Abstract

A set of differential equations is derived which specifies the shape of two aspherical surfaces of a lens system that will convert an incident plane wave with an arbitrary energy profile into collimated radiation with a uniform energy distribution. As an example, a lens system is designed that converts a laser beam with a Gaussian energy profile into an expanded beam with a uniform energy distribution. Off-axis rays are then traced through the lenses in order to analyze the performance of the lens system.

© 1980 Optical Society of America

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References

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  1. D. R. Herriott, Sci. Am. 219 (3), 140 (1968).
    [CrossRef]
  2. T. E. Horton, J. H. McDermit, J. Heat Transfer Trans. ASME C94, 453 (1972).
    [CrossRef]
  3. A. J. Glass, A. Greenbaum, J. Trenholme, in Laser Program Annual Report UCRL-50021-74, J. I. Davis, W. Clements, Eds., Lawrence Livermore Laboratory, 1974, pp. 234–242.
  4. H. M. Haskal, Appl. Opt. 18, 2143 (1979).
    [CrossRef] [PubMed]
  5. A. D. Berg, Proc. Soc. Photo-Opt. Instrum. Eng. 69, 14 (1975).
  6. H. M. Haskal, D. Chen, “Optical Data Storage,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1977).
  7. W. H. Southwell, Appl. Opt. 18, 1240 (1979).
    [CrossRef] [PubMed]
  8. P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456A (1979).
  9. D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 66, 1114A (1976).
  10. R. Brehm et al., Precis. Eng. 1.4, 207 (1979).
    [CrossRef]
  11. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).
  12. Scientific Subroutine Package, IBM Application Program (IBM, White Plains, N.Y., 1970).
  13. L. F. Shampine, R. C. Allen, Numerical Computing: An Introduction (Saunders, Philadelphia, 1973).
  14. A. L. Schawlow, Sci. Am. 219 (3), 120 (1968).
    [CrossRef]
  15. W. Koechner, Proc. Soc. Photo-Opt. Instrum. Eng. 69, 148 (1975).
  16. B. R. Frieden, Appl. Opt. 4, 1400 (1965).
    [CrossRef]
  17. P. W. Rhodes, “Design and Analysis of Refractive Optical Systems for Irradiance Redistribution of Collimated Radiation,” M.S. Thesis, U. Alabama in Birmingham (1979).
  18. D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
    [CrossRef]
  19. International Mathematical & Statistical Libraries, Houston, Texas (1979).
  20. R. S. Millman, G. D. Parker, Elements of Differential Geometry (Prentice-Hall, Englewood Cliffs, N.J., 1977).

1979 (4)

H. M. Haskal, Appl. Opt. 18, 2143 (1979).
[CrossRef] [PubMed]

W. H. Southwell, Appl. Opt. 18, 1240 (1979).
[CrossRef] [PubMed]

P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456A (1979).

R. Brehm et al., Precis. Eng. 1.4, 207 (1979).
[CrossRef]

1976 (1)

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 66, 1114A (1976).

1975 (3)

A. D. Berg, Proc. Soc. Photo-Opt. Instrum. Eng. 69, 14 (1975).

W. Koechner, Proc. Soc. Photo-Opt. Instrum. Eng. 69, 148 (1975).

D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
[CrossRef]

1972 (1)

T. E. Horton, J. H. McDermit, J. Heat Transfer Trans. ASME C94, 453 (1972).
[CrossRef]

1968 (2)

D. R. Herriott, Sci. Am. 219 (3), 140 (1968).
[CrossRef]

A. L. Schawlow, Sci. Am. 219 (3), 120 (1968).
[CrossRef]

1965 (1)

Allen, R. C.

L. F. Shampine, R. C. Allen, Numerical Computing: An Introduction (Saunders, Philadelphia, 1973).

Berg, A. D.

A. D. Berg, Proc. Soc. Photo-Opt. Instrum. Eng. 69, 14 (1975).

Brehm, R.

R. Brehm et al., Precis. Eng. 1.4, 207 (1979).
[CrossRef]

Burkhard, D. G.

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 66, 1114A (1976).

D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
[CrossRef]

Chen, D.

H. M. Haskal, D. Chen, “Optical Data Storage,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1977).

Frieden, B. R.

Glass, A. J.

