Abstract

The problem of design of a two-lens optical system for reshaping the irradiance distribution of a laser beam in a prescribed manner is considered in the geometrical optics approximation. The presented design equations are derived in a rigorous manner and are applicable to free-form two-lens optical systems without any a priori symmetry assumptions on radiance profiles and beam cross sections. The obtained equations are applied to derive an equation defining sensitivity of the output radiation intensity to figure errors. This equation is applied to analyze sensitivity in several cases, including rotationally symmetric reshapers with nonrotationally symmetric figure error. The presented approach shows also that even in the general case two different designs are available for the same data. In one of these designs one lens is always concave or convex and the second is convex or concave, while in the second design the lenses may be neither convex nor concave. Since, in general, the surface lenses are aspherical, the availability of a design with convex/concave lenses is particularly important for fabrication.

© 2008 Optical Society of America

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References

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  1. F. Dickey and S. Holswade, Laser Beam Shaping: Theory and Techniques (Dekker, 2000).
    [CrossRef]
  2. L. A. Romero and F. Dickey, “The mathematical and physical theory of lossless beam shaping,” in Laser Beam Shaping: Theory and Techniques, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 21-118.
  3. D. Shealy, “Geometrical methods,” in Laser Beam Shaping: Theory and Techniques, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 164-213.
  4. F. Dickey and S. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, F.Dickey, S.Holswade, and D.Shealy, eds. (CRC Press, 2006), pp. 269-305.
  5. J. Hoffnagle, “Sensitivity of a refractive beam reshaper to figure error,” Proc. SPIE 4770, 67-74 (2002).
    [CrossRef]
  6. J. Hoffnagle, IBM Almaden Research Center, San Jose, California 95120-6099, email: hoffnagl@almaden.ibm.com (personal communication, January 2008).
  7. B. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform radiation,” Appl. Opt. 4, 1400-1403 (1965).
    [CrossRef]
  8. J. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. Patent No. 3,476,463, November 4, 1969.
  9. P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545-3553 (1980).
    [CrossRef] [PubMed]
  10. J. Hoffnagle and C. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39, 5488-5499 (2000).
    [CrossRef]
  11. J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
    [CrossRef]
  12. V. I. Oliker and L. Prussner, “A new technique for synthesis of offset dual reflector systems,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, 1994), pp. 45-52.
  13. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590-595 (2002).
    [CrossRef]
  14. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005).
  15. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24, 463-469 (2007).
    [CrossRef]
  16. V. Oliker, “Optical design of freeform two-mirror beam-shaping systems,” J. Opt. Soc. Am. A 24, 3741-3752 (2007).
    [CrossRef]

2007

2003

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
[CrossRef]

2002

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19, 590-595 (2002).
[CrossRef]

J. Hoffnagle, “Sensitivity of a refractive beam reshaper to figure error,” Proc. SPIE 4770, 67-74 (2002).
[CrossRef]

2000

1980

1965

Benítez, P.

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005).

Dickey, F.

F. Dickey and S. Holswade, Laser Beam Shaping: Theory and Techniques (Dekker, 2000).
[CrossRef]

L. A. Romero and F. Dickey, “The mathematical and physical theory of lossless beam shaping,” in Laser Beam Shaping: Theory and Techniques, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 21-118.

F. Dickey and S. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, F.Dickey, S.Holswade, and D.Shealy, eds. (CRC Press, 2006), pp. 269-305.

Frieden, B.

Hoffnagle, J.

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
[CrossRef]

J. Hoffnagle, “Sensitivity of a refractive beam reshaper to figure error,” Proc. SPIE 4770, 67-74 (2002).
[CrossRef]

J. Hoffnagle and C. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39, 5488-5499 (2000).
[CrossRef]

J. Hoffnagle, IBM Almaden Research Center, San Jose, California 95120-6099, email: hoffnagl@almaden.ibm.com (personal communication, January 2008).

Holswade, S.

F. Dickey and S. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, F.Dickey, S.Holswade, and D.Shealy, eds. (CRC Press, 2006), pp. 269-305.

