Abstract

The problem of design of a two-mirror 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 two-mirror optical systems not limited to radiance profiles and beam cross sections that are rotational or rectangular symmetric. The resulting mirrors are free-form surfaces not restricted by a priori constraints. Moreover, 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 the first mirror is always concave and the second is convex, while in the second design the resulting mirrors may be neither convex nor concave. Since, in general, the surface mirrors are aspherical, the availability of a design with convex and concave mirrors is particularly important for fabrication.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. F. Dickey and S. Holswade, Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
    [CrossRef]
  2. L. A. Romero and F. Dickey, "The mathematical and physical theory of lossless beam shaping," in , pp. 21-118.
  3. D. Shealy, "Geometrical methods," in , pp. 164-213.
  4. J. W. Ogland, "Mirror system for uniform beam transformation in high-power annular lasers," Appl. Opt. 17, 2917-2923 (1978).
    [CrossRef] [PubMed]
  5. P. H. Malyak, "Two-mirror unobscured optical system for reshaping irradiance distribution of a laser beam," Appl. Opt. 31, 4377-4383 (1992).
    [CrossRef] [PubMed]
  6. K. Nemoto, T. Fujii, N. Goto, H. Takino, T. Kobayashi, N. Shibata, K. Yamamura, and Y. Mori, "Laser beam intensity profile transformation with a fabricated mirror," Appl. Opt. 36, 551-557 (1997).
    [CrossRef] [PubMed]
  7. D. F. Cornwell, "Intensity redistribution," in "Program of the 1979 Annual Meeting of the Optical Society of America," J. Opt. Soc. Am. 69, 1456 (1979) (abstract).
  8. P. Scott and W. Southwell, "Reflective optics for irradiance redistribution of laser beams: design," 20, 1606-1610 (1981).
  9. D. Shealy and S. Chao, "Design and analysis of an elliptical Gaussian laser beam shaping system," Proc. SPIE 4443, 24-35 (2001).
    [CrossRef]
  10. D. Shealy and S. Chao, "Geometric optics-based design of laser beam shapers," Opt. Eng. 42, 1-16 (2003).
    [CrossRef]
  11. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. (Springer-Verlag, 1987).
  12. V. I. Oliker and L. Prussner, "A new technique for synthesis of offset dual reflector systems," in 1994 Conference Proceeding, 10th Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics Society, 1994), Vol. I, pp. 45-52.
  13. O. Bryngdahl, "Geometrical transformations in optics," J. Opt. Soc. Am. 64, 1092-1099 (1974).
    [CrossRef]
  14. L. A. Romero and F. Dickey, "Lossless laser beam shaping," J. Opt. Soc. Am. A 13, 751-760 (1996).
    [CrossRef]
  15. H. Ries and J. Muschaweck, "Tailored freeform optical surfaces," J. Opt. Soc. Am. A 19, 590-595 (2002).
    [CrossRef]
  16. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics (Elsevier Academic, 2005).
  17. V. Oliker, "Mathematical aspects of design of beam shaping surfaces in geometrical optics," in Trends in Nonlinear Analysis, M.Kirkilionis, S.Krömker, R.Rannacher, and F.Tomi, eds. (Springer-Verlag, 2002), pp. 191-222.
  18. T. Glimm and V. Oliker, "Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle," Indiana Univ. Math. J. 53, 1255-1278 (2004).
    [CrossRef]
  19. V. Oliker, "Optical design of two-mirror beam-shaping systems. Convex and non-convex solutions for symmetric and non-symmetric data," Proc. SPIE 5876, 203-214 (2005).
  20. R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, 1962), Vol. 2.
  21. 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]
  22. J. Hoffnagle and C. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
    [CrossRef]
  23. L. Caffarelli, "The regularity of mappings with convex potentials," J. Am. Math. Soc. 5, 99-104 (1992).
    [CrossRef]
  24. L. Caffarelli, "Boundary regularity of maps with convex potentials," Commun. Pure Appl. Math. 45, 1141-1151 (1992).
    [CrossRef]
  25. L. Caffarelli, "Boundary regularity of maps with convex potentials. II," Ann. Math. 144, 453-496 (1996).
    [CrossRef]
  26. J. Benamou and Y. Brenier, "A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem," Numer. Math. 84, 375-393 (2000).
    [CrossRef]
  27. V. Oliker is preparing a paper, "A fast method for solving the second boundary value for the Monge-Ampère equation." Contact oliker@mathcs.emory.edu.
  28. V. I. Oliker and L. Prussner, "On the numerical solution of the equation (∂2z/∂x2)(∂2z/∂y2)−(∂2z/∂x∂y)2=f and its discretizations, I," Numer. Math. 54, 271-293 (1988).
    [CrossRef]

