Abstract

An approach to efficient axially symmetric focusing of a light beam to an extended line segment is studied wherein the on-axis intensity and the scale of transverse intensity distribution can be controlled precisely and simultaneously at any point of the segment. Schematic realization is based on employing two phase-only optical components in tandem, the first of which, with a disk form, performs prior central light beam shaping and marginal correction to smooth the edge slopes, while the second one, with an annular form, is used for the appropriate axicon-type focusing of a light beam to the segment. An instance is given of converting a collimated Gaussian laser beam into an oscillation-free nondiffracting zero-order Bessel beam reproduced on a finite interval.

© 1998 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2–TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
    [CrossRef]
  4. M. Rioux, R. Tremblay, P. A. Bélanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, 1532–1536 (1978).
    [CrossRef] [PubMed]
  5. G. Haüsler, W. Heckel, “Light sectioning with large depth and high resolution,” Appl. Opt. 27, 5165–5169 (1988).
    [CrossRef] [PubMed]
  6. S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).
  7. S. Klewitz, F. Brinkmann, S. Herminghaus, P. Leiderer, “Bessel-beam-pumped turnable distributed-feedback laser,” Appl. Opt. 34, 7670–7673 (1995).
    [CrossRef] [PubMed]
  8. J. A. Davis, D. M. Cottrell, “Range-finding by triangulation with nondiffracting beams,” Appl. Opt. 35, 2159–2161 (1996).
    [CrossRef] [PubMed]
  9. J. A. Kim, K. I. Lee, H. R. Noh, W. Jhe, M. Ohtsu, “Atom trap in an axicon mirror,” Opt. Lett. 22, 117–119 (1997).
    [CrossRef] [PubMed]
  10. B. Hafizi, E. Esarey, P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539–3545 (1997).
    [CrossRef]
  11. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  12. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general non-diffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  13. R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
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  14. A. J. Cox, D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. 30, 1330–1332 (1991).
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    [CrossRef]
  19. V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 13, 178–183 (1986).
    [CrossRef]
  20. V. P. Koronkevich, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 92–93 (1991).
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  22. J. Sochacki, Z. Jaroszewicz, L. R. Staroński, A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993).
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  23. N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
    [CrossRef]
  24. R. M. Herman, T. A. Wiggins, “High-efficiency diffractionless beams of constant size and intensity,” Appl. Opt. 33, 7297–7306 (1994).
    [CrossRef] [PubMed]
  25. S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
    [CrossRef]
  26. J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 843–845 (1994).
    [CrossRef] [PubMed]
  27. 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]
  28. B. Z. Dong, G. Z. Yang, B. Y. Gu, O. K. Ersoy, “Iterative optimization approach for designing an axicon with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 13, 97–103 (1996).
    [CrossRef]
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    [CrossRef] [PubMed]
  30. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [CrossRef]
  31. R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992).
    [CrossRef] [PubMed]
  32. A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).
    [CrossRef] [PubMed]
  33. S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
    [CrossRef]
  34. H. Bartelt, “Applications of the tandem component: an element with optimum light efficiency,” Appl. Opt. 24, 3811–3816 (1985).
    [CrossRef] [PubMed]
  35. A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7.
  36. V. P. Koronkevich, I. A. Mikhaltsova, E. G. Churin, Yu. I. Yurlov, “Lensacon,” Appl. Opt. 34, 5761–5772 (1995).
    [CrossRef] [PubMed]
  37. O. Bryngdahl, “Geometrical transformations in Optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]
  38. C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beam with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  39. N. C. Roberts, “Beam shaping by holographic filters,” Appl. Opt. 28, 31–32 (1989).
    [CrossRef] [PubMed]

1998 (1)

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

1997 (2)

B. Hafizi, E. Esarey, P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539–3545 (1997).
[CrossRef]

J. A. Kim, K. I. Lee, H. R. Noh, W. Jhe, M. Ohtsu, “Atom trap in an axicon mirror,” Opt. Lett. 22, 117–119 (1997).
[CrossRef] [PubMed]

1996 (4)

1995 (3)

1994 (3)

1993 (3)

1992 (4)

A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992).
[CrossRef] [PubMed]

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 31, 5326–5330 (1992).

