [CrossRef] [PubMed]
  16. H. M. Lai, F. C. Cheng, and W. K. Tang, "Goos-Hänchen effect around and off the critical angle," J. Opt. Soc. Am. A 3, 550-557 (1986).
    [CrossRef]
  17. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
    [CrossRef]
  18. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, 1971), pp. 304-321.
  19. T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by cylinder," J. Appl. Phys. 50, 41-46 (1979).
    [CrossRef]
  20. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1971), Chap. 3.
  21. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1974).
    [CrossRef]
  22. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  23. G. Hass, in American Institute of Physics Handbook, 3rd ed., D.E.Gray, ed. (McGraw-Hill, 1972), p. 6-135.
  24. R. F. Potter, in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), p. 474.
  25. H. R. Philipp and E. A. Taft, "Optical constants of germanium in the region 1to10 eV," Phys. Rev. 113, 1002-1005 (1959).
    [CrossRef]
  26. W. J. Wild and C. Lee Giles, "Goos-Hänchen shifts from absorbing media," Phys. Rev. A 25, 2099-2101 (1982).
    [CrossRef]
  27. D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
    [CrossRef]

2005 (2)

L.-G. Wang and S.-Y. Zhu, "Large and negative Goos-Hänchen shifts from a weakly absorbing left-handed slab," J. Appl. Phys. 98, 043522 (2005).
[CrossRef]

L.-G. Wang and S.-Y. Zhu, "Large negative lateral shifts from Kretschmann-Raether configuration with left-handed materials," Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

2004 (1)

C.-F. Li and Q. Wang, "Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration," Phys. Rev. E 69, 055601(R) (2004).
[CrossRef]

2003 (1)

C.-F. Li, "Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects," Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

2002 (1)

2000 (1)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

1993 (1)

1986 (2)

1985 (1)

1983 (1)

D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

1982 (1)

W. J. Wild and C. Lee Giles, "Goos-Hänchen shifts from absorbing media," Phys. Rev. A 25, 2099-2101 (1982).
[CrossRef]

1979 (2)

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

1977 (2)

1974 (2)

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1974).
[CrossRef]

Y. M. Antar and W. M. Boerner, "Gaussian beam interaction with a planar dielectric interface," Can. J. Phys. 52, 962-972 (1974).

1973 (1)

J. W. Ra, H. L. Bertoni, and L. B. Felsen, "Reflection and transmission of beams at a dielectric interface," SIAM J. Appl. Math. 24, 396-413 (1973).
[CrossRef]

1959 (1)

H. R. Philipp and E. A. Taft, "Optical constants of germanium in the region 1to10 eV," Phys. Rev. 113, 1002-1005 (1959).
[CrossRef]

1949 (1)

F. Goos and H. Hänchen, "Neumessung des Strahlversetzungseffeketes bei Totalreflexion," Ann. Phys. 5, 251-252 (1949).
[CrossRef]

1948 (1)

K. Artmann, "Berechnung der Seitenversetzung des totalreflektierten Strahles," Ann. Phys. 2, 87-102 (1948).
[CrossRef]

1947 (1)

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 1, 333-334 (1947).
[CrossRef]

Antar, Y. M.

Y. M. Antar and W. M. Boerner, "Gaussian beam interaction with a planar dielectric interface," Can. J. Phys. 52, 962-972 (1974).

Artmann, K.

K. Artmann, "Berechnung der Seitenversetzung des totalreflektierten Strahles," Ann. Phys. 2, 87-102 (1948).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Baylard, C.

Bertoni, H. L.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, "Reflection and transmission of beams at a dielectric interface," SIAM J. Appl. Math. 24, 396-413 (1973).
[CrossRef]

Boerner, W. M.

Y. M. Antar and W. M. Boerner, "Gaussian beam interaction with a planar dielectric interface," Can. J. Phys. 52, 962-972 (1974).

Brownstein, K. R.

Carniglia, C. K.

Chan, C. C.

Chan, S. W.

Cheng, F. C.

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Felsen, L. B.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, "Reflection and transmission of beams at a dielectric interface," SIAM J. Appl. Math. 24, 396-413 (1973).
[CrossRef]

Giles, C. Lee

W. J. Wild and C. Lee Giles, "Goos-Hänchen shifts from absorbing media," Phys. Rev. A 25, 2099-2101 (1982).
[CrossRef]

Goos, F.