A. J. Glass, A. Greenbaum, J. Trenholme, in Laser Program Annual Report UCRL-50021-74, J. I. Davis, W. Clements, Eds., Lawrence Livermore Laboratory, 1974, pp. 234–242.

Greenbaum, A.

A. J. Glass, A. Greenbaum, J. Trenholme, in Laser Program Annual Report UCRL-50021-74, J. I. Davis, W. Clements, Eds., Lawrence Livermore Laboratory, 1974, pp. 234–242.

Haskal, H. M.

H. M. Haskal, Appl. Opt. 18, 2143 (1979).
[CrossRef] [PubMed]

H. M. Haskal, D. Chen, “Optical Data Storage,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1977).

Herriott, D. R.

D. R. Herriott, Sci. Am. 219 (3), 140 (1968).
[CrossRef]

Horton, T. E.

T. E. Horton, J. H. McDermit, J. Heat Transfer Trans. ASME C94, 453 (1972).
[CrossRef]

Koechner, W.

W. Koechner, Proc. Soc. Photo-Opt. Instrum. Eng. 69, 148 (1975).

McDermit, J. H.

T. E. Horton, J. H. McDermit, J. Heat Transfer Trans. ASME C94, 453 (1972).
[CrossRef]

Millman, R. S.

R. S. Millman, G. D. Parker, Elements of Differential Geometry (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Parker, G. D.

R. S. Millman, G. D. Parker, Elements of Differential Geometry (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Rhodes, P. W.

P. W. Rhodes, “Design and Analysis of Refractive Optical Systems for Irradiance Redistribution of Collimated Radiation,” M.S. Thesis, U. Alabama in Birmingham (1979).

Schawlow, A. L.

A. L. Schawlow, Sci. Am. 219 (3), 120 (1968).
[CrossRef]

Scott, P. W.

P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456A (1979).

Shampine, L. F.

L. F. Shampine, R. C. Allen, Numerical Computing: An Introduction (Saunders, Philadelphia, 1973).

Shealy, D. L.

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 66, 1114A (1976).

D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
[CrossRef]

Southwell, W. H.

P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456A (1979).

W. H. Southwell, Appl. Opt. 18, 1240 (1979).
[CrossRef] [PubMed]

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

Trenholme, J.

A. J. Glass, A. Greenbaum, J. Trenholme, in Laser Program Annual Report UCRL-50021-74, J. I. Davis, W. Clements, Eds., Lawrence Livermore Laboratory, 1974, pp. 234–242.

Appl. Opt. (3)

J. Heat Transfer Trans. ASME (1)

T. E. Horton, J. H. McDermit, J. Heat Transfer Trans. ASME C94, 453 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

P. W. Scott, W. H. Southwell, J. Opt. Soc. Am. 69, 1456A (1979).

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 66, 1114A (1976).

Opt. Acta (1)

D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
[CrossRef]

Precis. Eng. (1)

R. Brehm et al., Precis. Eng. 1.4, 207 (1979).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

A. D. Berg, Proc. Soc. Photo-Opt. Instrum. Eng. 69, 14 (1975).

W. Koechner, Proc. Soc. Photo-Opt. Instrum. Eng. 69, 148 (1975).

Sci. Am. (2)

D. R. Herriott, Sci. Am. 219 (3), 140 (1968).
[CrossRef]

A. L. Schawlow, Sci. Am. 219 (3), 120 (1968).
[CrossRef]

Other (8)

P. W. Rhodes, “Design and Analysis of Refractive Optical Systems for Irradiance Redistribution of Collimated Radiation,” M.S. Thesis, U. Alabama in Birmingham (1979).

International Mathematical & Statistical Libraries, Houston, Texas (1979).

R. S. Millman, G. D. Parker, Elements of Differential Geometry (Prentice-Hall, Englewood Cliffs, N.J., 1977).

H. M. Haskal, D. Chen, “Optical Data Storage,” in Laser Applications, M. Ross, Ed. (Academic, New York, 1977).

A. J. Glass, A. Greenbaum, J. Trenholme, in Laser Program Annual Report UCRL-50021-74, J. I. Davis, W. Clements, Eds., Lawrence Livermore Laboratory, 1974, pp. 234–242.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

Scientific Subroutine Package, IBM Application Program (IBM, White Plains, N.Y., 1970).

L. F. Shampine, R. C. Allen, Numerical Computing: An Introduction (Saunders, Philadelphia, 1973).

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Figures (10)

Fig. 1
Fig. 1

Geometrical configuration of beam expander.

