Abstract

An early result of optical focusing theory is the Lommel field, resulting from a uniformly illuminated lens; the dark rings in the focal plane, the Airy rings, have been recognized as phase singularities. On the other hand, it is well known that Gaussian illumination leads to a Gaussian beam in the focal region without phase singularities. We report a theoretical and experimental study of the transition between the two cases. Theoretically, we studied this transition both within and outside the paraxial limit by means of diffraction theory. We show that in the gradual transition from uniform toward Gaussian illumination, the Airy rings reorganize themselves by means of a creation/annihilation process of the singularities. The most pronounced effect is the occurrence of extra dark rings (phase singularities) in front of and behind the focal plane. We demonstrate theoretically that one can bring these rings arbitrarily close together, thus leading to structures on a scale arbitrarily smaller than 1 wavelength, although at low intensities. Experimentally, we have studied the consequences of the reorganization process in the paraxial limit at optical wavelengths. To this end, we developed a technique to measure the three-dimensional intensity (3D) distribution of a focal field. We applied this technique in the study of truncated Gaussian beams; the experimentally obtained 3D intensity distributions confirm the existence and the reorganization of extra dark rings outside the focal plane.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
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    [CrossRef]
  10. G. P. Karman, A. van Duijl, M. W. Beijersbergen, J. P. Woerdman, “Measurement of the 3D intensity distribution in the neighbourhood of a paraxial focus,” Appl. Opt. 36, 8091–8095 (1997).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1997 (3)

1995 (2)

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

I. Freund, “Saddles, singularities, and extreme in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

1994 (4)

M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A 27, L391 (1994).
[CrossRef]

W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

W. P. Ambrose, P. M. Goodwin, J. C. Martin, R. A. Keller, “Single molecule detection and photochemistry on a surface using near-field optical excitation,” Phys. Rev. Lett. 72, 160–163 (1994).
[CrossRef] [PubMed]

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign-principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

1988 (1)

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

1984 (1)

1983 (1)

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London, Ser. A 387, 105–132 (1983).
[CrossRef]

1981 (2)

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

1980 (1)

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

1967 (1)

Ambrose, W. P.

W. P. Ambrose, P. M. Goodwin, J. C. Martin, R. A. Keller, “Single molecule detection and photochemistry on a surface using near-field optical excitation,” Phys. Rev. Lett. 72, 160–163 (1994).
[CrossRef] [PubMed]

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Beijersbergen, M. W.

Berry, M.

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kléman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981).

Berry, M. V.

M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A 27, L391 (1994).
[CrossRef]

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality, J. S. Anandan, J. L. Safko, eds. (World Scientific, Singapore, 1994), pp. 55–65.

Boivin, A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

Collett, E.

Dow, J.

Erikson, W. L.

W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

Freund, I.

I. Freund, “Saddles, singularities, and extreme in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign-principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

Goodwin, P. M.

W. P. Ambrose, P. M. Goodwin, J. C. Martin, R. A. Keller, “Single molecule detection and photochemistry on a surface using near-field optical excitation,” Phys. Rev. Lett. 72, 160–163 (1994).
[CrossRef] [PubMed]

Hajnal, J. V.

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

Hannay, J. H.

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964).

Karman, G. P.

Keller, R. A.

W. P. Ambrose, P. M. Goodwin, J. C. Martin, R. A. Keller, “Single molecule detection and photochemistry on a surface using near-field optical excitation,” Phys. Rev. Lett. 72, 160–163 (1994).
[CrossRef] [PubMed]

Li, Y.

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Martin, J. C.

W. P. Ambrose, P. M. Goodwin, J. C. Martin, R. A. Keller, “Single molecule detection and photochemistry on a surface using near-field optical excitation,” Phys. Rev. Lett. 72, 160–163 (1994).
[CrossRef] [PubMed]

Nye, J. F.

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London, Ser. A 387, 105–132 (1983).
[CrossRef]

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964).

Shvartsman, N.

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign-principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Singh, S.

W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

Soskin, M. S.

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, UK, 1986).

Tiziani, H. J.

M. Totzeck, H. J. Tiziani, “Phase singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

Totzeck, M.

M. Totzeck, H. J. Tiziani, “Phase singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

van Duijl, A.

Vasnetsov, M. V.

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

Visser, T. D.

Wiersma, S. H.

Woerdman, J. P.

