Abstract

Imaging by near-field scanning optical microscopy (NSOM) with a plasmonic gap probe (PGP) is simulated to confirm the operation of the recently proposed PGP. The simulations demonstrate that the probe works in illumination, collection-reflection and collection mode, and that is it not necessary to vibrate the probe tip in order to remove background noise. The resolution of the scanned image is also shown to be approximately equal to the diameter of the probe tip. Furthermore, the throughput of the probe is much higher than conventional aperture probes providing similar resolution. The proposed probe thus has the advantages of both aperture probes and scattering probes, and is expected to have excellent characteristics for use as a scanning probe for NSOM.

© 2006 Optical Society of America

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  1. K. Tanaka, M. Tanaka, and T. Sugiyama, "Metallic tip-probe providing high intensity and small spot size with a small background light in near-field optics," Appl. Phys. Lett. 87, 151116 (2005).
    [CrossRef]
  2. K. Tanaka, M. Tanaka, and T. Sugiyama, "Creation of strongly localized and strongly enhanced optical near-field on metallic probe-tip with surface plasmon polaritons," Opt. Express 14, 832-846 (2006).
    [CrossRef] [PubMed]
  3. C. Girard, A. Dereux, and O. J. F. Martin, "Importance of confined fields in near-field optical imaging of subwavelength objects," Phys. Rev. B 50, 14467-14473 (1994).
    [CrossRef]
  4. M. Xiao, "Theoretical treatment for scattering scanning near-field optical microscopy," J. Opt. Soc. Am. A 14, 2977-2984 (1997).
    [CrossRef]
  5. S. I. Bozhevolnyi, M. Xiao, and J. M. Hvan, "Polarization-resolved imaging with a reflection near-field optical microscope," J. Opt. Soc. Am. A 16, 2649-2657 (1999).
    [CrossRef]
  6. T. Setala, M. Kaivola, and A. T. Friberg, "Evanescent and propagating electromagnetic fields in scattering from point-dipole structure," J. Opt. Soc. Am. A 18, 678-688 (2001).
    [CrossRef]
  7. K. Tanaka, M. Yan, and M. Tanaka, "A simulation of near field optics by three dimensional volume integral equation of classical electromagnetic theory," Opt. Rev. 8, 43-53 (2001).
    [CrossRef]
  8. G. von Freymann, T. Schimmel, and M. Wegener, "Computer simulations on near-field scanning optical microscopy: Can subwavelength resolution be obtained using uncoated optical fiber probe?" Appl. Phys. Lett. 73, 1170-1172 (1998).
    [CrossRef]
  9. Y. Sasaki and H. Sasaki, "Probe design optimization for a high-resolution scattering-type scanning near-field optical microscope," J. Microscopy 202, 347-350 (2001).
    [CrossRef]
  10. K. Tanaka and M. Tanaka, "Simulation of an aperture in the thick metallic screen that gives high intensity and small spot size using surface plasmon polariton," J. Microscopy 210, 294-300 (2003).
    [CrossRef]
  11. K. Tanaka and M. Tanaka, "Optimized computer-aided design of I-shaped subwavelength aperture for high intensity and small spot size," Opt. Comm. 233, 231-244 (2004).
    [CrossRef]
  12. P. Zwamborn and P. M. van den Berg, "The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems," IEEE Trans. Microwave Theory Tech. 40, 1757-1766 (1992).
    [CrossRef]
  13. R. Barrett, T. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, (Society for Industrial and Applied Mathematics, New York, 1994).
    [CrossRef]
  14. E. K. Miller, L. Medgyesi-Mitschnag, and E. H. Newsman, ed., Computational Electromagnetics Frequency-Domain Method of Moments, (The Institute of Electrical and Electronics Engineers, 1992).
  15. G. S. Smith, An introduction to classical electromagnetic radiation (Cambridge Uni. Press, 1997).
  16. M. Ohtsu, ed., Near-field Nano/Atom Optics and Technology, (Springer-Verlag, Tokyo, 1998), Chap. 4.
    [CrossRef]
  17. S. Kawata, M. Ohtsu, and M. Irie, eds., Nano-Optics, (Springer- Verlag, Berlin Heidelberg, 2002), Chap. 5.
  18. J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
    [CrossRef]
  19. M. Ohtsu, ed., Near-field Nano/Atom Optics and Technology, (Springer-Verlag, Tokyo, 1998) Chap. 2.
    [CrossRef]
  20. F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution," Science 269, 1083-1085 (1995).
    [CrossRef] [PubMed]
  21. Y. Inoue and S. Kawata, "A scanning near-field optical microscope having scanning electron tunneling microscope capability using a single metallic probe tip," J. Microscopy 178, 14-19 (1994).
    [CrossRef]
  22. J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, "Tip-enhanced fluorescent microscopy at 10 nanometer resolution," Phys. Rev. Lett. 93, 18080 (2004).
    [CrossRef]

