Abstract

The problem of designing optical systems that contain free-form surfaces is a challenging one, even in the case of designing a single surface. Here we present a method for the coupled design of two free-form reflective surfaces that will have a prescribed distortion. On one hand, the method can be described using traditional vectors and matrices, which we do, but it is motivated by viewing the problem in the language of distributions from differential geometry and makes use of the exterior differential systems, which we relegate to an appendix. Example applications are given to the design of a mirror pair that increases the field of view of an observer, a similar mirror pair that also rotates the observer's view, and a pair of mirrors that give the observer a traditional panoramic strip view of the scene.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
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  17. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24, 463–469 (2007).
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  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, 1255–1277 (2004).
    [CrossRef]
  19. T. Ivey and J. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (American Mathematical Society, 2003).

2009

J. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[CrossRef]

2008

2007

2006

O. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).
[CrossRef]

2005

R. Winston, J. Miñano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

R. Hicks and R. Perline, “The blind-spot problem for motor vehicles,” Appl. Opt. 44, 3893–3897 (2005).
[CrossRef] [PubMed]

2004

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

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, 1255–1277 (2004).
[CrossRef]

B. Creber, “Fabrication of freeform optics,” in ASPE Proceedings, Free-Form Optics: Design, Fabrication, Metrology, Assembly (American Society for Precision Engineering, 2004).

2003

T. Ivey and J. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (American Mathematical Society, 2003).

2000

R. A. Hicks and R. Bajcsy, “Catadioptic sensors that approximate wide-angle perspective projections,” in Proceedings of Computer Vision Pattern Recognition (IEEE Computer Society, 2000), pp. 545–551.

1998

S. Kochengin and V. Oliker, “Determination of reflector surfaces from near-field scattering data. II. Numerical solution,” Numer. Math. 79, 553–568 (1998).
[CrossRef]

1997

E. Kreifeldt, “Darpa turns researchers loose on new class of optics,” Opt. Photonics News 8, 6–10 (1997).

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363–373 (1997).
[CrossRef]

1993

H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).

1992

1982

1959

C. Kanolt, “Multifocal opthalmic lenses,” U.S. patent 2,878,721 (March 24, 1959).

Bajcsy, R.

R. A. Hicks and R. Bajcsy, “Catadioptic sensors that approximate wide-angle perspective projections,” in Proceedings of Computer Vision Pattern Recognition (IEEE Computer Society, 2000), pp. 545–551.

Benitez, P.

J. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[CrossRef]

R. Winston, J. Miñano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

Blen, J.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Buchdahl, H.

H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).

Chaves, J.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Creber, B.

B. Creber, “Fabrication of freeform optics,” in ASPE Proceedings, Free-Form Optics: Design, Fabrication, Metrology, Assembly (American Society for Precision Engineering, 2004).

Dross, O.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Falicoff, W.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Forbes, G.

Gimenez-Benitez, P.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Glimm, T.

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, 1255–1277 (2004).
[CrossRef]

Hernandez, M.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Hicks, R.

Hicks, R. A.

R. A. Hicks, “Controlling a ray bundle with a free-form reflector,” Opt. Lett. 33, 1672–1674 (2008).
[CrossRef] [PubMed]

R. A. Hicks and R. Bajcsy, “Catadioptic sensors that approximate wide-angle perspective projections,” in Proceedings of Computer Vision Pattern Recognition (IEEE Computer Society, 2000), pp. 545–551.

Ivey, T.

T. Ivey and J. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (American Mathematical Society, 2003).

Kanolt, C.

C. Kanolt, “Multifocal opthalmic lenses,” U.S. patent 2,878,721 (March 24, 1959).

Kochengin, S.

S. Kochengin and V. Oliker, “Determination of reflector surfaces from near-field scattering data. II. Numerical solution,” Numer. Math. 79, 553–568 (1998).
[CrossRef]

Kochengin, S. A.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363–373 (1997).
[CrossRef]

Kreifeldt, E.

E. Kreifeldt, “Darpa turns researchers loose on new class of optics,” Opt. Photonics News 8, 6–10 (1997).

Landsberg, J.

T. Ivey and J. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (American Mathematical Society, 2003).

Miñano, J.

J. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[CrossRef]

R. Winston, J. Miñano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

Miñano, J. C.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Mohedano Arroyo, R.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Oliker, V.

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, 1255–1277 (2004).
[CrossRef]

S. Kochengin and V. Oliker, “Determination of reflector surfaces from near-field scattering data. II. Numerical solution,” Numer. Math. 79, 553–568 (1998).
[CrossRef]

Oliker, V. I.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363–373 (1997).
[CrossRef]

Perline, R.

Plummer, W. T.

Rubinstein, J.

Santamaria, A.

J. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[CrossRef]

Stavroudis, O.

O. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).
[CrossRef]

Stone, B.

Winston, R.

R. Winston, J. Miñano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

Wolansky, G.

Appl. Opt.

Indiana Univ. Math. J.

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, 1255–1277 (2004).
[CrossRef]

Inverse Probl.

S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363–373 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Numer. Math.

S. Kochengin and V. Oliker, “Determination of reflector surfaces from near-field scattering data. II. Numerical solution,” Numer. Math. 79, 553–568 (1998).
[CrossRef]

Opt. Eng. (Bellingham)

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. Mohedano Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. (Bellingham) 43, 1489–1502 (2004).
[CrossRef]

Opt. Lett.

