Abstract

We have developed a fast algorithm to design two-dimensional reflector surfaces that ties together the supporting paraboloids, linear programming, and numerical integration methods. The algorithm builds upon the properties of conics and is shown to be several orders of magnitude faster than the supporting paraboloids and linear programming methods. The scalability and ease of implementation of the algorithm are discussed.

© 2012 Optical Society of America

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References

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  1. L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, Contemp Math. 226, 13 (1999).
    [CrossRef]
  2. X.-J. Wang, Calc. Var. 20, 329 (2004).
    [CrossRef]
  3. T. Glimm and V. Oliker, J. Math. Sci. 117, 4096 (2003).
    [CrossRef]
  4. W. B. Elmer, The Optical Design of Reflectors, 2nd ed.(Wiley, 1980).
  5. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, Proc. SPIE 7423, 742302 (2009).
    [CrossRef]
  6. C. Canavesi, W. J. Cassarly, and J. P. Rolland, Opt. Express 20, 4050 (2012).
    [CrossRef]
  7. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, Opt. Express 18, 5295 (2010).
    [CrossRef]

2012 (1)

2010 (1)

2009 (1)

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, Proc. SPIE 7423, 742302 (2009).
[CrossRef]

2004 (1)

X.-J. Wang, Calc. Var. 20, 329 (2004).
[CrossRef]

2003 (1)

T. Glimm and V. Oliker, J. Math. Sci. 117, 4096 (2003).
[CrossRef]

1999 (1)

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, Contemp Math. 226, 13 (1999).
[CrossRef]

Caffarelli, L. A.

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, Contemp Math. 226, 13 (1999).
[CrossRef]

Canavesi, C.

Cassarly, W. J.

Elmer, W. B.

W. B. Elmer, The Optical Design of Reflectors, 2nd ed.(Wiley, 1980).

Fournier, F. R.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, Opt. Express 18, 5295 (2010).
[CrossRef]

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, Proc. SPIE 7423, 742302 (2009).
[CrossRef]

Glimm, T.

T. Glimm and V. Oliker, J. Math. Sci. 117, 4096 (2003).
[CrossRef]

Kochengin, S. A.

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, Contemp Math. 226, 13 (1999).
[CrossRef]

Oliker, V.

T. Glimm and V. Oliker, J. Math. Sci. 117, 4096 (2003).
[CrossRef]

Oliker, V. I.

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, Contemp Math. 226, 13 (1999).
[CrossRef]

Rolland, J. P.

Wang, X.-J.

X.-J. Wang, Calc. Var. 20, 329 (2004).
[CrossRef]

Calc. Var. (1)

X.-J. Wang, Calc. Var. 20, 329 (2004).
[CrossRef]

Contemp Math. (1)

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, Contemp Math. 226, 13 (1999).
[CrossRef]

J. Math. Sci. (1)

T. Glimm and V. Oliker, J. Math. Sci. 117, 4096 (2003).
[CrossRef]

Opt. Express (2)

Proc. SPIE (1)

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, Proc. SPIE 7423, 742302 (2009).
[CrossRef]

Other (1)

W. B. Elmer, The Optical Design of Reflectors, 2nd ed.(Wiley, 1980).

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Figures (6)

Fig. 1.
Fig. 1.

Evaluation with 22 rays of the reflector surfaces obtained (a) with the supporting paraboloids method and (b) with the linear programming method for three target directions run with a low number of rays (three and four, respectively). The rays are color coded to indicate which paraboloid they are hitting.

Fig. 2.
Fig. 2.

Parabola with focus at the origin, having focal parameter fj and with its axis along the direction described by Tj. The distance to the reflector along direction Si, ρi,j, is given by Eq. (1).

Fig. 3.
Fig. 3.

Steps describing the algorithm for direct calculation in the case of paraboloids expressed by Eq. (1).

Fig. 4.
Fig. 4.

Construction of the reflector.

Fig. 5.
Fig. 5.

Computation time for linear programming and direct calculation methods. A 1.87 GHz Intel processor was used.

Fig. 6.
Fig. 6.

(a) Reflector made of patches of ellipsoids obtained with the direct calculation method and (b) irradiance at the target plane calculated tracing 40 rays in LightTools.

Equations (1)

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ρi,j=fj1Si·Tj,

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