2006 (1)

2005 (1)

K. Tanaka, M. Tanaka, and T. Sugiyama, "Metallic tip-probe providing high intensity and small spot size with a small background light in near-field optics," Appl. Phys. Lett. 87, 151116 (2005).
[CrossRef]

2004 (2)

J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, "Tip-enhanced fluorescent microscopy at 10 nanometer resolution," Phys. Rev. Lett. 93, 18080 (2004).
[CrossRef]

K. Tanaka and M. Tanaka, "Optimized computer-aided design of I-shaped subwavelength aperture for high intensity and small spot size," Opt. Comm. 233, 231-244 (2004).
[CrossRef]

2003 (1)

K. Tanaka and M. Tanaka, "Simulation of an aperture in the thick metallic screen that gives high intensity and small spot size using surface plasmon polariton," J. Microscopy 210, 294-300 (2003).
[CrossRef]

2001 (3)

Y. Sasaki and H. Sasaki, "Probe design optimization for a high-resolution scattering-type scanning near-field optical microscope," J. Microscopy 202, 347-350 (2001).
[CrossRef]

K. Tanaka, M. Yan, and M. Tanaka, "A simulation of near field optics by three dimensional volume integral equation of classical electromagnetic theory," Opt. Rev. 8, 43-53 (2001).
[CrossRef]

T. Setala, M. Kaivola, and A. T. Friberg, "Evanescent and propagating electromagnetic fields in scattering from point-dipole structure," J. Opt. Soc. Am. A 18, 678-688 (2001).
[CrossRef]

1999 (1)

1998 (2)

G. von Freymann, T. Schimmel, and M. Wegener, "Computer simulations on near-field scanning optical microscopy: Can subwavelength resolution be obtained using uncoated optical fiber probe?" Appl. Phys. Lett. 73, 1170-1172 (1998).
[CrossRef]

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

1997 (1)

1995 (1)

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution," Science 269, 1083-1085 (1995).
[CrossRef] [PubMed]

1994 (2)

Y. Inoue and S. Kawata, "A scanning near-field optical microscope having scanning electron tunneling microscope capability using a single metallic probe tip," J. Microscopy 178, 14-19 (1994).
[CrossRef]

C. Girard, A. Dereux, and O. J. F. Martin, "Importance of confined fields in near-field optical imaging of subwavelength objects," Phys. Rev. B 50, 14467-14473 (1994).
[CrossRef]

1992 (1)

P. Zwamborn and P. M. van den Berg, "The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems," IEEE Trans. Microwave Theory Tech. 40, 1757-1766 (1992).
[CrossRef]

Bozhevolnyi, S. I.

Dereux, A.

C. Girard, A. Dereux, and O. J. F. Martin, "Importance of confined fields in near-field optical imaging of subwavelength objects," Phys. Rev. B 50, 14467-14473 (1994).
[CrossRef]

Friberg, A. T.

Gerton, J. M.

J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, "Tip-enhanced fluorescent microscopy at 10 nanometer resolution," Phys. Rev. Lett. 93, 18080 (2004).
[CrossRef]

Girard, C.

C. Girard, A. Dereux, and O. J. F. Martin, "Importance of confined fields in near-field optical imaging of subwavelength objects," Phys. Rev. B 50, 14467-14473 (1994).
[CrossRef]

Hvan, J. M.

Inoue, Y.

Y. Inoue and S. Kawata, "A scanning near-field optical microscope having scanning electron tunneling microscope capability using a single metallic probe tip," J. Microscopy 178, 14-19 (1994).
[CrossRef]

Kaivola, M.

Kawata, S.

Y. Inoue and S. Kawata, "A scanning near-field optical microscope having scanning electron tunneling microscope capability using a single metallic probe tip," J. Microscopy 178, 14-19 (1994).
[CrossRef]

Kuipers, L.

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

Lessard, G. A.

J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, "Tip-enhanced fluorescent microscopy at 10 nanometer resolution," Phys. Rev. Lett. 93, 18080 (2004).
[CrossRef]

Ma, Z.

J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, "Tip-enhanced fluorescent microscopy at 10 nanometer resolution," Phys. Rev. Lett. 93, 18080 (2004).
[CrossRef]

Martin, O. J. F.

C. Girard, A. Dereux, and O. J. F. Martin, "Importance of confined fields in near-field optical imaging of subwavelength objects," Phys. Rev. B 50, 14467-14473 (1994).
[CrossRef]

Martin, Y.