F. Goos and H. Hänchen, "Neumessung des Strahlversetzungseffeketes bei Totalreflexion," Ann. Phys. 5, 251-252 (1949).
[CrossRef]

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 1, 333-334 (1947).
[CrossRef]

Greffet, J.-J.

Hänchen, H.

F. Goos and H. Hänchen, "Neumessung des Strahlversetzungseffeketes bei Totalreflexion," Ann. Phys. 5, 251-252 (1949).
[CrossRef]

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 1, 333-334 (1947).
[CrossRef]

Hass, G.

G. Hass, in American Institute of Physics Handbook, 3rd ed., D.E.Gray, ed. (McGraw-Hill, 1972), p. 6-135.

Kojima, T.

T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Lai, H. M.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1974).
[CrossRef]

Li, C.-F.

C.-F. Li and Q. Wang, "Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration," Phys. Rev. E 69, 055601(R) (2004).
[CrossRef]

C.-F. Li, "Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects," Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1974).
[CrossRef]

McGuirk, M.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1974).
[CrossRef]

Philipp, H. R.

H. R. Philipp and E. A. Taft, "Optical constants of germanium in the region 1to10 eV," Phys. Rev. 113, 1002-1005 (1959).
[CrossRef]

Potter, R. F.

R. F. Potter, in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), p. 474.

Ra, J. W.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, "Reflection and transmission of beams at a dielectric interface," SIAM J. Appl. Math. 24, 396-413 (1973).
[CrossRef]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, 1971), pp. 304-321.

Studna, A. A.

D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Taft, E. A.

H. R. Philipp and E. A. Taft, "Optical constants of germanium in the region 1to10 eV," Phys. Rev. 113, 1002-1005 (1959).
[CrossRef]

Tamir, T.

Tang, W. K.

Wang, L.-G.

L.-G. Wang and S.-Y. Zhu, "Large and negative Goos-Hänchen shifts from a weakly absorbing left-handed slab," J. Appl. Phys. 98, 043522 (2005).
[CrossRef]

L.-G. Wang and S.-Y. Zhu, "Large negative lateral shifts from Kretschmann-Raether configuration with left-handed materials," Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

Wang, Q.

C.-F. Li and Q. Wang, "Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration," Phys. Rev. E 69, 055601(R) (2004).
[CrossRef]

Wild, W. J.

W. J. Wild and C. Lee Giles, "Goos-Hänchen shifts from absorbing media," Phys. Rev. A 25, 2099-2101 (1982).
[CrossRef]

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Yanagiuchi, Y.

T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1971), Chap. 3.

Zhu, S.-Y.

L.-G. Wang and S.-Y. Zhu, "Large negative lateral shifts from Kretschmann-Raether configuration with left-handed materials," Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

L.-G. Wang and S.-Y. Zhu, "Large and negative Goos-Hänchen shifts from a weakly absorbing left-handed slab," J. Appl. Phys. 98, 043522 (2005).
[CrossRef]

Ann. Phys. (3)

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. 1, 333-334 (1947).
[CrossRef]

F. Goos and H. Hänchen, "Neumessung des Strahlversetzungseffeketes bei Totalreflexion," Ann. Phys. 5, 251-252 (1949).
[CrossRef]

K. Artmann, "Berechnung der Seitenversetzung des totalreflektierten Strahles," Ann. Phys. 2, 87-102 (1948).
[CrossRef]

Appl. Phys. Lett. (1)

L.-G. Wang and S.-Y. Zhu, "Large negative lateral shifts from Kretschmann-Raether configuration with left-handed materials," Appl. Phys. Lett. 87, 221102 (2005).
[CrossRef]

Can. J. Phys. (1)

Y. M. Antar and W. M. Boerner, "Gaussian beam interaction with a planar dielectric interface," Can. J. Phys. 52, 962-972 (1974).

J. Appl. Phys. (2)

T. Kojima and Y. Yanagiuchi, "Scattering of an offset two-dimensional Gaussian beam wave by cylinder," J. Appl. Phys. 50, 41-46 (1979).
[CrossRef]

L.-G. Wang and S.-Y. Zhu, "Large and negative Goos-Hänchen shifts from a weakly absorbing left-handed slab," J. Appl. Phys. 98, 043522 (2005).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Lett. (3)

Phys. Rev. (1)

H. R. Philipp and E. A. Taft, "Optical constants of germanium in the region 1to10 eV," Phys. Rev. 113, 1002-1005 (1959).
[CrossRef]

Phys. Rev. A (3)

