Abstract

The geometric law of energy conservation is utilized in evaluating the phase transmittance function for axicons with arbitrary distribution of the on-axis intensity. Several simple analytical solutions are presented, and a computer-generated holographic version of the uniform-intensity axicon is examined.

© 1992 Optical Society of America

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References

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  1. J. H. McLeod, “Axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  2. J. H. McLeod, “Axicons and their uses,” J. Opt. Soc. Am. 50, 166–169 (1960).
    [CrossRef]
  3. M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction pattern and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
    [CrossRef]
  4. L. W. Casperson, M. S. Shekhani, “Air breakdown in a radial-mode focusing element,” Appl. Opt. 13, 104–108 (1974).
    [CrossRef] [PubMed]
  5. P. A. Bélanger, M. Rioux, “Ring pattern of a lens–axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17, 1080–1086 (1978).
    [CrossRef] [PubMed]
  6. M. Rioux, P. A. Bélanger, “Linear, annular, and radial focusing with axicons and application to laser machining,” Appl. Opt. 17, 1532–1536 (1978).
    [CrossRef] [PubMed]
  7. G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
  8. G. Häusler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
    [CrossRef] [PubMed]
  9. L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 27, pp. 109–160.
    [CrossRef]
  10. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  11. 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]
  12. D. Marcuse, Principles of Optical Fiber Measurements (Academic, New York, 1981), pp. 167–169.
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.
  14. L. B. Felsen, “Real spectra, complex spectra, compact spectra,” J. Opt. Soc. Am. A 3, 486–496 (1986).
    [CrossRef]
  15. H. L. Kazansky, “Numerical procedure for correcting phase function of focusators,” Komputernaya Optika 1 (1987), in Russian; we hope that this paper is available in the English translation of this journal, Computer Optics (Pergamon, New York). We are unable to specify issue, however.
  16. V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).
  17. See, for example, D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. 28, 976–983 (1989).
    [CrossRef] [PubMed]
  18. J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axicon,” Opt. Lett. 17, 7–9 (1992).
    [CrossRef] [PubMed]
  19. A. Kołodziejczyk, S. Bará, Z. Jaroszewicz, M. Sypek, “The light sword optical element—a new diffractive structure with extended depth of focus,” J. Mod. Opt. 37, 1283–1286 (1990).
    [CrossRef]
  20. Ref. 13, p. 124.

1992 (1)

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axicon,” Opt. Lett. 17, 7–9 (1992).
[CrossRef] [PubMed]

1990 (1)

A. Kołodziejczyk, S. Bará, Z. Jaroszewicz, M. Sypek, “The light sword optical element—a new diffractive structure with extended depth of focus,” J. Mod. Opt. 37, 1283–1286 (1990).
[CrossRef]

1989 (2)

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]

See, for example, D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. 28, 976–983 (1989).
[CrossRef] [PubMed]

1988 (2)

1987 (1)

H. L. Kazansky, “Numerical procedure for correcting phase function of focusators,” Komputernaya Optika 1 (1987), in Russian; we hope that this paper is available in the English translation of this journal, Computer Optics (Pergamon, New York). We are unable to specify issue, however.

1986 (2)

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction pattern and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

L. B. Felsen, “Real spectra, complex spectra, compact spectra,” J. Opt. Soc. Am. A 3, 486–496 (1986).
[CrossRef]

1985 (1)

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

1978 (2)

1974 (1)

1960 (1)

1954 (1)

Bará, S.

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axicon,” Opt. Lett. 17, 7–9 (1992).
[CrossRef] [PubMed]

A. Kołodziejczyk, S. Bará, Z. Jaroszewicz, M. Sypek, “The light sword optical element—a new diffractive structure with extended depth of focus,” J. Mod. Opt. 37, 1283–1286 (1990).
[CrossRef]

Bélanger, P. A.

Bickel, G.

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

Buralli, D. A.

Casperson, L. W.

Churin, E. G.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Cuadrado, J. M.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction pattern and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Felsen, L. B.

Friberg, A. T.

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]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

Gómez-Reino, C.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction pattern and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.

Häusler, G.

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

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

Heckel, W.

Jaroszewicz, Z.

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axicon,” Opt. Lett. 17, 7–9 (1992).
[CrossRef] [PubMed]

A. Kołodziejczyk, S. Bará, Z. Jaroszewicz, M. Sypek, “The light sword optical element—a new diffractive structure with extended depth of focus,” J. Mod. Opt. 37, 1283–1286 (1990).
[CrossRef]

Kazansky, H. L.

H. L. Kazansky, “Numerical procedure for correcting phase function of focusators,” Komputernaya Optika 1 (1987), in Russian; we hope that this paper is available in the English translation of this journal, Computer Optics (Pergamon, New York). We are unable to specify issue, however.

Kolodziejczyk, A.

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axicon,” Opt. Lett. 17, 7–9 (1992).
[CrossRef] [PubMed]

A. Kołodziejczyk, S. Bará, Z. Jaroszewicz, M. Sypek, “The light sword optical element—a new diffractive structure with extended depth of focus,” J. Mod. Opt. 37, 1283–1286 (1990).
[CrossRef]

Korolkov, V. P.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Koronkevitch, V. P.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Marcuse, D.

D. Marcuse, Principles of Optical Fiber Measurements (Academic, New York, 1981), pp. 167–169.

Maul, M.

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

McLeod, J. H.

Mikhaltsova, I. A.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Morris, G. M.

Palchikova, I. G.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Pérez, M. V.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction pattern and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Poleschuk, A. G.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Rioux, M.

Rogers, J. R.

Sedukhin, A. G.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Shekhani, M. S.

