Abstract

The SQM is a versatile methodology for designing freeform optics for irradiance redistribution. Recently it has often been applied in design of freeform lenses, mirrors and diffractive optical elements. Still, many questions regarding theory and performance of optics designed with the SQM are open. Here we investigate theoretically plano-freeform refractive lenses designed with the SQM when an incident collimated beam must be transformed into a beam illuminating with prescribed irradiances a large number of pixels on a flat screen. It is shown that a lens designed for such task with the SQM operates as a multifocal lens segmented into subapertures with focal lengths providing accurate control of the irradiance distribution between pixels. These subapertures are patches of hyperboloids of revolution. Two different designs are possible, one of which defines a concave lens. Eikonal function for such lenses is also derived. As a proof of concept, we numerically analyze performance of a plano-freeform lens designed with the SQM for transforming a uniform circular parallel light into an image of A. Einstein represented by gray values at ≈ 38K pixels.

© 2017 Optical Society of America

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References

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  1. R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. Bortz, Nonimaging Optics (Elsevier, 2005).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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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]
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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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2016 (1)

P. M. M. de Castro, Q. Mérigot, and B. Thibert, “Far-field reflector problem and intersection of paraboloids,” Numer. Math. 134(2), 389–411 (2016).
[Crossref]

2015 (4)

W. Song, D. Cheng, Y. Liu, and Y. Wang, “Free-form illumination of a refractive surface using multiple-faceted refractors,” Appl. Opt. 54(28), E1–E7 (2015).
[Crossref] [PubMed]

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light,” Adv. in Appl. Math. 62, 160–183 (2015).
[Crossref]

L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. I. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015).
[Crossref] [PubMed]

M. N. Ricketts, R. Winston, and V. Oliker, “Diffraction effects in freeform optics,” Proc. SPIE 9572, 957200 (2015).

2014 (2)

V. I. Oliker and B. V. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105(2014).
[Crossref]

V. I. Oliker, “Differential equations for design of a freeform single lens with prescribed irradiance properties,” Opt. Eng. 53(3), 031302 (2014).
[Crossref]

2013 (1)

2012 (1)

2011 (2)

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011).
[Crossref] [PubMed]

V.I. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Rational Mech. Anal. 201, 1013–1045 (2011).
[Crossref]

2008 (1)

L.A. Caffarelli and V.I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 37–46 (2008).

2005 (2)

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

J.A. Hoffnagle, “A new derivation of the Dickey-Romero-Holswade phase function,” Proc. SPIE 5876, 587606 (2005).
[Crossref]

2002 (1)

1998 (1)

S. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data II. Numerical solution,” Numer. Math. 79(4), 553–568 (1998).
[Crossref]

Benítez, P.

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

Bezus, E. A.

Born, M.

M. Born and E. Wolf, Principle of Optics, 7th (expanded) edition (Cambridge University, 1999).
[Crossref]

Bortz, J.

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

Bräuer, A.

Caffarelli, L.A.

L.A. Caffarelli and V.I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 37–46 (2008).

Canavesi, C.

C. Canavesi, Subaperture conics and geometric concepts applied to freeform reflector design for illumination (Ph.D. Diss., University of Rochester, 2014).

Cheng, D.

Cherkasskiy, B. V.

V. I. Oliker and B. V. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105(2014).
[Crossref]

de Castro, P. M. M.

P. M. M. de Castro, Q. Mérigot, and B. Thibert, “Far-field reflector problem and intersection of paraboloids,” Numer. Math. 134(2), 389–411 (2016).
[Crossref]

Doskolovich, L. L.

Doskolovich, L.L.

V. A. Soifer, V.V Kotlar, and L.L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (CRC, 1997).

Feßler, R.

Fournier, F. R.

F. R. Fournier, Freeform reflector design with extended sources (Ph.D. Diss., University of Central Florida, 2010).

Hoffnagle, J. A.

D. L. Shealy and J. A. Hoffnagle, “Geometrical methods,”), Laser Beam Shaping: Theory and Techniques, 2-nd edition, F.M. Dickey, ed. (CRC, 2014), Chap. 6.

Hoffnagle, J.A.

J.A. Hoffnagle, “A new derivation of the Dickey-Romero-Holswade phase function,” Proc. SPIE 5876, 587606 (2005).
[Crossref]

Jegorov, J.

Kochengin, S.

S. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data II. Numerical solution,” Numer. Math. 79(4), 553–568 (1998).
[Crossref]

Kotlar, V.V

V. A. Soifer, V.V Kotlar, and L.L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (CRC, 1997).

Li, H.