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992).
[CrossRef] [PubMed]

1991 (3)

1989 (2)

1988 (1)

1987 (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1986 (1)

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 13, 178–183 (1986).
[CrossRef]

1985 (1)

1984 (1)

I. A. Mikhaltsova, V. I. Nalivaiko, I. S. Soldatenkov, “Kinoform axicons,” Optik 67, 267–278 (1984).

1983 (1)

1980 (1)

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

1979 (1)

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2–TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

1978 (1)

1974 (1)

1962 (1)

1960 (1)

1954 (1)

Bará, S.

J. Sochacki, S. Bará, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 31, 5326–5330 (1992).

Bartelt, H.

Bélanger, P. A.

Blanchard, M.

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2–TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

Brinkmann, F.

Brinkmann, S.

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Bryngdahl, O.

Churin, E. G.

Cottrell, D. M.

Cox, A. J.

D’Anna, J.

D’Astous, Y.

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2–TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

Davidson, N.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

Davis, J. A.

Dibble, D. C.

Dong, B. Z.

Dresel, T.

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Ersoy, O. K.

Esarey, E.

B. Hafizi, E. Esarey, P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539–3545 (1997).
[CrossRef]

Friberg, A. T.

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general non-diffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

Friesem, A. A.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

Fujiwara, S.

Gu, B. Y.

Hafizi, B.

B. Hafizi, E. Esarey, P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539–3545 (1997).
[CrossRef]

Han, C.-Y.

Hasman, E.

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

Haüsler, G.

Heckel, W.

Herman, R. M.

Herminghaus, S.

Ishii, Y.

Jaroszewicz, Z.

Jhe, W.

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]

Kim, J. A.

Klewitz, S.

Kolodziejczyk, A.

Korobkin, V. V.

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 13, 178–183 (1986).
[CrossRef]

Koronkevich, V. P.

V. P. Koronkevich, I. A. Mikhaltsova, E. G. Churin, Yu. I. Yurlov, “Lensacon,” Appl. Opt. 34, 5761–5772 (1995).
[CrossRef] [PubMed]

V. P. Koronkevich, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 92–93 (1991).

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]

Lee, K. I.

Leiderer, P.

McLeod, J. H.

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Mikhaltsova, I. A.

V. P. Koronkevich, I. A. Mikhaltsova, E. G. Churin, Yu. I. Yurlov, “Lensacon,” Appl. Opt. 34, 5761–5772 (1995).
[CrossRef] [PubMed]

I. A. Mikhaltsova, V. I. Nalivaiko, I. S. Soldatenkov, “Kinoform axicons,” Optik 67, 267–278 (1984).

Murata, K.

Nalivaiko, V. I.

I. A. Mikhaltsova, V. I. Nalivaiko, I. S. Soldatenkov, “Kinoform axicons,” Optik 67, 267–278 (1984).

Noh, H. R.

Ohtsu, M.

Palchikova, I. G.

V. P. Koronkevich, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 92–93 (1991).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7.

Polonskii, L. Ya.

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 13, 178–183 (1986).
[CrossRef]

Poponin, V. P.

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 13, 178–183 (1986).
[CrossRef]

Popov, S. Yu.

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

Pyatnitskii, L. N.

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 13, 178–183 (1986).
[CrossRef]

Rioux, M.

Roberts, N. C.

Rosen, J.

Roy, G.

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2–TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

Schreiner, R.

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Schwider, J.

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

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]

Soldatenkov, I. S.

I. A. Mikhaltsova, V. I. Nalivaiko, I. S. Soldatenkov, “Kinoform axicons,” Optik 67, 267–278 (1984).

Sprangle, P.

B. Hafizi, E. Esarey, P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539–3545 (1997).
[CrossRef]

Staronski, L. R.

Tremblay, R.

