Abstract

The geometrical method for constructing optical surfaces for illumination purpose developed by Oliker and co-workers [Trends in Nonlinear Analysis (Springer, 2003)] is generalized in order to obtain freeform designs in arbitrary optical systems. The freeform is created by a set of primitive surface elements, which are generalized Cartesian ovals adapted to the given optical system. Those primitives are determined by Hamiltonian theory of ray optics. The potential of this approach is demonstrated by some examples, e.g., freeform lenses with collimating front elements.

© 2011 Optical Society of America

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References

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  1. J. C. Minano, P. Benitez, and A. Santamaria, Opt. Rev. 16, 99 (2009).
    [CrossRef]
  2. K. Mantel, D. Bachstein, and U. Peschel, Opt. Lett. 36, 199 (2011).
    [CrossRef] [PubMed]
  3. V. I. Oliker, in Trends in Nonlinear Analysis, M.Kirkilionis, ed. (Springer, 2003), pp. 193–224.
  4. D. L. Shealy, in Laser Beam Shaping, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 164–213.
  5. H. Ries and J. Muschaweck, J. Opt. Soc. Am. A 19, 590 (2002).
    [CrossRef]
  6. W. Cassarly, in Handbook of Optics (McGraw-Hill, 2001), pp. 2.23–2.42.
  7. F. R. Fournier, W. J. Cassarly, and J. P. Rollanda, Proc. SPIE 7103, 71030I (2008).
    [CrossRef]
  8. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).
  9. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

2011

2009

J. C. Minano, P. Benitez, and A. Santamaria, Opt. Rev. 16, 99 (2009).
[CrossRef]

2008

F. R. Fournier, W. J. Cassarly, and J. P. Rollanda, Proc. SPIE 7103, 71030I (2008).
[CrossRef]

2002

Bachstein, D.

Benitez, P.

J. C. Minano, P. Benitez, and A. Santamaria, Opt. Rev. 16, 99 (2009).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Cassarly, W.

W. Cassarly, in Handbook of Optics (McGraw-Hill, 2001), pp. 2.23–2.42.

Cassarly, W. J.

F. R. Fournier, W. J. Cassarly, and J. P. Rollanda, Proc. SPIE 7103, 71030I (2008).
[CrossRef]

Fournier, F. R.

F. R. Fournier, W. J. Cassarly, and J. P. Rollanda, Proc. SPIE 7103, 71030I (2008).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

Mantel, K.

Minano, J. C.

J. C. Minano, P. Benitez, and A. Santamaria, Opt. Rev. 16, 99 (2009).
[CrossRef]

Muschaweck, J.

Oliker, V. I.

V. I. Oliker, in Trends in Nonlinear Analysis, M.Kirkilionis, ed. (Springer, 2003), pp. 193–224.

Peschel, U.

Ries, H.

Rollanda, J. P.

F. R. Fournier, W. J. Cassarly, and J. P. Rollanda, Proc. SPIE 7103, 71030I (2008).
[CrossRef]

Santamaria, A.

J. C. Minano, P. Benitez, and A. Santamaria, Opt. Rev. 16, 99 (2009).
[CrossRef]

Shealy, D. L.

D. L. Shealy, in Laser Beam Shaping, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 164–213.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

J. Opt. Soc. Am. A

Opt. Lett.

Opt. Rev.

J. C. Minano, P. Benitez, and A. Santamaria, Opt. Rev. 16, 99 (2009).
[CrossRef]

Proc. SPIE

F. R. Fournier, W. J. Cassarly, and J. P. Rollanda, Proc. SPIE 7103, 71030I (2008).
[CrossRef]

Other

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

V. I. Oliker, in Trends in Nonlinear Analysis, M.Kirkilionis, ed. (Springer, 2003), pp. 193–224.

D. L. Shealy, in Laser Beam Shaping, F.Dickey and S.Holswade, eds. (Dekker, 2000), pp. 164–213.

W. Cassarly, in Handbook of Optics (McGraw-Hill, 2001), pp. 2.23–2.42.

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Figures (3)

Fig. 1
Fig. 1

Scheme of freeform design. The freeform is constructed by segments of primitive surfaces. Here ordinary COs are utilized for illustration, (a) COs with the same foci but decreasing path length parameter V a to V c , (b), (c) simple illustration of modification of the discrete power distribution P i by a three-segment optics made from three COs due to changing the path length parameter of the middle CO.

Fig. 2
Fig. 2

Scheme for determination of the family of COs as primitives for the construction of freeforms. An arbitrary system (here N front surfaces S 1 S N 1 separating N-regions with different refractive indices n i ) is described by the optical path length V 0 or the mixed characteristics W 0 from the source point F 0 to a dummy plane at z = 0 (bold dashed line). By adding the additional path length n N r D r N or mixed characteristics z N m N up to the CO ( S N ) correspondingly and considering the final path length V end for (a) collimation or (b) focusing the CO—surface is obtained parametrically. The ray intersection at the surface S i is denoted by r i and the ray vectors in the regions with the index n i are called g i = ( p i , q i , m i ) T with g i = n i .

Fig. 3
Fig. 3

Examples of illumination freeforms with CO- representation. (a) Freeform lens with differentiable profile and flat entrance face converting light from a Lambertian source ( NA 0.996 ) into the Lena picture with 15 ° × 15 ° output divergence; discretization, ~20,000 CO-primitives; inset, cross section of a typical CO-primitive. (b) Nonsteady differentiable freeform lens creating a hemicycle with an angular radius of 19 ° and a width of 4 ° ; source, Lambertian with NA 0.996 . (c) Telecentric spot generator consisting of a collimating lens and a steady, segmented splitting optics where the whole system is constructed by nine COs; the three cross sections show typical CO examples with different path length parameters; source, Lambertian with NA 0.996 . (d) Cross-line generation due to light collimation, total internal reflection inside a prism, and freeform light shaping; source, isotropic with NA = 0.5 , (e) Typical example of a CO-primitive and the freeform profile corresponding to (d).

Equations (2)

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r N ( p N , q N ) = W 0 + g N V i , k W 0 k g N | W 0 n N 2 k | g N .
z N 2 n N 2 n N 4 m N 2 2 z N 1 { g N | F i , k + ( 1 n N 2 ) W 0 n N 2 ( V i , k W 0 ) m N } + W 0 + F i , k 2 ( V i , k W 0 + g N | W 0 ) 2 = 0

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