Wolf, E.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A 27, L391 (1994).
[CrossRef]

Opt. Commun. (4)

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

M. Totzeck, H. J. Tiziani, “Phase singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138, 365–382 (1997).
[CrossRef]

I. V. Basistiy, M. S. Soskin, M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

I. Freund, N. Shvartsman, “Wave-field phase singularities: the sign-principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (2)

I. Freund, “Saddles, singularities, and extreme in random phase fields,” Phys. Rev. E 52, 2348–2360 (1995).
[CrossRef]

W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

W. P. Ambrose, P. M. Goodwin, J. C. Martin, R. A. Keller, “Single molecule detection and photochemistry on a surface using near-field optical excitation,” Phys. Rev. Lett. 72, 160–163 (1994).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (3)

J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. London, Ser. A 387, 105–132 (1983).
[CrossRef]

J. F. Nye, J. V. Hajnal, J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London, Ser. A 417, 7–20 (1988).
[CrossRef]

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Other (7)

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kléman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, UK, 1986).

M. W. Beijersbergen, “Phase singularities in optical beams,” Ph.D. thesis (Leiden University, The Netherlands, 1996).

P. Jacquinot, B. Roizen-Dossier, “Apodization,” in Progress in Optics III, E. Wolf, ed. (North-Holland, Amsterdam, 1964).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1986).

M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality, J. S. Anandan, J. L. Safko, eds. (World Scientific, Singapore, 1994), pp. 55–65.

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Figures (22)

Fig. 1
Fig. 1

Schematic focusing configuration. The lens is assumed to be aberration free; the focal distance f and the aperture radius a are assumed to be large as compared with the wavelength. The origin of the coordinate system is placed in the geometrical focal point. The incoming wave is assumed to be x polarized and propagates in the positive z direction. Refraction at the lens causes the E vector to rotate toward the focal point. ES is the field on the wave front S after refraction. The aperture is placed at z=-f. The wave vector of the incoming beam is denoted by k.

Fig. 2
Fig. 2

Lommel field, which describes the focus of a lens with uniform illumination in the paraxial limit for the case NA=0.1 and f=1000λ. (a) Amplitude u(ρ) in the focal plane z=0 according to relation (9). (b) Intensity distribution in the (ρ, z) plane according to relation (8). The intensity (|u|2) contours indicate intensities of 10-1, 10-2, ; the intensity in the geometrical focus (0, 0) is normalized to 1.

Fig. 3
Fig. 3

Enlargement of Fig. 2(b). Shown is the region close to the first dark Airy ring at ρ6.098λ. Thick curves are contours of constant intensity, with adjacent lines differing by a factor of 10 (and normalized to 1 in the focal point ρ=z=0). Thin curves are phase contours; the phase difference between adjacent phase contours is π/4. The point through which all phase contours cross (the first Airy ring) is a phase singularity, and the point S slightly above it is a phase saddle point. The fact that eight phase contours spaced by π/4 collapse into the dark Airy ring shows that the charge of the singularity is +1.

Fig. 4
Fig. 4

(a) Amplitude of a paraxial Gaussian beam in the focal plane, with w0/λ=10; (b) (ρ, z) plane of this paraxial Gaussian beam. The intensity contours in (b) are, from the bottom, 10-1, 10-2, , relative to the focal point intensity.

Fig. 5
Fig. 5

(a) Field in the focal plane (z=0) in the case of a truncated Gaussian, with the use of Fraunhofer diffraction. (b) Close look at the disappearance of two zero points in the Airy pattern; the first and second dark Airy rings are marked A and B, respectively. NA=0.1. The various curves in (a) and (b) correspond to different values of the ratio a/w.

Fig. 6
Fig. 6

Two examples of truncated Gaussian beams at NA=0.1. For clarity, phase contours have been omitted, and only intensity contours are shown, in the order 0.1, 0.01, …, and normalized to 1 in the focal point. (a) a/w=1.515, enlarged in (b); (c) a/w=1.818, enlarged in (d). The arrows in the enlargements point to phase singularities. The topological charges of the singularities marked A, B, C, and D are, respectively, +1, -1, +1, and +1. This follows from considering the phase contours (not shown, but see Fig. 13 below for a similar nonparaxial situation).