Liu, P.

Liu, X.

Liu, Y.

Luneburg, R.

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

Mérigot, Q.

P. M. M. de Castro, Q. Mérigot, and B. Thibert, “Far-field reflector problem and intersection of paraboloids,” Numer. Math. 134(2), 389–411 (2016).
[Crossref]

Michaelis, D.

Miñano, J. C.

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

Moiseev, M. A.

Muschaweck, J.

Notni, G.

Oliker, V.

M. N. Ricketts, R. Winston, and V. Oliker, “Diffraction effects in freeform optics,” Proc. SPIE 9572, 957200 (2015).

Oliker, V. I.

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light,” Adv. in Appl. Math. 62, 160–183 (2015).
[Crossref]

L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. I. Oliker, “On the use of the supporting quadric method in the problem of the light field eikonal calculation,” Opt. Express 23(15), 19605–19617 (2015).
[Crossref] [PubMed]

V. I. Oliker, “Differential equations for design of a freeform single lens with prescribed irradiance properties,” Opt. Eng. 53(3), 031302 (2014).
[Crossref]

V. I. Oliker and B. V. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105(2014).
[Crossref]

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

S. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data II. Numerical solution,” Numer. Math. 79(4), 553–568 (1998).
[Crossref]

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, 2003), pp. 191–222.

Oliker, V.I.

V.I. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Rational Mech. Anal. 201, 1013–1045 (2011).
[Crossref]

L.A. Caffarelli and V.I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 37–46 (2008).

Ricketts, M. N.

M. N. Ricketts, R. Winston, and V. Oliker, “Diffraction effects in freeform optics,” Proc. SPIE 9572, 957200 (2015).

Ries, H.

Rockafellar, R. T.

R. T. Rockafellar and R. J-B Wets, Variational Analysis, Grundlehren der Math. Wiss. 317 (Springer, 2009).

Rubinstein, J.

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light,” Adv. in Appl. Math. 62, 160–183 (2015).
[Crossref]

Schreiber, P.

Shatz, N.

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

Shealy, D. L.

D. L. Shealy and J. A. Hoffnagle, “Geometrical methods,”), Laser Beam Shaping: Theory and Techniques, 2-nd edition, F.M. Dickey, ed. (CRC, 2014), Chap. 6.

Soifer, V. A.

V. A. Soifer, V.V Kotlar, and L.L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (CRC, 1997).

Song, W.

Thibert, B.

P. M. M. de Castro, Q. Mérigot, and B. Thibert, “Far-field reflector problem and intersection of paraboloids,” Numer. Math. 134(2), 389–411 (2016).
[Crossref]

Wang, Y.

Wets, R. J-B

R. T. Rockafellar and R. J-B Wets, Variational Analysis, Grundlehren der Math. Wiss. 317 (Springer, 2009).

Winston, R.

M. N. Ricketts, R. Winston, and V. Oliker, “Diffraction effects in freeform optics,” Proc. SPIE 9572, 957200 (2015).

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

Wolansky, G.

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light,” Adv. in Appl. Math. 62, 160–183 (2015).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principle of Optics, 7th (expanded) edition (Cambridge University, 1999).
[Crossref]

Wu, R.

Xu, L.

Zhang, Y.

Zheng, Z.

Zwick, S.

Adv. in Appl. Math. (1)

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light,” Adv. in Appl. Math. 62, 160–183 (2015).
[Crossref]

Appl. Opt. (1)

Arch. Rational Mech. Anal. (1)

V.I. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Rational Mech. Anal. 201, 1013–1045 (2011).
[Crossref]

J. Math. Sci. (1)

L.A. Caffarelli and V.I. Oliker, “Weak solutions of one inverse problem in geometric optics,” J. Math. Sci. 154(1), 37–46 (2008).