F. Dickey and S. Holswade, Laser Beam Shaping: Theory and Techniques (Dekker, 2000).
[CrossRef]

Jefferson, C.

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
[CrossRef]

J. Hoffnagle and C. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39, 5488-5499 (2000).
[CrossRef]

Kreuzer, J.

J. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. Patent No. 3,476,463, November 4, 1969.

Miñano, J. C.

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005).

Muschaweck, J.

Oliker, V.

Oliker, V. I.

V. I. Oliker and L. Prussner, “A new technique for synthesis of offset dual reflector systems,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, 1994), pp. 45-52.

Prussner, L.

V. I. Oliker and L. Prussner, “A new technique for synthesis of offset dual reflector systems,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, 1994), pp. 45-52.

Rhodes, P. W.

Ries, H.

Romero, L. A.

L. A. Romero and F. Dickey, “The mathematical and physical theory of lossless beam shaping,” in Laser Beam Shaping: Theory and Techniques, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 21-118.

Rubinstein, J.

Shealy, D.

D. Shealy, “Geometrical methods,” in Laser Beam Shaping: Theory and Techniques, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 164-213.

Shealy, D. L.

Winston, R.

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005).

Wolansky, G.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng. (Bellingham) 42, 3090-3099 (2003).
[CrossRef]

Proc. SPIE

J. Hoffnagle, “Sensitivity of a refractive beam reshaper to figure error,” Proc. SPIE 4770, 67-74 (2002).
[CrossRef]

Other

J. Hoffnagle, IBM Almaden Research Center, San Jose, California 95120-6099, email: hoffnagl@almaden.ibm.com (personal communication, January 2008).

J. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. Patent No. 3,476,463, November 4, 1969.

F. Dickey and S. Holswade, Laser Beam Shaping: Theory and Techniques (Dekker, 2000).
[CrossRef]

L. A. Romero and F. Dickey, “The mathematical and physical theory of lossless beam shaping,” in Laser Beam Shaping: Theory and Techniques, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 21-118.

D. Shealy, “Geometrical methods,” in Laser Beam Shaping: Theory and Techniques, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 164-213.

F. Dickey and S. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, F.Dickey, S.Holswade, and D.Shealy, eds. (CRC Press, 2006), pp. 269-305.

V. I. Oliker and L. Prussner, “A new technique for synthesis of offset dual reflector systems,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, 1994), pp. 45-52.

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier, 2005).

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Figures (1)

Fig. 1
Fig. 1

Two-lens system.

Tables (1)

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Table 1 Conclusions of Section 4

Equations (109)

