Abstract

A refracting system consisting of two lenses is designed to transform an annular Gaussian laser beam into a circular Bessel beam. The slopes of the input and output surfaces fit well with a sixth-order polynomial. A smooth variation of the radii of curvature of the resulting aspheric surfaces is very attractive for easy machining of the surfaces. The diffraction-free length for the designed system is 59.4735 m at 633 nm.

© 1998 Optical Society of America

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References

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  1. J. E. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  2. J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 54, 1499–1501 (1987).
    [CrossRef]
  3. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  4. A. J. Cox, D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. 30, 1330–1332 (1991).
    [CrossRef] [PubMed]
  5. J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
    [CrossRef]
  6. M. A. Karim, A. K. Cherri, A. A. S. Awwal, A. Basit, “Refracting system for annular laser transformation,” Appl. Opt. 26, 2446–2449 (1987).
    [CrossRef] [PubMed]
  7. S. R. Jahan, M. A. Karim, “Refracting system for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
    [CrossRef]
  8. K. Thews, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
    [CrossRef]
  9. K. M. Iftekharuddin, A. A. S. Awwal, M. A. Karim, “Gaussian-to-Bessel beam transformation using a split refracting system,” Appl. Opt. 32, 2252–2256 (1993).
    [CrossRef]
  10. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  11. M. Arif, M. M. Hossain, A. A. S. Awwal, M. N. Islam, “Refracting system for annular Gaussian-to-Bessel beam transformation,” Appl. Opt. 37, 649–652 (1998).
    [CrossRef]
  12. E. Kreyszig, Advanced Engineering Mathematics (Wiley, New York, 1983).

1998

1993

1991

K. Thews, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
[CrossRef]

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

1990

J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

1989

S. R. Jahan, M. A. Karim, “Refracting system for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

1988

1987

Arif, M.

Awwal, A. A. S.

Basit, A.

Cherri, A. K.

Cox, A. J.

Dibble, D. C.

Durnin, J. E.

J. E. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

Friberg, A. T.

Hossain, M. M.

Iftekharuddin, K. M.

Islam, M. N.

Jabczynski, J. K.

J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

Jahan, S. R.

S. R. Jahan, M. A. Karim, “Refracting system for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

Karim, M. A.

K. M. Iftekharuddin, A. A. S. Awwal, M. A. Karim, “Gaussian-to-Bessel beam transformation using a split refracting system,” Appl. Opt. 32, 2252–2256 (1993).
[CrossRef]

K. Thews, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
[CrossRef]

S. R. Jahan, M. A. Karim, “Refracting system for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

M. A. Karim, A. K. Cherri, A. A. S. Awwal, A. Basit, “Refracting system for annular laser transformation,” Appl. Opt. 26, 2446–2449 (1987).
[CrossRef] [PubMed]

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics (Wiley, New York, 1983).

Miceli, J. J.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

Siegman, A. E.

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

Thews, K.

K. Thews, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
[CrossRef]

Turunen, J.

Vasara, A.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

Opt. Laser Technol.

S. R. Jahan, M. A. Karim, “Refracting system for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

K. Thews, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
[CrossRef]

Phys. Rev. Lett.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beam,” Phys. Rev. Lett. 54, 1499–1501 (1987).
[CrossRef]

Other

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

E. Kreyszig, Advanced Engineering Mathematics (Wiley, New York, 1983).

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Figures (6)

Fig. 1
Fig. 1

Half of the axially symmetric two-element refracting system.

Fig. 2
Fig. 2

Intensity profile for (a) annular Gaussian and (b) Bessel beams.

Fig. 3
Fig. 3

Plot of (a) dy i /dr i versus r i , (b) dy o /dr o versus r o for the system.

Fig. 4
Fig. 4

Plot of (a) y i versus r i , (b) y o versus r o for the system.

Fig. 5
Fig. 5

Plot of R versus r for (a) input and (b) output surfaces.

Fig. 6
Fig. 6

Ray tracing through the refractive system.

Equations (15)

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I g r = I g 0 1 - R o exp - 2 r 2 / w o 2 exp - 2 r 2 / w 2 ,
I b ρ = I b 0 J 0 2 α ρ ,
0 r i   r 1 - R o exp - 2 r 2 / w o 2 exp - 2 r 2 / w 2 d r = k   0 r o   ρ J 0 2 α ρ d ρ ,
tan θ ii - θ ri = tan θ io - θ ro = r i - r o / D - y i - y o ,
ny i + r i - r o 2 + D - y i - y o 2 1 / 2 + ny o = f .
w 2 4 1 - exp - 2 r i 2 / w 2 - R o 4 w o 2 w 2 w 0 2 + w 2 × 1 - exp - 2 r i 2 1 w o 2 + 1 w 2 = k r o 2 2   J 0 2 α r o + r o 2 2   J 1 2 α r o .
f = f - nD ,
r i - r o 1 - n   cos θ ii - θ ri / sin θ ii - θ ri = f .
d y i / d r i = - tan   θ ii = - tan   θ io = d y o / d r o .
d y i / d r i = d y o / d r o = - 1 / f / r i - r o 2 + n 2 - 1 1 / 2 .
d y i d r i = 0.0002 - 0.2477 r i + 0.3853 r i 2 + 0.3444 r i 3 - 2.1948 r i 4 + 1.3202 r i 5 + 0.7853 r i 6 ,
d y o d r o = 0.0002 - 0.8014 r o + 8.8474 r o 2 - 64.5471 r o 3 + 270.4415 r o 4 - 606.7529 r o 5 + 551.0820 r o 6 .
y i = 0.0002 r i - 0.1239 r i 2 + 0.1284 r i 3 + 0.0861 r i 4 - 0.4390 r i 5 + 0.2200 r i 6 + 0.1122 r i 7 ,
y o = 0.0002 r o - 0.4007 r o 2 + 2.9491 r o 3 - 16.1368 r o 4 + 54.0883 r o 5 - 101.1255 r o 6 + 78.7257 r o 7 .
R = 1 + d y / d r 2 3 / 2 / | d 2 y / d r 2 | .

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