Abstract

The freeform optical system plays a key role in illumination engineering, and several methods have been reported to manage the design of such system. In this paper, an approach to generate the polar-grids based flux transportation mapping for an arbitrarily-shaped target is proposed based on the conventional variable separation method. The source emitting grid is divided along the azimuth angle and the zenith angle respectively under the spherical coordinate system. Then, the target grid is achieved by solving the flux integral equations in polar coordinates using separation of variables method. When establishing the target grid along the polar radius, a strategy based on uniformly scaling down the external contour of the target is introduced. According to the mapping, a smooth freeform surface is then generated using the geometric construction method according to Snell’s law. Finally, an iterative feedback process is adopted to compensate the deterioration of the target distribution caused by surface construction errors and the extension of a real source. Based on this method, a series of freeform lenses are designed for a 1 × 1 mm2 LED source to generate uniform, Gaussian and multiple-rings illumination distributions within different target regions. High-performance optical systems with the light utilization efficiency η over 0.8 and the relative standard deviation (RSD) of the simulated illumination distribution less than 0.1 are obtained simultaneously for all the cases.

© 2015 Optical Society of America

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References

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  1. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. C. Bortz, Nonimaging Optics (Elsevier, 2005), Chap. 7.
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  12. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012).
    [Crossref] [PubMed]
  13. W. A. Parkyn, “Segmented illumination lenses for steplighting and wall-washing,” Proc. SPIE 3779, 363–370 (1999).
    [Crossref]
  14. 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]
  15. K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
    [Crossref]
  16. L. Hongtao, C. Shichao, H. Yanjun, and L. Yi, “A fast feedback method to design easy-molding freeform optical system with uniform illuminance and high light control efficiency,” Opt. Express 21(1), 1258–1269 (2013).
    [Crossref] [PubMed]
  17. K. Wang, S. Liu, F. Chen, Z. Liu, X. Luo, and X. Luo, “Effect of manufacturing defects on optical performance of discontinuous freeform lenses,” Opt. Express 17(7), 5457–5465 (2009).
    [Crossref] [PubMed]
  18. Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010).
    [Crossref] [PubMed]
  19. X. L. Mao, H. T. Li, Y. J. Han, and Y. Luo, “Two-step design method for highly compact three-dimensional freeform optical system for LED surface light source,” Opt. Express 22(Suppl 6), A1491–A1506 (2014).
    [PubMed]
  20. W. C. Situ, Y. J. Han, H. T. Li, and Y. Luo, “Combined feedback method for designing a free-form optical system with complicated illumination patterns for an extended LED source,” Opt. Express 19(S5), A1022–A1030 (2011).
    [Crossref] [PubMed]
  21. S. C. Shen, J. S. Li, and M. C. Huang, “Design a light pattern of multiple concentric circles for LED fishing lamps using Fourier series and an energy mapping method,” Opt. Express 22(11), 13460–13471 (2014).
    [PubMed]

2014 (2)

2013 (3)

2012 (1)

2011 (2)

2010 (2)

2009 (2)

K. Wang, S. Liu, F. Chen, Z. Liu, X. Luo, and X. Luo, “Effect of manufacturing defects on optical performance of discontinuous freeform lenses,” Opt. Express 17(7), 5457–5465 (2009).
[Crossref] [PubMed]

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

2008 (1)

2007 (1)

2005 (1)

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

2004 (1)

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

2002 (1)

1999 (1)

W. A. Parkyn, “Segmented illumination lenses for steplighting and wall-washing,” Proc. SPIE 3779, 363–370 (1999).
[Crossref]

1994 (1)

1972 (1)

J. S. Schruben, “Formulation a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. A 62(12), 1498–1501 (1972).
[Crossref]

Bäuerle, A.

Benítez, P.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Blen, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Bräuer, A.

Bruneton, A.

Canavesi, C.

Cassarly, W. J.

Chaves, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Chen, F.

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

K. Wang, S. Liu, F. Chen, Z. Liu, X. Luo, and X. Luo, “Effect of manufacturing defects on optical performance of discontinuous freeform lenses,” Opt. Express 17(7), 5457–5465 (2009).
[Crossref] [PubMed]

Ding, Y.

Dross, O.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Falicoff, W.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Feng, Z.

Fournier, F. R.

Gu, P. F.

Han, Y.

Han, Y. J.

Hernández, M.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Hongtao, L.

Huang, M. C.

Li, H.

Li, H. F.

Li, H. T.

Li, J. S.

Liu, P.

Liu, S.

K. Wang, S. Liu, F. Chen, Z. Liu, X. Luo, and X. Luo, “Effect of manufacturing defects on optical performance of discontinuous freeform lenses,” Opt. Express 17(7), 5457–5465 (2009).
[Crossref] [PubMed]

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

Liu, X.

Liu, Z.

K. Wang, S. Liu, F. Chen, Z. Liu, X. Luo, and X. Luo, “Effect of manufacturing defects on optical performance of discontinuous freeform lenses,” Opt. Express 17(7), 5457–5465 (2009).
[Crossref] [PubMed]

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

Loosen, P.