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L ( P d ( x ) ) J ( P d ( x ) ) = I ( x ) , x Ω .
z 1 = z x 1 , z 2 = z x 2 , z = ( z 1 , z 2 ) .
n = ( z , 1 ) 1 + z 2 .
n sin θ i = sin θ r ,
k = ( k n ) n + ( k n ) n ,
ω = ( ω n ) n + ( ω n ) n .
ω k = ( k n ) ( ω n ) + ( k n ) ( ω n ) = cos θ i cos θ r + sin θ i sin θ r = cos θ i cos θ r + n sin 2 θ i .
ω = a k + b n
ω k = a + b ( n k ) ,
ω n = a ( k n ) + b .
( ω n ) ( k n ) ω k = a [ ( k n ) 2 1 ] .
a [ ( k n ) 2 1 ] = ( ω n ) ( k n ) cos θ i cos θ r n sin 2 θ i = n [ 1 ( k n ) 2 ] .
b = ( ω n ) n ( k n ) = 1 sin 2 θ r n ( k n ) = 1 n 2 sin 2 θ i n ( k n ) = 1 n 2 [ 1 ( k n ) 2 ] n ( k n ) = n + M 1 + z 2 ,
M 1 + ( 1 n 2 ) z 2 = ( ω n ) 1 + z 2 > 0 .
ω = n k + n + M 1 + z 2 n .
R 2 : r 2 ( x ) = r 1 ( x ) + t ( x ) ω ( x ) , x Ω ¯ ,
P d ( x ) = r 1 ( x ) + t ( x ) ω ( x ) + s ( x ) k , x Ω ¯ .
P ( x ) = r 2 ( x ) ( r 2 ( x ) k ) k = x + t ( x ) [ ω ( ω k ) k ] .
ω ( ω k ) k = n + M 1 + z 2 z .
P ( x ) = x t ( x ) n + M ( x ) 1 + z ( x ) 2 z ( x ) .
l ( x ) = l ( = const. ) for all x Ω ¯ .
β = l n d .
t ( x ) = β n [ w ( P ( x ) ) z ( x ) ] .
w ( P ( x ) ) z ( x ) = r 2 ( x ) k z ( x ) = t ( x ) ( ω ( x ) k ) = t ( x ) n z ( x ) 2 + M ( x ) 1 + z ( x ) 2 .
t ( x ) = β ( 1 + z ( x ) 2 ) M ( x ) ( M ( x ) n ) .
P ( x ) = x β z ( x ) M ( x ) , x Ω ¯ .
w ( P ( x ) ) = z ( x ) + β 1 n 2 [ n + 1 M ( x ) ] , x Ω ¯ .
P d ( x ) = P ( x ) + d k ,
L ( P ( x ) ) J ( P ( x ) ) = I ( x ) , x Ω .
E [ δ i j ] [ 1 0 0 1 ] ,
A i j ( z ) δ i j + n 2 1 M 2 ( z ) z i z j , A ( z ) [ A i j ( z ) ] ,
z i j 2 z x i x j , Hess ( z ) [ z i j ] ,
γ i j ( z ) δ i j β M ( z ) s A i s ( z ) z s j ,
γ ( z ) [ γ i j ( z ) ] ,
B i j ( z ) δ i j ( n 2 1 ) z i z j , B ( z ) [ B i j ( z ) ] ,
Q i j ( z ) M ( z ) B i j ( z ) β z i j , Q ( z ) [ Q i j ( z ) ]
det A ( z ) = 1 M 2 ( z ) , A ( z ) B ( z ) = E ,
γ ( z ) = A ( z ) ( B ( z ) β M ( z ) Hess ( z ) ) .
J ( P ( z ) ) = det γ ( z ) = det A ( z ) det ( B ( z ) β M ( z ) Hess ( z ) ) = det Q ( z ) M 4 ( z ) .
β = ( 1 n ) d .
T L ( p ) d p = Ω I ( x ) d x ,
J ( P ( x ) ) > 0 in Ω ¯ .
P λ ( x ) P ( z λ ( x ) ) = x β z λ ( x ) M λ ( x ) ,
w λ ( x ) z λ ( x ) + β 1 n 2 [ n + 1 M λ ( x ) ] , x Ω ¯ .