W. J. Wild and C. Lee Giles, "Goos-Hänchen shifts from absorbing media," Phys. Rev. A 25, 2099-2101 (1982).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1974).
[CrossRef]

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Phys. Rev. B (1)

D. E. Aspnes and A. A. Studna, "Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV," Phys. Rev. B 27, 985-1009 (1983).
[CrossRef]

Phys. Rev. E (2)

C.-F. Li and Q. Wang, "Prediction of simultaneously large and opposite generalized Goos-Hänchen shifts for TE and TM light beams in an asymmetric double-prism configuration," Phys. Rev. E 69, 055601(R) (2004).
[CrossRef]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

C.-F. Li, "Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects," Phys. Rev. Lett. 91, 133903 (2003).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (1)

J. W. Ra, H. L. Bertoni, and L. B. Felsen, "Reflection and transmission of beams at a dielectric interface," SIAM J. Appl. Math. 24, 396-413 (1973).
[CrossRef]

Other (4)

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, 1971), pp. 304-321.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart and Winston, 1971), Chap. 3.

G. Hass, in American Institute of Physics Handbook, 3rd ed., D.E.Gray, ed. (McGraw-Hill, 1972), p. 6-135.

R. F. Potter, in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Academic, 1985), p. 474.

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Figures (9)

Fig. 1
Fig. 1

Original X Y Z coordinate system, the incident X Y Z coordinate system, and the reflection (left-handed) X Y Z coordinate system, all with the same y axis directed into the plane of the figure and the same origin O. Note that the x y plane separates the two media of real ϵ 1 and complex ϵ 2 = ϵ r + i ϵ i .

Fig. 2
Fig. 2

Reflection of a beam from an absorbing medium ( z > 0 ) is shown schematically by the incident beam axis and the reflected beam axis, with the GH shift S, the focal shift Δ f , the angular shift Δ θ , and the focus F indicated. (Size not to scale.)

Fig. 3
Fig. 3

GH shifts in units of wavelength versus angle (in radians) for ϵ ̃ r = 1.8 and ϵ ̃ i = 0.36 (curve 1), 0.18 (curve 2), and 0.09 (curve 3) in the TM case, where θ b 0.931 (in radians) 53.3 ° . Larger negative maximum shift for smaller ϵ ̃ i is apparent. Curve a, which is slightly above the abscissa, is the already tenfold magnified shift for ϵ ̃ i = 0.18 in the TE case.

Fig. 4
Fig. 4

GH shifts in units of wavelength versus angle (in radians) for ϵ ̃ r = 0.44 and ϵ ̃ i = 0.088 (curve 4), 0.044 (curve 5), and 0.022 (curve 6), where θ b 0.586 (in radians) 33.6 ° and θ c 0.725 (in radians) 41.6 ° . Positive shifts above θ c are the usual GH shifts in total internal reflection now modified by the presence of absorption. Larger negative shift around θ b for smaller ϵ ̃ i is apparent.

Fig. 5
Fig. 5

Computed GH shifts for reflection from germanium: curve 7 is for ϵ 2 = 23.17 + 18.85 i (i.e., refractive index 5.15 + 1.83 i ) at 0.562 μ m ,[23] and curve 8 is for ϵ 2 = 30.36 + 10.43 i (i.e., refractive index 5.588 + 0.933 i ) at 0.6199 μ m ,[24] ϵ 1 = 1 being assumed.

Fig. 6
Fig. 6

Γ versus angle for the eight ϵ ̃ ’s, indicated by curve 1 to curve 8, corresponding exactly to those in the previous three figures. Odd-functioned shape and large values near each Brewster’s dip are noted. Curves 7 and 8 are already tenfold magnified.

Fig. 7
Fig. 7

Negative inverse-linear curve gives the shift according to the simple formula, and the eight crosses give the computed maximum shifts for the eight cases in Subsection 3A. Also, shown as small triangles (against the right-hand ordinate) are the fractional deviations of S b from S m (in percentage) for the eight ϵ ̃ ’s.

Fig. 8
Fig. 8

Curves 7f and 8f give the focal shifts in units of wavelength versus angle while curves 7g and 8g, already magnified tenfold, are the two corresponding GH shifts in Fig. 5 merely reproduced here for reference.

Fig. 9
Fig. 9

Net lateral shift of the reflected beam spot versus angle for k σ = 10 3 and ϵ 2 ϵ 1 = 1.8 + 0.18 i at four different distances: z z ¯ R = 0.1 (curve 1), 1 (curve 2), 5 (curve 3), and 10 (curve 4).