2005 (1)

V. Oliker, "Optical design of two-mirror beam-shaping systems. Convex and non-convex solutions for symmetric and non-symmetric data," Proc. SPIE 5876, 203-214 (2005).

2004 (1)

T. Glimm and V. Oliker, "Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle," Indiana Univ. Math. J. 53, 1255-1278 (2004).
[CrossRef]

2003 (2)

D. Shealy and S. Chao, "Geometric optics-based design of laser beam shapers," Opt. Eng. 42, 1-16 (2003).
[CrossRef]

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

2002 (1)

2001 (1)

D. Shealy and S. Chao, "Design and analysis of an elliptical Gaussian laser beam shaping system," Proc. SPIE 4443, 24-35 (2001).
[CrossRef]

2000 (2)

J. Benamou and Y. Brenier, "A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem," Numer. Math. 84, 375-393 (2000).
[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]

1997 (1)

1996 (2)

L. A. Romero and F. Dickey, "Lossless laser beam shaping," J. Opt. Soc. Am. A 13, 751-760 (1996).
[CrossRef]

L. Caffarelli, "Boundary regularity of maps with convex potentials. II," Ann. Math. 144, 453-496 (1996).
[CrossRef]

1992 (3)

L. Caffarelli, "The regularity of mappings with convex potentials," J. Am. Math. Soc. 5, 99-104 (1992).
[CrossRef]

L. Caffarelli, "Boundary regularity of maps with convex potentials," Commun. Pure Appl. Math. 45, 1141-1151 (1992).
[CrossRef]

P. H. Malyak, "Two-mirror unobscured optical system for reshaping irradiance distribution of a laser beam," Appl. Opt. 31, 4377-4383 (1992).
[CrossRef] [PubMed]

1988 (1)

V. I. Oliker and L. Prussner, "On the numerical solution of the equation (∂2z/∂x2)(∂2z/∂y2)−(∂2z/∂x∂y)2=f and its discretizations, I," Numer. Math. 54, 271-293 (1988).
[CrossRef]

1981 (1)

P. Scott and W. Southwell, "Reflective optics for irradiance redistribution of laser beams: design," 20, 1606-1610 (1981).

1979 (1)

D. F. Cornwell, "Intensity redistribution," in "Program of the 1979 Annual Meeting of the Optical Society of America," J. Opt. Soc. Am. 69, 1456 (1979) (abstract).

1978 (1)

1974 (1)

Ann. Math. (1)

L. Caffarelli, "Boundary regularity of maps with convex potentials. II," Ann. Math. 144, 453-496 (1996).
[CrossRef]

Appl. Opt. (4)

Commun. Pure Appl. Math. (1)

L. Caffarelli, "Boundary regularity of maps with convex potentials," Commun. Pure Appl. Math. 45, 1141-1151 (1992).
[CrossRef]

Indiana Univ. Math. J. (1)

T. Glimm and V. Oliker, "Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle," Indiana Univ. Math. J. 53, 1255-1278 (2004).
[CrossRef]

J. Am. Math. Soc. (1)

L. Caffarelli, "The regularity of mappings with convex potentials," J. Am. Math. Soc. 5, 99-104 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

D. F. Cornwell, "Intensity redistribution," in "Program of the 1979 Annual Meeting of the Optical Society of America," J. Opt. Soc. Am. 69, 1456 (1979) (abstract).