Fig. 2
Fig. 2

Laser beam expander with lens 2 at z = 20.

Fig. 3
Fig. 3

Aspherical lens surfaces from Fig. 2.

Fig. 4
Fig. 4

Aspherical lens surfaces from Fig. 2 for a lens separation of 38 units.

Fig. 5
Fig. 5

Laser beam reshaper with lens 2 at Z = 20.

Fig. 6
Fig. 6

Aspherical lens surfaces from Fig. 5.

Fig. 7
Fig. 7

Input/output intensity profiles for the laser beam expander with α = 1. Lens system designed for α = 2.

Fig. 8
Fig. 8

Input/output intensity profiles for the laser beam expander with the position of the second lens at Z = 15 units. Lens system designed for Z = 20.

Fig. 9
Fig. 9

Same as Fig. 8 with the second lens at Z = 25.

Fig. 10
Fig. 10

Irradiance distribution on a test plane at Z = 60 for the laser beam expander with the input beam tilted 5° off-axis. Dots are not decimal points but specify ray intercept with the target plane. The integer number to the right of the period, divided by 100, is the flux density associated with the ray.

Tables (2)

Tables Icon

Table I Beam Expander Parameters

Tables Icon

Table II Beam Reshaper Parameters

Equations (46)

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0 2 π 0 r 0 σ ( r ) r d r d θ = 0 2 π 0 R 0 ( r ) r d r d θ = π R 0 2 ,
R = [ ( 2 / ) 0 r σ ( x ) x d x ] 1 / 2 ,
B = γ A + Ω n ˆ ,
γ = n 1 / n 0 , A = k ˆ , Ω = γ A · n ˆ + { 1 γ 2 [ 1 ( A · n ˆ ) 2 ] } 1 / 2 = γ + { 1 + ( z ) 2 [ 1 γ 2 ] } 1 / 2 [ 1 + ( z ) 2 ] 1 / 2 ,
n ˆ = [ z i ˆ + k ˆ ] / [ 1 + ( z ) 2 ] 1 / 2 ,
z = d z ( r ) / d r .
( R r ) B z = ( Z z ) B x ,
( z ) 4 [ γ 2 ( R r ) 2 + ( γ 2 1 ) ( Z z ) 2 ] + ( z ) 3 [ 2 ( R r ) ( Z z ) ] + ( z ) 2 { ( 1 γ 2 ) [ ( R r ) 2 + ( Z z ) 2 ] } + z [ 2 ( R r ) ( Z z ) ] + ( R r ) 2 = 0.
OPL = n 1 t + ( Z 0 t ) n 0 + n 2 T ,
OPL = n 1 z + n 0 [ ( R r ) 2 + ( Z z ) 2 ] 1 / 2 + n 2 [ Z 0 + T Z ] ,
n 1 t + n 0 ( Z 0 t ) + n 2 T n 2 ( Z 0 + T ) K ( const ) = n 1 z n 2 Z + n 0 [ ( R r ) 2 + ( Z z ) 2 ] 1 / 2 .
Z 2 ( n 2 2 n 0 2 ) + 2 Z ( n 0 2 z n 2 K n 1 n 2 z ) + [ n 1 2 z 2 + K 2 + 2 n 1 z K n 0 2 z 2 n 0 2 ( R r ) 2 ] = 0.
Z = ( n 1 n 2 n 0 2 ) z + n 2 K n 2 2 n 0 2 + n 0 [ ( K + n 1 n 2 z ) 2 + ( n 2 2 n 0 2 ) ( R r ) 2 ] 1 / 2 n 2 2 n 0 2 ,
C = γ 2 B + Ω 2 N ,
γ 2 = n 0 / n 2 , N = Z i ˆ + k ˆ [ 1 + ( Z ) 2 ] 1 / 2 ,
