Abstract

Here we present a method for the coupled design of four freeform reflective surfaces that will control a bundle of rays. By this, we mean that given an input bundle of rays, we can construct an optical system that will map it to a given output bundle, where a ray-to-ray correspondence is realized as per the prescribed data. The method makes use of the Cartan–Kähler theorem of exterior differential systems. Sample imaging applications are given.

© 2014 Optical Society of America

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References

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  1. 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.
  2. D. G. Burkhard and D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
    [CrossRef]
  3. J. H. McDermit and T. E. Horton, “Reflective optics for obtaining prescribed irradiative distributions from collimated sources,” Appl. Opt. 13, 1444–1450 (1974).
    [CrossRef]
  4. R. A. Hicks and R. Perline, “The blind-spot problem for motor vehicles,” Appl. Opt. 44, 3893–3897 (2005).
    [CrossRef]
  5. R. A. Hicks, “Controlling a ray bundle with a free-form reflector,” Opt. Lett. 33, 1672–1674 (2008).
    [CrossRef]
  6. R. A. Hicks and C. Croke, “Designing coupled free-form surfaces,” J. Opt. Soc. Am. A 27, 2132–2137 (2010).
    [CrossRef]
  7. S. A. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data,” Inverse Probl. 13, 363–373 (1997).
    [CrossRef]
  8. S. Kochengin and V. Oliker, “Determination of reflector surfaces from near-field scattering data. II. Numerical solution,” Numer. Math. 79, 553–568 (1998).
    [CrossRef]
  9. J. S. Schruben, “Formulation of a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. 62, 1498–1501 (1972).
    [CrossRef]
  10. C. W. Kanolt, “Multifocal ophthalmic lenses,” U.S. Patent2,878,721 (March24, 1959).
  11. W. T. Plummer, “Unusual optics of the Polaroid SX-70 land camera,” Appl. Opt. 21, 196–208 (1982).
    [CrossRef]
  12. E. H. Land, “Reflex camera,” U.S. Patent3,672,281 (June27, 1972).
  13. J. G. Baker, “Compact folding reflex camera,” U.S. Patent3,678,831 (July25, 1972).
  14. W. T. Plummer, “Reflective imaging apparatus,” U.S. Patent3,735,685 (May29, 1973).
  15. E. Kreifeldt, “DARPA turns researchers loose on new class of optics,” Opt. Photon. News 8(1), 6–10 (1997).
  16. R. R. Shannon, J. P. Mills, H. M. Pollicove, P. A. Trotta, and L. N. Durvasula, “Optics that fit,” Photonics Spectra 97, 86–88 (2001).
  17. B. D. Stone and G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–109 (1992).
    [CrossRef]
  18. B. D. Stone and G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992).
    [CrossRef]
  19. J. Rubinstein and G. Wolansky, “Intensity control with a free-form lens,” J. Opt. Soc. Am. A 24, 463–469 (2007).
    [CrossRef]
  20. R. Winston, J. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).
  21. D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” Proc. SPIE 4095, 1–15 (2000).
    [CrossRef]
  22. R. Winston, Selected Papers on Nonimaging Optics (SPIE Optical Engineering1995).
  23. W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy (Academic, 1978).
  24. T. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (American Mathematical Society, 2003).

2010

2008

2007

2005

2001

R. R. Shannon, J. P. Mills, H. M. Pollicove, P. A. Trotta, and L. N. Durvasula, “Optics that fit,” Photonics Spectra 97, 86–88 (2001).

2000

D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” Proc. SPIE 4095, 1–15 (2000).
[CrossRef]

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

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

E. Kreifeldt, “DARPA turns researchers loose on new class of optics,” Opt. Photon. News 8(1), 6–10 (1997).

1992

1982

1975

D. G. Burkhard and D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
[CrossRef]

1974

1972

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.

Baker, J. G.

J. G. Baker, “Compact folding reflex camera,” U.S. Patent3,678,831 (July25, 1972).

Benitez, P.

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

Burkhard, D. G.

D. G. Burkhard and D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
[CrossRef]

Croke, C.

Durvasula, L. N.