O. Bryngdahl, "Geometrical transformations in optics," J. Opt. Soc. Am. 64, 1092-1099 (1974).
[CrossRef]

J. Opt. Soc. Am. A (2)

Numer. Math. (2)

V. I. Oliker and L. Prussner, "On the numerical solution of the equation (∂2z/∂x2)(∂2z/∂y2)−(∂2z/∂x∂y)2=f and its discretizations, I," Numer. Math. 54, 271-293 (1988).
[CrossRef]

J. Benamou and Y. Brenier, "A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem," Numer. Math. 84, 375-393 (2000).
[CrossRef]

Opt. Eng. (2)

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

D. Shealy and S. Chao, "Geometric optics-based design of laser beam shapers," Opt. Eng. 42, 1-16 (2003).
[CrossRef]

Proc. SPIE (2)

D. Shealy and S. Chao, "Design and analysis of an elliptical Gaussian laser beam shaping system," Proc. SPIE 4443, 24-35 (2001).
[CrossRef]

V. Oliker, "Optical design of two-mirror beam-shaping systems. Convex and non-convex solutions for symmetric and non-symmetric data," Proc. SPIE 5876, 203-214 (2005).

Other (10)

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley Interscience, 1962), Vol. 2.

V. Oliker is preparing a paper, "A fast method for solving the second boundary value for the Monge-Ampère equation." Contact oliker@mathcs.emory.edu.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed. (Springer-Verlag, 1987).

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

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

V. Oliker, "Mathematical aspects of design of beam shaping surfaces in geometrical optics," in Trends in Nonlinear Analysis, M.Kirkilionis, S.Krömker, R.Rannacher, and F.Tomi, eds. (Springer-Verlag, 2002), pp. 191-222.

P. Scott and W. Southwell, "Reflective optics for irradiance redistribution of laser beams: design," 20, 1606-1610 (1981).

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

L. A. Romero and F. Dickey, "The mathematical and physical theory of lossless beam shaping," in , pp. 21-118.

D. Shealy, "Geometrical methods," in , pp. 164-213.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Two-mirror system.

Fig. 2
Fig. 2

Two-mirror beam shaper. Input (bottom) and output beams cross sections.

Fig. 3
Fig. 3

Two-mirror beam shaper (rectangle-to-circular disk). Galilean design, first mirror.

Fig. 4
Fig. 4

Two-mirror beam shaper (rectangle-to-circular disk). Keplerian design, first mirror.

Fig. 5
Fig. 5

Two-mirror beam shaper (rectangle-to-circular disk). Keplerian design.

Fig. 6
Fig. 6

Two-mirror beam shaper (rectangle-to-rectangle), Galilean design, first mirror.

Fig. 7
Fig. 7

Two-mirror beam shaper (rectangle-to-rectangle), Keplerian design.

Equations (83)

Equations on this page are rendered with MathJax. Learn more.

