Abstract

Axicons in oblique illumination produce broadened focal lines, a problem, e.g., in scanning applications. A compact mathematical description of the focal segment is presented, for the first time, to our knowledge, and the results are compared with elliptical axicons in normal illumination. In both cases, analytical expressions in the form of asteroid curves are obtained from asymptotic wave theory and caustic surfaces. The results are confirmed by direct diffraction simulations and by experiments. In addition we show that at a fixed angle an elliptical axicon can be used to compensate for the adverse effects of oblique illumination.

© 2003 Optical Society of America

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References

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  1. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  2. L. M. Soroko, Meso-Optics—Foundations and Applications (World Scientific, Singapore, 1996), Chap. 2 and references therein.
  3. Z. Jaroszewicz, Axicons: Design and Propagation Properties, Research & Development Treatises, Vol. 5 (SPIE Polish Chapter, Warsaw, 1997).
  4. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef] [PubMed]
  5. J. A. Davis, E. Carcole, D. M. Cottrell, “Range-finding by triangulation with nondiffracting beams,” Appl. Opt. 35, 2159–2161 (1996).
    [CrossRef] [PubMed]
  6. G. Bickel, G. Haüsler, M. Haul, “Triangulation with extended range of depth,” Opt. Eng. 24, 975–977 (1985).
    [CrossRef]
  7. G. Haüsler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
    [CrossRef] [PubMed]
  8. I. Golub, R. Tremblay, “Light focusing and guiding by an axicon-pair generated tubular light beam,” J. Opt. Soc. Am. B 7, 1264–1267 (1990).
    [CrossRef]
  9. J. W. Ogland, “Mirror system for uniform beam transformation in high-power annular lasers,” Appl. Opt. 17, 2917–2923 (1978).
    [CrossRef] [PubMed]
  10. R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
    [CrossRef]
  11. J. Arlt, T. Hitomi, K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
    [CrossRef]
  12. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  13. J. Arlt, K. Dholakia, “Generation of high-order Besel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
    [CrossRef]
  14. R. Arimoto, C. Saloma, T. Tanaka, S. Kawata, “Imaging properties of axicon in a scanning optical system,” Appl. Opt. 31, 6653–6657 (1992).
    [CrossRef] [PubMed]
  15. Z. Bin, L. Zhu, “Diffraction property of an axicon in oblique illumination,” Appl. Opt. 37, 2563–2568 (1998).
    [CrossRef]
  16. T. Tanaka, S. Yamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
    [CrossRef]
  17. Z. Jaroszewicz, A. Thaning, A. T. Friberg, “Focal segments of obliquely illuminated axicons,” European Optical Society, Topical Meetings Digest Series 30, 82–83 (2001).
  18. A. Thaning, Z. Jaroszewicz, A. T. Friberg, “Axicon focusing in oblique illumination,” ICO XIX: Optics for the Quality of Life, A. Consortini, G. C. Righini, eds., Proc. SPIE4829, 295–296 (2002).
  19. S. Bara, Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, “Method for scaling the output focal curves by computer generated zone plates,” Opt. Laser Technol. 23, 303–307 (1991).
    [CrossRef]
  20. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Chaps. 8 and 9, Sec. 10.3.
  21. M. Born, E. Wolf, Principles of Optics, 7th edition (Cambridge University, Cambridge, UK, 1999), Sec. 8.3.2 and Appendix III.
  22. J. J. Stamnes, “Diffraction, asymptotics and catastrophes,” Opt. Acta 29, 823–842 (1982).
    [CrossRef]
  23. N. C. Albertsen, P. Balling, N. E. Jensen, “Caustics and caustic corrections to a field diffracted by a curved edge,” IEEE Trans. Antennas Propag. 25, 297–303 (1977).
    [CrossRef]
  24. D. Leseberg, C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, 3020–3024 (1988).
    [CrossRef] [PubMed]
  25. Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, C. Gomez-Reino, “Nonparaxial analytical solution for the generation of focal curves,” J. Mod. Opt. 43, 617–637 (1996).
    [CrossRef]
  26. B. Lunitz, “Strahlformung und-führung mit mikrooptischen Elementen und Systemen,” Ph.D dissertation (Fern Universität, Hagen, Germany, 2000) (in German).

