Abstract

Refractive systems using two gradient-index lenses have been designed to convert a collimated Gaussian-profile laser beam into a plane wave with a uniform intensity distribution. The axial gradient-index distribution for two lenses is determined by using the energy conservation condition and the constant optical path-length condition. The design consideration and theoretical analysis are presented along with several applications.

© 1993 Optical Society of America

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References

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  1. J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent3,476,463 (4November1969).
  2. P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
    [CrossRef] [PubMed]
  3. D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
    [CrossRef]
  4. Y. Ozaki, K. Takamoto, “Cylindrical fly’s eye lens for intensity redistribution of an excimer laser beam,” Appl. Opt. 28, 106–110 (1989).
    [CrossRef] [PubMed]
  5. W. B. Veldkamp, “Laser beam profile shaping with interlaced binary diffraction gratings,” Appl. Opt. 21, 3209–3212 (1982).
    [CrossRef] [PubMed]
  6. C. Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  7. C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
    [CrossRef]
  8. P. J. Sands, “Inhomogeneous lenses. IV: Aberrations of lenses with axial index distribution,” J. Opt. Soc. Am. 61, 1086–1091 (1971).
    [CrossRef]
  9. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 5.
  10. Y. Koike, H. Hidaka, Y. Ohtsuka, “Plastic axial gradient-index lens,” Appl. Opt. 24, 4321–4325 (1985).
    [CrossRef] [PubMed]
  11. P. O. Mclaughlin, M. Toyama, I. Kitano, “Gradient-index objective lens for the compact disk system,” Appl. Opt. 25, 3340–3344 (1986).
    [CrossRef]
  12. D. S. Kindred, D. T. Moore, “Design, fabrication, and testing of a gradient-index binocular objective,” Appl. Opt. 27, 492–495 (1988).
    [CrossRef] [PubMed]
  13. I. Kitano, “Current status of aplanatic gradient-index lens systems,” Appl. Opt. 29, 3992–3997 (1990).
    [CrossRef] [PubMed]
  14. D. S. Kindred, J. Bentley, D. T. Moore, “Axial and radial gradient-index titania flint glasses,” Appl. Opt. 29, 4036–4041 (1990).
    [CrossRef] [PubMed]
  15. J. E. Samuels, D. T. Moore, “Gradient-index profile control from mixed molten salt baths,” Appl. Opt. 29, 4042–4050 (1990).
    [CrossRef] [PubMed]
  16. K. Shingyouchi, S. Konishi, “Gradient-index doped silica rod lenses produced by a solgel method,” Appl. Opt. 29, 4061–4063 (1990).
    [CrossRef] [PubMed]

1991 (1)

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

1990 (4)

1989 (1)

1988 (1)

1986 (1)

1985 (1)

1983 (1)

1982 (2)

W. B. Veldkamp, “Laser beam profile shaping with interlaced binary diffraction gratings,” Appl. Opt. 21, 3209–3212 (1982).
[CrossRef] [PubMed]

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

1980 (1)

1971 (1)

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Bentley, J.

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Han, C. Y.

Hidaka, H.

Ishii, Y.

Kindred, D. S.

Kitano, I.

Koike, Y.

Konishi, S.

Kreuzer, J. L.

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent3,476,463 (4November1969).

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 5.

Mclaughlin, P. O.

Moore, D. T.

Murata, K.

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Ohtsuka, Y.

Ozaki, Y.

Rhodes, P. W.

Samuels, J. E.

Sands, P. J.

Shafer, D.

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Shealy, D. L.

Shingyouchi, K.

Takamoto, K.

Toyama, M.

Veldkamp, W. B.

Appl. Opt. (11)

Y. Ozaki, K. Takamoto, “Cylindrical fly’s eye lens for intensity redistribution of an excimer laser beam,” Appl. Opt. 28, 106–110 (1989).
[CrossRef] [PubMed]

W. B. Veldkamp, “Laser beam profile shaping with interlaced binary diffraction gratings,” Appl. Opt. 21, 3209–3212 (1982).
[CrossRef] [PubMed]

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

P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
[CrossRef] [PubMed]

Y. Koike, H. Hidaka, Y. Ohtsuka, “Plastic axial gradient-index lens,” Appl. Opt. 24, 4321–4325 (1985).
[CrossRef] [PubMed]

P. O. Mclaughlin, M. Toyama, I. Kitano, “Gradient-index objective lens for the compact disk system,” Appl. Opt. 25, 3340–3344 (1986).
[CrossRef]

D. S. Kindred, D. T. Moore, “Design, fabrication, and testing of a gradient-index binocular objective,” Appl. Opt. 27, 492–495 (1988).
[CrossRef] [PubMed]

I. Kitano, “Current status of aplanatic gradient-index lens systems,” Appl. Opt. 29, 3992–3997 (1990).
[CrossRef] [PubMed]

D. S. Kindred, J. Bentley, D. T. Moore, “Axial and radial gradient-index titania flint glasses,” Appl. Opt. 29, 4036–4041 (1990).
[CrossRef] [PubMed]

J. E. Samuels, D. T. Moore, “Gradient-index profile control from mixed molten salt baths,” Appl. Opt. 29, 4042–4050 (1990).
[CrossRef] [PubMed]

K. Shingyouchi, S. Konishi, “Gradient-index doped silica rod lenses produced by a solgel method,” Appl. Opt. 29, 4061–4063 (1990).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Opt. Laser Technol. (1)

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Other (2)

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 5.

