Abstract

The Monge–Ampère (MA) equation arising in illumination design is highly nonlinear so that the convergence of the MA method is strongly determined by the initial design. We address the initial design of the MA method in this paper with the L2 Monge-Kantorovich (LMK) theory. An efficient approach is proposed to find the optimal mapping of the LMK problem. The characteristics of the new approach are introduced and the limitations of the LMK theory in illumination design are presented. Three examples, including the beam shaping of collimated beam and point light source, are given to illustrate the potential benefits of the LMK theory in the initial design. The results show the MA method converges more stably and faster with the application of the LMK theory in the initial design.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  14. R. M. Wu, K. Li, P. Liu, Z. R. Zheng, H. F. Li, and X. Liu, “Conceptual design of dedicated road lighting for city park and housing estate,” Appl. Opt. 52(21), 5272–5278 (2013).
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    [Crossref]
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    [Crossref]
  17. 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]
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    [Crossref]
  20. B. D. Froese, “A numerical method for the elliptic Monge-Ampere equation with transport boundary conditions,” SIAM J. Sci. Comput. 34(3), A1432–A1459 (2012).
    [Crossref]
  21. M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
    [Crossref]
  22. A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “Limitations of the ray mapping approach in freeform optics design,” Opt. Lett. 38(11), 1945–1947 (2013).
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    [Crossref]

2013 (6)

2012 (1)

B. D. Froese, “A numerical method for the elliptic Monge-Ampere equation with transport boundary conditions,” SIAM J. Sci. Comput. 34(3), A1432–A1459 (2012).
[Crossref]

2011 (1)

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

2010 (1)

2008 (3)

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]

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

G. L. Delzanno, L. Chacon, J. M. Finn, Y. Chung, and G. Lapenta, “An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization,” J. Comput. Phys. 227(23), 9841–9864 (2008).
[Crossref]

2007 (2)

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

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
[Crossref]

2006 (1)

E. J. Dean and R. Glowinski, “Numerical methods for fully nonlinear elliptic equations of the Monge Ampere type,” Comput. Methods Appl. Mech. Eng. 195(13–16), 1344–1386 (2006).
[Crossref]

2004 (3)

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]

T. L. R. Davenport, T. A. Hough, and W. J. Cassarly, “Optimization for illumination systems: the next level of design,” Proc. SPIE 5456, 81–90 (2004).
[Crossref]

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60(3), 225–240 (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)

Angenent, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60(3), 225–240 (2004).
[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]

Bortz, J.

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
[Crossref]

Bruneton, A.

Cassarly, W. J.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freeform reflector generation using source-target maps,” Opt. Express 18(5), 5295–5304 (2010).
[Crossref] [PubMed]

T. L. R. Davenport, T. A. Hough, and W. J. Cassarly, “Optimization for illumination systems: the next level of design,” Proc. SPIE 5456, 81–90 (2004).
[Crossref]

Chacon, L.

G. L. Delzanno, L. Chacon, J. M. Finn, Y. Chung, and G. Lapenta, “An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization,” J. Comput. Phys. 227(23), 9841–9864 (2008).
[Crossref]

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]

Chung, Y.

G. L. Delzanno, L. Chacon, J. M. Finn, Y. Chung, and G. Lapenta, “An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization,” J. Comput. Phys. 227(23), 9841–9864 (2008).
[Crossref]

Davenport, T. L. R.

T. L. R. Davenport, T. A. Hough, and W. J. Cassarly, “Optimization for illumination systems: the next level of design,” Proc. SPIE 5456, 81–90 (2004).
[Crossref]

Dean, E. J.

E. J. Dean and R. Glowinski, “Numerical methods for fully nonlinear elliptic equations of the Monge Ampere type,” Comput. Methods Appl. Mech. Eng. 195(13–16), 1344–1386 (2006).
[Crossref]

Delzanno, G. L.

G. L. Delzanno, L. Chacon, J. M. Finn, Y. Chung, and G. Lapenta, “An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization,” J. Comput. Phys. 227(23), 9841–9864 (2008).
[Crossref]

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. X.

Finn, J. M.

G. L. Delzanno, L. Chacon, J. M. Finn, Y. Chung, and G. Lapenta, “An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization,” J. Comput. Phys. 227(23), 9841–9864 (2008).
[Crossref]

Fournier, F.

Fournier, F. R.

Froese, B. D.

B. D. Froese, “A numerical method for the elliptic Monge-Ampere equation with transport boundary conditions,” SIAM J. Sci. Comput. 34(3), A1432–A1459 (2012).
[Crossref]

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]

Glowinski, R.

E. J. Dean and R. Glowinski, “Numerical methods for fully nonlinear elliptic equations of the Monge Ampere type,” Comput. Methods Appl. Mech. Eng. 195(13–16), 1344–1386 (2006).
[Crossref]

Gong, M.

