Abstract

We propose a fast algorithm to estimate the flux collected by conic reflector patches, based on the calculation of intersections between neighboring patches. The algorithm can be employed in conjunction with the supporting ellipsoids algorithm for freeform reflector design and is shown to be orders of magnitude faster and more scalable than the commonly used Monte Carlo ray tracing approach.

© 2013 Optical Society of America

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  1. H. Ries and J. Muschaweck, J. Opt. Soc. Am. A 19, 590 (2002).
    [CrossRef]
  2. X.-J. Wang, Calc. Var. Partial Differential Equation 20, 329 (2004).
    [CrossRef]
  3. T. Glimm and V. Oliker, J. Math. Sci. 117, 4096 (2003).
    [CrossRef]
  4. V. Oliker, Proc. SPIE 5942, 594207 (2005).
    [CrossRef]
  5. C. Canavesi, W. J. Cassarly, and J. P. Rolland, Opt. Express 20, 4050 (2012).
    [CrossRef]
  6. T. Graf and V. I. Oliker, Inverse Probl. 28, 025001 (2012).
    [CrossRef]
  7. V. I. Oliker, in Trends in Nonlinear Analysis, M. Kirkilionis, S. Kromker, R. Rannacher, and F. Tomi, eds. (Springer-Verlag, 2003), p. 191.
  8. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, Proc. SPIE 7423, 742302 (2009).
    [CrossRef]
  9. S. A. Kochengin and V. O. Oliker, Comput. Visualization Sci. 6, 15 (2003).
    [CrossRef]
  10. P. Shirley and R. K. Morley, Realistic Ray Tracing, 2nd ed. (A K Peters, 2003).
  11. D. G. Koch, Proc. SPIE 1780, 226 (1992).
  12. L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).
  13. S. A. Kochengin and V. I. Oliker, Numer. Math. 79, 553 (1998).
    [CrossRef]

2012 (2)

2009 (1)

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

2008 (1)

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).

2005 (1)

V. Oliker, Proc. SPIE 5942, 594207 (2005).
[CrossRef]

2004 (1)

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

2003 (2)

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

S. A. Kochengin and V. O. Oliker, Comput. Visualization Sci. 6, 15 (2003).
[CrossRef]

2002 (1)

1998 (1)

S. A. Kochengin and V. I. Oliker, Numer. Math. 79, 553 (1998).
[CrossRef]

1992 (1)

D. G. Koch, Proc. SPIE 1780, 226 (1992).

Canavesi, C.

Cassarly, W. J.

C. Canavesi, W. J. Cassarly, and J. P. Rolland, Opt. Express 20, 4050 (2012).
[CrossRef]

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

Dupont, L.

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).

Fournier, F. R.

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]

Graf, T.

T. Graf and V. I. Oliker, Inverse Probl. 28, 025001 (2012).
[CrossRef]

Koch, D. G.

D. G. Koch, Proc. SPIE 1780, 226 (1992).

Kochengin, S. A.

S. A. Kochengin and V. O. Oliker, Comput. Visualization Sci. 6, 15 (2003).
[CrossRef]

S. A. Kochengin and V. I. Oliker, Numer. Math. 79, 553 (1998).
[CrossRef]

Lazard, D.

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).

Lazard, S.

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).

Morley, R. K.

P. Shirley and R. K. Morley, Realistic Ray Tracing, 2nd ed. (A K Peters, 2003).

Muschaweck, J.

Oliker, V.

V. Oliker, Proc. SPIE 5942, 594207 (2005).
[CrossRef]

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

Oliker, V. I.

T. Graf and V. I. Oliker, Inverse Probl. 28, 025001 (2012).
[CrossRef]

S. A. Kochengin and V. I. Oliker, Numer. Math. 79, 553 (1998).
[CrossRef]

V. I. Oliker, in Trends in Nonlinear Analysis, M. Kirkilionis, S. Kromker, R. Rannacher, and F. Tomi, eds. (Springer-Verlag, 2003), p. 191.

Oliker, V. O.

S. A. Kochengin and V. O. Oliker, Comput. Visualization Sci. 6, 15 (2003).
[CrossRef]

Petitjean, S.

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).

Ries, H.

Rolland, J. P.

C. Canavesi, W. J. Cassarly, and J. P. Rolland, Opt. Express 20, 4050 (2012).
[CrossRef]

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

Shirley, P.

P. Shirley and R. K. Morley, Realistic Ray Tracing, 2nd ed. (A K Peters, 2003).

Wang, X.-J.

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

Calc. Var. Partial Differential Equation (1)

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

Comput. Visualization Sci. (1)

S. A. Kochengin and V. O. Oliker, Comput. Visualization Sci. 6, 15 (2003).
[CrossRef]

Inverse Probl. (1)

T. Graf and V. I. Oliker, Inverse Probl. 28, 025001 (2012).
[CrossRef]

J. Math. Sci. (1)

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

J. Opt. Soc. Am. A (1)

J. Symb. Comput. (1)

L. Dupont, D. Lazard, S. Lazard, and S. Petitjean, J. Symb. Comput. 43, 168 (2008).

Numer. Math. (1)

S. A. Kochengin and V. I. Oliker, Numer. Math. 79, 553 (1998).
[CrossRef]

Opt. Express (1)

Proc. SPIE (3)

V. Oliker, Proc. SPIE 5942, 594207 (2005).
[CrossRef]

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

D. G. Koch, Proc. SPIE 1780, 226 (1992).

Other (2)

V. I. Oliker, in Trends in Nonlinear Analysis, M. Kirkilionis, S. Kromker, R. Rannacher, and F. Tomi, eds. (Springer-Verlag, 2003), p. 191.

P. Shirley and R. K. Morley, Realistic Ray Tracing, 2nd ed. (A K Peters, 2003).

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Figures (6)

Fig. 1.
Fig. 1.

Reflector surface obtained as a solution of the supporting ellipsoids method with flux evaluation performed by Monte Carlo ray tracing for a 4×4 discrete target, as rendered in LightTools. The vertices and boundaries of one ellipsoid patch are highlighted.

Fig. 2.
Fig. 2.

(a) Two-ellipsoid and (b) three-ellipsoid intersections calculated with the intersection method. The intersection curves (ellipses) between two ellipsoids are shown with a solid line; the dots in (b) represent the two intersection points that occur in space.

Fig. 3.
Fig. 3.

Relative flux estimation time for the intersection method (cross) and Monte Carlo ray tracing with 10 rays/ellipsoid (circle) or 100 rays/ellipsoid (square), normalized to the computation time for 1024 target points with the intersection method.

Fig. 4.
Fig. 4.

Projection in the x–y plane of the rays hitting the 10×10 reflector obtained with the supporting ellipsoids algorithm (a) with Monte Carlo flux estimation for 100 iterations, and (b) switching to the intersection method until the final solution is obtained. Rays hitting different ellipsoids have different colors for visualization purposes; the vertices and boundaries calculated with the intersection method are shown in black.

Fig. 5.
Fig. 5.

Map of the normalized flux collected by the 10×10 reflector of Fig. 4(b) as calculated by (a) Monte Carlo ray tracing with 1000 rays/ellipsoid and (b) intersection method. The scale of the images is set to display the entire range of values of the Monte Carlo ray tracing result.

Fig. 6.
Fig. 6.

(a) Projection in the x–y plane of a reflector that produces a target with 45° twist, and (b) normalized flux estimated with the intersection method (the scale for the visualization is set at ±5%).

Equations (1)

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ρ=f1e(S·T),

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