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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References

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  1. F. Fournier and J. Rolland, “Design methodology for high brightness projectors,” J. Disp. Technol. 4(1), 86–91 (2008).
    [CrossRef]
  2. L. B. W. Jolley, J. M. Waldram, and G. H. Wilson, The theory and design of illuminating engineering equipment (Chapman & Hall, London, 1930).
  3. W. B. Elmer, The optical design of reflectors, 2d ed. (Wiley, New York, 1980).
  4. V. I. Oliker, “Mathematical aspects of design of beam shaping surfaces in geometrical optics,” in Trends in Nonlinear Analysis (2002), pp. 191–222.
  5. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
    [CrossRef]
  6. 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.
  7. 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]
  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]
  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]
  10. L. Noakes, and R. Kozera, “2D leapfrog algorithm for optimal surface reconstruction,” in Vision Geometry VIII, (SPIE, 1999), 317–328.
  11. 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.
  12. W. J. Cassarly, “Nonimaging Optics: Concentration and Illumination,” in Handbook of optics, 2nd ed. (McGraw-Hill, New York, 1995).
  13. F. Fournier and J. Rolland, “Optimization of freeform lightpipes for light-emitting-diode projectors,” Appl. Opt. 47(7), 957–966 (2008).
    [CrossRef] [PubMed]

2008 (3)

2007 (1)

2002 (1)

1988 (1)

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]

Chellappa, R.

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]

Ding, Y.

Fournier, F.

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

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

Frankot, R.

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]

Gu, P. F.

Liu, X.

Luo, Y.

Muschaweck, J.

Qian, K.

Ries, H.

Rolland, J.

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

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

Wang, L.

Zheng, Z. R.

Appl. Opt. (2)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

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]

J. Disp. Technol. (1)

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

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

Opt. Express (1)

Other (7)

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.

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).

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.

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Figures (10)

Fig. 1
Fig. 1

Source flux is mapped onto the target by matching equi-flux grids. In this example, the solid angle in red contains the same flux than the red area on the target. The equi-flux grid on the right was derived analytically based on the radial symmetry of the system but does not satisfy the integrability condition and thus leads to a discontinuous reflector surface. The equi-flux grid on the left also corresponds to a uniform square target, but it fulfills the integrability condition and can therefore be achieved with a smooth continuous surface.

Fig. 2
Fig. 2

Magnitude of η (residual projected curl) for the two target maps presented in Fig. 1. η is plotted as a function of the target map location. The analytical target map on the right results in large residual curl as we move away from the axis. In comparison, the alternate mapping on the left minimizes η over the entire target and can therefore be achieved with a smooth continuous reflector surface.

Fig. 3
Fig. 3

(a) Target points are mapped to the center of their corresponding facets. In order to preserve edges, points at the edge of the target are mapped to the middle of the outer facet instead of the facet center. (b) The discrete mapping relationships x = f(θ,ϕ) and y = g(θ,ϕ) can then be approximated by a surface in order to get a continuous mapping function. (c) The generated map plots the contours corresponding to constant θ and constant ϕ values.

Fig. 4
Fig. 4

Target maps for three different system geometries: square, rectangle and off-axis square. In all cases the target is uniform, the point source is isotropic, and the collection angle of the reflector is a hemisphere.

Fig. 5
Fig. 5

Target maps for reflector facets with three different azimuth angles ϕ (0°, 22.5° and 45°) and a uniform square target. In all cases the elevation θ is equal to 60° and the collection angle subtended by each facet is 20° by 20°. 225 ellipsoids were used to generate the maps.

Fig. 6
Fig. 6

Illuminance distributions produced by smooth reflectors obtained by direct interpolation of the initial reflectors made of ellipsoid facets, for the geometry shown in Fig. 4a. We use the same set of points that we used to generate maps. Artifacts typically remain at the edges, even for relatively large number of target points.

Fig. 7
Fig. 7

Overview of the reflector generation process. The original desired target distribution is discretized into a small number of target points and a corresponding reflector shape is generated using the Oliker algorithm. A continuous target map is then created from the finite set of target points and their corresponding ellipsoid facets. Finally, a smooth reflector shape is generated by numerical integration along the contour lines of the map.

Fig. 8
Fig. 8

Reflectors generated from maps. Below each reflector we show the target illuminance and a cross-section of the illuminance plot at y = 0. (a) Uniform 18m × 18m square target at z = 9m. Light is collected over a hemisphere and the reflector diameter is 30cm. (b) Uniform 36m × 18m rectangular target at z = 9m. Light is collected over a hemisphere and the reflector diameter is 30cm. (c) Uniform 1m × 1m square target at z = 3m. The collection angle is 20° × 20° centered on θ = 60° and ϕ = 45°. The facet size is 6cm × 5cm.

Fig. 9
Fig. 9

Target non-uniformity and relative computation time vs. number of ellipsoids, for the uniform square target example shown in Fig. 8a. The statistical noise for these simulations is 0.7%. Computation time is normalized to the time it takes to generate 400 ellipsoids (10 seconds with a 2.5GHz Intel quad core computer). A performance plateau is reached at about 600 ellipsoids. Using an initial reflector with more ellipsoids would increase computation time without any significant performance improvement.

Fig. 10
Fig. 10

Sag difference between an initial reflector surface made of 149 ellipsoids and the smooth reflector surface generated from its corresponding map, shown in Fig. 7a.

Equations (6)

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I S ( θ , ϕ ) sin θ d θ d ϕ = E T ( x , y ) d x d y
( x , y ) = ( f ( θ , ϕ ) , g ( θ , ϕ ) )
η = N ( × N ) = 0
C N d l = 0
C N d l = S ( × N ) N d s = 0
N ( × N ) = 0

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