2001 (1)

Z. Jaroszewicz, A. Thaning, A. T. Friberg, “Focal segments of obliquely illuminated axicons,” European Optical Society, Topical Meetings Digest Series 30, 82–83 (2001).

2000 (3)

J. Arlt, T. Hitomi, K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

T. Tanaka, S. Yamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
[CrossRef]

J. Arlt, K. Dholakia, “Generation of high-order Besel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

1998 (1)

1996 (2)

J. A. Davis, E. Carcole, D. M. Cottrell, “Range-finding by triangulation with nondiffracting beams,” Appl. Opt. 35, 2159–2161 (1996).
[CrossRef] [PubMed]

Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, C. Gomez-Reino, “Nonparaxial analytical solution for the generation of focal curves,” J. Mod. Opt. 43, 617–637 (1996).
[CrossRef]

1992 (2)

1991 (1)

S. Bara, Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, “Method for scaling the output focal curves by computer generated zone plates,” Opt. Laser Technol. 23, 303–307 (1991).
[CrossRef]

1990 (1)

1989 (1)

1988 (2)

1985 (1)

G. Bickel, G. Haüsler, M. Haul, “Triangulation with extended range of depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

1982 (1)

J. J. Stamnes, “Diffraction, asymptotics and catastrophes,” Opt. Acta 29, 823–842 (1982).
[CrossRef]

1979 (1)

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

1978 (1)

1977 (1)

N. C. Albertsen, P. Balling, N. E. Jensen, “Caustics and caustic corrections to a field diffracted by a curved edge,” IEEE Trans. Antennas Propag. 25, 297–303 (1977).
[CrossRef]

1954 (1)

Albertsen, N. C.

N. C. Albertsen, P. Balling, N. E. Jensen, “Caustics and caustic corrections to a field diffracted by a curved edge,” IEEE Trans. Antennas Propag. 25, 297–303 (1977).
[CrossRef]

Arimoto, R.

Arlt, J.

J. Arlt, K. Dholakia, “Generation of high-order Besel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

J. Arlt, T. Hitomi, K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

Balling, P.

N. C. Albertsen, P. Balling, N. E. Jensen, “Caustics and caustic corrections to a field diffracted by a curved edge,” IEEE Trans. Antennas Propag. 25, 297–303 (1977).
[CrossRef]

Bara, S.

J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
[CrossRef] [PubMed]

S. Bara, Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, “Method for scaling the output focal curves by computer generated zone plates,” Opt. Laser Technol. 23, 303–307 (1991).
[CrossRef]

Bickel, G.

G. Bickel, G. Haüsler, M. Haul, “Triangulation with extended range of depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

Bin, Z.

Blanchard, M.

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th edition (Cambridge University, Cambridge, UK, 1999), Sec. 8.3.2 and Appendix III.

Carcole, E.

Cottrell, D. M.

D’Astous, Y.

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

Davis, J. A.

Dholakia, K.

J. Arlt, T. Hitomi, K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

J. Arlt, K. Dholakia, “Generation of high-order Besel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

Frère, C.

Friberg, A. T.

Z. Jaroszewicz, A. Thaning, A. T. Friberg, “Focal segments of obliquely illuminated axicons,” European Optical Society, Topical Meetings Digest Series 30, 82–83 (2001).

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

A. Thaning, Z. Jaroszewicz, A. T. Friberg, “Axicon focusing in oblique illumination,” ICO XIX: Optics for the Quality of Life, A. Consortini, G. C. Righini, eds., Proc. SPIE4829, 295–296 (2002).

Golub, I.

Gomez-Reino, C.

Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, C. Gomez-Reino, “Nonparaxial analytical solution for the generation of focal curves,” J. Mod. Opt. 43, 617–637 (1996).
[CrossRef]

Haul, M.