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent3,476,463 (4November1969).

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Figures (11)

Fig. 1
Fig. 1

Configuration of a two-lens laser beam expander.

Fig. 2
Fig. 2

Relations between vertex radii and depths of index gradients: (a) primary lens, (b) secondary lens.

Fig. 3
Fig. 3

Relations between gradient-index characteristics and layout parameters.

Fig. 4
Fig. 4

Layout of a laser beam reshaper.

Fig. 5
Fig. 5

Gradient-index profile of the primary lens.

Fig. 6
Fig. 6

Gradient-index profile of the secondary lens.

Fig. 7
Fig. 7

Intensity distributions of the incoming beam and exiting beam.

Fig. 8
Fig. 8

Layout of a laser beam expander.

Fig. 9
Fig. 9

Gradient-index profile of the primary lens.

Fig. 10
Fig. 10

Gradient-index profile of the secondary lens.

Fig. 11
Fig. 11

Intensity distributions of the incoming and exiting beams.

Tables (2)

Tables Icon

Table 1 Layout Parameters for a Laser Beam Reshaper

Tables Icon

Table 2 Layout Parameters for a 2× Laser Beam Expander

Equations (33)

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d d s ( n d r d s ) = n .
x = x 0 + K 0 z 0 z d z M ,
y = y 0 + L 0 z 0 z d z M ,
M = [ n 2 ( z ) K 2 L 2 ] 1 / 2 = [ n 2 ( z ) K 0 2 L 0 2 ] 1 / 2 .
σ ( r 1 ) = exp ( 2 r 1 2 / r 0 2 ) ,
E ( r 1 ) = 2 π 0 r 1 σ ( r ) r d r = π r 0 2 2 [ 1 exp ( 2 r 1 2 / r 0 2 ) ] .
0 2 π d θ 0 r 1 σ ( r ) r d r = 0 2 π d θ 0 r 2 Σ ( r ) r d r ,
π r 0 2 2 [ 1 exp ( 2 r 1 2 / r 0 2 ) ] = π Σ r 2 2 ,
r 2 = { r 0 2 2 Σ [ 1 exp ( 2 r 1 2 / r 0 2 ) ] } 1 / 2 ,
r 1 = [ r 0 2 2 log ( 1 1 2 Σ r 2 2 / r 0 2 ) ] 1 / 2 .
τ = r 2 max r 0 .
Σ = 1 τ 2 [ 1 2 ( 1 1 / e 2 ) ] = 1 τ 2 Σ 1 .
r 1 2 = 2 z 1 R 1 z 1 2
z 1 = r 1 2 R 1 1 + ( 1 r 1 2 / R 1 2 ) 1 / 2
r 2 2 = 2 z 2 R 2 z 2 2
z 2 = r 2 2 / R 2 1 + ( 1 r 2 2 / R 2 2 ) 1 / 2 .
tan α = r 2 r 1 d z 1 + z 2 ,
cot θ 1 = R 1 z 1 r 1 = R 1 z 1 ( 2 R 1 z 1 z 1 2 ) 1 / 2 ,
cot θ 2 = R 2 z 2 r 2 = R 2 z 2 ( 2 R 2 z 2 z 2 2 ) 1 / 2 .
n 1 ( z 1 ) sin θ 1 = n 0 sin ( θ 1 + α ) ,
n 2 ( z 2 ) sin θ 2 = n 0 sin ( θ 2 + α ) .
n 1 ( z 1 ) = n 0 ( cot θ 1 sin α + cos α ) ,
n 2 ( z 2 ) = n 0 ( cot θ 2 sin α + cos α ) .
r 2 = ( r 0 2 2 Σ { 1 exp [ 2 ( 2 R 1 z 1 z 1 2 ) / r 0 2 ] } ) 1 / 2 .
z 2 = ( r 0 2 2 Σ { 1 exp [ 2 ( 2 R 1 z 1 z 1 2 ) / r 0 2 ] } ) / R 2 1 + [ 1 ( r 0 2 2 Σ { 1 exp [ 2 ( 2 R 1 z 1 z 1 2 ) / r 0 2 ] } ) / R 2 2 ] 1 / 2 .
n 1 ( z 1 ) = n 0 { [ R 1 z 1 ( 2 R 1 z 1 z 1 2 ) 1 / 2 ] sin α ( z 1 ) + cos α ( z 1 ) } .
n 2 ( z 2 ) = n 0 { [ R 2 z 2 ( 2 R 2 z 2 z 2 2 ) 1 / 2 ] sin α ( z 2 ) + cos α ( z 2 ) } .
R 1 r 0 2 + ( Δ z 1 ) 2 2 ( Δ z 1 ) ,
R 2 r 2 max 2 + ( Δ z 2 ) 2 2 ( Δ z 2 ) .
n 1 = 1 . 537910 0 . 036171 z + 0 . 008827 z 2 .
n 2 = 1 . 525456 0 . 010882 z 0 . 000801 z 2 .
n 1 = 1 . 600350 0 . 059840 z + 0 . 012423 z 2 .
n 2 = 1 . 567088 0 . 009410 z .

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