Sochacki, J.

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axicon,” Opt. Lett. 17, 7–9 (1992).
[CrossRef] [PubMed]

Sokolov, A. P.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Soroko, L. M.

L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 27, pp. 109–160.
[CrossRef]

Sypek, M.

A. Kołodziejczyk, S. Bará, Z. Jaroszewicz, M. Sypek, “The light sword optical element—a new diffractive structure with extended depth of focus,” J. Mod. Opt. 37, 1283–1286 (1990).
[CrossRef]

Turunen, J.

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]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

Vasara, A.

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]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

Yurkov, Y. I.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

Appl. Opt. (6)

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

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]

J. Mod. Opt. (1)

A. Kołodziejczyk, S. Bará, Z. Jaroszewicz, M. Sypek, “The light sword optical element—a new diffractive structure with extended depth of focus,” J. Mod. Opt. 37, 1283–1286 (1990).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Komputernaya Optika (1)

H. L. Kazansky, “Numerical procedure for correcting phase function of focusators,” Komputernaya Optika 1 (1987), in Russian; we hope that this paper is available in the English translation of this journal, Computer Optics (Pergamon, New York). We are unable to specify issue, however.

Opt. Acta (1)

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction pattern and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Opt. Lett. (1)

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axicon,” Opt. Lett. 17, 7–9 (1992).
[CrossRef] [PubMed]

Opt. Eng. (1)

G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).

Other (5)

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, A. G. Sedukhin, “Kinoforms: technologies, new elements, and optical systems,” preprint IAE SB (USSR Academy of Science, Moscow, 1989), Vol. 421, p 15 (in English).

D. Marcuse, Principles of Optical Fiber Measurements (Academic, New York, 1981), pp. 167–169.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.

L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 27, pp. 109–160.
[CrossRef]

Ref. 13, p. 124.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem and notation.

Fig. 2
Fig. 2

Differential relationship between phase φ and ray deflection θ. The light ray is assumed to be orthogonal to wave fronts.

Fig. 3
Fig. 3

Phase retardation introduced by the forward uniform-intensity axicon (curve 1) and its inverse analog (curve 2). In both cases the geometric parameters are the following: R = 10 mm, d1 = 20 mm, and d2 = 50 mm. Dashed curves 3 and 4 correspond to phase retardation introduced by perfect focusers of the focal lengths f = d1 and f = d2, respectively.

Fig. 4
Fig. 4

Experimentally determined on-axis intensity distribution of the holographic version of the forward axicon.

Fig. 5
Fig. 5

Comparison of the imaging capabilities of (a)–(c) the holographic Fresnel lens and (d)–(f) the holographic version of the uniform-intensity forward axicon (d)–(f).

Equations (26)

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t ( r ) = exp [ i k φ ( r ) ] , k = 2 π λ , r = ( x 2 + y 2 ) 1 / 2 ,
2 π P σ ( r ) r d r = ± P z ( z ) d z ,
2 π 0 r P σ ( r ) r d r = d 1 z ( r ) P z ( z ) d z .
2 π 0 r P σ ( r ) r d r = z ( r ) d 2 P z ( z ) d z .
d φ ( r ) d r = - sin θ = - r [ r 2 + z 2 ( r ) ] - 1 / 2
P σ ( r ) = P σ = const . , P z ( z ) = c z , c = const . , d 1 z d 2 .
z ( r ) = ( d 1 2 + a r 2 ) 1 / 2 ,
z ( r ) = ( d 2 2 - a r 2 ) 1 / 2 ,
a 2 π P σ c = d 2 2 - d 1 2 R 2 .
φ + ( r ) = - [ ( 1 + a ) r 2 + d 1 2 ] 1 / 2 1 + a + const . ,
φ - ( r ) = - [ ( 1 - a ) r 2 + d 2 2 ] 1 / 2 1 - a + const .
φ ( r ) = - ( 1 + d 2 2 R 2 ) - 1 / 2 r ,
z ( r ) = d 1 + a r 2 ,
z ( r ) = d 2 - a r 2 ,
a π P σ P z = d 2 - d 1 R 2 .
φ + ( r ) = - 1 2 a ln { 2 a [ a 2 r 4 + ( 1 + 2 a d 1 ) r 2 + d 1 2 ] 1 / 2 + 2 a 2 r 2 + 1 + 2 a d 1 } + const . ,
φ - ( r ) = - 1 2 a ln { 2 a [ a 2 r 4 + ( 1 - 2 a d 2 ) r 2 + d 2 2 ] 1 / 2 + 2 a 2 r 2 + 1 - 2 a d 2 } + const .
φ ˜ ( r ) = - 1 2 a ln ( d 1 + a r 2 ) + const .
d φ ( r ) d r = - sin θ - tan θ = - r z ( r ) = - r ( d 1 + a r 2 ) - 1 .
φ ( r ) = - ( f 2 + r 2 ) 1 / 2 + const
P σ ( r ) = P σ ( 1 + b 2 r 2 ) - 1 / 2 , P σ = const ,
z ( r ) = a [ ( 1 + b 2 r 2 ) 1 / 2 - 1 ] + d 1 ,
z ( r ) = a [ 1 - ( 1 + b 2 r 2 ) 1 / 2 ] + d 2
a 2 π P σ P z b 2 = d 2 - d 1 ( 1 + b 2 R 2 ) 1 / 2 - 1 .
b 2 = d 2 2 - d 1 2 d 1 2 R 2 .
φ + ( r ) = - { [ 1 + ( d 2 2 - d 1 2 ) / R 2 ] r 2 + d 1 2 } 1 / 2 1 + ( d 2 2 - d 1 2 ) / R 2 + const .

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