Abstract

Incoherent collimated beam has a wide application, and reshaping the collimated beam with freeform optics has become a popular and challenging topic of noniamging design. In this paper, we address this issue, embedded in three-dimensional space without any symmetry, with a freeform surface from a new perspective. A mathematical model is established for achieving the one-freeform surface design based on the problem of optimal mass transport. A numerical technique for solving this design model is disclosed for the first time, and boundary conditions for balancing light are presented. Besides, some key issues in achieving complex illuminations are addressed, and the influence of caustic surface on this design model is also discussed. Design examples are given to verify these theories. The results show elegance of the design model in tackling complex illumination tasks. The conclusions obtained in this paper can be generalized to achieve LED illumination and tackle multiple freeform surfaces illumination design.

© 2013 Optical Society of America

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References

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  1. T. Uemura, US Patent No. 2008/006, 254, 1 A1 (2008).
  2. H. Tanaka, US Patent No. 2009/000, 267, 1 A1 (2009).
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    [Crossref]
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  6. A. Bruneton, A. B¨auerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707, 816707-9 (2011).
    [Crossref]
  7. L. Wang, K. Y. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007).
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    [Crossref] [PubMed]
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  13. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
    [Crossref] [PubMed]
  14. X. J. Wang, “On the design of a reflector antenna,” Inverse Probl. 12(3), 351–375 (1996).
    [Crossref]
  15. S. A. Kochengin, V. I. Oliker, and O. Tempski, “On the design of reflectors with prespecified distribution of virtual sources and intensities,” Inverse Probl. 14(3), 661–678 (1998).
    [Crossref]
  16. 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]
  17. L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 39–49 (2008).
    [Crossref]
  18. R. M. Wu, L. Xu, P. Liu, Y. Q. Zhang, Z. R. Zheng, H. F. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
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    [Crossref]

2013 (1)

2012 (1)

2011 (3)

2010 (1)

2008 (3)

2007 (1)

2004 (1)

F. M. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004).
[Crossref]

2003 (1)

T. Glimm and V. I. 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)

S. A. Kochengin, V. I. Oliker, and O. Tempski, “On the design of reflectors with prespecified distribution of virtual sources and intensities,” Inverse Probl. 14(3), 661–678 (1998).
[Crossref]

1996 (1)

X. J. Wang, “On the design of a reflector antenna,” Inverse Probl. 12(3), 351–375 (1996).
[Crossref]

1972 (1)

1953 (1)

L. Nirenberg, “On nonlinear elliptic partial differential equations and hölder continuity,” Commun. Pure Appl. Math. 6(1), 103–156 (1953).
[Crossref]

B¨auerle, A.

A. Bruneton, A. B¨auerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707, 816707-9 (2011).
[Crossref]

Bruneton, A.

A. Bruneton, A. B¨auerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707, 816707-9 (2011).
[Crossref]

Caffarelli, L. A.

L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 39–49 (2008).
[Crossref]

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.

Ding, Y.

Fournier, F.

Fournier, F. R.

Glimm, T.

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

Gu, P. F.

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]

S. A. Kochengin, V. I. Oliker, and O. Tempski, “On the design of reflectors with prespecified distribution of virtual sources and intensities,” Inverse Probl. 14(3), 661–678 (1998).
[Crossref]

Li, H.

Li, H. F.

Liu, P.

Liu, X.

Loosen, P.

A. Bruneton, A. B¨auerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707, 816707-9 (2011).
[Crossref]

Luo, Y.

Muschaweck, J.

Nirenberg, L.

L. Nirenberg, “On nonlinear elliptic partial differential equations and hölder continuity,” Commun. Pure Appl. Math. 6(1), 103–156 (1953).
[Crossref]

Oliker, V. I.

L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 39–49 (2008).
[Crossref]

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

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]

S. A. Kochengin, V. I. Oliker, and O. Tempski, “On the design of reflectors with prespecified distribution of virtual sources and intensities,” Inverse Probl. 14(3), 661–678 (1998).
[Crossref]

Qian, K. Y.

Ries, H.

Rolland, J.

Rolland, J. P.

Schellenberg, F. M.

F. M. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004).
[Crossref]

Schruben, J. S.

Tempski, O.

S. A. Kochengin, V. I. Oliker, and O. Tempski, “On the design of reflectors with prespecified distribution of virtual sources and intensities,” Inverse Probl. 14(3), 661–678 (1998).
[Crossref]

Wang, L.

Wang, X. J.

X. J. Wang, “On the design of a reflector antenna,” Inverse Probl. 12(3), 351–375 (1996).
[Crossref]

Wester, R.