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 .
η = k 2 ( k n ) n = k 2 ( z , 1 ) 1 + z 2 ,
R 2 : r 2 ( x ) = r 1 ( x ) + t ( x ) η ( x ) , x Ω ¯ ,
P d ( x ) = r 1 ( x ) + t ( x ) η ( x ) + s ( x ) k , x Ω ¯ .
x P d ( x ) p = P ( P d ( x ) ) ,
P ( x ) x = t ( x ) η α ( x ) ,
η α ( x ) = 2 z ( x ) 1 + z ( x ) 2 .
P d ( x ) = P ( x ) + d k .
P d ( x ) k = d .
β = t ( x ) ( 1 η ( x ) k ) = 2 t ( x ) 1 + z ( x ) 2 ,
t ( x ) = β ( 1 + z ( x ) 2 ) 2 .
P ( x ) = x + β z ( x ) , x Ω ¯ .
z ( x ) w ( P ( x ) ) = β t ( x ) .
L ( P d ( x ) ) J ( P d ( x ) ) = I ( x ) , x Ω .
L ( P ( x ) ) J ( P ( x ) ) = I ( x ) , x Ω .
L ( x + β z ) det [ I d + β H e s s ( z ) ] = I ( x ) , x Ω .
T L ( p ) d p = Ω I ( x ) d x ,
w ( P ( x ) ) = z ( x ) + β 2 ( z 2 1 ) .
J ( P ( x ) ) = det [ I d + β H e s s ( z ( x ) ) ] 0 ( but 0 )
J ( P ( x ) ) = det [ I d + β H e s s ( z ( x ) ) ] 0 ( but 0 ) .
L ( x + β z ( x ) ) det [ I d + β H e s s ( z ( x ) ) ] = I ( x ) , x Ω .
V ( x ) = x 2 2 + β z ( x ) β 2 2 .
P ( x ) = V ( x ) , x Ω ¯ ,
P ( x ) = V : Ω ¯ T ¯ ,
L ( V ) det [ H e s s ( V ) ] = I in Ω .
z c c ( x ) = 1 β [ V c c ( x ) x 2 2 + β 2 2 ] ,
z c v ( x ) = 1 β [ V c v ( x ) x 2 2 + β 2 2 ] .
V c c ( x ) β , x 2 2 β , β 2 .
Adm + ( Ω , T ) { ζ ( x ) + ( ζ ( x ) η ( p ) ) 2 + x p 2 η ( p ) β }
F ( ζ , ω ) = Ω ζ ( x ) I d x T η ( p ) L d p .
Adm ( Ω , T ) { ζ ( x ) + ( ζ ( x ) η ( p ) ) 2 + x p 2 η ( p ) β }
Ω ¯ = { x = ( x 1 , x 2 ) α a 1 x 1 b 1 , a 2 x 2 b 2 } ,
T ¯ = { p = ( p 1 , p 2 ) α c 1 p 1 d 1 , c 2 p 2 d 2 } ,
I ( x ) = I 1 ( x 1 ) I 2 ( x 2 ) ,
L ( p ) = L 1 ( p 1 ) L 2 ( p 2 ) ,
c 1 d 1 L ( p 1 ) d p 1 = a 1 b 1 I 1 ( x 1 ) d x 1 ,
c 2 d 2 L ( p 2 ) d p 2 = a 2 b 2 I 2 ( x 2 ) d x 2 .
L 1 ( V ̇ 1 ( x 1 ) ) L 2 ( V ̇ 2 ( x 2 ) ) V ̈ 1 ( x 1 ) V ̈ 2 ( x 2 ) = I 1 ( x 1 ) I 2 ( x 2 ) ,
L 1 ( V ̇ 1 ( x 1 ) ) V ̈ 1 ( x 1 ) = I 1 ( x 1 ) , a 1 < x 1 b 1 ,
V 1 ( a 1 ) = A 1 , V ̇ 1 ( a 1 ) = c 1 ,
L 2 ( V ̇ 2 ( x 2 ) ) V ̈ 2 ( x 2 ) = I 2 ( x 2 ) , a 2 < x 2 b 2 ,
V 2 ( a 2 ) = A 2 , V ̇ 2 ( a 2 ) = c 2 ,