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2–TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

M. Rioux, R. Tremblay, P. A. Bélanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, 1532–1536 (1978).
[CrossRef] [PubMed]

Turunen, J.

Vasara, A.

Wiggins, T. A.

Yang, G. Z.

Yariv, A.

Yurlov, Yu. I.

Appl. Opt. (11)

R. M. Herman, T. A. Wiggins, “High-efficiency diffractionless beams of constant size and intensity,” Appl. Opt. 33, 7297–7306 (1994).
[CrossRef] [PubMed]

S. Klewitz, F. Brinkmann, S. Herminghaus, P. Leiderer, “Bessel-beam-pumped turnable distributed-feedback laser,” Appl. Opt. 34, 7670–7673 (1995).
[CrossRef] [PubMed]

V. P. Koronkevich, I. A. Mikhaltsova, E. G. Churin, Yu. I. Yurlov, “Lensacon,” Appl. Opt. 34, 5761–5772 (1995).
[CrossRef] [PubMed]

J. A. Davis, D. M. Cottrell, “Range-finding by triangulation with nondiffracting beams,” Appl. Opt. 35, 2159–2161 (1996).
[CrossRef] [PubMed]

M. Rioux, R. Tremblay, P. A. Bélanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, 1532–1536 (1978).
[CrossRef] [PubMed]

C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beam with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
[CrossRef] [PubMed]

H. Bartelt, “Applications of the tandem component: an element with optimum light efficiency,” Appl. Opt. 24, 3811–3816 (1985).
[CrossRef] [PubMed]

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

N. C. Roberts, “Beam shaping by holographic filters,” Appl. Opt. 28, 31–32 (1989).
[CrossRef] [PubMed]

A. J. Cox, D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. 30, 1330–1332 (1991).
[CrossRef] [PubMed]

R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992).
[CrossRef] [PubMed]

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. (4)

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

N. Davidson, A. A. Friesem, E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[CrossRef]

R. Tremblay, Y. D’Astous, G. Roy, M. Blanchard, “Laser plasmas optically pumped by focusing with an axicon a CO2–TEA laser beam in a high-pressure gas,” Opt. Commun. 28, 193–196 (1979).
[CrossRef]

G. Roy, R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

Opt. Eng. (1)

S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

Opt. Lett. (6)

Optik (3)

I. A. Mikhaltsova, V. I. Nalivaiko, I. S. Soldatenkov, “Kinoform axicons,” Optik 67, 267–278 (1984).

V. P. Koronkevich, I. G. Palchikova, “Kinoforms with increased depth of focus,” Optik 87, 92–93 (1991).

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Phys. Rev. E (1)

B. Hafizi, E. Esarey, P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539–3545 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. V. Korobkin, L. Ya. Polonskii, V. P. Poponin, L. N. Pyatnitskii, “Focusing of Gaussian and super-Gaussian laser beams by axicons to obtain continuous laser sparks,” Sov. J. Quantum Electron. 13, 178–183 (1986).
[CrossRef]

Other (1)

A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7.

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Figures (8)

Fig. 1
Fig. 1

Formation of an interference pattern by a local annular portion of a converging axially symmetric wave front.

Fig. 2
Fig. 2

Illustration of two stages of beam-preshaping axicon focusing. Smoothing and broadening of I(R) and I(z) is shown as dashed lines.

Fig. 3
Fig. 3

Example of a diffractive compound axicon.

Fig. 4
Fig. 4

Variations of the phase functions (solid curves) of compound-axicon components, in focusing a collimated Gaussian beam, with the e-2 radius r0=0.75 mm and λ=0.6328 µm, to a constant-axial-intensity line segment of length L=20 mm (z1=8.10 mm, z2=28.10 mm) and constant central-spot diameter d0=2 µm. The design axicon’s parameters are np=1.5183, zp/2=15 mm, ra=1.5 mm, R1=2 mm, and R2=6.94 mm. The dashed curve corresponds to the corrected phase function φ2(R), which was obtained by direct calculation of the phase distortion before the second component (through the Fresnel transform), compensating this distortion, and adding the conical term.

