Abstract

A singular-value-decomposition analysis of the imaging kernel for three-dimensional fluorescent laser scanning microscopy at a high numerical aperture (NA) is presented. The design and superresolving performance of image-plane binary optical masks are then derived, and new computational techniques for calculating these masks are given. Initial experimental results with a microscope equipped with such a mask at NA=1.3 are presented. The improvement in both contrast and resolution over the confocal and type 1 instruments is demonstrated.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 10, 59–64 (1993).
    [CrossRef]
  2. M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
    [CrossRef]
  3. M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
    [CrossRef]
  4. J. Grochmalicki, E. R. Pike, J. G. Walker, “Experimental confirmation of superresolution in incoherent scanning microscopy,” Pure Appl. Opt. 2, 565–568 (1993).
    [CrossRef]
  5. M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and superresolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
    [CrossRef]
  6. M. Bertero, P. Boccacci, F. Malfanti, E. R. Pike, “Superresolution in confocal scanning microscopy: V. Axial superresolution in the incoherent case,” Inverse Probl. 10, 1059–1077 (1994).
    [CrossRef]
  7. V. S. Ignatowsky, “Diffraction by an objective lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd 1(4), 1–36 (1921) (in Russian).
  8. B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems: II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  9. M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
    [CrossRef]
  10. M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
    [CrossRef]
  11. A. Papoulis, “Hankel transforms,” in Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  12. H. E. Fettis, “Lommel-type integrals involving three Bessel functions,” J. Math. Phys. (Paris) 36, 88–95 (1957).
  13. J. Grochmalicki, E. R. Pike, J. G. Walker, M. Bertero, P. Boccacci, R. E. Davies, “Superresolving masks for incoherent scanning microscopy,” J. Opt. Soc. Am. A 10, 1074–1077 (1993).
    [CrossRef]
  14. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  15. A. E. Siegman, “Quasi-fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef]

1994

M. Bertero, P. Boccacci, F. Malfanti, E. R. Pike, “Superresolution in confocal scanning microscopy: V. Axial superresolution in the incoherent case,” Inverse Probl. 10, 1059–1077 (1994).
[CrossRef]

1993

1992

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

1990

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and superresolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

1989

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
[CrossRef]

1977

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems: II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

1957

H. E. Fettis, “Lommel-type integrals involving three Bessel functions,” J. Math. Phys. (Paris) 36, 88–95 (1957).

1921

V. S. Ignatowsky, “Diffraction by an objective lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd 1(4), 1–36 (1921) (in Russian).

Bertero, M.

M. Bertero, P. Boccacci, F. Malfanti, E. R. Pike, “Superresolution in confocal scanning microscopy: V. Axial superresolution in the incoherent case,” Inverse Probl. 10, 1059–1077 (1994).
[CrossRef]

J. Grochmalicki, E. R. Pike, J. G. Walker, M. Bertero, P. Boccacci, R. E. Davies, “Superresolving masks for incoherent scanning microscopy,” J. Opt. Soc. Am. A 10, 1074–1077 (1993).
[CrossRef]

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 10, 59–64 (1993).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and superresolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, F. Malfanti, E. R. Pike, “Superresolution in confocal scanning microscopy: V. Axial superresolution in the incoherent case,” Inverse Probl. 10, 1059–1077 (1994).
[CrossRef]

J. Grochmalicki, E. R. Pike, J. G. Walker, M. Bertero, P. Boccacci, R. E. Davies, “Superresolving masks for incoherent scanning microscopy,” J. Opt. Soc. Am. A 10, 1074–1077 (1993).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and superresolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

Brakenhoff, G. J.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 10, 59–64 (1993).
[CrossRef]

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and superresolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

Davies, R. E.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 10, 59–64 (1993).
[CrossRef]

J. Grochmalicki, E. R. Pike, J. G. Walker, M. Bertero, P. Boccacci, R. E. Davies, “Superresolving masks for incoherent scanning microscopy,” J. Opt. Soc. Am. A 10, 1074–1077 (1993).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

De Mol, C.

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

Defrise, M.

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

Fettis, H. E.

H. E. Fettis, “Lommel-type integrals involving three Bessel functions,” J. Math. Phys. (Paris) 36, 88–95 (1957).

Grochmalicki, J.