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution," Science 269, 1083-1085 (1995).
[CrossRef] [PubMed]

Otter, A. M.

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

Quake, S. R.

J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, "Tip-enhanced fluorescent microscopy at 10 nanometer resolution," Phys. Rev. Lett. 93, 18080 (2004).
[CrossRef]

Sasaki, H.

Y. Sasaki and H. Sasaki, "Probe design optimization for a high-resolution scattering-type scanning near-field optical microscope," J. Microscopy 202, 347-350 (2001).
[CrossRef]

Sasaki, Y.

Y. Sasaki and H. Sasaki, "Probe design optimization for a high-resolution scattering-type scanning near-field optical microscope," J. Microscopy 202, 347-350 (2001).
[CrossRef]

Schimmel, T.

G. von Freymann, T. Schimmel, and M. Wegener, "Computer simulations on near-field scanning optical microscopy: Can subwavelength resolution be obtained using uncoated optical fiber probe?" Appl. Phys. Lett. 73, 1170-1172 (1998).
[CrossRef]

Setala, T.

Sugiyama, T.

K. Tanaka, M. Tanaka, and T. Sugiyama, "Creation of strongly localized and strongly enhanced optical near-field on metallic probe-tip with surface plasmon polaritons," Opt. Express 14, 832-846 (2006).
[CrossRef] [PubMed]

K. Tanaka, M. Tanaka, and T. Sugiyama, "Metallic tip-probe providing high intensity and small spot size with a small background light in near-field optics," Appl. Phys. Lett. 87, 151116 (2005).
[CrossRef]

Tanaka, K.

K. Tanaka, M. Tanaka, and T. Sugiyama, "Creation of strongly localized and strongly enhanced optical near-field on metallic probe-tip with surface plasmon polaritons," Opt. Express 14, 832-846 (2006).
[CrossRef] [PubMed]

K. Tanaka, M. Tanaka, and T. Sugiyama, "Metallic tip-probe providing high intensity and small spot size with a small background light in near-field optics," Appl. Phys. Lett. 87, 151116 (2005).
[CrossRef]

K. Tanaka and M. Tanaka, "Optimized computer-aided design of I-shaped subwavelength aperture for high intensity and small spot size," Opt. Comm. 233, 231-244 (2004).
[CrossRef]

K. Tanaka and M. Tanaka, "Simulation of an aperture in the thick metallic screen that gives high intensity and small spot size using surface plasmon polariton," J. Microscopy 210, 294-300 (2003).
[CrossRef]

K. Tanaka, M. Yan, and M. Tanaka, "A simulation of near field optics by three dimensional volume integral equation of classical electromagnetic theory," Opt. Rev. 8, 43-53 (2001).
[CrossRef]

Tanaka, M.

K. Tanaka, M. Tanaka, and T. Sugiyama, "Creation of strongly localized and strongly enhanced optical near-field on metallic probe-tip with surface plasmon polaritons," Opt. Express 14, 832-846 (2006).
[CrossRef] [PubMed]

K. Tanaka, M. Tanaka, and T. Sugiyama, "Metallic tip-probe providing high intensity and small spot size with a small background light in near-field optics," Appl. Phys. Lett. 87, 151116 (2005).
[CrossRef]

K. Tanaka and M. Tanaka, "Optimized computer-aided design of I-shaped subwavelength aperture for high intensity and small spot size," Opt. Comm. 233, 231-244 (2004).
[CrossRef]

K. Tanaka and M. Tanaka, "Simulation of an aperture in the thick metallic screen that gives high intensity and small spot size using surface plasmon polariton," J. Microscopy 210, 294-300 (2003).
[CrossRef]

K. Tanaka, M. Yan, and M. Tanaka, "A simulation of near field optics by three dimensional volume integral equation of classical electromagnetic theory," Opt. Rev. 8, 43-53 (2001).
[CrossRef]

van den Berg, P. M.

P. Zwamborn and P. M. van den Berg, "The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems," IEEE Trans. Microwave Theory Tech. 40, 1757-1766 (1992).
[CrossRef]

van Hulst, N. F.

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

Veerman, J. A.

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

von Freymann, G.

G. von Freymann, T. Schimmel, and M. Wegener, "Computer simulations on near-field scanning optical microscopy: Can subwavelength resolution be obtained using uncoated optical fiber probe?" Appl. Phys. Lett. 73, 1170-1172 (1998).
[CrossRef]

Wade, L. A.

J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, "Tip-enhanced fluorescent microscopy at 10 nanometer resolution," Phys. Rev. Lett. 93, 18080 (2004).
[CrossRef]

Wegener, M.

