Abstract

For the problem of point source forming prescribed irradiance, a new, to the best of our knowledge, method-variable separation mapping method is presented, which establishes separately the correspondence between variables on the light source and the target plane. The role played by the optical surfaces is then to redirect the light rays to their corresponding target points. The surface of the lens is determined by first calculating the surface points and then their normal vectors. Considering that normal deviations are produced in the surface construction process, a normal deviation control method is also presented to restrict the deviation. With this normal deviation control method, discontinuities are introduced onto the lens surface. From these mapping and normal control methods, a fast and efficient algorithm has been developed for several prescribed irradiance problems with simple nonrotational shape of the illuminated region.

© 2007 Optical Society of America

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References

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  1. R. Winston, J. C. Miñano, and P. Benítez, eds., with contributions by N.Shatz and J.C.Bortz, eds., Nonimaging Optics (Elsevier, 2005).
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  3. J. S. Schruben, "Formulation a reflector-design problem for a lighting fixture," J. Opt. Soc. Am. A 6, 1498-1501 (1972).
  4. H. Ries and J. A. Muschaweck, "Tailoring freeform lenses for illuminations, in Novel Optical Systems Design and Optimization, Proc. SPIE 4442, 43-50 (2001).
    [CrossRef]
  5. A. Timinger, J. Muschaweck, and H. Ries, "Designing tailored free-form surfaces for general illumination," in Design of Efficient Illumination Systems, Proc. SPIE 5186, 128-132 (2003).
    [CrossRef]
  6. V. Oliker, "Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties," in Nonimaging Optics and Efficient Illumination Systems II, Proc. SPIE 594207 (2005).
    [CrossRef]
  7. L. Caffarelli, S. Kochengin, and V. Oliker, "On the numerical solution of the problem of reflector design with given far-field scattering data," Contemp. Math. 226, 13-32 (1999).
  8. X.-J. Wang, "On design of a reflector antenna II," Calculus Var. Partial Differ. Equ. 20, 329-341 (2004).
    [CrossRef]
  9. W. A. Parkyn, "The design of illumination lenses via extrinsic differential geometry," in International Optical Design Conference, Proc. SPIE 3482, 191-193 (1998).
  10. 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, 1489-1502 (2004).
    [CrossRef]
  11. L. Piegl and W. Tiller, The NURBS Book, 2nd ed. (Springer-Verlag, 1997).
  12. G. Farin, J. Hoschek, and M. Kim, Handbook of Computer Aided Geometric Design (Elsevier, 2002).

2005 (1)

V. Oliker, "Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties," in Nonimaging Optics and Efficient Illumination Systems II, Proc. SPIE 594207 (2005).
[CrossRef]

2004 (2)

X.-J. Wang, "On design of a reflector antenna II," Calculus Var. Partial Differ. Equ. 20, 329-341 (2004).
[CrossRef]

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, 1489-1502 (2004).
[CrossRef]

2003 (1)

A. Timinger, J. Muschaweck, and H. Ries, "Designing tailored free-form surfaces for general illumination," in Design of Efficient Illumination Systems, Proc. SPIE 5186, 128-132 (2003).
[CrossRef]

2001 (1)

H. Ries and J. A. Muschaweck, "Tailoring freeform lenses for illuminations, in Novel Optical Systems Design and Optimization, Proc. SPIE 4442, 43-50 (2001).
[CrossRef]

1999 (1)

L. Caffarelli, S. Kochengin, and V. Oliker, "On the numerical solution of the problem of reflector design with given far-field scattering data," Contemp. Math. 226, 13-32 (1999).

1998 (1)

W. A. Parkyn, "The design of illumination lenses via extrinsic differential geometry," in International Optical Design Conference, Proc. SPIE 3482, 191-193 (1998).

1972 (1)

J. S. Schruben, "Formulation a reflector-design problem for a lighting fixture," J. Opt. Soc. Am. A 6, 1498-1501 (1972).

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, 1489-1502 (2004).
[CrossRef]

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

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, 1489-1502 (2004).
[CrossRef]

Caffarelli, L.

L. Caffarelli, S. Kochengin, and V. Oliker, "On the numerical solution of the problem of reflector design with given far-field scattering data," Contemp. Math. 226, 13-32 (1999).