G. Bickel, G. Haüsler, M. Haul, “Triangulation with extended range of depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

Haüsler, G.

G. Haüsler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
[CrossRef] [PubMed]

G. Bickel, G. Haüsler, M. Haul, “Triangulation with extended range of depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

Heckel, W.

Hitomi, T.

J. Arlt, T. Hitomi, K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

Jaroscewicz, Z.

Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, C. Gomez-Reino, “Nonparaxial analytical solution for the generation of focal curves,” J. Mod. Opt. 43, 617–637 (1996).
[CrossRef]

S. Bara, Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, “Method for scaling the output focal curves by computer generated zone plates,” Opt. Laser Technol. 23, 303–307 (1991).
[CrossRef]

Jaroszewicz, Z.

Z. Jaroszewicz, A. Thaning, A. T. Friberg, “Focal segments of obliquely illuminated axicons,” European Optical Society, Topical Meetings Digest Series 30, 82–83 (2001).

J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
[CrossRef] [PubMed]

Z. Jaroszewicz, Axicons: Design and Propagation Properties, Research & Development Treatises, Vol. 5 (SPIE Polish Chapter, Warsaw, 1997).

A. Thaning, Z. Jaroszewicz, A. T. Friberg, “Axicon focusing in oblique illumination,” ICO XIX: Optics for the Quality of Life, A. Consortini, G. C. Righini, eds., Proc. SPIE4829, 295–296 (2002).

Jensen, N. E.

N. C. Albertsen, P. Balling, N. E. Jensen, “Caustics and caustic corrections to a field diffracted by a curved edge,” IEEE Trans. Antennas Propag. 25, 297–303 (1977).
[CrossRef]

Kawata, S.

Kolodziejczyk, A.

Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, C. Gomez-Reino, “Nonparaxial analytical solution for the generation of focal curves,” J. Mod. Opt. 43, 617–637 (1996).
[CrossRef]

J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
[CrossRef] [PubMed]

S. Bara, Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, “Method for scaling the output focal curves by computer generated zone plates,” Opt. Laser Technol. 23, 303–307 (1991).
[CrossRef]

Leseberg, D.

Lunitz, B.

B. Lunitz, “Strahlformung und-führung mit mikrooptischen Elementen und Systemen,” Ph.D dissertation (Fern Universität, Hagen, Germany, 2000) (in German).

McLeod, J. H.

Ogland, J. W.

Roy, G.

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

Saloma, C.

Sochacki, J.

Soroko, L. M.

L. M. Soroko, Meso-Optics—Foundations and Applications (World Scientific, Singapore, 1996), Chap. 2 and references therein.

Stamnes, J. J.

J. J. Stamnes, “Diffraction, asymptotics and catastrophes,” Opt. Acta 29, 823–842 (1982).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Chaps. 8 and 9, Sec. 10.3.

Sypek, M.

Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, C. Gomez-Reino, “Nonparaxial analytical solution for the generation of focal curves,” J. Mod. Opt. 43, 617–637 (1996).
[CrossRef]

S. Bara, Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, “Method for scaling the output focal curves by computer generated zone plates,” Opt. Laser Technol. 23, 303–307 (1991).
[CrossRef]

Tanaka, T.

Thaning, A.

Z. Jaroszewicz, A. Thaning, A. T. Friberg, “Focal segments of obliquely illuminated axicons,” European Optical Society, Topical Meetings Digest Series 30, 82–83 (2001).

A. Thaning, Z. Jaroszewicz, A. T. Friberg, “Axicon focusing in oblique illumination,” ICO XIX: Optics for the Quality of Life, A. Consortini, G. C. Righini, eds., Proc. SPIE4829, 295–296 (2002).

Tremblay, R.

I. Golub, R. Tremblay, “Light focusing and guiding by an axicon-pair generated tubular light beam,” J. Opt. Soc. Am. B 7, 1264–1267 (1990).
[CrossRef]

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

Turunen, J.