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

Numer. Math. (2)

S. Kochengin and V. I. Oliker, “Determination of reflector surfaces from near-field scattering data II. Numerical solution,” Numer. Math. 79(4), 553–568 (1998).
[Crossref]

P. M. M. de Castro, Q. Mérigot, and B. Thibert, “Far-field reflector problem and intersection of paraboloids,” Numer. Math. 134(2), 389–411 (2016).
[Crossref]

Opt. Eng. (1)

V. I. Oliker, “Differential equations for design of a freeform single lens with prescribed irradiance properties,” Opt. Eng. 53(3), 031302 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Proc. SPIE (4)

V. I. Oliker and B. V. Cherkasskiy, “Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties,” Proc. SPIE 9191, 919105(2014).
[Crossref]

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

M. N. Ricketts, R. Winston, and V. Oliker, “Diffraction effects in freeform optics,” Proc. SPIE 9572, 957200 (2015).

J.A. Hoffnagle, “A new derivation of the Dickey-Romero-Holswade phase function,” Proc. SPIE 5876, 587606 (2005).
[Crossref]

Other (9)

V. A. Soifer, V.V Kotlar, and L.L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (CRC, 1997).

C. Canavesi, Subaperture conics and geometric concepts applied to freeform reflector design for illumination (Ph.D. Diss., University of Rochester, 2014).

M. Born and E. Wolf, Principle of Optics, 7th (expanded) edition (Cambridge University, 1999).
[Crossref]

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

R. T. Rockafellar and R. J-B Wets, Variational Analysis, Grundlehren der Math. Wiss. 317 (Springer, 2009).

D. L. Shealy and J. A. Hoffnagle, “Geometrical methods,”), Laser Beam Shaping: Theory and Techniques, 2-nd edition, F.M. Dickey, ed. (CRC, 2014), Chap. 6.

F. R. Fournier, Freeform reflector design with extended sources (Ph.D. Diss., University of Central Florida, 2010).

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, 2003), pp. 191–222.

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

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Figures (7)

Fig. 1
Fig. 1 Schematic drawing of a lens to be determined in an ICP. A collimated light beam with cross section Ω ¯ α and irradiance I(x), x ∈ Ω, propagates in direction z. The goal is to determine a refracting plano-freeform lens R which intercepts the beam and redistributes the light over a given region T ¯ d on a plane α′ parallel to α. In this paper it is assumed that α′ is in the near-field. The irradiance on Td is a given function L(p, d), pT, where T is the projection of Td on α. The lens is defined by the function z(x) to be determined. The refractive index n = const of the lens is given. In the figure the input and output irradiance patterns are not shown. Only the (active) curved side of the lens is shown. Only two complete paths of light are shown.
Fig. 2
Fig. 2 Any horizontal ray hitting F p ¯ , f ¯ from the left is refracted and passes through the right focus ( p ¯ , d ).
Fig. 3
Fig. 3 (a) The target T ¯ d is a discrete set of points in the plane α′ with given discrete irradiance distribution L(p), ( p , d ) T ¯ d. (b) The graph of the function z defined by Eq. (7) as a pointwise minimum is the lens Rz. The foci p1, …, p5 are not shown.
Fig. 4
Fig. 4 The designed lens realizes a Keplerian configuration [22] in which the refracted rays crossover and the images of the segments in (a) are switched in (b) relative to the origin in (a) (indicated by a small bright square). The outer rectangle in (a) is a graphical artifact.
Fig. 5
Fig. 5 (a) synthetic image of A. Einstein produced with the designed freeform lens shown in (b) (planar side is not shown); (c) original photo; (d) plot of the gray values of the photo.
Fig. 6
Fig. 6 Distribution of pixels as a function of DSDi.
Fig. 7
Fig. 7 Color coded distributions of DSD, D S x 1 and D S x 2 for the image in Fig. 5(a); cyan: DSD, D S x 1, D S x 2 < 0.3333 mm; magenta: 0.3333mm ≤ DSD, D S x 1, D S x 2 0.6666 mm; black: DSD, D S x 1, D S x 2 > 0.6666 mm.

Tables (2)