R. R. Shannon, J. P. Mills, H. M. Pollicove, P. A. Trotta, and L. N. Durvasula, “Optics that fit,” Photonics Spectra 97, 86–88 (2001).

Forbes, G. W.

Hicks, R. A.

Horton, T. E.

Ivey, T.

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

Kanolt, C. W.

C. W. Kanolt, “Multifocal ophthalmic lenses,” U.S. Patent2,878,721 (March24, 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. Photon. News 8(1), 6–10 (1997).

Land, E. H.

E. H. Land, “Reflex camera,” U.S. Patent3,672,281 (June27, 1972).

Landsberg, J. M.

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

McDermit, J. H.

Mills, J. P.

R. R. Shannon, J. P. Mills, H. M. Pollicove, P. A. Trotta, and L. N. Durvasula, “Optics that fit,” Photonics Spectra 97, 86–88 (2001).

Minano, J.

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

Oliker, V.

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.

W. T. Plummer, “Unusual optics of the Polaroid SX-70 land camera,” Appl. Opt. 21, 196–208 (1982).
[CrossRef]

W. T. Plummer, “Reflective imaging apparatus,” U.S. Patent3,735,685 (May29, 1973).

Pollicove, H. M.

R. R. Shannon, J. P. Mills, H. M. Pollicove, P. A. Trotta, and L. N. Durvasula, “Optics that fit,” Photonics Spectra 97, 86–88 (2001).

Rubinstein, J.

Schruben, J. S.

Shannon, R. R.

R. R. Shannon, J. P. Mills, H. M. Pollicove, P. A. Trotta, and L. N. Durvasula, “Optics that fit,” Photonics Spectra 97, 86–88 (2001).

Shealy, D. L.

D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” Proc. SPIE 4095, 1–15 (2000).
[CrossRef]

D. G. Burkhard and D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
[CrossRef]

Stone, B. D.

Trotta, P. A.

R. R. Shannon, J. P. Mills, H. M. Pollicove, P. A. Trotta, and L. N. Durvasula, “Optics that fit,” Photonics Spectra 97, 86–88 (2001).

Welford, W. T.

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy (Academic, 1978).

Winston, R.

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy (Academic, 1978).

R. Winston, Selected Papers on Nonimaging Optics (SPIE Optical Engineering1995).

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

Wolansky, G.

Appl. Opt.

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.

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. Lett.

Opt. Photon. News

E. Kreifeldt, “DARPA turns researchers loose on new class of optics,” Opt. Photon. News 8(1), 6–10 (1997).

Photonics Spectra

R. R. Shannon, J. P. Mills, H. M. Pollicove, P. A. Trotta, and L. N. Durvasula, “Optics that fit,” Photonics Spectra 97, 86–88 (2001).

Proc. SPIE

D. L. Shealy, “Theory of geometrical methods for design of laser beam shaping systems,” Proc. SPIE 4095, 1–15 (2000).
[CrossRef]

Sol. Energy

D. G. Burkhard and D. L. Shealy, “Design of reflectors which will distribute sunlight in a specified manner,” Sol. Energy 17, 221–227 (1975).
[CrossRef]

Other

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.

C. W. Kanolt, “Multifocal ophthalmic lenses,” U.S. Patent2,878,721 (March24, 1959).

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

E. H. Land, “Reflex camera,” U.S. Patent3,672,281 (June27, 1972).

J. G. Baker, “Compact folding reflex camera,” U.S. Patent3,678,831 (July25, 1972).

W. T. Plummer, “Reflective imaging apparatus,” U.S. Patent3,735,685 (May29, 1973).

R. Winston, Selected Papers on Nonimaging Optics (SPIE Optical Engineering1995).

W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy (Academic, 1978).

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

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Figures (14)

Fig. 1.
Fig. 1.

Schematic description of our problem—we wish to design an optical system consisting of mirrors that maps a ray bundle to a ray bundle at the ray level.

Fig. 2.
Fig. 2.

Surfaces M1-M4 must all be perpendicular to the ray path. This imposes conditions on their derivatives.

Fig. 3.
Fig. 3.

Given a curve on our surface, we use the known equations that it must satisfy to compute the tangent space at a discrete set of points on the curve, and then extend the surface discretely by moving out infinitesimally on the tangent spaces.

