Abstract

An iterative optimization approach for designing the phase of the uniform-intensity axicon is presented that is based on the general theory of amplitude-phase retrieval in optical systems. We extend previous theoretical formulas to deal with the linear imaging system with multiple output planes. We carry out numerical simulation calculations for designing the axicon for cases of both uniform and Gaussian beam illuminations. The numerical results are in good agreement with the desired performance of the axicon, for instance, with considerable uniformity and smoothness of the on-axis intensity and energy flow as well as with high lateral resolution. The influence of phase quantization on the designed result is also investigated.

© 1996 Optical Society of America

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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 zero 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]
  6. M. Rioux, P. A. Bélanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, 1532–1536 (1978).
    [CrossRef] [PubMed]
  7. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  8. 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]
  9. G. Bickel, G. Häusler, M. Maul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
    [CrossRef]
  10. G. Häusler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
    [CrossRef] [PubMed]
  11. L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), Vol. 27, pp. 109–160.
    [CrossRef]
  12. A. T. Friberg, S. Y. Popov, “Partially coherently illuminated uniform-intensity holographic axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.
  13. Z. Jaroszewicz, J. F. R. Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicon,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.
  14. N. Davidson, A. A. Friesem, E. Hasman, “Holographic axicon: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991).
    [CrossRef] [PubMed]
  15. J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 17, 7–9 (1992).
    [CrossRef] [PubMed]
  16. J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, S. Bará, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [CrossRef] [PubMed]
  17. L. F. Staroński, J. Sochacki, Z. Jaroszewicz, A. Kołodziejczyk, “Lateral distribution and flow of energy in uniform-intensity axicons,” J. Opt. Soc. Am. A 9, 2091–2094 (1992).
    [CrossRef]
  18. J. Sochacki, Z. Jaroszewicz, L. R. Staroński, A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993).
    [CrossRef]
  19. Z. Jaroszewicz, J. Sochacki, A. Kołodziejczyk, L. R. Staroński, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
    [CrossRef] [PubMed]
  20. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
    [CrossRef]
  21. R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
    [CrossRef] [PubMed]
  22. J. Rosen, A. Yariv, “Synsthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 843–845 (1994).
    [CrossRef] [PubMed]
  23. J. Rosen, “Synsthesis of nondiffracting beams in free space,” Opt. Lett. 19, 369–371 (1994).
    [PubMed]
  24. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).
  25. B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981) (in Chinese).
  26. G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta. Phys. Sin. 30, 410–413 (1981) (in Chinese).
  27. G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involving nonunitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).
  28. G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3153–3224 (1993).
    [CrossRef]
  29. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
    [CrossRef] [PubMed]
  30. G. Z. Yang, B. Y. Gu, X. Tan, M. P. Chang, B. Z. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. 11, 1632–1640 (1994).
    [CrossRef]
  31. B. Y. Gu, G. Z. Yang, B. Z. Dong, M. P. Chang, O. K. Ersoy, “Diffractive-phase-element design that implements several optical functions,” Appl. Opt. 34, 2564–2570 (1995).
    [CrossRef] [PubMed]
  32. B. Dong, G. Yang, B. Gu, “Phase retardation for a uniform-intensity axicon: a new method of design,” submitted to Opt. Lett.
  33. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.

1995 (1)

1994 (5)

1993 (4)

J. Sochacki, Z. Jaroszewicz, L. R. Staroński, A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993).
[CrossRef]

Z. Jaroszewicz, J. Sochacki, A. Kołodziejczyk, L. R. Staroński, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993).
[CrossRef] [PubMed]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

1992 (3)

1991 (1)

1989 (1)

1988 (2)

1987 (1)

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involving nonunitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

1986 (1)

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

1985 (1)

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

1981 (2)

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981) (in Chinese).

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta. Phys. Sin. 30, 410–413 (1981) (in Chinese).

1978 (2)

1974 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).

1960 (1)

1954 (1)

Bará, S.

Bélanger, P. A.

Bickel, G.

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

Casperson, L. W.

Chang, M. P.