A. Bruneton, A. B¨auerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707, 816707-9 (2011).
[Crossref]

Wu, R. M.

Xu, L.

Zhang, Y. Q.

Zheng, Z.

Zheng, Z. R.

Appl. Opt. (5)

Commun. Pure Appl. Math. (1)

L. Nirenberg, “On nonlinear elliptic partial differential equations and hölder continuity,” Commun. Pure Appl. Math. 6(1), 103–156 (1953).
[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]

Inverse Probl. (2)

X. J. Wang, “On the design of a reflector antenna,” Inverse Probl. 12(3), 351–375 (1996).
[Crossref]

S. A. Kochengin, V. I. Oliker, and O. Tempski, “On the design of reflectors with prespecified distribution of virtual sources and intensities,” Inverse Probl. 14(3), 661–678 (1998).
[Crossref]

J. Math. Sci. (2)

L. A. Caffarelli and V. I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 39–49 (2008).
[Crossref]

T. Glimm and V. I. 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. (1)

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

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (2)

F. M. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE 5377, 1–20 (2004).
[Crossref]

A. Bruneton, A. B¨auerle, P. Loosen, and R. Wester, “Freeform lens for an efficient wall washer,” Proc. SPIE 8167, 816707, 816707-9 (2011).
[Crossref]

Other (2)

T. Uemura, US Patent No. 2008/006, 254, 1 A1 (2008).

H. Tanaka, US Patent No. 2009/000, 267, 1 A1 (2009).

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Figures (16)

Fig. 1
Fig. 1 Schematic illustration of the collimated beam shaping process.

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Fig. 2
Fig. 2 The geometrical design layout of the freeform surface.

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Fig. 3
Fig. 3 The discretization of the domain S1.

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Fig. 4
Fig. 4 (a) The 9-point finite difference scheme for the interior points and (b) the forward (or backward) difference approximation with second-order error for the boundary points.

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Fig. 5
Fig. 5 The target illumination pattern.

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Fig. 6
Fig. 6 (a) There is no caustic surface between the freeform lens and the target plane. (b) There is a caustic surface between the freeform lens and the target plane.

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Fig. 7
Fig. 7 (a) The model of the freeform lens and (b) the pseudocolor plot of the irradiance distribution obtained from the simulation for the first optical configuration.

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Fig. 8
Fig. 8 Irradiance distribution along the line y = −3.4 mm. This figure shows the influence of the spread angle of the collimated beam on the irradiance distribution. The red solid line, the black dashed line and the blue dot line represent that the spread angle of the collimated beam is 0 mrad, 3 mrad and 5 mrad, respectively. The irradiance ratio is almost 4 (the letters) to 1 (background) to zero (outside). This design has a large tolerance to the spread angle.

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Fig. 9
Fig. 9 The change of the position of the boundary ray (x = 2,y = 2) on the target plane. In the initial design, the position is (49.871,25.1194). Its position is automatically adjusted to meet the design requirements during the iterative process by the numerical technique presented above.

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Fig. 10
Fig. 10 The influence of lighting distance on the illumination. The lighting distance (a) tz = 200mm and (b) tz = 400mm.

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Fig. 11
Fig. 11 The influence of the spread angle on the illumination pattern. The spread angle is (a) 3 mrad; (b) 5 mrad; (c) 10 mrad; (d) 15 mrad. This design has a large tolerance to spread angle.

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Fig. 12
Fig. 12 (a) m = n = 39, and h1 = h2 = 0.1025mm; (b) m = n = 59, and h1 = h2 = 0.0678 mm. The irradiance curve represents the irradiance distribution along the line y = −3.4 mm. The optical performance of a design is strongly determined by the spacing.

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Fig. 13
Fig. 13 The irradiance ratio is (a) 2:1, (b) 3:1, (c) 5:1 and (d) 6:1, respectively. The irradiance curve represents the irradiance distribution along the line y = −3.4 mm.

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Fig. 14
Fig. 14 (a) The model of the freeform lens and (b) the pseudocolor plot of the irradiance distribution obtained from simulation for the second optical configuration.

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Fig. 15
Fig. 15 The influence of the lighting distance on the illumination for the second configuration. The lighting distance (a) tz = 200mm and (b) tz = 400mm.

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Fig. 16
Fig. 16 The influence of the spread angle on the illumination pattern for the second configuration. The spread angle is (a) 3 mrad; (b) 5 mrad; (c) 10 mrad; (d) 15 mrad. There are few differences in the results shown in Figs. 14(b), 16(a) and 16(b). The illumination pattern becomes a little blurred, when the spread angle increases to 15 mrad.