Vasara, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th edition (Cambridge University, Cambridge, UK, 1999), Sec. 8.3.2 and Appendix III.

Yamoto, S.

T. Tanaka, S. Yamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
[CrossRef]

Zhu, L.

Appl. Opt. (7)

Appl. Phys. B (1)

J. Arlt, T. Hitomi, K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71, 549–554 (2000).
[CrossRef]

European Optical Society, Topical Meetings Digest Series (1)

Z. Jaroszewicz, A. Thaning, A. T. Friberg, “Focal segments of obliquely illuminated axicons,” European Optical Society, Topical Meetings Digest Series 30, 82–83 (2001).

IEEE Trans. Antennas Propag. (1)

N. C. Albertsen, P. Balling, N. E. Jensen, “Caustics and caustic corrections to a field diffracted by a curved edge,” IEEE Trans. Antennas Propag. 25, 297–303 (1977).
[CrossRef]

J. Mod. Opt. (1)

Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, C. Gomez-Reino, “Nonparaxial analytical solution for the generation of focal curves,” J. Mod. Opt. 43, 617–637 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Acta (1)

J. J. Stamnes, “Diffraction, asymptotics and catastrophes,” Opt. Acta 29, 823–842 (1982).
[CrossRef]

Opt. Commun. (3)

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2-TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

J. Arlt, K. Dholakia, “Generation of high-order Besel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

T. Tanaka, S. Yamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
[CrossRef]

Opt. Eng. (1)

G. Bickel, G. Haüsler, M. Haul, “Triangulation with extended range of depth,” Opt. Eng. 24, 975–977 (1985).
[CrossRef]

Opt. Laser Technol. (1)

S. Bara, Z. Jaroscewicz, A. Kolodziejczyk, M. Sypek, “Method for scaling the output focal curves by computer generated zone plates,” Opt. Laser Technol. 23, 303–307 (1991).
[CrossRef]

Other (6)

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Chaps. 8 and 9, Sec. 10.3.

M. Born, E. Wolf, Principles of Optics, 7th edition (Cambridge University, Cambridge, UK, 1999), Sec. 8.3.2 and Appendix III.

A. Thaning, Z. Jaroszewicz, A. T. Friberg, “Axicon focusing in oblique illumination,” ICO XIX: Optics for the Quality of Life, A. Consortini, G. C. Righini, eds., Proc. SPIE4829, 295–296 (2002).

L. M. Soroko, Meso-Optics—Foundations and Applications (World Scientific, Singapore, 1996), Chap. 2 and references therein.

Z. Jaroszewicz, Axicons: Design and Propagation Properties, Research & Development Treatises, Vol. 5 (SPIE Polish Chapter, Warsaw, 1997).

B. Lunitz, “Strahlformung und-führung mit mikrooptischen Elementen und Systemen,” Ph.D dissertation (Fern Universität, Hagen, Germany, 2000) (in German).

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Figures (11)

Fig. 1
Fig. 1

Geometry of an annular-aperture axicon and illustration of the notations. The focal-line segment indicated on the optical axis extends from z = d 1 to z = d 2. The inner and outer radii of the annular aperture are R 1 and R 2 and the convergence angle is θ. The origins of both the (x′, y′, z) and the (x, y, z) coordinate systems are at the center of the axicon.

Fig. 2
Fig. 2

Example of the central part of an elliptical, linear axicon, displaying the phase function ϕ(x′, y′) modulo 2π. Bright areas indicate high values and dark areas low values.

Fig. 3
Fig. 3

Caustic surface describing the focal segment for an elliptical axicon, where α = 0.02 and β = 0.8 for 20 mm ≤ z ≤ 40 mm. The surface was evaluated from Eq. (11).

Fig. 4
Fig. 4

Geometry and notations for oblique illumination. The axicon is in the (x′, y′) plane, which has been rotated by an angle ϕ about the x′ axis. The distance between the two points (x′, y′, 0) and (x, y, z) is r.

