Abstract

A two-element refracting system is designed to transform a Gaussian laser beam into a diffraction-free Bessel beam. The resulting input and output surfaces are almost spherical, which makes for easy implementation of the system.

© 1993 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, “Difraction-free beam,” Phys. Rev. Lett. 54, 1499–1501 (1987).
    [CrossRef]
  3. J. E. Durnin, J. J. Miceli, J. H. Eberly, “Reply to DeBeer, S.R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
    [CrossRef] [PubMed]
  4. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959– 3962 (1988).
    [CrossRef] [PubMed]
  5. A. J. Cox, D. C. Dibble, “Holographic reproduction of a diffraction-free beam,” Appl. Opt. 30, 1330–1332 (1991).
    [CrossRef] [PubMed]
  6. J. K. Jabczynski, “A diffraction-free resonator,” Opt. Commun. 77, 292–294 (1990).
    [CrossRef]
  7. M. A. Karim, A. K. Cherri, A. A. S. Awwal, A. Basit, “Refracting system for annular laser beam transformation,” Appl. Opt. 26, 2446–2449 (1987).
    [CrossRef] [PubMed]
  8. K. Thewes, M. A. Karim, A. A. S. Awwal, “Diffraction-free beam generation using refracting system,” Opt. Laser Technol. 23, 105–108 (1991).
    [CrossRef]
  9. M. A. Karim, Electro-Optical Devices and Systems (PWS-Kent, Boston, Mass., 1990).
  10. E. Kreysig, Advance Engineering Mathematics (Wiley, New York, 1983).
  11. S. R. Jahan, M. A. Karim, “Refracting systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
    [CrossRef]
  12. P. Z. Peebles, Probability, Random Variables and Random Signal Principles (McGraw-Hill, New York, 1980).

1991 (2)

K. Thewes, 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 (1)

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

1989 (1)

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

1988 (1)

1987 (4)

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

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Reply to DeBeer, S.R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

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

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

Awwal, A. A. S.

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

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

Basit, A.

Cherri, A. K.

Cox, A. J.

Dibble, D. C.

Durnin, J. E.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Reply to DeBeer, S.R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

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

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

Eberly, J. H.

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

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Reply to DeBeer, S.R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

Friberg, A. T.

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 systems for Gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21, 27–30 (1989).
[CrossRef]

Karim, M. A.

K. Thewes, 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 systems 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 beam transformation,” Appl. Opt. 26, 2446–2449 (1987).
[CrossRef] [PubMed]

M. A. Karim, Electro-Optical Devices and Systems (PWS-Kent, Boston, Mass., 1990).

Kreysig, E.

E. Kreysig, Advance Engineering Mathematics (Wiley, New York, 1983).

Miceli, J. J.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Reply to DeBeer, S.R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

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

Peebles, P. Z.

P. Z. Peebles, Probability, Random Variables and Random Signal Principles (McGraw-Hill, New York, 1980).

Thewes, K.

K. Thewes, 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. (3)

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

Opt. Commun. (1)

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

Opt. Laser Technol. (2)

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

K. Thewes, 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. (2)

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

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Reply to DeBeer, S.R. Hartmann and R. Friedberg,” Phys. Rev. Lett. 59, 2612 (1987).
[CrossRef] [PubMed]

Other (3)

M. A. Karim, Electro-Optical Devices and Systems (PWS-Kent, Boston, Mass., 1990).

E. Kreysig, Advance Engineering Mathematics (Wiley, New York, 1983).

P. Z. Peebles, Probability, Random Variables and Random Signal Principles (McGraw-Hill, New York, 1980).

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Figures (9)

Fig. 1
Fig. 1

Plot of J02(αρ) versus ρ.

Fig. 2
Fig. 2

Half of the axially symmetric two-element refracting system.

Fig. 3
Fig. 3

(a) dyi/dri versus ri, and (b) dy0/dr0 versus r0 data for the system.

Fig. 4
Fig. 4

Curve-fitted plot of (a) dyi/dr0 versus ri and (b) dy0/dr0 versus r0.

Fig. 5
Fig. 5

Plot of (a) yi versus ri and (b) y0 versus r0.

Fig. 6
Fig. 6

Plot of y0 versus r0 that was obtained by using the Runge–Katta approximation.

Fig. 7
Fig. 7

Plot of R versus ri for (a) the first aspheric surface and (b) the second aspheric surface.

Fig. 8
Fig. 8

Number distribution at the (a) input and (b) output.

Fig. 9
Fig. 9

Ray tracing through the refracting system.

Equations (12)

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2 π I 0 0 r i r exp [ 2 ( r / w 0 ) 2 ] d r = k 2 0 r 0 2 π ρ J 0 2 ( α ρ ) d ρ ,
n y i + [ ( r i r 0 ) 2 + ( D y i + y 0 ) 2 ] 1 / 2 n y 0 = f ,
tan ( θ i i θ r i ) = ( r i r 0 ) / ( D y i + y 0 ) = tan ( θ i 0 θ r 0 ) ,
r i = [ ( w 0 2 / 2 ) × ln ( 1 m n { ( 1 ) m + n × π k 2 ( α / 2 ) 2 ( m + n ) r 0 2 ( m + n + 1 ) / [ ( m ! n ! ) 2 2 ( m + n + 1 ) ] } ) ] 1 / 2 ,
r 0 , max { π k 2 m n [ ( 1 ) m + n ( α r 0 , max / 2 ) 2 ( m + n ) / ( m ! n ! ) 2 2 ( m + n + 1 / 2 ) ] } 1 / 2 .
n y i n y 0 n D + [ ( r i r 0 ) 2 + ( D y i + y 0 ) 2 ] 1 / 2 = f n D ,
n tan ( θ i i θ r i ) + sec ( θ i i θ r i ) tan ( θ i i θ r i ) = c r i r 0 ,
( r i r 0 ) [ 1 n cos ( θ i i θ r i ) ] sin ( θ i i θ r i ) = c .
d y i d r i = n ( r i r 0 ) c .
y i = 2.907 r i + 0.725 r i 2 + 0.262 r i 3 + 0.023 r i 4 + 0.052 r i 5 + 0.001 r i 6 + 0.002 r i 7 + 0.000 r i 8 ,
y 0 = 2.907 r 0 + 1.041 r 0 2 + 0.024 r 0 3 + 0.027 r 0 4 + 0.003 r 0 5 + 0.003 r 0 6 + 0.001 r 0 7 + 0.002 r 0 8 ,
R = [ 1 + ( dy / d r ) 2 ] 3 / 2 / | ( d 2 y / d r 2 ) | .

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