Gu, P. F.

Haker, S.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60(3), 225–240 (2004).
[Crossref]

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]

Hough, T. A.

T. L. R. Davenport, T. A. Hough, and W. J. Cassarly, “Optimization for illumination systems: the next level of design,” Proc. SPIE 5456, 81–90 (2004).
[Crossref]

Huang, L.

Jin, G.

Lapenta, G.

G. L. Delzanno, L. Chacon, J. M. Finn, Y. Chung, and G. Lapenta, “An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization,” J. Comput. Phys. 227(23), 9841–9864 (2008).
[Crossref]

Li, H. F.

Li, K.

Liu, P.

Liu, X.

Loosen, P.

Luo, Y.

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. I.

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]

Qian, K. Y.

Ries, H.

Rolland, J.

Rolland, J. P.

Russell, R. D.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Shatz, N.

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
[Crossref]

Stollenwerk, J.

Sulman, M. M.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Tannenbaum, A.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60(3), 225–240 (2004).
[Crossref]

Wang, L.

Wester, R.

Williams, J. F.

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Wu, R. M.

Xu, L.

Zhang, Y. Q.

Zheng, Z. R.

Zhu, L.

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60(3), 225–240 (2004).
[Crossref]

Appl. Numer. Math. (1)

M. M. Sulman, J. F. Williams, and R. D. Russell, “An efficient approach for the numerical solution of Monge-Ampère equation,” Appl. Numer. Math. 61(3), 298–307 (2011).
[Crossref]

Appl. Opt. (3)

Comput. Methods Appl. Mech. Eng. (1)

E. J. Dean and R. Glowinski, “Numerical methods for fully nonlinear elliptic equations of the Monge Ampere type,” Comput. Methods Appl. Mech. Eng. 195(13–16), 1344–1386 (2006).
[Crossref]

Int. J. Comput. Vis. (1)

S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal mass transport for registration and warping,” Int. J. Comput. Vis. 60(3), 225–240 (2004).
[Crossref]

J. Comput. Phys. (1)

G. L. Delzanno, L. Chacon, J. M. Finn, Y. Chung, and G. Lapenta, “An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization,” J. Comput. Phys. 227(23), 9841–9864 (2008).
[Crossref]

J. Math. Sci. (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]

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

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 (5)

Opt. Lett. (2)

Proc. SPIE (2)

J. Bortz and N. Shatz, “Iterative generalized functional method of nonimaging optical design,” Proc. SPIE 6670, 66700A (2007).
[Crossref]

T. L. R. Davenport, T. A. Hough, and W. J. Cassarly, “Optimization for illumination systems: the next level of design,” Proc. SPIE 5456, 81–90 (2004).
[Crossref]

SIAM J. Sci. Comput. (1)

B. D. Froese, “A numerical method for the elliptic Monge-Ampere equation with transport boundary conditions,” SIAM J. Sci. Comput. 34(3), A1432–A1459 (2012).
[Crossref]

Other (3)

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, 2000), pp. 193–224.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier, 2005).

N. S. Trudinger and X.-J. Wang, “The Monge-Ampère equations and its geometric applications,” in Handbook of Geometric Analysis (International Press, 2008), pp. 467–524.

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Figures (16)