J. Grochmalicki, E. R. Pike, J. G. Walker, M. Bertero, P. Boccacci, R. E. Davies, “Superresolving masks for incoherent scanning microscopy,” J. Opt. Soc. Am. A 10, 1074–1077 (1993).
[CrossRef]

J. Grochmalicki, E. R. Pike, J. G. Walker, “Experimental confirmation of superresolution in incoherent scanning microscopy,” Pure Appl. Opt. 2, 565–568 (1993).
[CrossRef]

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by an objective lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd 1(4), 1–36 (1921) (in Russian).

Jiang, S. H.

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

Malfanti, F.

M. Bertero, P. Boccacci, F. Malfanti, E. R. Pike, “Superresolution in confocal scanning microscopy: V. Axial superresolution in the incoherent case,” Inverse Probl. 10, 1059–1077 (1994).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and superresolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

Papoulis, A.

A. Papoulis, “Hankel transforms,” in Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Pike, E. R.

M. Bertero, P. Boccacci, F. Malfanti, E. R. Pike, “Superresolution in confocal scanning microscopy: V. Axial superresolution in the incoherent case,” Inverse Probl. 10, 1059–1077 (1994).
[CrossRef]

J. Grochmalicki, E. R. Pike, J. G. Walker, “Experimental confirmation of superresolution in incoherent scanning microscopy,” Pure Appl. Opt. 2, 565–568 (1993).
[CrossRef]

J. Grochmalicki, E. R. Pike, J. G. Walker, M. Bertero, P. Boccacci, R. E. Davies, “Superresolving masks for incoherent scanning microscopy,” J. Opt. Soc. Am. A 10, 1074–1077 (1993).
[CrossRef]

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 10, 59–64 (1993).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems: II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Siegman, A. E.

van der Voort, H. T. M.

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and superresolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

Walker, J. G.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 10, 59–64 (1993).
[CrossRef]

J. Grochmalicki, E. R. Pike, J. G. Walker, “Experimental confirmation of superresolution in incoherent scanning microscopy,” Pure Appl. Opt. 2, 565–568 (1993).
[CrossRef]

J. Grochmalicki, E. R. Pike, J. G. Walker, M. Bertero, P. Boccacci, R. E. Davies, “Superresolving masks for incoherent scanning microscopy,” J. Opt. Soc. Am. A 10, 1074–1077 (1993).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems: II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Young, M. R.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 10, 59–64 (1993).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

Inverse Probl.

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks,” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. Bertero, P. Boccacci, F. Malfanti, E. R. Pike, “Superresolution in confocal scanning microscopy: V. Axial superresolution in the incoherent case,” Inverse Probl. 10, 1059–1077 (1994).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
[CrossRef]

J. Math. Phys. (Paris)

H. E. Fettis, “Lommel-type integrals involving three Bessel functions,” J. Math. Phys. (Paris) 36, 88–95 (1957).

J. Microsc.

M. Bertero, P. Boccacci, G. J. Brakenhoff, F. Malfanti, H. T. M. van der Voort, “Three-dimensional image restoration and superresolution in fluorescence confocal microscopy,” J. Microsc. 157, 3–20 (1990).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of superresolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Proc. R. Soc. London, Ser. A

B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems: II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Pure Appl. Opt.

J. Grochmalicki, E. R. Pike, J. G. Walker, “Experimental confirmation of superresolution in incoherent scanning microscopy,” Pure Appl. Opt. 2, 565–568 (1993).
[CrossRef]

Trans. Opt. Inst. Petrograd

V. S. Ignatowsky, “Diffraction by an objective lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd 1(4), 1–36 (1921) (in Russian).

Other

A. Papoulis, “Hankel transforms,” in Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

First eight transverse singular functions uk(ρ, 0) in the z=0 object plane corresponding to the singular values given in Table 1.

Fig. 2
Fig. 2

First eight transverse singular functions vk(ρ) in the image plane corresponding to the singular values given in Table 1.

Fig. 3
Fig. 3

Continuous-mask function for the superresolving microscope calculated with the singular functions of Figs. 1 and 2.

