## Fast freeform reflector generation using source-target maps

Optics Express, Vol. 18, Issue 5, pp. 5295-5304 (2010)

http://dx.doi.org/10.1364/OE.18.005295

Acrobat PDF (1250 KB)

### Abstract

We propose a freeform reflector design method based on the mapping of equi-flux grids between a point source and a target. This method imposes no restriction on the target distribution, the reflector collection angle or the source intensity pattern. Source-target maps are generated from a small number of target points using the Oliker algorithm. Such maps satisfy the surface integrability condition and can thus be used to quickly generate reflectors that produce continuous illuminance distributions.

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## 1. Background

1. F. Fournier and J. Rolland, “Design methodology for high brightness projectors,” J. Disp. Technol. **4**(1), 86–91 (2008). [CrossRef]

5. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A **19**(3), 590–595 (2002). [CrossRef]

8. L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. **46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

*I*is the intensity distribution of the point source in spherical coordinates and

_{S}*E*the desired illuminance distribution on a flat target perpendicular to the

_{T}*z*axis, as shown in Fig. 1 . Ideally, one would want to obtain from Eq. (1) relationships between the source emission angle and the target location, so thatwhere

*f*and

*g*are continuous functions that map uniquely each source ray direction (

*θ*,

*ϕ*) to a target location (

*x*,

*y*). Equation (2) cannot be solved directly in the general case, but mappings can be obtained analytically in some cases where the system geometry allows separation of variables.

8. L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. **46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

*θ*and

*ϕ*directions. Since the source is isotropic, the amount of flux is the same in each solid angle. The target can also be partitioned into equal area cells in the same fashion as the source by first slicing the target concentrically (concentric blue contours in Fig. 1b) and then radially (green lines in Fig. 1b). By comparing the source grid and the target grid, we can determine where each ray emitted by the source must hit the target. We can therefore compute the normals to the reflector surface required to aim source rays to their corresponding target locations. However, there is no guarantee that a continuous surface can fit all surface normals. In fact, a field of surface normals

**N**cannot be compatible with a smooth continuous surface unless it satisfies the integrability conditionThis smoothness constraint is often used for surface reconstruction algorithms in computer graphics [9

9. R. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. **10**(4), 439–451 (1988). [CrossRef]

*η*, the residual projected curl of

**N**as defined in Eq. (3), for the two target maps shown in Fig. 1. The analytical mapping presented in Fig. 2b, albeit intuitive, does not satisfy the integrability condition, and therefore cannot yield a smooth continuous surface. One possible solution is to introduce step discontinuities in the surface [6–8

8. L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. **46**(18), 3716–3723 (2007). [CrossRef] [PubMed]

*η*[10], but a satisfactory result may be difficult to obtain if the residual curl is too large or spreads unevenly. Alternatively, the plot in Fig. 2a shows that the mapping method we propose minimizes

*η*over the entire target, and can therefore yield smooth reflector surfaces. We will describe in section 2 how such maps are generated, and in section 3 how maps can be used to quickly generate reflector shapes.

## 2. Generating maps that satisfy the integrability condition

*a posteriori*retrieve the mapping between the source and the target. Figure 3a describes the map retrieval process. In this example, the target is a square made of 25 target points. The center of each ellipsoid facet is associated to its corresponding target point, except for target edges where the middle of the facet outer edge is used instead of the center (red dots in Fig. 3a). Target values on the edge must thus be weighted appropriately to prevent edge effects. This way, we map the coordinates (

*x*,

_{i}*y*) of each target point to a source direction (

_{i}*θ*,

_{i}*ϕ*). In other words, we obtain discrete versions of the mapping relationships

_{i}*x*=

*f(θ*,

*ϕ*) and

*y*=

*g*(

*θ*,

*ϕ*). This set of data points is then approximated by a surface in order to generate a continuous mapping function, as shown in Fig. 3b. The maps shown in Fig. 1, 4 and 5 can finally be generated by plotting the (

*x*,

*y*) contours corresponding to constant

*θ*values (green lines) or constant

*ϕ*values (blue lines).

*θ*contours (blue lines) are nearly circular at the center of the target and gradually become squarer closer to the edges of the target. A good fit for these contours can be obtained with superellipses (also referred to as Lamé curves). Interestingly, it appears that the rectangular map matches closely a stretched version of the square map.

*z*, the target size, the reflector size, the source distribution, or the solid angle over which light is collected (collection angle). As a general rule, maps are invariant under moderate changes, although the corresponding reflector shapes may be dramatically different. For instance, the same map can be used if a Lambertian source is used instead of an isotropic source, as long as the source equi-flux grid is modified accordingly. The collection angle appears to be the most sensitive parameter, while target size, target distance and reflector size can sometimes be changed by an order of magnitude with minimal impact on the map.