L 1 ( V ̇ 1 ( x 1 ) ) V ̈ 1 ( x 1 ) = I 1 ( x 1 ) , a 1 < x 1 b 1 ,
V 1 ( a 1 ) = A 1 , V ̇ 1 ( a 1 ) = d 1 ,
L 2 ( V ̇ 2 ( x 2 ) ) V ̈ 2 ( x 2 ) = I 2 ( x 2 ) , a 2 < x 2 b 2 ,
V 2 ( a 2 ) = A 2 , V ̇ 2 ( a 2 ) = d 2 .
Ω ¯ = { x = ( x 1 , x 2 ) x 1 2 + x 2 2 ρ 1 2 } ,
T ¯ = { p = ( p 1 , p 2 ) p 1 2 + p 2 2 ρ 2 2 } ,
I ( x ) I ( x ) > 0 , x Ω ,
L ( p ) L ( p ) > 0 , p T ,
L ( v ̇ ) v ̈ v ̇ = s I ( s ) ,
v ̇ ( 0 ) = 0 , v ̇ ( ρ 1 ) = ρ 2 .
a ( p ) = j k e j k z i 2 π z i a ( x ) exp { j k [ ϕ ( x ) k + x p 2 2 z i ] } d x
z ( x ) = p x 2 z i , where z ( x ) = ϕ ( x ) 2 k .
a ( p ) 2 [ ( 2 z i z x x + 1 ) ( 2 z i z y y + 1 ) 4 z i 2 z x y 2 ] = a ( x ) 2 ,
x Ω ,
z ( x ) = p x β ,
Ω ¯ = { 1 x 1 1 , 3 x 2 3 } ,
I I input = exp [ ( 2 2 x 1 ) 2 ] exp [ ( 2 2 x 2 3 ) 2 ] .
T ¯ = { x 1 2 + ( x 2 5 ) 2 4 } ,
L I output = C { 1 + exp [ x 1 2 + ( x 2 5 ) 2 2 16 ] } 1 ,
T ¯ = { 2 x 1 2 , 3 x 2 7 } ,
I output = 0.073622 .
k = η ( x ) 2 ( η ( x ) N ( x ) ) N ( x ) .
r 2 ( x ) x i = r 1 ( x ) x i + t ( x ) x i η ( x ) + t ( x ) η ( x ) x i , i = 1 , 2 .
r 2 ( x ) x i N ( x ) = 0 , η ( x ) x i η ( x ) = 0 , i = 1 , 2 ,
r 2 ( x ) x i k = r 2 ( x ) x i η ( x ) = r 1 ( x ) x i η ( x ) + t ( x ) x i , i = 1 , 2 .
r 1 ( x ) x i η ( x ) = r 1 ( x ) x i k = z i , i = 1 , 2 .
r 2 ( x ) x i k = [ z ( x ) + t ( x ) ] x i , i = 1 , 2 .
l ( x ) x i = [ z ( x ) + t ( x ) + s ( x ) ] x i = [ r 2 ( x ) x i + s ( x ) x i k ] k = 0 , i = 1 , 2 .
t ( x ) = β 2 [ 1 + P ( x ) x 2 β 2 ] .
w ( P ( x ) ) = P ( x ) x 2 β 2 2 β + z ( x ) .
z V ( x 0 ) = [ P ( x 0 ) ] ( x x 0 ) ,
Q ( x , p ) = x p + β w ( p ) p 2 2 , x Ω ¯ , p T ¯ ,
V ( x ) = Q ( x , P ( x ) ) , V ( x 0 ) + [ P ( x 0 ) ] ( x x 0 ) = Q ( x , P ( x 0 ) ) .
V ( x ) = Q ( x , P ( x ) ) V ( x 0 ) + [ P ( x 0 ) ] ( x x 0 ) = Q ( x , P ( x 0 ) ) .
Q ( x , P ( x ) ) Q ( x , p ) ,
Q ( x , p ) Q ( x , p ) for any p T ¯ ,
β [ w ( p ) w ( p ) ] x ( p p ) + p 2 p 2 2 = [ p + p 2 x ] ( p p ) = ( p x ) ( p p ) + ( p p ) 2 2 .
x = p β p w ( p ) ,
w ( p ) w ( p ) + [ p w ( p ) ] ( p p ) for all p T ¯ .

Metrics