Fig. 4
Fig. 4

(a) Approximate binary form of the superresolving mask of Fig. 3. (b) Design of the binary image-plane mask to be placed at 45° to the optical axis at the focal point.

Fig. 5
Fig. 5

Comparison of transverse sections of the PSF’s at z=0 between the superresolving designs, i.e., continuous (short-dashed curve) and binary (long-dashed curve) mask (indistinguishable on this scale) and the ordinary confocal (pinhole radius π/2) (dotted–dashed curve) and type 1 microscopes (solid curve). An increase of resolution of 57% at HWHM is achieved.

Fig. 6
Fig. 6

Comparison of lateral transfer functions corresponding to the PSF’s of Fig. 5. Improvement in the high-frequency response demonstrates the superresolving performance of the mask design. Type 1 microscope (solid curve), confocal microscope (dotted–dashed curve) and continuous (short-dashed curve) and binary mask (long-dashed curve) microscope.

Fig. 7
Fig. 7

Comparison of axial sections of the PSF’s at ρ=0 between the superresolving designs, i.e., the continuous (short-dashed curve) and binary (long-dashed curve) mask and the confocal (pinhole size π/2) (dotted–dashed curve) and type 1 (solid curve) microscopes. While the functions for the continuous and binary masks are virtually identical, both maintain the good axial performance of the confocal microscope (see Fig. 8).

Fig. 8
Fig. 8

(a) Transverse sections of the PSF at z=0 and (b) axial sections at ρ=0 for values of truncation number K from 0 to 6. A comparison with Fig. 7 reveals axial superresolution for K=2 and K=4, where K=0 (short-dashed curve), K=2 (dotted–dashed curve), K=4 (long-dashed curve), and K=6 (solid curve).

Fig. 9
Fig. 9

xz diagrams at y=0 of the PSF’s of (a) a K=6 binary-mask superresolving microscope and (b) a type 1 microscope. NA=1.3.

Fig. 10
Fig. 10

Schematic diagram of the experimental microscope. EP, eyepiece; DM, dichroic mirror; TL, tube lens; OL, objective lens.

Fig. 11
Fig. 11

Calculated image of a set of parallel bars. Bars are 125 nm wide, separated by another 125 nm. The upper image corresponds to a type 1 microscope, and the lower to our superresolving microscope.

Fig. 12
Fig. 12

Experimental images of the set of parallel bars of Fig. 11 obtained with our superresolving microscope. Upper image shows pixel intensities in type 1 mode. Lower image is the in-band superresolved image.

Tables (1)

Tables Icon

Table 1 First Nine Singular Values of the NA=1.3 Fluorescence Microscope System

Equations (58)