Fig. 1
Fig. 1 Some practical considerations for the convergence of the MA method. x0 and x1, respectively, represent the initial value and the x-intercept obtained from the first iteration. (a) The iteration diverges at the first iteration. In this case, x1 may deteriorate the optical performance of the surface. For example, one will uaually encounter complications associated with singularities of the freeform surface, and the method cannot be iterated (This will be demonstrated in the first design example). (b) Assume F1(x0) = F1(x1) and |F1’(x0)| = |F1’(x1)|. The iteration will go into an infinite loop in this case. (c) x0 may be a good initial guess of a pure mathematical problem in this case. The convergence of the MA method, however, still depends on the optical performance of the freeform surface. For a freeform lens design, the MA method can converge successfully only and if only the total internal reflection will not take place on the freeform surface in the initial design (This will be demonstrated in the second design example).
Fig. 2
Fig. 2 Define the initial value for the LMK problem.
Fig. 3
Fig. 3 Comparison between the new approach and one kind of method which relies on the gradient of time. (a) Convergence of the new approach, (b) a gray-scale plot of |F2| on Ω0 and (c) convergence of the Sulman’s method.
Fig. 4
Fig. 4 (a) Influence of spacing on the LMK problem. (b) Convergence of the new approach.
Fig. 5
Fig. 5 (a) Some discrete points predefined on Ω02, and the distribution of these discrete points obtained from (b) the initial design, (c) the first iteration and (d) the second iteration.
Fig. 6
Fig. 6 Collimated beam shaping with (a) freeform reflector and (b) freeform lens. (c) The beam shaping of a point source. l denotes the distance between the source and the target plane.
Fig. 7
Fig. 7 (a) The optimal mapping of the LMK problem, and the mappings (b) obtained from the reflector designed by the LMK and (c) the reflector designed by the MA method.
Fig. 8
Fig. 8 The illumination patterns obtained from (a) the LMK and (b) the MA method. (c) The difference between the intercept point Tai of the i-th ray and its target point Tti.
Fig. 9
Fig. 9 The relationship between the maximum Disti and the lighting distance for (a) the MA method and (b) the LMK.
Fig. 10
Fig. 10 (a) The optimal mapping of the LMK problem, and the mappings (b) obtained from the lens designed by the LMK and (c) the lens designed by the MA method.
Fig. 11
Fig. 11 The illumination patterns obtained from (a) the LMK and (b) the MA method. (c) Design flow charts of SMA and LMA methods.
Fig. 12
Fig. 12 The relationship between the maximum Disti and the lighting distance for (a) the MA method and (b) the LMK.
Fig. 13
Fig. 13 (a) The convergence of the LMK problem and (b) the illumination pattern obtained from the LMK. (c) The rapid convergence of the LMA method and the obtained smooth reflector, and (d) the illumination pattern obtained from the LMA method. (e) The illumination pattern obtained from a variable separation mapping, (f) the divergence of the SMA method and the singularities of the freeform reflector. Five million rays are traced.
Fig. 14
Fig. 14 (a) The illumination pattern obtained from the LMK, and (b) the irradiance curves along the line x = 0mm. (c) The rapid convergence of the LMA method and (d) the illumination pattern obtained from the LMA method. (e) A variable separation mapping which is not the optimal mapping of the LMK problem, (f) the pattern produced by the initial design with five million rays traced and (g) illustration of the total internal reflection taking place on the freeform surface. The unit vector of the incident ray depicted in (g) is (0.8480,0,0.5299) .
Fig. 15
Fig. 15 (a) The target illumination pattern, and the illumination patterns obtained from (b) the LMK, (c) the LMA method and (d) the SMA method. Five million rays are traced.
Fig. 16
Fig. 16 Illustration of the convergence of the LMA method and the SMA method at (a) l = 8.2mm and (b) l = 10mm.

Tables (2)

Tables Icon

Table 1 Design parameters (unit: millimeter).

Tables Icon

Table 2 Design parameters (unit: millimeter).

Equations (16)

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{ A 1 ( z x x z y y z x y 2 ) + A 2 z x x + A 3 z y y + A 4 z x y + A 5 = 0 B C : { 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
Ω 0 ρ 0 ( ξ ) dξ= Ω 1 ρ 1 ( η ) dη
det( ϕ( ξ ) ) ρ 1 ( ϕ( ξ ) )= ρ 0 ( ξ )
C( ϕ )= R d | ϕ( ξ )ξ | 2 ρ 0 ( ξ )dξ
ρ 1 ( w( ξ ) )det 2 w( ξ )= ρ 0 ( ξ )
log [ ρ 1 ( w ( ξ ) ) det 2 w ( ξ ) ρ 0 ( ξ ) + 1 ] = 0
f ( w ( ξ ) ) = 0 ξ Ω 0
{ log [ ρ 1 ( w ( ξ ) ) det 2 w ( ξ ) ρ 0 ( ξ ) + 1 ] = 0 ξ Ω ¯ 0 B C : f ( w ( ξ ) ) = 0 ξ Ω 0
F 2 ( Y ) = 0
F 2 ( Y k ) + F 2 ( Y k ) ( Y k + 1 Y k ) = 0
Y 0 = 1 2 [ X ¯ min + x x min x max x min ( X ¯ max X ¯ min ) ] 2 x max x min X ¯ max X ¯ min + 1 2 [ Y ¯ min + y y min y max y min ( Y ¯ max Y ¯ min ) ] 2 × y max y min Y ¯ max Y ¯ min
{ ρ 0 ( x,y )=exp[ 2( x 2 + y 2 ) / 0.5 2 ], ξ=( x,y ) Ω 0 = [ 0.5,0.5 ] 2 ρ 1 ( t x , t y )= I 0 , η=( t x , t y ) Ω 1 = [ 2,2 ] 2
{ ρ 0 ( x , y ) = { exp [ 2 ( x 2 + y 2 ) / 0.5 2 ] , if x 2 + y 2 0.5 2 0 , elsewise , ξ = ( x , y ) Ω 0 = [ 0.5 , 0.5 ] 2 ρ 1 ( t x , t y ) = I 0 , η = ( t x , t y ) Ω 1
Dis t i = ( t x i T x i ) 2 + ( t y i T y i ) 2 + ( t z i T z i ) 2
E ( x , y ) = C 3 + cos [ 2 π ( x 0.7 ) 2 + ( y 0.4 ) 2 ]
R M S = 1 M i = 1 M ( E a i E t i E t i ) 2

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