Luo, X.

Luo, Y.

Mao, X. L.

Michaelis, D.

Miñano, J. C.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Mohedano, R.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[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 (2005).
[Crossref]

Parkyn, W. A.

W. A. Parkyn, “Segmented illumination lenses for steplighting and wall-washing,” Proc. SPIE 3779, 363–370 (1999).
[Crossref]

Qian, K.

Qin, Z.

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

Ries, H.

Ries, H. R.

Rolland, J. P.

Schreiber, P.

Schruben, J. S.

J. S. Schruben, “Formulation a reflector-design problem for a lighting fixture,” J. Opt. Soc. Am. A 62(12), 1498–1501 (1972).
[Crossref]

Shen, S. C.

Shichao, C.

Situ, W. C.

Stollenwerk, J.

Wang, K.

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

K. Wang, S. Liu, F. Chen, Z. Liu, X. Luo, and X. Luo, “Effect of manufacturing defects on optical performance of discontinuous freeform lenses,” Opt. Express 17(7), 5457–5465 (2009).
[Crossref] [PubMed]

Wang, L.

Wester, R.

Winston, R.

Wu, R. M.

Xu, L.

Yanjun, H.

Yi, L.

Zhang, Y. Q.

Zheng, Z. R.

Appl. Opt. (1)

J. Opt. A, Pure Appl. Opt. (1)

K. Wang, S. Liu, F. Chen, Z. Qin, Z. Liu, and X. Luo, “Freeform LED lens for rectangularly prescribed illumination,” J. Opt. A, Pure Appl. Opt. 11(10), 105501 (2009).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Opt. Express (9)

L. Hongtao, C. Shichao, H. Yanjun, and L. Yi, “A fast feedback method to design easy-molding freeform optical system with uniform illuminance and high light control efficiency,” Opt. Express 21(1), 1258–1269 (2013).
[Crossref] [PubMed]

K. Wang, S. Liu, F. Chen, Z. Liu, X. Luo, and X. Luo, “Effect of manufacturing defects on optical performance of discontinuous freeform lenses,” Opt. Express 17(7), 5457–5465 (2009).
[Crossref] [PubMed]

Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010).
[Crossref] [PubMed]

X. L. Mao, H. T. Li, Y. J. Han, and Y. Luo, “Two-step design method for highly compact three-dimensional freeform optical system for LED surface light source,” Opt. Express 22(Suppl 6), A1491–A1506 (2014).
[PubMed]

W. C. Situ, Y. J. Han, H. T. Li, and Y. Luo, “Combined feedback method for designing a free-form optical system with complicated illumination patterns for an extended LED source,” Opt. Express 19(S5), A1022–A1030 (2011).
[Crossref] [PubMed]

S. C. Shen, J. S. Li, and M. C. Huang, “Design a light pattern of multiple concentric circles for LED fishing lamps using Fourier series and an energy mapping method,” Opt. Express 22(11), 13460–13471 (2014).
[PubMed]

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. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012).
[Crossref] [PubMed]

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]

Opt. Lett. (3)

Proc. SPIE (2)

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

W. A. Parkyn, “Segmented illumination lenses for steplighting and wall-washing,” Proc. SPIE 3779, 363–370 (1999).
[Crossref]

Other (1)

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. C. Bortz, Nonimaging Optics (Elsevier, 2005), Chap. 7.

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Figures (17)