B. Y. Gu, G. Z. Yang, B. Z. Dong, M. P. Chang, O. K. Ersoy, “Diffractive-phase-element design that implements several optical functions,” Appl. Opt. 34, 2564–2570 (1995).
[CrossRef] [PubMed]

G. Z. Yang, B. Y. Gu, X. Tan, M. P. Chang, B. Z. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. 11, 1632–1640 (1994).
[CrossRef]

Cuadrado, J. M.

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

Davidson, N.

Dong, B.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involving nonunitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

B. Dong, G. Yang, B. Gu, “Phase retardation for a uniform-intensity axicon: a new method of design,” submitted to Opt. Lett.

Dong, B. Z.

B. Y. Gu, G. Z. Yang, B. Z. Dong, M. P. Chang, O. K. Ersoy, “Diffractive-phase-element design that implements several optical functions,” Appl. Opt. 34, 2564–2570 (1995).
[CrossRef] [PubMed]

G. Z. Yang, B. Y. Gu, X. Tan, M. P. Chang, B. Z. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. 11, 1632–1640 (1994).
[CrossRef]

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
[CrossRef] [PubMed]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

Dopazo, J. F. R.

Z. Jaroszewicz, J. F. R. Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicon,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.

Ersoy, O. K.

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]

A. T. Friberg, S. Y. Popov, “Partially coherently illuminated uniform-intensity holographic axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.

Friesem, A. A.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).

Gómez-Reino, C.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction pattern and zero 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.

Gu, B.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involving nonunitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta. Phys. Sin. 30, 410–413 (1981) (in Chinese).

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981) (in Chinese).

B. Dong, G. Yang, B. Gu, “Phase retardation for a uniform-intensity axicon: a new method of design,” submitted to Opt. Lett.

Gu, B. Y.

B. Y. Gu, G. Z. Yang, B. Z. Dong, M. P. Chang, O. K. Ersoy, “Diffractive-phase-element design that implements several optical functions,” Appl. Opt. 34, 2564–2570 (1995).
[CrossRef] [PubMed]

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
[CrossRef] [PubMed]

G. Z. Yang, B. Y. Gu, X. Tan, M. P. Chang, B. Z. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. 11, 1632–1640 (1994).
[CrossRef]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

Hasman, E.

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).
[CrossRef]

Heckel, W.

Jaroszewicz, Z.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

Kolodziejczyk, A.

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

Maul, M.

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

McLeod, J. H.

Pérez, M. V.

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

Piestun, R.

Popov, S. Y.

A. T. Friberg, S. Y. Popov, “Partially coherently illuminated uniform-intensity holographic axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.

Rioux, M.

Rosen, J.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).

Shamir, J.

Shekhani, M. S.

Sochacki, J.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

Soroko, L. M.

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

Staronski, L. F.

Staronski, L. R.

Tan, X.

G. Z. Yang, B. Y. Gu, X. Tan, M. P. Chang, B. Z. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. 11, 1632–1640 (1994).
[CrossRef]

Turunen, J.

Vasara, A.

Wang, L.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involving nonunitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Yang, G.

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involving nonunitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981) (in Chinese).

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta. Phys. Sin. 30, 410–413 (1981) (in Chinese).

B. Dong, G. Yang, B. Gu, “Phase retardation for a uniform-intensity axicon: a new method of design,” submitted to Opt. Lett.

Yang, G. Z.

B. Y. Gu, G. Z. Yang, B. Z. Dong, M. P. Chang, O. K. Ersoy, “Diffractive-phase-element design that implements several optical functions,” Appl. Opt. 34, 2564–2570 (1995).
[CrossRef] [PubMed]

G. Z. Yang, B. Y. Gu, X. Tan, M. P. Chang, B. Z. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. 11, 1632–1640 (1994).
[CrossRef]

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
[CrossRef] [PubMed]

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

Yariv, A.

Zhuang, J. Y.

Acta Opt. Sin. (1)

B. Gu, G. Yang, “On the phase retrieval problem in optical and electronic microscopy,” Acta Opt. Sin. 1, 517–522 (1981) (in Chinese).

Acta. Phys. Sin. (1)

G. Yang, B. Gu, “On the amplitude-phase retrieval problem in the optical system,” Acta. Phys. Sin. 30, 410–413 (1981) (in Chinese).

