## Observations on the linear programming formulation of the single reflector design problem |

Optics Express, Vol. 20, Issue 4, pp. 4050-4055 (2012)

http://dx.doi.org/10.1364/OE.20.004050

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### 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.

© 2012 OSA

## 1. Introduction

## 2. Linear programming formulation of the reflector problem

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

*N*source ray directions

_{S}*s*and

*N*target directions

_{T}*t*, it is possible to find a reflector solution consisting of

*N*patches of paraboloids, with each paraboloid directing light from the source into one target direction.

_{T}**c**

*is the transpose of*

^{T}**c**, and the vectors

**c**,

**x**and

**b**are defined as

**A**has size

*N*× (

_{S}N_{T}*N*and is defined according to Eq. (1) to have only two non-zero elements per row. The size of

_{S}+ N_{T})**A**is the main limitation of the implementation of the linear programming algorithm, making it more suited to solve the starting point problem. As the number of rays and targets increases, the size of

**A**quickly becomes very large. For

*N*100 and

_{S}=*N*100, for example, the size of

_{T}=**A**is 10,000 × 200.

**A**is a matrix of size

_{eq}*N*× (

_{S}N_{T}*N*whose only non-zero element is the first one and

_{S}+ N_{T})**b**is a zero vector of length

_{eq}*N*.

_{S}N_{T}## 3. 2D reflector design

*N*= 4) uniformly distributed between 0 and −10°. Figures 1(a) -1(c) shows the reflectors obtained as a solution of the linear programming method with 5, 21 and 4 rays, respectively.

_{T}*nN*1 rays are used, with

_{T}+*n*integer, then each reflector collects

*n +*1 rays, with

*N*1 rays falling at the intersections between neighboring reflectors. Additionally, the

_{T}-*N*paraboloids generated with

_{T}*nN*+ 1 rays are identical even for different

_{T}*n*, i.e., they have the same focal parameters and polar radii, as shown in Figs. 1(a) and 1(b) for an isotropic source with 5 and 21 rays respectively, making it possible to obtain the solution with a low number of rays (

*N*1). If a different number of rays is used,

_{T}+*N*for example, the solution is generally not feasible, as reflectors that only collect one ray are generated, as shown in Fig. 1(c) for 4 rays.

_{S}= N_{T}*N*1) will apply.

_{T}+## 4. 3D reflector design

*N*= (

_{S}*N*+ 1)(

_{T,x}*N*+ 1), where

_{T,y}*N*and

_{T,x}*N*are the number of target points in the

_{T,y}*x*and

*y*directions (

*z*being the optical axis).

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]

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]

*T*represents the flux directed to the

_{i}*i*-th target. Tracing 16,000 rays to evaluate the flux collected by the reflectors of Fig. 4, the merit function obtained with the linear programming reflector was 0.18, while the merit function for the ellipsoid algorithm was 0.66.

## 5. Discussion

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]

**A**with number of rays and targets does not make it currently feasible to exploit the algorithm to get a point source solution. Instead, the starting point obtained with the linear programming can be directly optimized to produce the point source solution.

## 6. Conclusion

## Acknowledgments

## References and links

1. | W. B. Elmer, |

2. | W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE |

3. | L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. |

4. | H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A |

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. |

6. | V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in |

7. | X.-J. Wang, “On the design of a reflector antenna II,” Calculus Var. Partial Differ. Eq. |

8. | T. Glimm and V. Oliker, “Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem,” J. Math. Sci. |

9. | V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties,” Proc. SPIE |

10. | F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” Proc. SPIE |

11. | D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, and A. Bräuer, “Incoherent beam shaping with freeform mirror,” Proc. SPIE |

12. | F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation usingsource-target maps,” Opt. Express |

**OCIS Codes**

(080.1753) Geometric optics : Computation methods

(220.2945) Optical design and fabrication : Illumination design

(080.4228) Geometric optics : Nonspherical mirror surfaces

(080.4298) Geometric optics : Nonimaging optics

**History**

Original Manuscript: December 19, 2011

Revised Manuscript: January 30, 2012

Manuscript Accepted: January 30, 2012

Published: February 2, 2012

**Citation**

Cristina Canavesi, William J. Cassarly, and Jannick P. Rolland, "Observations on the linear programming formulation of the single reflector design problem," Opt. Express **20**, 4050-4055 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4050

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### References

- W. B. Elmer, The Optical Design of Reflectors, 2nd ed. (Wiley, 1980).
- W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE3482, 389–396 (1998). [CrossRef]
- L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt.46(18), 3716–3723 (2007). [CrossRef] [PubMed]
- H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19(3), 590–595 (2002). [CrossRef] [PubMed]
- 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]
- 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.
- X.-J. Wang, “On the design of a reflector antenna II,” Calculus Var. Partial Differ. Eq.20(3), 329–341 (2004). [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]
- V. Oliker, “Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties,” Proc. SPIE5942, 594207, 594207-12 (2005). [CrossRef]
- F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” Proc. SPIE7423, 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. SPIE7059, 705905, 705905-6 (2008). [CrossRef]
- F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation usingsource-target maps,” Opt. Express18(5), 5295–5304 (2010). [CrossRef] [PubMed]

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