Fig. 4.
Fig. 4.

Above figure depicts the four initial curves in three-dimensional space, and below we can see the trajectories of the rays.

Fig. 5.
Fig. 5.

Four-reflector system (not plotted to scale), numbered as in Fig. 4.

Fig. 6.
Fig. 6.

Test pattern (image) used in the test.

Fig. 7.
Fig. 7.

Three tests with the test pattern at (240, 210, 200), (640, 610, 600), and (940, 910, 900). They should be very similar due to collimation of the output beam. Note that the mirrors are simulated in a virtual room with checkerboard walls of various colors and some sample text.

Fig. 8.
Fig. 8.

Image of the test pattern via a rotated collimated bundle with r=5.

Fig. 9.
Fig. 9.

Image of the test pattern via a rotated collimated bundle with r=10.

Fig. 10.
Fig. 10.

Images of the test pattern (placed at different positions) via a collimated bundle with r=3 and no twist. We can see from the distortion in the lower image that the bundle is not collimated perfectly.

Fig. 11.
Fig. 11.

Image of the test pattern via a rotated collimated bundle with r=0.5.

Fig. 12.
Fig. 12.

Here, the y scaling is 4 and the z scaling is 8, with the image plane at (240, 210, 200) (top) and (940, 910, 900) (bottom).

Fig. 13.
Fig. 13.

From top left to bottom right: taking r=3, a series of POVRay tests with the test pattern centered at (140, 110, 100), (240, 210, 200), (640, 610, 600), and (840, 810, 800) for the output beam from a virtual point source at (40,10,0)203(1,1,1).

Fig. 14.
Fig. 14.

From top to bottom: taking r=3, a series of POVRay tests with the test pattern centered at (140, 110, 100), (440, 410, 400), which is near the focal point, and (940, 910, 900) for the output beam from a virtual point source at (40,10,0)+7003(1,1,1).

Equations (12)

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

Mi:(s,t)(xi(s,t),yi(s,t),zi(s,t)).
V1(s,t)=VR(s,t)+M2(s,t)M1(s,t)|M2(s,t)M1(s,t)|,V2(s,t)=M1(s,t)M2(s,t)|M1(s,t)M2(s,t)|+M3(s,t)M2(s,t)|M3(s,t)M2(s,t)|,V3(s,t)=M2(s,t)M3(s,t)|M2(s,t)M3(s,t)|+M4(s,t)M3(s,t)|M4(s,t)M3(s,t)|,V4(s,t)=VRout(s,t)+M2(s,t)M1(s,t)|M3(s,t)M4(s,t)|.
Γ(s,t)=(x1(s,t),y1(s,t),z1(s,t),x2(s,t),y2(s,t),z2(s,t),x3(s,t),,z4(s,t)).
(X2,Y2,Z2)=(x1,x2,x3)(x4,x5,x6)|(x1,x2,x3)(x4,x5,x6)|+(x7,x8,x9)(x4,x5,x6)|(x7,x8,x9)(x4,x5,x6)|.
V˜2=(0,0,0,X2,Y2,Z2,0,0,0,0,0,0).
(X1,Y1,Z1)=VR(s(x1,x2,x3),t(x1,x2,x3))+(x4,x5,x6)(x1,x2,x3)|(x4,x5,x6)(x1,x2,x3)|,
V˜1=(X1,Y1,Z1,0,0,0,0,0,0,0,0,0),
(X4,Y4,Z4)=(x7,x8,x9)(x10,x11,x12)|(x7,x8,x9)(x10,x11,x12)|+VRout(s(x1,x2,x3),t(x1,x2,x3)),
V˜4=(0,0,0,0,0,0,0,0,0,X4,Y4,Z4).
C(t)=(C1(t),C2,(t),C3(t),C4(t),,C12(t))
θi(W)=θ2(W)=θ3(W)=θ4(W)=0,dθ1(V,W)=dθ2(V,W)=dθ3(V,W)=dθ4(V,W)=0,
(0,t,0,0,t,10+0.05t,C7(t),C8(t),C9(t),40+rYout(1)t,10+rYout(2)t,rYout(3)t).

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