Equations (33)

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

S = d ϕ k d θ ,
S b λ = ϵ r + 1 π ϵ r ϵ r ϵ i ,
r = ν k z k z ( t ) ν k z + k z ( t ) = ν cos θ ϵ ̃ sin 2 θ ν cos θ + ϵ ̃ sin 2 θ ,
ν = 1 or ν = ϵ ̃
F ( i ) ( x , z ) = k k G ( k x ) exp ( i k x x + i k z z ) d k x y ̂ for z 0
F ( r ) ( x , z ) = k k r ( k x ) G ( k x ) exp ( i k x x + i k z z ) d k x y ̂
for z 0
k x = k x cos θ 0 + k z sin θ 0 , k z = k x sin θ 0 + k z cos θ 0 ,
G ( k x ) = exp ( k x 2 σ 2 2 )
k σ 1 or λ σ 2 π
F ( i ) ( x , z ) = 2 π σ σ z exp ( x 2 2 σ z 2 ) exp [ i Ψ ( x , z ) ] exp ( i k z ) y ̂
σ z σ 1 + ( z z R ) 2
z R k σ 2
Ψ ( x , z ) z x 2 2 z R σ z 2 1 2 tan 1 z z R
r = exp ( ln ρ + i ϕ ) = ρ 0 exp ( i ϕ 0 ) exp ( α k x + β k x 2 2 ) exp ( i ϕ 1 k x + i ϕ 2 k x 2 2 ) ,
α d d k x ln ρ , β d 2 d k x 2 ln ρ , ϕ 1 d ϕ d k x , ϕ 2 d 2 ϕ d k x 2
1 6 d ρ ρ d θ [ 2 ( d ρ ρ d θ ) 2 3 d 2 ρ ρ d θ 2 ] + d 3 ρ ρ d θ 3 ( k σ ) 3
1 6 d 3 ϕ d θ 3 ( k σ ) 3
F ( r ) ( x , z ) = ρ 0 exp ( α 2 2 σ ¯ 2 ) 2 π σ ¯ σ ¯ z exp ( i ϕ 0 ) exp [ ( x ¯ α z ¯ z ¯ R ) 2 2 σ ¯ z 2 ] exp [ i Ψ ¯ ( x ¯ , z ¯ ) ] exp ( i k z ) y ̂
x ¯ x + ϕ 1 , z ¯ z k ϕ 2
σ ¯ σ 1 β σ 2 , σ ¯ z σ ¯ 1 + ( z ¯ z ¯ R ) 2
z ¯ R k σ ¯ 2
Ψ ¯ ( x ¯ , z ¯ ) z ¯ ( x ¯ 2 α 2 ) + 2 α z ¯ R x ¯ 2 z ¯ R σ ¯ z 2 1 2 tan 1 z ¯ z ¯ R
x ¯ = α z ¯ R z ¯ or x = ϕ 1 + α z ¯ R ( z k ϕ 2 )
S x z ¯ x ¯ + α z ¯ R z ¯ R σ ¯ z 2 , S z k [ 1 + O ( 1 k 2 σ ¯ 2 ) ] ,
S x S z = α z ¯ R
S ϕ 1 = λ 2 π d ϕ d θ ,
Δ θ tan 1 α z ¯ R α z ¯ R = 1 k 2 σ ¯ 2 d ρ ρ 0 d θ ,
Δ f k ϕ 2 = λ 2 π d 2 ϕ d θ 2
k 2 σ 2 Δ θ i k S = 2 ν ( 1 ϵ ̃ ) sin θ ϵ ̃ sin 2 θ ( ν 2 cos 2 θ + sin 2 θ ϵ ̃ ) ,
Δ f = Im { 2 ν ( 1 ϵ ̃ ) cos θ [ ϵ ̃ ( ν 2 ϵ ̃ ) + ( ν 2 1 ) ( ϵ ̃ 2 sin 2 θ ) sin 2 θ ] k ( ϵ ̃ sin 2 θ ) 3 2 ( ν 2 cos 2 θ + sin 2 θ ϵ ̃ ) 2 } ,
Σ m π ϵ ̃ r ϵ ̃ r + 1 S m λ , Σ b π ϵ ̃ r ϵ ̃ r + 1 S b λ ,
S N = S + Δ θ z = S + α z z ¯ R .

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