Fig. 7
Fig. 7

Creation/annihilation process. Shown is the ρz plane in the focal region near the first two Airy rings (horizontally z/λ, vertically ρ/λ). NA=0.1. Intensity contours in the order 0.1, 0.01, … are normalized to 1 in the focal point. From top left to bottom right rowwise, the ratio a/w is increased from 1.370 to 1.667, showing the gradual transition from uniform toward Gaussian illumination. The labeling of the singularities is as in the other figures: A and B denote the Airy rings or its remnants, and C and D denote additionally created rings. At a/w=1.471 creation of C and D occurs; at a/w=1.621 annihilation of A and B occurs.

Fig. 8
Fig. 8

Behavior of the saddle points during the annihilation of two singularities. The vertical line is the ρ axis, filled dots indicate the singularities A and B, and the crosses indicate the saddle points. From left to right, A and B approach and annihilate in the point indicated by the open circle.

Fig. 9
Fig. 9

Location of the zero points in the focal plane as a function of the truncation ratio a/w (with a kept constant), as calculated with scalar paraxial Debye theory. NA=0.1. At the points at which two curves join, two phase singularities annihilate. For example, at a/w=1.621, the singularities A and B from Figs. 5(b), 6(b), and 7 annihilate. The left side of the figure (a/w0) corresponds to uniform illumination, giving the Airy pattern, and the right side (a/w) corresponds to a Gaussian beam without singularities.

Fig. 10
Fig. 10

Field distribution in the xz plane in the case of uniform illumination: NA=0.9, f=1000λ, and a=2065λ. (a) Contour lines of total energy density (E·E*+B·B*), in the order 0.5, 0.2, 0.1, etc., normalized to 1 in the focal point. (b) Ex: thick curves are curves of constant intensity |Ex|2, with adjacent curves differing by a factor of 10 and normalized to 1 in the focal point; thin curves are curves of constant phase, with adjacent curves differing by π/4.

Fig. 11
Fig. 11

Distribution of Ex in the xz plane in the case of Gaussian illumination: NA=0.9, f=1000λ, a=2065λ, w=200λ, and a/w=10.3. Shown are curves of constant intensity (|Ex|2), with adjacent curves differing by a factor of 10 and normalized to 1 in the focal point.

Fig. 12
Fig. 12

Distribution of Ex in the xz plane in the case of a partially truncated Gaussian beam: NA=0.9, f=1000λ, a=2065λ, w=570λ, and a/w=3.62. Thick curves are curves of constant intensity (|Ex|2), with adjacent curves differing by a factor of 10 and normalized to 1 in the focal point. Thin curves are curves of constant phase, with adjacent curves differing by π/4. A series of dark (Airy) rings appears in the focal plane; the region around the first dark ring is enlarged in Fig. 13 below, showing the reorganization process.

Fig. 13
Fig. 13

Enlargement of a part of Fig. 12. The four phase singularities are labeled A, B, C, and D, as in Fig. 6(b), and are separated by distances of 0.15λ. Singularities A and B are remnants of the Airy rings, and singularities C and D are newly created singularities and have moved away from the focal plane. The vorticity of the phase contour lines around the singularities indicates the topological charge; the charges of A, B, C, and D are, respectively, +1, -1, +1, and +1.

Fig. 14
Fig. 14

Positions of the singularities (zeros of Ex) along the x axis as a function of the truncation ratio a/w, for different values of NA, as calculated with vector Debye theory. The top left plot corresponds to the paraxial limit (NA=0.1, as in Fig. 9); the other plots show the results for NA=0.3, 0.7, and 0.9. As in Fig. 9, the joining of two curves indicates annihilation of two singularities.

Fig. 15
Fig. 15

Different field components along the x axis for the case depicted in Figs. 12 and 13: the x component of the electric field, |Ex|2 (dashed curve); the z component |Ez|2 (solid curve); and the total energy density E·E*+B·B* (dotted curve). E·E* has been normalized to 1 in the focal point. The first two zeros of Ex are separated by a distance of 0.15λ and correspond to the singularities A and B in Fig. 13.

Fig. 16
Fig. 16

Calculated intensity I|Ex|2 in the saddle point between the neighboring phase singularities A and B as a function of their distance Δx. The distance was changed by slightly changing the width w of the beam (while keeping NA=0.9 constant). Each intensity value was divided by the intensity in the focal point, so the values plotted are relative intensities. Note the double logarithmic scale. The line is a linear fit to the calculated points and has a slope of 4.02±0.03.