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

x=2n sin αλx0,y=2n sin αλy0,
z=2n sin2 αλz0,
NA=n sin α
g(ρ)=W2(|ρ-ρ|, z)W1(|ρ|, z)f(ρ, z)dρdz,
W1(ρ, z)=W2(ρ, z)W(ρ, z).
W(ρ, z)=|I0(ρ, z)|2+2|I1(ρ, z)|2+|I2(ρ, z)|2,
I0(ρ, z)=0α(cos θ)1/2 sin θ(1+cos θ)×J0sin θsin αρexpicos θsin2 αzdθ,
I1(ρ, z)=0α(cos θ)1/2 sin2 θJ1sin θsin αρ×expicos θsin2 αzdθ,
I2(ρ, z)=0α(cos θ)1/2 sin θ(1-cos θ)J2sin θsin αρ×expicos θsin2 αzdθ.
Auk=αkvk,A*vk=αkuk,
(A*g)(ρ, z)=W(ρ, z)W(|ρ-ρ|, z)g(ρ)dρ.
f(0, 0)=k=0K-11αk(g, vk)uk(0, 0).
(g, vk)=g(ρ)vk(ρ)dρ.
f(0, 0)=M(ρ)g(ρ)dρ,
M(ρ)=k=0K-11αkuk(0, 0)vk(ρ).
Wˆ(ω, η)=R3W(ρ, z)exp(-i(ρ, ω)-iηz)dρdz=2π0+ρdρ-+dzJ0(ωρ)exp(-iηz)W(ρ, z),
|ω|Ω,|η|Ω,
Ω=2π,Ω=π1+cos α.
g0(ρ)=12π02πg(ρ, ϕ)dϕ,
f0(ρ, z)=12π02πf(ρ, ϕ; z)dϕ
g0(ρ)=(A0f0)(ρ)=0-+W0(ρ, ρ; z)W(ρ, z)×f0(ρ, z)ρdρdz,
W0(ρ, ρ; z)=02πW[(ρ2+ρ2-2ρρ cos β)1/2, z]dβ.
h˜(ω)=0ρJ0(ωρ)h(ρ)dρ,
A0u0, k=α0, kv0, k,A0*v0, k=α0, ku0, k,
(A0*g0)(ρ)=W(ρ, z)0W0(ρ, ρ; z)g0(ρ)ρdρ.
h(ρ)=n=12xnJ1(xn)h(xn/Ω)J0(Ωρ)xn2-(Ωρ)2.
Sn(Ω, ρ)=Ω2 xnJ0(Ωρ)xn2-(Ωρ)2,n=1, 2,
h(ρ)=n=12ΩJ1(xn)h(xn/Ω)Sn(Ω, ρ).
0Sn(Ω, ρ)Sm(Ω, ρ)ρdρ=δnm,
0ρSn(Ω, ρ)h(ρ)dρ=2ΩJ1(xn)h(xn/Ω).
f0(ρ, z)=m=122ΩJ1(xm)×f0(xm/2Ω, z)Sm(2Ω, ρ),
g0xnΩ=m=12(2Ω)2J12(xm)-+W0xnΩ, xm2Ω; z×Wxm2Ω, zf0xm2Ω, zdz.
f0xm2Ω, z=l=-+f0xm2Ω, zl×sinc2Ωπ(z-zl),m=1, 2,,
zl=π2Ωl,l=0, ±1, ±2,
sinc2Ωπ(z-zl)=π2Ωsin[2Ω(z-zl)]π(z-zl).
-+W0xnΩ, xm2Ω; zWxm2Ω, z×sinc2Ωπ(z-zl)dz
=π2ΩW0xnΩ, xm2Ω; zlWxm2Ω, zl.
bn=m=1l=-+An;m,lam,l,n=1, 2,,
bn=2ΩJ1(xn)g0xnΩ,
am,l=π2Ω1/2 22ΩJ1(xm)f0xm2Ω, zl,
An;m,l=π2Ω1/2 W0xnΩ, xm2Ω; zlWxm2Ω, zlΩ2J1(xn)J1(xm).
n=1, 2,, N0,m=1, 2,, M0,
l=0, ±1, ±2,, ±L0.
vk(ρ)=2π2n=1N0Vn(k) xnJ0(2πρ)xn2-(2πρ)2,
uk(ρ, z)=42π1+cos α1/2m=1M0l=-L0L0×Um,l(k) xmJ0(4πρ)xm2-(4πρ)2sinc2(z-zl)1+cos α.
W(ρ, z)=k=1Wk(z)Sk(2π, ρ),
Wk(z)=22πJ1(xn)Wxk2π, z.
02πSk[2π, (ρ2+ρ2-2ρρ cos θ)1/2]dθ=1ρSk(2π, ρ) * δ(ρ-ρ),
Sk(2π, ρ) * δ(ρ-ρ)=2 4π2ρxkJ1(xk)Iρ, ρ, xk2π,
I(a, b, c)=02πJ0(ax)J0(bx)J0(cx)xdx.
W0(ρ, ρ; z)=4πk=1 Wxk2π, z[J1(xk)]2Iρ, ρ, xk2π.
Ixn2π, xm4π, xk2π,k=1, 2,.
qxn2+xm2/4-xk2xnxm,
Ixn2π, xm4π, xk2π=8π2 xkJ1(xk)xnxm×p=1Jp(xn)Jp(xm/2)exp(-pθ)sinh(θ).
Ixn2π, xm4π, xk2π=8π2 xkJ1(xk)xnxm×p=1Jp(xn)Jp(xm/2)sin(pθ)sin(θ).
f(0, 0)=T(x, z)f(x, z)dxdz,
T(x, z)=W(x, z)W(|x-x|)M(x)dx.
f(0, 0)=(M, g)=(M, Af)=(A*M, f).

Metrics