Appl. Opt. (8)

Int. J. Mod. Phys. B (1)

G. Z. Yang, B. Y. Gu, B. Z. Dong, “Theory of the amplitude-phase retrieval in any linear transform system and its applications,” Int. J. Mod. Phys. B 7, 3153–3224 (1993).
[CrossRef]

J. Mod. Opt. (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, “Calculation of the focusators into a longitudinal line-segment and study of a focal area,” J. Mod. Opt. 40, 761–769 (1993).
[CrossRef]

J. Opt. Soc. Am. (3)

G. Z. Yang, B. Y. Gu, X. Tan, M. P. Chang, B. Z. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. 11, 1632–1640 (1994).
[CrossRef]

J. H. McLeod, “Axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
[CrossRef]

J. H. McLeod, “Axicons and their uses,” J. Opt. Soc. Am. 50, 166–169 (1960).
[CrossRef]

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

Opt. Acta (1)

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

Opt. Eng. (1)

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

Opt. Lett. (6)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane picture,” Optik 35, 237–246 (1972).

Optik (Stuttgart) (1)

G. Yang, L. Wang, B. Dong, B. Gu, “On the amplitude-phase retrieval problem in an optical system involving nonunitary transformation,” Optik (Stuttgart) 75, 68–74 (1987).

Other (5)

B. Dong, G. Yang, B. Gu, “Phase retardation for a uniform-intensity axicon: a new method of design,” submitted to Opt. Lett.

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, New York, 1989), Vol. 27, pp. 109–160.
[CrossRef]

A. T. Friberg, S. Y. Popov, “Partially coherently illuminated uniform-intensity holographic axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.

Z. Jaroszewicz, J. F. R. Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicon,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.

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Figures (6)

Fig. 1
Fig. 1

Schematic of a diffractive optical system for designing an axicon.

Fig. 2
Fig. 2

Three-dimensional plots for the axial and lateral intensity distributions produced by an axicon with three different phase-retardation functions: (a) ϕ1c(r1), (b) ϕ1a(r1), and (c) ϕ1b(r1). The relevant parameters are as follows: the central position of the focal region is located at fc = 80 mm, the focal depth is assumed to be δzg = 8 mm, the radius of the axicon is R = 1.5 mm, and the wavelength of the illuminating light beam is λ = 780 nm; the numbers of sampling points are N1r = 128, N2r = 8, and Nz = 40. The incident light beam has a uniform-intensity profile.

Fig. 3
Fig. 3

Effect of phase quantization of the axicon on the on-axis intensity distribution. The phase of the axicon is designed by the YG algorithm, and the illuminating light beam has a Gaussian profile with width parameter σ = 0.843 mm. The relevant parameters are as follows: fc = 80 mm, δzg = 6 mm, R = 1.9 mm, N1r = 128, N2r = 8, and Nz = 40, λ = 780 nm. Solid curve, continuous phase value; dashed curve, quantized phase with 16 levels (of the order of k = 4); dotted curve, quantized phase with 8 levels (of the order of k = 3).

Fig. 4
Fig. 4

Three-dimensional plots of the axial and lateral intensity distributions generated by the axicon with continuous phase and with quantized phase distribution: (a) continuous phase, (b) quantization phase of the order of k = 4. The phase of the axicon is designed by the YG algorithm, and the illuminating light beam has a Gaussian profile. The relevant parameters are the same as for Fig. 3.

Fig. 5
Fig. 5

Calculated quantization phase distribution of the axicon of the order of k = 4. The relevant parameters are the same as for Fig. 3.

Fig. 6
Fig. 6

Energy flow around the focal range of the axicon designed by the YG algorithm. The solid curve corresponds to the case of the uniform illuminating axicon. The radial size R2 of the limited aperture takes 18.75 μm in evaluating the energy flow integral of Eq. (17). The other parameters are the same as for Fig. 2. Dotted curve corresponds to the case of the Gaussian-profile beam illumination; the radial size R2 is 35.65 μm. The other parameters are the same as for Fig. 3.