Tables Icon

Table 1 Comparison of prescribed with computed irradiances

Tables Icon

Table 2 RMS value of OPL and its variation

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

P ( x ) = x ( d z ( x ) ) ( 1 n 2 ) 1 + n M ( x ) z ( x ) , x Ω ¯ .
L ( P d ( x ) ) | det J ( P d ) ( x ) | = I ( x ) , x Ω ,
L ( P ( x ) , d ) | det J ( P ( z ( x ) ) ) | = I in Ω .
P d ( Ω ¯ ) = T ¯ d or , equivalently , P ( Ω ¯ ) = T ¯ .
z p ¯ , f ¯ = f 2 + ( n 2 1 ) ( x p ¯ ) 2 + f ¯ n n 2 1 + d , x α .
F l = ( p ¯ , h n | f ¯ | n 2 1 ) , F r = ( p ¯ , h + n | f ¯ | n 2 1 = d ) .
P d 1 ( p ¯ , d ) : = { x Ω ¯ | P d ( x ) = ( p ¯ , d ) }
P 1 ( p ¯ ) : = { x Ω ¯ | P ( x ) = p ¯ } . P 1 ( p ¯ ) : = { x Ω ¯ | P ( x ) = p ¯ } .
G ( z p ¯ , f ¯ , p ) = P 1 ( p ¯ ) I ( x ) d x if p = p ¯ , G ( z p ¯ , f ¯ , p ) ) = 0 if p p ¯ and μ : = Ω ¯ I ( x ) d x ,
G ( z p ¯ , f ¯ , p ) = μ δ ( p p ¯ ) , P ( z p ¯ , f ¯ ( x ) ) ) = p ¯ for all x Ω ¯ ,
z ( x ) = min 1 i K z p i , f i ( x ) , x Ω ¯ .
P z ( x ) : = { p i T ¯ | z ( x ) = z p i , f i ( x ) } , x Ω ¯ , and P z 1 ( p i ) : = { x Ω ¯ | z ( x ) = z p i , f i ( x ) } , p i T ¯ .
G ( z , p i ) : = P z 1 ( p i ) I ( x ) d x , i = 1 , 2 , , K .
Ω ¯ I ( x ) d x = i = 1 K μ i > 0 .
G ( z , p i ) = μ i δ ( p p i ) , i = 1 , 2 , , K , T ¯ = P ( z ( Ω ¯ ) ) .
f i = n 2 1 n sup x Ω ¯ [ z ( x ) d + f i 2 + ( n 2 1 ) ( x p i ) 2 n 2 1 ] ,
z ( x ) = x p i f i 2 + ( n 2 1 ) ( x p i ) 2 .
| n i ( x ) n j ( x ) | C | p i p j |
Ψ p i , f i ( x ) = k n z p i , f i ( x ) + k [ z p i , f i ( p i ) z p i , f i ( x ) ] = k ( n 1 ) z p i , f i ( x ) + k ( f i n 1 + d ) .
Ψ R z ( x ) = k ( n 1 ) z ( x ) + k ( f i n 1 + d ) , x Ω ¯ i , i = 1 , 2 , , K .
| f i f j | n 1 n + 1 | p i p j | .
OPL i ( x ) = n z p i , f i ( x ) + ( x p i ) 2 + ( z p i , f i ( x ) d ) 2 .
OPL i ( x ¯ ) = lim x x ¯ , x Ω i OPL i ( x ) .
OPL i ( x ) = n d + f i for all x Ω ¯ i , i = 1 , 2 , , K .
z p i , f i ( x ) = ( x p i ) 2 2 | f i | | f i | n 1 + d + O ( | x p i | 4 | f i | 4 ) .
Ψ p i , f i ( x ) = k ( n 1 ) [ ( x p i ) 2 2 | f i | | f i | n 1 + d ] + O ( | x p i | 4 | f i | 4 ) .
Ψ ^ R z ( x ) = k ( n 1 ) ( x p i ) 2 2 | f i | k | f i | + k ( n 1 ) d , x Ω ¯ i , i = 1 , , K .
S k : = f k 2 + ( n 2 1 ) ( x ¯ p k ) 2 , S ˜ k = f k 2 + n 2 ( x ¯ p k ) 2 .
S ˜ i n i S ˜ j n j = ( p j p i , S i S j ) .
| f i f j | = ( n 1 ) | z ( p i ) z ( p j ) | .
| z ( p i ) z ( p j ) | | p i p j | n 2 1 .
| f i f j | n 1 n + 1 | p i p j | .
| f i f j | σ n 1 n + 1 .
( x p k ) 2 = [ ( d m ) ( n 2 1 ) n | f k | ] 2 | f k | 2 n 2 1 , k = i , j .
( n 2 1 ) [ ( x p i ) 2 ( x p j ) 2 ] = [ ( d m ) ( n 2 1 ) n | f i | ] 2 [ ( d m ) ( n 2 1 ) n | f j | ] 2 | f i | 2 + | f j | 2 .
LHS = ( n 2 1 ) ( 2 x + p i + p j ) ( p i p j ) , RHS = ( n 2 1 ) [ 2 ( d m ) n ( | f i | + | f j | ) + | f i | 2 | f j | 2 ] .
( x + p i + p j 2 ) ( p i p j ) = | f i | 2 | f j | 2 2 + ( d m ) n ( | f j | | f i | ) .
d x ( m ) d m ( p j p i ) = n ( | f j | | f i | ) ,

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