Fig. 5
Fig. 5

Light intensity distributions of the compound axicon (a) at the plane of first component, (b) at the plane of second component, and (c) within the focal segment for the uncorrected phase functions and parameters of Fig. 4.

Fig. 6
Fig. 6

(a) Corrective phase function of the first component and (b) the displacements of the points of the radial coordinate R and the axial coordinate z after correction.

Fig. 7
Fig. 7

Light intensity distributions of the compound axicon (a) at the plane of the second component and (b) within the focal segment after correction; (c) is the central-spot-diameter variation.

Fig. 8
Fig. 8

Comparison of the focal regions of (a) an equivalent lens and (b) the compound axicon with the phase functions and parameters of Figs. 4 and 6(a). [The scale along the z axis in (b) is reduced with respect to that of (a) by a factor of 2×103.]

Equations (35)

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U(R)=U0(R)exp[-iφ(R)],
dφ(R)/dR=-kn sin θ(R)-knθ(R),
U(ρ, z)=kiz RU0(R)exp[iΦ(R, ρ, z)]J0kRρzRdR,
Φ(R, ρ, z)=k[z+(R2+ρ2)/(2z)]+φ(R).
R0=-z0k dφ(R0)dRz0θ(R0)z0θ(z0).
U(ρ, z0)kiz0U0[R0(z0)]J0kR0(z0)ρz0R0(z0)2πz0k+z0d2φ[R0(z0)]/dR21/2 expikz0+R02(z0)+ρ22z0+φ[R0(z0)]k+i π4.
|U(ρ, z)|U0[zθ(z)]J0[kθ(z)ρ]θ(z)(2πkz)1/21+zθ(z) θ(z)z-1/2.
d(z)=2j1/[kθ(z)],
φ(R)=-2j10R dRd(R).
|U0(z)|U(0, z)θ(z)(2πkz)-1/21+zθ(z) θ(z)z1/2.
2π0rI(r)rdr=2πRI(R)RdRZI(z)dz,
R1R or RR2 or -R-R1 or -R2-R.
I(z)=I0=const.z[z1, z2]0z[z1, z2],
d(z)=d0=const.z[z1, z2],
I(r)=exp(-2r2/r02)0rra0r>ra,
θ(z)=θ(R)=2j1/(kd0)=θ0=const.
I(R)=I0/R,
1-exp(-2r2/r02)1-exp(-2ra2/r02)=R2-RR2-R1.
dφ1(r)dr=-knp sin γ(r)knp R(r)-rzp,
φ1(r)=knpzp R2r-R2-R11-exp(-2ra2/r02)×r-π8 r0 erf2r0r-r22,
φ2(R)=-φ1[r(R)]-knp{zp2+[R-r(R)]2}1/2-kRθ0,
ΔR(r)=R(r)-R=[zp/(knp)]dφ1cor(r)/dr,
τL(r)=exp[-ikr2/(2 f)],
IL(ρ, ΔzL)=kf 0ra[I(r)]1/2 exp-ikr2ΔzLf2×J0krρfrdr2.
IL(ρ, 0)=2kSπΔf exp-k2θ02ρ22,
IL(0, Δz)=2kSπΔf 1+2kSπΔf Δzf2-1,
Δf=4/(kθ02)
dL(0.5)=8 ln 2/(kθ0)2.355/(kθ0).
I(ρ, z)=(kS/L)J02(kθ0ρ),z[z1,z2],
I(0, z)=(kS/L)=I0,z[z1,z2],
d0(0.5)2.253/(kθ0),
I0/IL(0, 0)=(π/2)(Δf/L).
I(R)=knpzp 0ra[I(r)]1/2 exp[iφ1(r)]J0knpRrzprdr2.
I(ρ, z)=kz R1R2[I(R)]1/2 expikR22z-Rθ0×J0kRρzRdR2
φ1cor(r)=-18r0r+1.04r048+r0r-3.16r010,

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