L λ ( P λ ( x ) ) I ( x ) J ( P λ ( x ) ) and Δ L ( x ) L λ ( P λ ( x ) ) L ( P ( x ) ) .
M d M ( z λ ) d λ λ = 0 = ( 1 n 2 ) s z s h s 1 + ( 1 n 2 ) z 2 = n 2 1 M s z s h s .
B d B ( z λ ) d λ λ = 0 = ( n 2 1 ) s ( δ i s z j + δ j s z i ) h s ,
Q d Q ( z λ ) d λ λ = 0 = M B ( z ) + M ( z ) B β Hess ( h ) .
d d λ 1 J ( P λ ) λ = 0 = d d λ ( M λ ) 4 det Q λ λ = 0 = 4 M 3 ( z ) M det Q ( z ) M 4 ( z ) tr [ Q 1 ( z ) Q ] det Q ( z ) ,
Δ L L ( P ( z ) ) = { 4 M M ( z ) tr [ Q 1 ( z ) Q ] } λ + o ( λ ) = { ϴ 1 + ϴ 2 } λ + o ( λ ) ,
ϴ 1 β tr [ Q 1 ( z ) Hess ( h ) ] ,
ϴ 2 n 2 1 M 2 ( z ) i , s [ ( M ( z ) tr [ Q 1 ( z ) B ( z ) ] 4 ) E + 2 M 3 ( z ) Q 1 ( z ) ] i s z i h s ,
ϴ 1 = β ( 1 β z 22 ) h 11 2 β z 12 h 12 + ( 1 β z 11 ) h 22 ( 1 β z 11 ) ( 1 β z 22 ) β 2 z 12 2 .
I ( x ) I ( x ) > 0 , x Ω ,
L ( p ) L ( p ) > 0 , p T .
z ρ d z d ρ , q ρ β z ρ M ( z ) , q ρ d q d ρ .
M ( z ) = 1 + ( 1 n 2 ) z ρ 2 , P ( ρ , ϕ ) = q τ ( ϕ ) ,
q ρ = 1 β z ρ ρ M 3 ( z ) , and J ( P ) = q q ρ ρ ,
B 11 ( z ) = M 2 ( z ) cos 2 ϕ + sin 2 ϕ ,
B 12 ( z ) = ( M 2 1 ) cos ϕ sin ϕ ,
B 21 ( z ) = B 12 ( z ) , B 22 ( z ) = M 2 ( z ) sin 2 ϕ + cos 2 ϕ .
Q 11 ( z ) = M ( z ) B 11 ( z ) β ( z ρ ρ cos 2 ϕ + z ρ ρ sin 2 ϕ ) = M ( z ) ρ [ M 2 ( z ) q ρ ρ cos 2 ϕ + q sin 2 ϕ ] ,
Q 12 ( z ) = M ( z ) B 12 ( z ) β ( z ρ ρ z ρ ρ ) cos ϕ sin ϕ = M ( z ) ρ [ M 2 ( z ) q ρ ρ q ] cos ϕ sin ϕ ,
Q 21 ( z ) = Q 12 ( z ) ,
Q 22 ( z ) = M ( z ) B 22 ( z ) β ( z ρ ρ sin 2 ϕ + z ρ ρ cos 2 ϕ ) = M ( z ) ρ [ M 2 ( z ) q ρ ρ sin 2 ϕ + q cos 2 ϕ ] ,
Q 1 ( z ) = 1 det Q ( z ) [ Q 22 ( z ) Q 21 ( z ) Q 12 ( z ) Q 11 ( z ) ] ,
det Q ( z ) = M 4 ( z ) q q ρ ρ .
ϴ 1 = β ρ M 4 ( z ) q q ρ [ Q 22 ( z ) h 11 2 Q 12 ( z ) h 12 + Q 11 ( z ) h 22 ] = β M 3 ( z ) q q ρ { [ M 2 ( z ) q ρ ρ sin 2 ϕ + q cos 2 ϕ ] h 11 2 [ M 2 ( z ) q ρ ρ q ] cos ϕ sin ϕ h 12 + [ M 2 ( z ) q ρ ρ cos 2 ϕ + q sin 2 ϕ ] h 22 } ,
ϴ 1 = β M ( z ) [ h ρ ρ M 2 ( z ) q ρ + h ϕ ϕ q ρ + h ρ q ] ,
ϴ 1 = β tr [ S T Λ S Hess ( h ) ] ,
S = [ sin ϕ cos ϕ cos ϕ sin ϕ ] , S T = transpose of S ,
Λ = 1 M 3 ( z ) [ M 2 ( z ) ρ q 0 0 1 q ρ . ] .
M ( z ) tr [ Q 1 ( z ) B ( z ) ] = ρ q + 1 q ρ .
h = h ρ τ ( ϕ ) + h ϕ ρ d τ ( ϕ ) d ϕ , τ ( ϕ ) d τ ( ϕ ) d ϕ = 0 ,