G. von Freymann, T. Schimmel, and M. Wegener, "Computer simulations on near-field scanning optical microscopy: Can subwavelength resolution be obtained using uncoated optical fiber probe?" Appl. Phys. Lett. 73, 1170-1172 (1998).
[CrossRef]

Wickramasinghe, H. K.

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution," Science 269, 1083-1085 (1995).
[CrossRef] [PubMed]

Xiao, M.

Yan, M.

K. Tanaka, M. Yan, and M. Tanaka, "A simulation of near field optics by three dimensional volume integral equation of classical electromagnetic theory," Opt. Rev. 8, 43-53 (2001).
[CrossRef]

Zenhausern, F.

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution," Science 269, 1083-1085 (1995).
[CrossRef] [PubMed]

Zwamborn, P.

P. Zwamborn and P. M. van den Berg, "The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems," IEEE Trans. Microwave Theory Tech. 40, 1757-1766 (1992).
[CrossRef]

Appl. Phys. Lett. (3)

G. von Freymann, T. Schimmel, and M. Wegener, "Computer simulations on near-field scanning optical microscopy: Can subwavelength resolution be obtained using uncoated optical fiber probe?" Appl. Phys. Lett. 73, 1170-1172 (1998).
[CrossRef]

J. A. Veerman, A. M. Otter, L. Kuipers, and N. F. van Hulst, "High definition aperture probes for near-field optical microscopy fabricated by focused ion beam milling," Appl. Phys. Lett. 72, 3115-3117 (1998).
[CrossRef]

K. Tanaka, M. Tanaka, and T. Sugiyama, "Metallic tip-probe providing high intensity and small spot size with a small background light in near-field optics," Appl. Phys. Lett. 87, 151116 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. Zwamborn and P. M. van den Berg, "The three-dimensional weak form of the conjugate gradient FFT method for solving scattering problems," IEEE Trans. Microwave Theory Tech. 40, 1757-1766 (1992).
[CrossRef]

J. Microscopy (3)

Y. Sasaki and H. Sasaki, "Probe design optimization for a high-resolution scattering-type scanning near-field optical microscope," J. Microscopy 202, 347-350 (2001).
[CrossRef]

K. Tanaka and M. Tanaka, "Simulation of an aperture in the thick metallic screen that gives high intensity and small spot size using surface plasmon polariton," J. Microscopy 210, 294-300 (2003).
[CrossRef]

Y. Inoue and S. Kawata, "A scanning near-field optical microscope having scanning electron tunneling microscope capability using a single metallic probe tip," J. Microscopy 178, 14-19 (1994).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Comm. (1)

K. Tanaka and M. Tanaka, "Optimized computer-aided design of I-shaped subwavelength aperture for high intensity and small spot size," Opt. Comm. 233, 231-244 (2004).
[CrossRef]

Opt. Express (1)

Opt. Rev. (1)

K. Tanaka, M. Yan, and M. Tanaka, "A simulation of near field optics by three dimensional volume integral equation of classical electromagnetic theory," Opt. Rev. 8, 43-53 (2001).
[CrossRef]

Phys. Rev. B (1)

C. Girard, A. Dereux, and O. J. F. Martin, "Importance of confined fields in near-field optical imaging of subwavelength objects," Phys. Rev. B 50, 14467-14473 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

J. M. Gerton, L. A. Wade, G. A. Lessard, Z. Ma, and S. R. Quake, "Tip-enhanced fluorescent microscopy at 10 nanometer resolution," Phys. Rev. Lett. 93, 18080 (2004).
[CrossRef]

Science (1)

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, "Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution," Science 269, 1083-1085 (1995).
[CrossRef] [PubMed]

Other (6)

M. Ohtsu, ed., Near-field Nano/Atom Optics and Technology, (Springer-Verlag, Tokyo, 1998) Chap. 2.
[CrossRef]

R. Barrett, T. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, (Society for Industrial and Applied Mathematics, New York, 1994).
[CrossRef]

E. K. Miller, L. Medgyesi-Mitschnag, and E. H. Newsman, ed., Computational Electromagnetics Frequency-Domain Method of Moments, (The Institute of Electrical and Electronics Engineers, 1992).

G. S. Smith, An introduction to classical electromagnetic radiation (Cambridge Uni. Press, 1997).

M. Ohtsu, ed., Near-field Nano/Atom Optics and Technology, (Springer-Verlag, Tokyo, 1998), Chap. 4.
[CrossRef]

S. Kawata, M. Ohtsu, and M. Irie, eds., Nano-Optics, (Springer- Verlag, Berlin Heidelberg, 2002), Chap. 5.

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Figures (8)

Fig. 1.
Fig. 1.