Fig. 5
Fig. 5

Numerical simulations of the intensity in the focal region of an elliptical axicon for α = 0.02, β = 0.8, and z = 20 mm. The white outline is the shape predicted in Eq. (11). The simulations were performed with the Fresnel diffraction integral in expression (3).

Fig. 6
Fig. 6

Same as Fig. 5 but for z = 40 mm.

Fig. 7
Fig. 7

Experimental transverse intensity distribution for the elliptical axicon illuminated perpendicularly. The annular diffractive axicon, manufactured by photolithography in two discrete phase levels, has α = 0.0076 and β = 0.94, and the distance is z = 1000 mm.

Fig. 8
Fig. 8

Experimental transverse intensity distribution for the same axicon as in Fig. 7 but illuminated at an angle of 19.3°.

Fig. 9
Fig. 9

Numerical simulations of the transverse intensity profiles in the observation plane for a circular axicon in oblique illumination. The parameters are z = 100 mm and α = 0.036, and the angle of illumination is 5.0°. The result is calculated from the diffraction integral in Eq. (17).

Fig. 10
Fig. 10

Same as Fig. 9, but the angle of illumination is 10.0°.

Fig. 11
Fig. 11

CCD images of the focal segment for a circular, linear axicon of α = 0.036 taken at z = 100 mm. The axicon is illuminated obliquely at four different angles: (a) 0°, (b) 5.0°, (c) 7.5°, and (d) 10.0°. The white outlines indicate the analytical description of the focus, according to Eq. (18). The axicon is a binary-phase element manufactured by photolithography.

Tables (1)

Tables Icon

Table 1 Different Terms of the Taylor Expansion (1 + γ)1/2 = 1 + γ/2 - γ2/8 + γ3/16 - … with γ Given by Eq. (B1), their Approximations when r0z0, and their Dependence on z0

Equations (30)

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

ϕx, y=αx2+y21/2.
ϕx, y=αx2+β2y21/2,
Ex, y, z  1zA expikx2+y22z-xx+yyz-αx2+β2y21/2dxdy,
|Ex, y, z|  2πk|H|1/2,
H=1z1z-αβ2xc2+yc2xc2+β2yc23/2.
xc-x=αzxcxc2+β2yc21/2,
yc-y=αzβ2ycxc2+β2yc21/2.
yc=Cxc=±Czαβ21+C2/1+β2C23/2.
x=±zαβ2-1/1+β2C23/2,
y=±C3zαβ21-β2/1+β2C23/2,
x2/3+βy2/3=1-β2αz2/3,
Ex, y, z  A expiky sin φexpikrx, y, x, yrx, y, x, y×exp-ikϕx, ydxdy,
r=z0-y sin φ2+x-x2+y-y cos φ21/2,
r0=z02+x2+y21/2,
r=r01+x2+y2r02-2z0y sin φ+2xx+2yy cos φr021/2,
rr0+x2+y22r0-xx+y cos φ+z0 sin φyr0,
Ex, y, z  1z  expikx2+y2 cos2 φ2z0-xx+yy cos φz0-αx2+y21/2dxdy.
x2/3+y2/3=1-1/cos2 φαz1/3,
Ex, y, z  1z  expikX2+Y22z0-xX+yYz0-αX2+Y21/2dXdY,
fx, y=x2+y2/2z-xx+yy/z-αx2+β2y21/2,
fnm=1n!m!nxmyxnymfx, y
f10=x-xz-αxx2+β2y21/2,
f01=y-yz-αβ2yx2+β2y21/2,
f11=αβ2xyx2+β2y23/2,
2f20=1z-αβ2y2x2+β2y23/2,
2f02=1z-αβ2x2x2+β2y23/2.
H=4f02f20-f112=1z1z-αβ2x2+y2x2+β2y23/2,
xc-x=αzxcxc2+β2yc21/2,
yc-y=αzβ2ycxc2+β2yc21/2.
γ=x2+y2r02-2z0y sin φ+2xx+2yy cos φr02

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