Abstract

We describe a technique for measuring the three-dimensional (3D) intensity distribution of a paraxial focus, based on scanning a CCD image sensor along the optical axis and on subsequently analyzing the data. We demonstrate the possibility of measuring high-resolution 3D intensity maps of the focal field, down to intensities of more than 5 orders of magnitude below that in the focal point, and show the excellent agreement with scalar diffraction theory. Further applications of the technique are indicated.

© 1997 Optical Society of America

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References

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  1. J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, UK, 1986), Sect. 12.5.
  2. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [CrossRef]
  3. C. A. Taylor, B. J. Thomson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am.48, 844–850 (1958); reprinted in SPIE Milestone SeriesSelected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., MS 51, 514–520 (1991).
    [CrossRef]
  4. G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can J. Phys. 36, 935–943 (1958).
    [CrossRef]
  5. J. J. Stamnes, B. Spelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  6. R. B. Johnson, Lenses in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 1.
  7. 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]
  8. W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. 119, 453–459 (1995).
    [CrossRef]
  9. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [CrossRef]
  10. J. J. Stamnes, University of Bergen, N-5007 Bergen, Norway (personal communication, 1997).
  11. G. Lenz, “Far-field diffraction of truncated higher-order Laguerre–Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
    [CrossRef]
  12. W. H. Carter, M. F. Aburdene, “Focal shift in Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 4, 1949–1952 (1987).
    [CrossRef]
  13. A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
    [CrossRef]
  14. J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
    [CrossRef]
  15. G. P. Karman, M. W. Beijersbergen, A. van Duijl, J. P. Woerdman, “Creation and annihilation of phase singularities in a focal field,” Opt. Lett. 22, 1503–1505 (1997).
    [CrossRef]
  16. P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
    [CrossRef] [PubMed]
  17. W. Wang, A. Friberg, E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–496 (1997).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1986), Chap. 8.
  19. E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966).
    [CrossRef]

1997 (3)

1996 (3)

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[CrossRef] [PubMed]

A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
[CrossRef]

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre–Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

1995 (1)

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. 119, 453–459 (1995).
[CrossRef]

1987 (1)

1984 (1)

1983 (1)

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

1982 (1)

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

1981 (1)

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

1966 (1)

1958 (1)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can J. Phys. 36, 935–943 (1958).
[CrossRef]

Aburdene, M. F.

Asakura, T.

A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
[CrossRef]

Beijersbergen, M. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1986), Chap. 8.

Carter, W. H.

Farnell, G. W.

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can J. Phys. 36, 935–943 (1958).
[CrossRef]

Friberg, A.

Johnson, R. B.

R. B. Johnson, Lenses in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 1.

Karman, G. P.

Lenz, G.

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre–Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

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]

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Marchand, E. W.

Platzer, H.

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Spelkavik, B.

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

Stamnes, J. J.

J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

J. J. Stamnes, B. Spelkavik, “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), Sect. 12.5.

J. J. Stamnes, University of Bergen, N-5007 Bergen, Norway (personal communication, 1997).

Taylor, C. A.

C. A. Taylor, B. J. Thomson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am.48, 844–850 (1958); reprinted in SPIE Milestone SeriesSelected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., MS 51, 514–520 (1991).
[CrossRef]

Thomson, B. J.

C. A. Taylor, B. J. Thomson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am.48, 844–850 (1958); reprinted in SPIE Milestone SeriesSelected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., MS 51, 514–520 (1991).
[CrossRef]

Török, P.

van Duijl, A.

Varga, P.

Wang, W.

W. Wang, A. Friberg, E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–496 (1997).
[CrossRef]

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. 119, 453–459 (1995).
[CrossRef]

Woerdman, J. P.

Wolf, E.

Yoshida, A.

A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
[CrossRef]

Can J. Phys. (1)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can J. Phys. 36, 935–943 (1958).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Opt. Commun. (5)

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

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre–Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

A. Yoshida, T. Asakura, “Propagation and focusing of Gaussian laser beams beyond conventional diffraction limit,” Opt. Commun. 123, 694–704 (1996).
[CrossRef]

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. 119, 453–459 (1995).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Opt. Lett. (2)

Pure Appl. Opt. (1)

J. J. Stamnes, “Focusing at low Fresnel numbers in the presence of cylindrical or spherical aberration,” Pure Appl. Opt. 6, 85–96 (1997).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1986), Chap. 8.

R. B. Johnson, Lenses in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 1.

C. A. Taylor, B. J. Thomson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am.48, 844–850 (1958); reprinted in SPIE Milestone SeriesSelected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., MS 51, 514–520 (1991).
[CrossRef]

J. J. Stamnes, University of Bergen, N-5007 Bergen, Norway (personal communication, 1997).

J. J. Stamnes, Waves in Focal Regions (Institute of Physics, Bristol, UK, 1986), Sect. 12.5.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. The origin of the coordinate system is placed at the geometric focal point L. The incoming wave propagates in the z direction. The He–Ne laser beam is linearly polarized and attenuated by rotation of the polarizer.

Fig. 2
Fig. 2

Experimental result. Conditions: f = 200 mm, a = 0.505 mm, w = 1.90 mm, NA = 2.53 × 10-3, N = 2.01, w/a = 3.8. Lines are shown of constant intensity (of the order of 1, 0.1, 0.01, etc.) in the rz plane. The intensity is normalized to 1 at the geometric focal point (z = r = 0). The region in the inset is shown in more detail in Fig. 3.

Fig. 3
Fig. 3

Enlargement of the inset showing the experimental intensity pattern region in Fig. 2. Shown are lines of constant intensity (here of the order of 1, 0.5, 0.2, 0.1, 0.05, etc.) The intensity is normalized to 1 in the geometric focal point (z = r = 0).

Fig. 4
Fig. 4

Theoretical result (after convolution, see text) for conditions as in Fig. 3. Shown are lines of constant intensity (of the order of 1, 0.5, 0.2, 0.1, 0.05, etc.). The intensity is normalized to 1 in the geometric focal point (z = r = 0). Note the excellent agreement with the experimental result in Fig. 3.

Equations (4)

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Ir1-u2πN01J0vtexp-γt2tdt2,
γaw2+12iu,
u2πNz/f1+z/f,
v2πNr/a1+z/f.

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