Geometry of image formation by NSOM with PGP. The dielectric F-shaped object (blue) is illuminated by the near field created by the probe tip and scanned in the x and y directions with respect to the probe. The incident wave is assumed to be Gaussian beam.

Fig. 2.
Fig. 2.

Nanometric 3D dielectric F-shaped object on the plane parallel to the x-y plane. The dielectric F-shaped object consists of eight equal-sized cubes with side length δ.

Fig. 3.
Fig. 3.

Cross section of the screen and pyramidal structure that contains probe tip which consists of one descretized cube of size on the plane parallel (a) to the x-z plane in the range -3.0 < k 0 x < 3.0 only and (b) to the y-z plane in the range -3.0 < k 0 y < 3.0 only.

Fig. 4.
Fig. 4.

Simulated image of F-shaped dielectric object for (a) illumination mode with P(0,π/2), and (b) collection-reflection mode with P(π/2,π). Grayscale represents the percentage of incident power determined by Eq. (12) (range is 2% of incident power).

Fig. 5.
Fig. 5.

Simulated image of single isolated dielectric object. (a) illumination mode with P(0,π/2), and (b) collection-reflection with P(π/2,π). Grayscale represents the percent of incident power determined by Eq. (12).

Fig. 6.
Fig. 6.

Simulated images of collection-mode scan with P(π/2,π) for a wave incident from the positive z direction (see Fig. 2). (a). Image of F-shaped dielectric object in a 9 × 9 pixel plane. (b). Image for a single dielectric cube in a 7 × 7 pixel plane.

Fig. 7.
Fig. 7.

Simulated images of illumination mode scan with P(0, π/2). (a) Probe tip consists of 2 × 2 × 1 discretized cubes. Distance between object plane and probe tip is k 0 d = 0. (b) Distance between object plane and probe tip is k 0 d = 0.1. Probe tip consists of one discretized cubes.

Fig. 8.
Fig. 8.

Simulated images of the object with the substrate of (a) illumination-mode scan with P(0, π/2), (b) collection-reflection mode scan with P(π/2,π) and (c) collection mode scan with P(π/2,π).

Equations (13)

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

E i ( x ) = D ( x ) / ε r ( x ) ( k 0 2 + ∇∇• ) A ( x )
A ( x ) = ( 1 / ε 0 ) V [ ( ε r ( x′ ) ε 0 ) / ε r ( x′ ) ] G ( x x′ ) D ( x′ ) d v′
G ( x x′ ) = exp ( j k 0 x x′ ) / ( 4 π x x′ )
E i ( x ) = E 0 W 0 / W ( z ) exp [ ( z ) ] { i x + i z j [ 1 + γ 2 ( z ) ] 1 / 2 γ ( z ) exp [ ( z ) ] }
× exp [ ( x 2 + y 2 ) / W 2 ( z ) ] exp { j k 0 ( x 2 + y 2 ) / [ 2 R ( z ) ] } exp ( j k 0 z ) ,
W ( z ) = W 0 [ 1 + γ 2 ( z ) ] 1 / 2 , ψ ( z ) = tan 1 γ ( z ) , R ( z ) = z [ 1 + 1 / γ 2 ( z ) ] ,
γ ( z ) = 2 z / ( k 0 W 0 2 ) , γ ( x ) = 2 x / ( k 0 W 0 2 ) .
E ( r , θ , ϕ ) [ exp ( j k 0 r ) / ( k 0 r ) ] [ F s ( θ , φ ) + F gauss ( θ , φ ) ]
F s ( θ , φ ) = j k 0 3 / ( 4 π ) i r × i r × { V [ ε r ( x′ ) 1 ] E ( x′ ) exp ( j k 0 x · i r ) d v′ }
F gauss ( π , ϕ ) = F θ gauss ( θ , ϕ ) i θ + F ϕ gauss ( θ , ϕ ) i ϕ
F θ gauss ( θ , ϕ ) = 2 j ( k 0 W 0 / 2 ) 2 E 0 exp { [ k 0 sin θ ( W 0 / 2 ) ] 2 } cos ϕ
F ϕ gauss ( π , ϕ ) = 2 j ( k 0 W 0 / 2 ) 2 E 0 exp { [ k 0 sin θ ( W 0 / 2 ) ] 2 } cos θ sin ϕ
P ( θ α , θ β ) = θ α θ β 0 2 π F s ( θ , ϕ ) + F gauss ( θ , ϕ ) 2 sin θ d θ d ϕ / ( π W 0 2 / 2 )

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