Abstract

We implemented the linear programming approach proposed by Oliker and by Wang to solve the single reflector problem for a point source and a far-field target. The algorithm was shown to produce solutions that aim the input rays at the intersections between neighboring reflectors. This feature makes it possible to obtain the same reflector with a low number of rays – of the order of the number of targets – as with a high number of rays, greatly reducing the computation complexity of the problem.

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  1. W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, 1980).
  2. W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3482, 389–396 (1998).
    [CrossRef]
  3. L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007).
    [CrossRef] [PubMed]
  4. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
    [CrossRef] [PubMed]
  5. L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. 226, 13–32 (1999).
    [CrossRef]
  6. V. I. 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, 2003), Chap. 4, pp. 193–222.
  7. X.-J. Wang, “On the design of a reflector antenna II,” Calculus Var. Partial Differ. Eq. 20(3), 329–341 (2004).
    [CrossRef]
  8. T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
    [CrossRef]
  9. V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties,” Proc. SPIE 5942, 594207, 594207-12 (2005).
    [CrossRef]
  10. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” Proc. SPIE 7423, 742302, 742302-12 (2009).
    [CrossRef]
  11. D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
    [CrossRef]
  12. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation usingsource-target maps,” Opt. Express 18(5), 5295–5304 (2010).
    [CrossRef] [PubMed]

2010 (1)

2009 (1)

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” Proc. SPIE 7423, 742302, 742302-12 (2009).
[CrossRef]

2008 (1)

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
[CrossRef]

2007 (1)

2005 (1)

V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties,” Proc. SPIE 5942, 594207, 594207-12 (2005).
[CrossRef]

2004 (1)

X.-J. Wang, “On the design of a reflector antenna II,” Calculus Var. Partial Differ. Eq. 20(3), 329–341 (2004).
[CrossRef]

2003 (1)

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[CrossRef]

2002 (1)

1999 (1)

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. 226, 13–32 (1999).
[CrossRef]

1998 (1)

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3482, 389–396 (1998).
[CrossRef]

Bräuer, A.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
[CrossRef]

Caffarelli, L. A.

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. 226, 13–32 (1999).
[CrossRef]

Cassarly, W. J.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation usingsource-target maps,” Opt. Express 18(5), 5295–5304 (2010).
[CrossRef] [PubMed]

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” Proc. SPIE 7423, 742302, 742302-12 (2009).
[CrossRef]

Fournier, F. R.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation usingsource-target maps,” Opt. Express 18(5), 5295–5304 (2010).
[CrossRef] [PubMed]

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” Proc. SPIE 7423, 742302, 742302-12 (2009).
[CrossRef]

Gebhardt, A.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
[CrossRef]

Glimm, T.

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[CrossRef]

Kochengin, S. A.

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. 226, 13–32 (1999).
[CrossRef]

Kudaev, S.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
[CrossRef]

Luo, Y.

Michaelis, D.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
[CrossRef]

Muschaweck, J.

Oliker, V.

V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties,” Proc. SPIE 5942, 594207, 594207-12 (2005).
[CrossRef]

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[CrossRef]

Oliker, V. I.

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. 226, 13–32 (1999).
[CrossRef]

Parkyn, W. A.

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3482, 389–396 (1998).
[CrossRef]

Qian, K.

Ries, H.

Rolland, J. P.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation usingsource-target maps,” Opt. Express 18(5), 5295–5304 (2010).
[CrossRef] [PubMed]

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” Proc. SPIE 7423, 742302, 742302-12 (2009).
[CrossRef]

Schreiber, P.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
[CrossRef]

Steinkopf, R.

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
[CrossRef]

Wang, L.

Wang, X.-J.

X.-J. Wang, “On the design of a reflector antenna II,” Calculus Var. Partial Differ. Eq. 20(3), 329–341 (2004).
[CrossRef]

Appl. Opt. (1)

Calculus Var. Partial Differ. Eq. (1)

X.-J. Wang, “On the design of a reflector antenna II,” Calculus Var. Partial Differ. Eq. 20(3), 329–341 (2004).
[CrossRef]

Contemp. Math. (1)

L. A. Caffarelli, S. A. Kochengin, and V. I. Oliker, “On the numerical solution of the problem of reflector design with given far-field scattering data,” Contemp. Math. 226, 13–32 (1999).
[CrossRef]

J. Math. Sci. (1)

T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci. 117(3), 4096–4108 (2003).
[CrossRef]

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

Opt. Express (1)

Proc. SPIE (4)

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3482, 389–396 (1998).
[CrossRef]

V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties,” Proc. SPIE 5942, 594207, 594207-12 (2005).
[CrossRef]

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” Proc. SPIE 7423, 742302, 742302-12 (2009).
[CrossRef]

D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE 7059, 705905, 705905-6 (2008).
[CrossRef]

Other (2)

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

V. I. 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, 2003), Chap. 4, pp. 193–222.

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