Fig. 1
Fig. 1 Two types of source-target ray mappings.
Fig. 2
Fig. 2 (a) (u, v) coordinate system and (b) (θ, φ) coordinate system.
Fig. 3
Fig. 3 A flow diagram of the design process.
Fig. 4
Fig. 4 Grid division of the source based on spherical coordinates (θ, φ). The black curves (equal-φ arcs) are the grid boundaries of the azimuth angles. The green curves (equal-θ arcs) are the grid boundaries of the zenith angles. The red curve is the circular edge of the source intensity distribution.
Fig. 5
Fig. 5 Grid division of a simply connected target field based on polar coordinates (ρ, γ). The black lines are the grid boundaries of the polar angles. The green curves are the grid boundaries of the polar radii. The red curve is the boundary of the target.
Fig. 6
Fig. 6 Grid division of a doubly connected target field based on polar coordinates (ρ, γ). The black lines are the grid boundaries of the polar angles. The green curves are the grid boundaries of the polar radii. The red curves are respectively the inner and outer boundaries of the target.
Fig. 7
Fig. 7 Geometric configuration of the freeform surface. The black curves are the lens profiles along the zenith angle. The green curves are the lens profiles along the azimuth angle.
Fig. 8
Fig. 8 Design parameters.
Fig. 9
Fig. 9 (a) The equal-flux source grid divided over the unit emitting hemisphere of the source; (b) The source grid projected onto the x-y plane. The black curves are equal-φ curves. The green curves are equal-θ curves. The red curve is the edge of the source intensity distribution.
Fig. 10
Fig. 10 (a)-(d) Equal-flux target grids; (e)-(h) Simulated illuminations without feedback modification; (i)-(l) Simulated illuminations with feedback modification; (m)-(p) Lens models.
Fig. 11
Fig. 11 (a) Equal-flux target grid; (b) Lens model; (c) Simulated illumination; (d) Cross-sections of the illumination at x = 0 (dashed red line) and y = 0 (black line).
Fig. 12
Fig. 12 (a) The desired illumination distribution (normalized) under the divided grid; (b) Lens model; (c) Simulated illumination distribution (normalized); (d) The cross-sections of the simulated distribution (normalized) at x = 0 (red line) and y = 0 (black line), and the corresponding cross-sections of the desired distribution (dashed blue line).
Fig. 13
Fig. 13 (a) Fit the discrete boundary points (the red points) using spline curve (the black curve) under polar coordinate system (ρ, γ); (b) Transform (a) to Cartesian coordinate system (x, y).
Fig. 14
Fig. 14 (a) Equal-flux target grid; (b) Simulated illumination; (c) Top view of the lens; (d) Front view of the lens.
Fig. 15
Fig. 15 (a) Equal-flux target grid; (b) Lens model; (c) Simulated illumination; (d) Cross-sections of the illumination at x = 0 (dashed red line) and y = 0 (black line).
Fig. 16
Fig. 16 (a) The target pattern; (b) The source grid and (c) the target grid are respectively divided corresponding to each ring of the target using the proposed method.
Fig. 17
Fig. 17 (a) Lens model; (b) Simulated illumination; (c) Cross-sections of the illumination at x = 0 (dashed red line) and y = 0 (black line).

Equations (19)

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RSD= 1 N P i=1 N P [ P S (i) P 0 (i) P 0 (i) ] 2 ,
Ω S I(θ)sinθdθdφ = Ω T P(ρ,γ)ρdρdγ ,
φ min φ k+1 dφ θ min θ max I(θ)sinθdθ φ min φ max dφ θ min θ max I(θ)sinθdθ = φ min φ k+1 dφ φ min φ max dφ = j=1 k i=1 M E(i,j) j=1 N i=1 M E(i,j) ,
φ k φ k+1 dφ θ min θ s+1 k+1 I(θ)sinθdθ φ k φ k+1 dφ θ min θ max I(θ)sinθdθ = θ min θ s+1 k+1 I(θ)sinθdθ θ min θ max I(θ)sinθdθ = i=1 s E(i,k) i=1 M E(i,k) ,
ρ SC =f(γ),
0 γ k+1 dγ 0 f(γ) P(ρ,γ)ρdρ 0 2π dγ 0 f(γ) P(ρ,γ)ρdρ = j=1 k i=1 M E(i,j) j=1 N i=1 M E(i,j) ,
γ k γ k+1 dγ 0 ω s+1 k+1 f(γ) P(ρ,γ)ρdρ γ k γ k+1 dγ 0 f(γ) P(ρ,γ)ρdρ = i=1 s E(i,k) i=1 M E(i,k) ρ s+1 k+1 = ω s+1 k+1 f( γ k+1 ) },
{ ρ DC1 = f 1 (γ) ρ DC2 = f 2 (γ) ,
0 γ k+1 dγ f 1 (γ) f 2 (γ) P(ρ,γ)ρdρ 0 2π dγ f 1 (γ) f 2 (γ) P(ρ,γ)ρdρ = j=1 k i=1 M E(i,j) j=1 N i=1 M E(i,j) ,
γ k γ k+1 dγ f 1 (γ) ω s+1 k+1 f 2 (γ) P(ρ,γ)ρdρ γ k γ k+1 dγ f 1 (γ) f 2 (γ) P(ρ,γ)ρdρ = i=1 s E(k,j) i=1 M E(k,j) ρ s+1 k+1 = ω s+1 k+1 f 2 ( γ k+1 ) },
( x s k , y s k )=( ρ s k cos γ k , ρ s k sin γ k ),s=1,2,,M+1,k=1,2,,N+1.
N= n o O n i I | n o O n i I | ,
β k (i,j)= { P 0 (i,j)/[ λ 1 P Sk (i,j)+(1 λ 1 ) P 0 (i,j)] } λ 2 ,i=1,2,,M,j=1,2,,N,
E k (i,j)= Π l=1 k β l (i,j)E(i,j),i=1,2,,M,j=1,2,,N.
| x a | n + | y b | n =1,
ρ SC = ( | cosγ a | n + | sinγ b | n ) 1/n .
ρ SC = R 1 (1sinγ),
P(ρ,γ)= P 0 exp{ 2[ (ρcosγ) 2 a 0 2 + (ρsinγ) 2 b 0 2 ] },
{ ρ DC1 = ( | cosγ a 1 | n 1 + | sinγ b 1 | n 1 ) 1/ n 1 ρ DC2 = ( | cosγ a 2 | n 2 + | sinγ b 2 | n 2 ) 1/ n 2 ,

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