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, 1489-1502 (2004).
[CrossRef]

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, 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, 1489-1502 (2004).
[CrossRef]

Farin, G.

G. Farin, J. Hoschek, and M. Kim, Handbook of Computer Aided Geometric Design (Elsevier, 2002).

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, 1489-1502 (2004).
[CrossRef]

Hoschek, J.

G. Farin, J. Hoschek, and M. Kim, Handbook of Computer Aided Geometric Design (Elsevier, 2002).

Kochengin, S.

L. Caffarelli, S. Kochengin, and V. Oliker, "On the numerical solution of the problem of reflector design with given far-field scattering data," Contemp. Math. 226, 13-32 (1999).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

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, 1489-1502 (2004).
[CrossRef]

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

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, 1489-1502 (2004).
[CrossRef]

Muschaweck, J.

A. Timinger, J. Muschaweck, and H. Ries, "Designing tailored free-form surfaces for general illumination," in Design of Efficient Illumination Systems, Proc. SPIE 5186, 128-132 (2003).
[CrossRef]

Muschaweck, J. A.

H. Ries and J. A. Muschaweck, "Tailoring freeform lenses for illuminations, in Novel Optical Systems Design and Optimization, Proc. SPIE 4442, 43-50 (2001).
[CrossRef]

Oliker, V.

V. Oliker, "Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties," in Nonimaging Optics and Efficient Illumination Systems II, Proc. SPIE 594207 (2005).
[CrossRef]

L. Caffarelli, S. Kochengin, and V. Oliker, "On the numerical solution of the problem of reflector design with given far-field scattering data," Contemp. Math. 226, 13-32 (1999).

Parkyn, W. A.

W. A. Parkyn, "The design of illumination lenses via extrinsic differential geometry," in International Optical Design Conference, Proc. SPIE 3482, 191-193 (1998).

Piegl, L.

L. Piegl and W. Tiller, The NURBS Book, 2nd ed. (Springer-Verlag, 1997).

Ries, H.

A. Timinger, J. Muschaweck, and H. Ries, "Designing tailored free-form surfaces for general illumination," in Design of Efficient Illumination Systems, Proc. SPIE 5186, 128-132 (2003).
[CrossRef]

H. Ries and J. A. Muschaweck, "Tailoring freeform lenses for illuminations, in Novel Optical Systems Design and Optimization, Proc. SPIE 4442, 43-50 (2001).
[CrossRef]

Schruben, J. S.

J. S. Schruben, "Formulation a reflector-design problem for a lighting fixture," J. Opt. Soc. Am. A 6, 1498-1501 (1972).

Tiller, W.

L. Piegl and W. Tiller, The NURBS Book, 2nd ed. (Springer-Verlag, 1997).

Timinger, A.

A. Timinger, J. Muschaweck, and H. Ries, "Designing tailored free-form surfaces for general illumination," in Design of Efficient Illumination Systems, Proc. SPIE 5186, 128-132 (2003).
[CrossRef]

Wang, X.-J.

X.-J. Wang, "On design of a reflector antenna II," Calculus Var. Partial Differ. Equ. 20, 329-341 (2004).
[CrossRef]

Winston, R.

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

Calculus Var. Partial Differ. Equ. (1)

X.-J. Wang, "On design of a reflector antenna II," Calculus Var. Partial Differ. Equ. 20, 329-341 (2004).
[CrossRef]

Contemp. Math. (1)

L. Caffarelli, S. Kochengin, and V. Oliker, "On the numerical solution of the problem of reflector design with given far-field scattering data," Contemp. Math. 226, 13-32 (1999).

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

J. S. Schruben, "Formulation a reflector-design problem for a lighting fixture," J. Opt. Soc. Am. A 6, 1498-1501 (1972).

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, 1489-1502 (2004).
[CrossRef]

Proc. SPIE (4)

H. Ries and J. A. Muschaweck, "Tailoring freeform lenses for illuminations, in Novel Optical Systems Design and Optimization, Proc. SPIE 4442, 43-50 (2001).
[CrossRef]

A. Timinger, J. Muschaweck, and H. Ries, "Designing tailored free-form surfaces for general illumination," in Design of Efficient Illumination Systems, Proc. SPIE 5186, 128-132 (2003).
[CrossRef]

V. Oliker, "Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties," in Nonimaging Optics and Efficient Illumination Systems II, Proc. SPIE 594207 (2005).
[CrossRef]

W. A. Parkyn, "The design of illumination lenses via extrinsic differential geometry," in International Optical Design Conference, Proc. SPIE 3482, 191-193 (1998).