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Tables (1)

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Table 1 Design Parameters for the Design Examples

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Equations (22)

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N= 1 z x 2 + z y 2 +1 (− z x ,− z y ,1)
n o O= n i I+ P 1 N
P 1 = n o (1− n i 2 n o 2 )( z x 2 + z y 2 )+1 − n i z x 2 + z y 2 +1
O= 1 n o ( z x 2 + z y 2 +1 ) ( O x , O y , O z ) where, { O x =− z x [ n o a( z x 2 + z y 2 )+1 − n i ] O y =− z y [ n o a( z x 2 + z y 2 )+1 − n i ] O z = n i ( z x 2 + z y 2 )+ n o a( z x 2 + z y 2 )+1 and a=1− n i 2 / n o 2
t x =x−( z− t z ) O x O z t y =y−( z− t z ) O y O z
t x = t x (x,y,z, z x , z y ) t y = t y (x,y,z, z x , z y )
d t x d t y =|J(T)|dxdy, |J(T)|=| ∂ t x ∂x ∂ t x ∂y ∂ t y ∂x ∂ t y ∂y |
|J(T)|E( t x (x,y), t y (x,y))=I(x,y)
∬ S 1 E( t x ( x,y ), t y ( x,y ) ) |J( T )|dxdy= ∬ S 1 I( x,y )dxdy
∬ S 2 E( t x , t y ) d t x d t y = ∬ S 1 I(x,y)dxdy
A 1 ( z xx z yy − z xy 2 )+ A 2 z xx + A 3 z yy + A 4 z xy + A 5 =0 where, b= a( z x 2 + z y 2 )+1 , A 1 = ( z− t z ) 2 n o b [ 1+( z x 2 + z y 2 ) ] ( n o b− n i ) 2 [ n o b+ n i ( z x 2 + z y 2 ) ] 3 , A 2 =( z− t z ) ( n o b− n i )[ n o b( z y 2 +1 )− n i z x 2 ]+ n o n i a b z x 2 ( z x 2 + z y 2 +1 ) [ n o b+ n i ( z x 2 + z y 2 ) ] 2 , A 3 =( z− t z ) ( n o b− n i )[ n o b( z x 2 +1 )− n i z y 2 ]+ n o n i a b z y 2 ( z x 2 + z y 2 +1 ) [ n o b+ n i ( z x 2 + z y 2 ) ] 2 , A 4 =2( z− t z ) z x z y [ n o n i a b ( z x 2 + z y 2 +1 )−( n o 2 b 2 − n i 2 ) ] [ n o b+ n i ( z x 2 + z y 2 ) ] 2 , A 5 = n o b( z x 2 + z y 2 +1 ) n o b+ n i ( z x 2 + z y 2 ) − I( x,y ) E( t x , t y )
{ t x = t x (x,y,z, z x , z y ) t y = t y (x,y,z, z x , z y ) :∂ S 1 →∂ S 2
{ A 1 ( z xx z yy − z xy 2 )+ A 2 z xx + A 3 z yy + A 4 z xy + A 5 =0 BC: { t x = t x (x,y,z, z x , z y ) t y = t y (x,y,z, z x , z y ) :∂ S 1 →∂ S 2
( t x − C x ) 2 a 2 + ( t y − C y ) 2 b 2 =1
z x = z i+1,j − z i−1,j 2 h 1 , z y = z i,j+1 − z i,j−1 2 h 2 , z xx = z i+1,j −2 z i,j + z i−1,j h 1 2 z yy = z i,j+1 −2 z i,j + z i,j−1 h 2 2 , z xy = z i+1,j+1 − z i+1,j−1 − z i−1,j+1 + z i−1,j−1 4 h 1 h 2
z x = 3 z m,j −4 z m−1,j + z m−2,j 2 h 1 z y = z m,j+1 − z m,j−1 2 h 2
F( X )=0
D= A 1 ( z xx z yy − z xy 2 )+ A 2 z xx + A 3 z yy + A 4 z xy + A 6
DE( t x , t y )=I( x,y )
Q=4 D z xx D z yy − D z xy 2 >0 where, D z xx = ∂D / ∂ z xx , D z yy = ∂D / ∂ z yy and D z xy = ∂D / ∂ z xy .
D z xx = A 1 z yy + A 2 , D z yy = A 1 z xx + A 3 , D z xy =−2 A 1 z xy + A 4
Q=4 A 1 D

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