i , s 2 M 3 ( z ) [ Q 1 ( z ) ] i s z i h s = 2 z ρ q ρ τ ( ϕ ) h = 2 z ρ h ρ q ρ .
ϴ 2 = β ( n 2 1 ) z ρ M 3 ( z ) q q ρ [ z ρ + ( 4 q ρ ) z ρ ρ M 2 ( z ) ] h ρ .
ϴ 1 + ϴ 2 = β M ( z ) [ h ρ ρ M 2 ( z ) q ρ + h ϕ ϕ q ρ + 1 M 2 ( z ) ( 1 q + 3 ( n 2 1 ) z ρ z ρ ρ M 2 ( z ) q ρ ) h ρ ] .
ϴ 1 + ϴ 2 = β M 3 ( z ) [ h ρ ρ q ρ + ( 1 q + 3 ( n 2 1 ) z ρ z ρ ρ M 2 ( z ) q ρ ) h ρ ] .
ϴ 1 + ϴ 2 = β [ h ρ ρ q ρ + h ρ q ] .
J ( P ( z ) ) 0 for all x Ω ¯ .
z 2 < 1 n 2 1 for all x Ω ¯ .
i , j Q i j ( z ) ξ i ξ j = M ( z ) [ ξ 2 ( n 2 1 ) ( z ξ ) 2 ] β i , j z i j ξ i ξ j { 0 0 } .
β i , j z i j ξ i ξ j M ( z ) [ ξ 2 + ( n 2 1 ) ( z ξ ) 2 ]
β i , j z i j ξ i ξ j M ( z ) [ ξ 2 + ( n 2 1 ) ( z ξ ) 2 ] .
β i , j z i j ξ i ξ j M 3 ( z ) ξ 2 ,
i , j z i j ξ i ξ j < 0 for all ξ ( 0 , 0 ) and all x Ω ¯ ;
β i , j z i j ξ i ξ j M ( z ) ξ 2 .
i , j z i j ξ i ξ j > 0 for all ξ ( 0 , 0 ) and all x Ω ¯ ;
i , j z i j ξ i ξ j > 0 for all ξ ( 0 , 0 ) and all x Ω ¯ ;
i , j z i j ξ i ξ j < 0 for all ξ ( 0 , 0 ) and all x Ω ¯ ;
β 2 z 2 M 2 = ( p x ) 2 .
β 2 M 2 = β 2 ( 1 n 2 ) ( p x ) 2 c 2 ( x , p ) .
p x 2 β 2 1 n 2 .
β M = { + c ( x , p ) if β > 0 c ( x , p ) if β < 0 } ,
J ( P ( x ) ) > 0 .
w ( p ) = z ( x ( p ) ) + 1 1 n 2 [ β n + sign ( β ) c ( x ( p ) , p ) ] ,
p T ¯ .
w p i = j = 1 2 z x j x j p i sign β c j = 1 2 ( p j x j ) ( δ j i x j p i ) = j = 1 2 z x j x j p i + sign β β c M j = 1 2 z x j ( δ j i x j p i ) = j = 1 2 z x j x j p i + j = 1 2 z x j ( δ j i x j p i ) = z x i .
2 w p i p j = s = 1 2 2 z x i x s x s p j , i , j = 1 , 2 .
k = a ( x ) ω ( x ) + b ( x ) N ( x ) ,
r 2 ( x ) x i = r 1 ( x ) x i + t ( x ) x i ω ( x ) + t ( x ) ω ( x ) x i , i = 1 , 2 .
r 2 ( x ) x i N ( x ) = 0 , ω ( x ) x i ω ( x ) = 0 ,
i = 1 , 2 , and ω ω = 1 .
r 2 ( x ) x i k = a ( x ) r 2 ( x ) x i ω ( x ) = a ( x ) [ r 1 ( x ) x i ω ( x ) + t ( x ) x i ] , i = 1 , 2 .
r 1 ( x ) x i ω ( x ) = n r 1 ( x ) x i k = n z i , i = 1 , 2 .
r 2 ( x ) x i k = a ( x ) [ n z ( x ) + t ( x ) ] x i , i = 1 , 2 .
s ( x ) x i = r 2 ( x ) x i k , i = 1 , 2 .
a ( x ) l ( x ) x i = a ( x ) n 1 [ n z ( x ) + t ( x ) + n s ( x ) ] x i = [ 1 a ( x ) n ] r 2 ( x ) x i , i = 1 , 2 .
ω = n k + n + M p 1 + p w 2 N ,

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