Fig. 17
Fig. 17

Calculated intensity I|Ex|2 in the saddle point between two neighboring phase singularities A and B as a function of NA (with Δx=0.15λ kept constant). Each intensity value was divided by the intensity in the focal point, so the values plotted are relative intensities. Note the double logarithmic scale. The line is a linear fit to the calculated points and has a slope of 4.1±0.1.

Fig. 18
Fig. 18

Experimental setup. The following acronyms are used: P=polarizer, SF=spatialfilter/telescope, L=lens, A=aperture, and CCD=CCD image sensor. The origin of the coordinate system is located at the geometrical focal point.

Fig. 19
Fig. 19

Experimental result: positions of the singularities in the focal plane as a function of the ratio a/w (with w kept constant). The conditions are λ=632.9 nm, f=1±0.02 m, w=1.74±0.04 mm, a=0.8 to 3 mm, NA2×10-3, and N6. The circles are the experimental results, and the curves are the corresponding theoretical results. The vertical axis shows aρ, where a is the aperture radius and ρ is the distance to the optical axis. The first two Airy rings are labeled A and B.

Fig. 20
Fig. 20

Experimental result: the intensity distribution in the neighborhood of the focal point. Shown are contours of constant intensity, normalized to 1 in the geometrical focal point (ρ=z=0). The conditions are f=0.6±0.01 m, a=1.8±0.02 mm, w=3.0±0.1 mm, NA=3×10-3, N=8.5, and a/w=0.59. Clearly, one sees the Airy rings in the focal plane (z=0). The reader should compare this figure with the theoretical result of Fig. 2. We concentrate on the boxed region near the first two Airy rings.

Fig. 21
Fig. 21

Experimental result: the boxed region of Fig. 20 for two values of a/w. The conditions are f=1±0.02 m and w=1.46±0.05 mm. (a) a/w=1.44 (a=2.1±0.05 mm). Beside the Airy rings A and B, the extra singularities C and D are clearly visible outside the focal plane. (b) a/w=1.6 (a=2.3±0.05 mm). Only the singularities C and D are present. A and B have annihilated. The reader should compare the plots in (a) and (b) with the theory in Fig. 7.

Fig. 22
Fig. 22

Intensity distribution in the case of the low Fresnel number N=3, where NA=0.1 and a/w=1.60. This is to be compared with the Debye result (N=) of Fig. 7. The intensity is normalized to 1 in the geometrical focal point, and the contour lines are shown in the order 1, 0.1, 0.01, … .

Equations (21)

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2-1c22t2V(r, t)=0
(2+k2)u(r)=0,
u(r)=exp(ik·r),
u(x, y, z)ΩU(kx, ky)×exp[i(kxx+kyy+kzz)]dkxdky,
U(kx, ky)ua(-fkx/kz,-fky/kz)kz2.
u(ρ, z)0k sin θ ua(fkt/k)kz2exp(ikzz)J0(ktρ)kt dkt,
kz=k2-kt2.
u(ρ, z)0kθua(fkt/k)exp(ikzz)J0(ktρ)kt dkt.
u(ρ, 0)2J1(kθρ)kθρ.
ua(ρ)=exp(-ρ2/w2)forρa0forρ>a.
E(r)0θ02πES(ϑ, φ)exp(ik·r)sin ϑ dϑdφ,
ES(ϑ, φ)=Ein(ρ)cos ϑ×sin2 φ+cos ϑ cos2 φ(cos φ sin φ)(cos ϑ-1)sin ϑ cos φ,
Ein(ρ)=exp(-ρ2/w2)withρ=f sin ϑandwλ.
ES(ϑ, φ)NA1Ein(ρ)×O(1)O(θ2)O(θ),
E(r, θ, ϕ)I0+I2 cos(2ϕ)I2 sin(2ϕ)-2iI1 cos ϕ,
I00θEin(ρ)cos ϑ(sin ϑ)(1+cos ϑ)×J0(kr sin θ sin ϑ)×exp(ikr cos θ cos ϑ)dϑ,
I10θEin(ρ)cos ϑ sin2 ϑJ1(kr sin θ sin ϑ)×exp(ikr cos θ cos ϑ)dϑ,
I20θEin(ρ)cos ϑ(sin ϑ)(1-cos ϑ)×J2(kr sin θ sin ϑ)×exp(ikr cos θ cos ϑ)dϑ,
Ex(x; y=z=0)NA10θEin(ρ)J0(kxϑ)ϑ d ϑ,
f(x, A, δ)=- exp[ik(u)x]exp-1δ2(u-iA)2du,
k(u)=cos (u),

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