Other (4)

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

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

L. Piegl and W. Tiller, The NURBS Book, 2nd ed. (Springer-Verlag, 1997).

G. Farin, J. Hoschek, and M. Kim, Handbook of Computer Aided Geometric Design (Elsevier, 2002).

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Figures (11)

Fig. 1
Fig. 1

Slicing of the source and target region. The source output solid angle Ω and the target region D are sliced into several strips. Each pair of corresponding energy strips have the same amount of energy.

Fig. 2
Fig. 2

Source output solid angle Ω and the target region D are sliced further into many corresponding cells. Each pair of corresponding energy cells has the same amount of energy.

Fig. 3
Fig. 3

Calculation of the first latitudinal curve step by step: P 2 is calculated at the tangent plane of P 1 ; normal vector N 2 is then determined by incident ray i 2 and refracted ray O 2 ; tangent vector V 1 is perpendicular to normal vector N 1 .

Fig. 4
Fig. 4

Calculation of points on curve C 2 at the tangent plane of points on curve C 1 . Normal vector N 2 is perpendicular to vector V 2 .

Fig. 5
Fig. 5

Calculation of normal vector deviation.

Fig. 6
Fig. 6

Design sketch: length units are arbitrary and the drawing is not to scale.

Fig. 7
Fig. 7

a, Division of the character E and the emitting solid angle of point source. b, u, v parameters in Cartesian coordinates.

Fig. 8
Fig. 8

Left, diagram of the normal deviation angle distribution before the deviation control; right, normal deviation angle distribution after control, when normal deviation angle reaches 4°, an interrupt occurs caused by replacing the unqualified curve with a new qualified curve.

Fig. 9
Fig. 9

Shape of the immersion lens that cast the character E on the target screen from a Lambertian light source. The optical surface is locally C1 continuous but generally discontinuous.

Fig. 10
Fig. 10

Irradiance distribution resembles the original design goal on the whole. However, at the edges of the character, where the light rays of large angle from the point source are involved, the normal deviations of the lens surface become larger and blur the edges. The result is obtained based on Monte Carlo ray tracing of 1 × 106 rays.

Fig. 11
Fig. 11

Vertical cross section through the irradiance distribution along lines 1 and 2 in Fig. 10. The fluctuation of the irradiance distribution is partly caused by statistical noise and partly caused by the uncontrolled rays.

Equations (10)

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Ω I ( i⃗ ) d Ω = D E ( p⃗ ) d s .
Ω I ( u , v ) | J ( u , v ) | d u d v = D E ( x , y ) | J ( x , y ) | d x d y .
I ( u , v ) | J ( u , v ) | d u d v = E ( x , y ) | J ( x , y ) | d x d y .
( I ( u , v ) | J ( u , v ) | d u ) d v = ( E ( x , y ) | J ( x , y ) | d x ) d y .
I ( u , v ) | J ( u , v ) | d u   = E ( x , y ) | J ( x , y ) | I ( u , v ) | J ( u , v ) | d u E ( x , y ) | J ( x , y ) | d x  d x .
I 0 sin 2 u cos v d u d v = E 0 d x d y .
I 0 ( u / 2 sin ( 2 u ) / 4 ) | u 1 u 2 cos v d v = E 0 y | 0 L 1 d x .
x = I 0 [ 1 2 ( u 2 u 1 ) 1 4 ( sin ( 2 u 2 ) sin ( 2 u 1 ) ) ] ( 1 + sin v ) E 0 L 1 .
sin 2 u d u 1 2 ( u 2 u 1 ) 1 4 [ sin ( 2 u 2 ) sin ( 2 u 1 ) ] = d y L 1 .
y = L 1 1 2 ( u u 1 ) 1 4 [ sin ( 2 u ) sin ( 2 u 1 ) ] 1 2 ( u 2 u 1 ) 1 4 [ sin ( 2 u 2 ) sin ( 2 u 1 ) ] .

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