Equations (21)

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

U 1 = U 1 ( X 1 ) = ρ 1 ( X 1 ) exp [ i ϕ 1 ( X 1 ) ] .
U 2 α = U 2 ( X 2 α , z α ) = ρ 2 ( X 2 α , z α ) exp [ i ϕ 2 ( X 2 α , z α ) ] .
U 2 ( X 2 α , z α ) = G ( X 2 α , X 1 , z α ) U 1 ( X 1 ) d X 1 .
U 2 ( X 2 α , z α ) = G ^ ( z α ) U 1 ( X 1 ) ,
U 1 n = ρ 1 n exp ( i ϕ 1 n ) ,             n = 1 , 2 , 3 N 1 r ,
U 2 m α = ρ 2 m α exp ( i ϕ 2 m α ) ,             m = 1 , 2 , 3 N 2 r , α = 1 , 2 , 3 N z .
U 2 m α = n = 1 N 1 r G m n ( z α ) U 1 n ,             m = 1 , 2 , 3 N 2 r , α = 1 , 2 , 3 N z .
D 2 = α [ U 2 α - G ^ ( z α ) U 1 ] 2 = ( 1 / N 2 ) α ( i ρ 2 i α 2 + i j ρ 1 i ρ 1 j A i j ( z α , z α ) × exp [ - i ( ϕ 1 i - ϕ 1 j ) ] - i j { ρ 2 i α ρ 1 j G i j ( z α ) exp [ - i ( ϕ 2 i α - ϕ 1 j ) ] + c . c . } ) ,
D 2 ϕ 1 k = i N 2 α ( j { ρ 1 j ρ 1 k A j k ( z α ) exp [ - i ( ϕ 1 j - ϕ 1 k ) ] - c . c . } - j { ρ 2 j α ρ 1 k G j k ( z α ) exp [ - i ( ϕ 2 j α - ϕ 1 k ) ] - c . c . } ) = 0.
Im [ Q ˜ k ρ 1 k exp ( i ϕ 1 k ) ] = 0 ,
Q ˜ k = α [ j k ρ 1 j exp ( - i ϕ 1 j ) A j k ( z α ) - j ρ 2 j α exp ( - i ϕ 2 j α ) G j k ( z α ) ] .
exp ( i ϕ 1 k ) = Q ˜ k * Q ˜ k ,             k = 1 , 2 , 3 N 1 r .
D 2 ϕ 2 k γ = i N 2 [ ρ 2 k γ exp ( - i ϕ 2 k γ ) j G k j ( z γ ) ρ 1 j exp ( i ϕ 1 j ) - c . c . ] = 0.
Im [ ρ 2 k γ exp ( - i ϕ 2 k γ ) j G k j ( z γ ) ρ 1 j exp ( i ϕ 1 j ) ] = 0 ,
exp ( i ϕ 2 k γ ) = j G k j ( z γ ) ρ 1 j exp ( i ϕ 1 j ) | j G k j ( z γ ) ρ 1 j exp ( i ϕ 1 j ) | , k = 1 , 2 , 3 N 2 r ,             γ = 1 , 2 , 3 N z .
G ( r 2 α , r 1 , z α ) = 2 π i λ z α exp ( i 2 π z α / λ ) × exp [ i π ( r 1 2 + r 2 α 2 ) / λ z α ] × J 0 ( 2 π r 1 r 2 α / λ z α ) r 1 ,
I ( z , r 2 ) = ( 2 π λ z ) 2 | 0 R exp { i [ π r 1 2 / λ + ϕ 1 ( r 1 ) ] } × J 0 ( 2 π r 1 r 2 / λ z ) r 1 ρ 1 ( r 1 ) d r 1 | 2 ,
ϕ 1 a ( r 1 ) = - ( π λ ) r 1 2 ( f 0 + δ z g r 1 2 / R 2 )             for 0 r 1 R ,
ϕ 1 b ( r 1 ) = - ( π λ ) ( R 2 δ z g ) ln ( 1 + δ z g r 1 2 f 0 R 2 )             for 0 r 1 R ,
ϕ 1 c ( r 1 ) = ( 2 π λ ) ( n s - 1 ) h 1 ( r 1 )             for 0 r 1 R .
E ( z ) = 2 π 0 R 2 I ( z , r 2 ) r 2 d r 2

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