*θ*= 60° and the collection angle is 20° × 20°. These maps were generated from 225 ellipsoids for each facet. We see that contours follow the facet orientation as the azimuth angle increases. The map is thus close to a standard rectangular grid when

*ϕ*= 0°, and becomes diamond shaped when

*ϕ*= 45°. It is worth noting that when

*ϕ*is different from 0°, the corners of the facet do not match the corner of the target. Corners cannot be mapped to corners when the azimuth

*ϕ*is different from 0°, because the resulting twist in the mapping would cause the curl to be non-zero and the surface to be non-integrable.

## 3. Generating smooth reflectors from maps

*θ*= 0°). From the starting point, reflector profiles are then generated for each

*ϕ*by integrating the surface normals along

*θ*. Surface normals can be determined at each point using the law of reflection so that each source ray reaches its corresponding target location, in accordance with the mapping. The set of surface points created this way is finally interpolated with a NURBS surface and exported to illumination design software for analysis.

13. F. Fournier and J. Rolland, “Optimization of freeform lightpipes for light-emitting-diode projectors,” Appl. Opt. **47**(7), 957–966 (2008). [CrossRef] [PubMed]

## 4. Concluding remarks

## Appendix

*C*is an arbitrary closed integration path within the domain over which the surface is defined (the collection angle of the reflector),

**N**is the field of desired surface normals and

**dl**is a differential displacement vector. Using Stokes’ theorem, Eq. (4) can be rewritten aswhere

*S*is a surface capping contour

*C*, and

*ds*is a differential surface area. Since this constraint applies for any arbitrary integration path, we must satisfy at any point on the surface

## Acknowledgments

## References and links

1. | F. Fournier and J. Rolland, “Design methodology for high brightness projectors,” J. Disp. Technol. |

2. | L. B. W. Jolley, J. M. Waldram, and G. H. Wilson, |

3. | W. B. Elmer, |

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

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

6. | W. A. Parkyn, “Segmented illumination lenses for step lighting and wall washing,” in |

7. | Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express |

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

9. | R. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. |

10. | L. Noakes, and R. Kozera, “2D leapfrog algorithm for optimal surface reconstruction,” in |

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

12. | W. J. Cassarly, “Nonimaging Optics: Concentration and Illumination,” in |

13. | F. Fournier and J. Rolland, “Optimization of freeform lightpipes for light-emitting-diode projectors,” Appl. Opt. |

**OCIS Codes**

(220.2945) Optical design and fabrication : Illumination design

(080.4228) Geometric optics : Nonspherical mirror surfaces

(080.4298) Geometric optics : Nonimaging optics

**History**

Original Manuscript: January 6, 2010

Revised Manuscript: February 13, 2010

Manuscript Accepted: February 13, 2010

Published: February 26, 2010

**Citation**

Florian R. Fournier, William J. Cassarly, and Jannick P. Rolland, "Fast freeform reflector generation using source-target maps," Opt. Express **18**, 5295-5304 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-5295

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### References

- F. Fournier and J. Rolland, “Design methodology for high brightness projectors,” J. Disp. Technol. 4(1), 86–91 (2008). [CrossRef]
- L. B. W. Jolley, J. M. Waldram, and G. H. Wilson, The theory and design of illuminating engineering equipment (Chapman & Hall, London, 1930).
- W. B. Elmer, The optical design of reflectors, 2d ed. (Wiley, New York, 1980).
- V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis (2002), pp. 191–222.
- H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef]
- W. A. Parkyn, “Segmented illumination lenses for step lighting and wall washing,” in Current Developments in Optical Design and Optical Engineering VIII, (SPIE, 1999), 363–370.
- Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef] [PubMed]
- L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef] [PubMed]
- R. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10(4), 439–451 (1988). [CrossRef]
- L. Noakes, and R. Kozera, “2D leapfrog algorithm for optimal surface reconstruction,” in Vision Geometry VIII, (SPIE, 1999), 317–328.
- F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Designing freeform reflectors for extended sources,” in Nonimaging Optics: Efficient Design for Illumination and Solar Concentration VI, (SPIE, 2009), 742302.
- W. J. Cassarly, “Nonimaging Optics: Concentration and Illumination,” in Handbook of optics, 2nd ed. (McGraw-Hill, New York, 1995).
- F. Fournier and J. Rolland, “Optimization of freeform lightpipes for light-emitting-diode projectors,” Appl. Opt. 47(7), 957–966 (2008). [CrossRef] [PubMed]

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