Opt. Photonics News

E. Kreifeldt, “Darpa turns researchers loose on new class of optics,” Opt. Photonics News 8, 6–10 (1997).

Opt. Rev.

J. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16, 99–102 (2009).
[CrossRef]

Other

B. Creber, “Fabrication of freeform optics,” in ASPE Proceedings, Free-Form Optics: Design, Fabrication, Metrology, Assembly (American Society for Precision Engineering, 2004).

R. Winston, J. Miñano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

O. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).
[CrossRef]

H. Buchdahl, An Introduction to Hamiltonian Optics (Dover, 1993).

C. Kanolt, “Multifocal opthalmic lenses,” U.S. patent 2,878,721 (March 24, 1959).

R. A. Hicks and R. Bajcsy, “Catadioptic sensors that approximate wide-angle perspective projections,” in Proceedings of Computer Vision Pattern Recognition (IEEE Computer Society, 2000), pp. 545–551.

T. Ivey and J. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (American Mathematical Society, 2003).

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Figures (11)

Fig. 1
Fig. 1

Given a transformation F that points from an image plane to points on an object surface, one can define a vector field W that will be normal to any mirror surface that realizes the correspondence. In and Out are unit vectors.

Fig. 2
Fig. 2

A two-mirror system for controlling a ray bundle emanating from a single source L. Likewise, this could be viewed as a system consisting of a pinhole camera and two reflectors.

Fig. 3
Fig. 3

Our numerical method for generating a solution surface from an initial curve.

Fig. 4
Fig. 4

A pair of mirrors that increases an observer’s field of view.

Fig. 5
Fig. 5

A ray tracing simulation of the mirror pair in Fig. 4. The wall is visible beyond the lower mirror, which presents a wide-angle view of the wall.

Fig. 6
Fig. 6

A wide-angle pair that also rotates the image by 45°.

Fig. 7
Fig. 7

A ray tracing simulation of the mirror pair in Fig. 6.

Fig. 8
Fig. 8

A pair of mirrors that give a panoramic view.

Fig. 9
Fig. 9

A ray tracing simulation of the mirror pair in Fig. 8.

Fig. 10
Fig. 10

The wide-angle mirror pair generated by the “straight line vertical” initial data. Compare to Fig. 4.

Fig. 11
Fig. 11

A ray tracing simulation of the mirror pair in Fig. 10.

Equations (21)

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ϕ ( x , y , z ) = const ,
P : ( s , t ) ( x ( s , t ) , y ( s , t ) , z ( s , t ) ) ,
Q : ( s , t ) ( u ( s , t ) , v ( s , t ) , w ( s , t ) ) ,
P s ( s , t ) W ( s , t ) = 0 ,     P t ( s , t ) W ( s , t ) = 0 ,
Q s ( s , t ) V ( s , t ) = 0 ,     Q t ( s , t ) V ( s , t ) = 0.
W = ( 0 , 0 , 0 ) P ( s , t ) | ( 0 , 0 , 0 ) P ( s , t ) | + Q ( s , t ) P ( s , t ) | Q ( s , t ) P ( s , t ) | ,
V = P ( s , t ) Q ( s , t ) | P ( s , t ) Q ( s , t ) | + F ( s , t ) Q ( s , t ) | F ( s , t ) Q ( s , t ) | .
Γ ( s , t ) = ( x ( s , t ) , y ( s , t ) , z ( s , t ) , u ( s , t ) , v ( s , t ) , w ( s , t ) ) .
( 0 , 0 , 0 ) ( x , y , z ) | ( 0 , 0 , 0 ) ( x , y , z ) | + ( u , v , w ) ( x , y , z ) | ( u , v , w ) ( x , y , z ) | ,
W ̃ ( x , y , z , u , v , w ) = ( W 1 , W 2 , W 3 , 0 , 0 , 0 ) .
V ̃ ( x , y , z , u , v , w ) = ( 0 , 0 , 0 , V 1 , V 2 , V 3 ) ,
( x , y , z ) ( u , v , w ) | ( x , y , z ) ( u , v , w ) | + F ( y / x , z / x ) ( u , v , w ) | F ( y / x , z / x ) ( u , v , w ) | .
C ( t ) = ( C 1 ( t ) , C 2 ( t ) , C 3 ( t ) , C 4 ( t ) , C 5 ( t ) , C 6 ( t ) ) ,
W ̃ ( C ( t 0 ) ) T = 0 ,
V ̃ ( C ( t 0 ) ) T = 0.
A i , j = W ̃ i x j W ̃ j x i ,
B i , j = V ̃ i x j V ̃ j x i .
S t A T = 0 ,     S t B T = 0.
W ̃ T = 0 ,     V ̃ T = 0 ,     C t A T = 0 ,     C t B T = 0.
C ( t ) = ( 9   cos ( t ) , 9   sin ( t ) , 0 , 9   cos ( t ) , 9   sin ( t ) , β ( t ) ) .
θ 1 ( Y ) = θ 2 ( Y ) = d θ 1 ( X , Y ) = d θ 2 ( X , Y ) = 0 ,

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