Ω 2 = γ 2 { γ ( z ) 2 + [ 1 + ( z ) 2 ( 1 γ 2 ) ] 1 / 2 [ 1 + z Z ] γ z Z } [ 1 + ( z ) 2 ] [ 1 + ( Z ) 2 ] 1 / 2 + [ 1 + ( Z ) 2 ] 1 / 2 .
γ 2 B x + Ω 2 N x = 0 ,
γ 2 B z + Ω 2 N z = 1.
Z = γ 2 z { γ [ 1 + ( z ) 2 ( 1 γ 2 ) ] 1 / 2 } 1 + ( z ) 2 γ 2 γ ( z ) 2 γ 2 [ 1 + ( z ) 2 ( 1 γ 2 ) ] 1 / 2 .
Z = z + n 2 K + n 1 [ K 2 + ( n 2 2 n 0 2 ) ( R r ) 2 ] 1 / 2 ( n 2 2 n 0 2 ) ,
Z = z .
σ ( r ) = P exp ( r 2 / 2 α 2 ) 2 π α 2 [ 1 exp ( r 0 2 ) / 2 α 2 ] .
Σ = P π R 0 2 .
R ( r ) = R 0 [ 1 exp ( r 2 / 2 α 2 ) ] 1 / 2 [ 1 exp ( r 0 2 / 2 α 2 ) ] 1 / 2 .
P = 2 π α 2 [ 1 exp ( r 0 2 / 2 α 2 ) ] ,
σ ( r ) = exp ( r 2 / 2 α 2 ) .
σ ( r 0 ) = σ ( 0 ) / e 2 = 1 / e 2 = 0.135 ,
F d S i d S i + 1 = σ i ρ i cos ϕ i cos ϕ i + 1 | L 0 i + r L 1 i + r 2 L 2 i | .
L 0 i = A i g i 1 / 2 · ( x i u i × x i υ i ) ;
L 1 i = A i g i 1 / 2 · [ ( x i u i × A i υ i ) + ( A i u i × x i υ i ) ] ;
L 2 i = A i g i 1 / 2 · ( A i u i × A i υ i ) ;
g i 1 / 2 = | x i u i × x i υ i | .
x i = x ( u i , υ i ) i ˆ + y ( u i , υ i ) j ˆ + z ( u i , υ i ) k ˆ .
x = i ˆ r cos ϕ + j ˆ r sin ϕ + k ˆ z ( r ) .
x r = i ˆ cos ϕ + j ˆ sin ϕ + k ˆ z ( r ) ,
x ϕ = i ˆ r sin ϕ + j ˆ r cos ϕ + 0.
g r r = 1 + ( z ) 2 , g ϕ ϕ = r 2 , g r ϕ = 0 , g ϕ r = 0 , g = r 2 [ 1 + ( z ) 2 ] ,
b r r = z [ 1 + ( z ) 2 ] 1 / 2 b ϕ ϕ = r z [ 1 + ( z ) 2 ] 1 / 2 . b r ϕ = 0 , b ϕ r = 0
x = i ˆ x + j ˆ y + k ˆ z ,
x x = i ˆ ; x y = j ˆ .
g x x = 1 ; g y y = 1 ; g x y = 0 ; g y x = 0 ; g = 1.
b x x = 0 ; b y y = 0 ; b x y = 0 ; b y x = 0.
z = { 2 ( r R ) ( d R / d r 1 ) + z [ 2 ( z Z ) ( d R / d r 1 ) + 2 ( r R ) ( d Z / d r z ) ] + ( z ) 2 2 ( γ 2 1 ) [ ( R r ) ( d R / d r 1 ) + ( Z z ) ( d Z / d r z ) ] + ( z ) 3 [ 2 ( z Z ) ( d R / d r 1 ) + 2 ( r R ) ( d Z / d r z ) ] + ( z ) 4 [ 2 γ 2 ( R r ) ( d R / d r 1 ) + 2 ( γ 2 1 ) ( Z z ) ( d Z / d r z ) ] } / { 4 ( z ) 3 [ γ 2 ( R r ) 2 + ( γ 2 1 ) ( Z z ) 2 ] + 3 ( z ) 2 [ 2 ( r R ) ( Z z ) ] 2 z ( 1 γ 2 ) [ ( R r ) 2 + ( Z z ) 2 ] ( R r ) 2 } .
d R / d r = r σ ( r ) R Σ
Z = γγ 2 δ γ 2 β δ γ 2 ( z ) 2 ( 1 γ 2 ) δ / β 1 + λ ( z ) 2 γ 2 + ( γγ 2 z γ 2 z β ) [ γ 2 ( 1 γ 2 ) z δ / β 2 λ z δ ] [ 1 + λ ( z ) 2 γ 2 β ] 2 ,
d r / d R = R Σ r σ ( r )

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