Abstract

A new iterative technique for calculating the eikonal function of a light field providing the focusing into a set of points is introduced. This technique is a modification of the supporting quadric method widely used for design of reflecting and refracting optical surfaces for generating prescribed illuminance distributions at given discrete set of points. As an example, we design a refractive optical element which focuses an incident beam into a set of points with energy pattern forming an image of a keyboard of a calculator. It is shown that the proposed technique is well-suited for the design of diffractive optical elements producing continuous intensity distributions within the scalar theory of diffraction. It is also shown that the calculated eikonal function is a good initial guess when designing diffractive optical elements using the iterative Gerchberg-Saxton algorithm.

© 2015 Optical Society of America

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References

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2015 (2)

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

X. Mao, H. Li, Y. Han, and Y. Luo, “Polar-grids based source-target mapping construction method for designing freeform illumination system for a lighting target with arbitrary shape,” Opt. Express 23(4), 4313–4328 (2015).
[Crossref] [PubMed]

2014 (2)

2013 (2)

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

L. L. Doskolovich, A. Yu. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (5)

2009 (1)

1998 (1)

S. A. 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]

1992 (1)

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).
[Crossref]

1982 (1)

Belkin, M.

Benítez, P.

Bich, A.

Bitterli, R.

Bräuer, A.

Chen, F.

de Rooij, N.

Dmitriev, A. Yu.

L. L. Doskolovich, A. Yu. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

Doskolovich, L. L.

M. A. Moiseev, S. V. Kravchenko, and L. L. Doskolovich, “Design of efficient LED optics with two free-form surfaces,” Opt. Express 22(S7), A1926–A1935 (2014).
[Crossref] [PubMed]

L. L. Doskolovich, A. Yu. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

M. A. Moiseev and L. L. Doskolovich, “Design of refractive spline surface for generating required irradiance distribution with large angular dimension,” J. Mod. Opt. 57(7), 536–544 (2010).
[Crossref]

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).
[Crossref]

Feng, Z.

Fienup, J. R.

Golub, M. A.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).
[Crossref]

Han, Y.

Herzig, H.-P.

Kazanskiy, N.

Kazanskiy, N. L.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).
[Crossref]

Kharitonov, S. I.

L. L. Doskolovich, A. Yu. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).
[Crossref]

Kochengin, S. A.

S. A. 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]

Kravchenko, S. V.

Lahav-Yacouel, K.

Li, H.

Liu, P.

Liu, S.

Liu, X.

Liu, Z.

Luo, X.

Luo, Y.

Mao, X.

Michaelis, D.

Miñano, J. C.

Moiseev, M. A.

M. A. Moiseev, S. V. Kravchenko, and L. L. Doskolovich, “Design of efficient LED optics with two free-form surfaces,” Opt. Express 22(S7), A1926–A1935 (2014).
[Crossref] [PubMed]

M. A. Moiseev and L. L. Doskolovich, “Design of refractive spline surface for generating required irradiance distribution with large angular dimension,” J. Mod. Opt. 57(7), 536–544 (2010).
[Crossref]

Noell, W.

Oliker, V.

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

Oliker, V. I.

S. A. 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]

Roth, S.

Rubinstein, J.

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

Scharf, T.

Schmidt, M.

Schreiber, P.

Skidanov, R.

Soifer, V. A.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).
[Crossref]

Voelkel, R.

Wang, K.

Weible, K. J.

Wolansky, G.

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

Wu, R.

Xu, L.

Yaish, S. B.

Yehezkel, O.

Zalevsky, Z.

Zhang, Y.

Zheng, Z.

Zimmermann, M.

Zlotnik, A.

Adv. Appl. Math. (1)

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

Appl. Opt. (2)

J. Mod. Opt. (2)

M. A. Moiseev and L. L. Doskolovich, “Design of refractive spline surface for generating required irradiance distribution with large angular dimension,” J. Mod. Opt. 57(7), 536–544 (2010).
[Crossref]

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).
[Crossref]

Numer. Math. (1)

S. A. 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]

Opt. Eng. (1)

L. L. Doskolovich, A. Yu. Dmitriev, and S. I. Kharitonov, “Analytic design of optical elements generating a line focus,” Opt. Eng. 52(9), 091707 (2013).
[Crossref]

Opt. Express (6)

Opt. Lett. (4)

Other (6)

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd., 1997).

L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces (Schwarz Press, 2008).

http://www.lambdares.com

http://www.rhino3d.com

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

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).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the problem of focusing into a set of focal points.

Fig. 2
Fig. 2

An image of a calculator on an equidistant mesh of 56×82 pixels.

Fig. 3
Fig. 3

The eikonal function to focus into a set of points in Fig. 2.

Fig. 4
Fig. 4

Illuminance distribution generated in the target plane z=100 mm using the eikonal function in Fig. 3.

Fig. 5
Fig. 5

A refractive optical element to focus into a set of points in Fig. 2 and the generated illuminance pattern obtained using the TracePro software.

Fig. 6
Fig. 6

(a)–(c) Fragments of the eikonal function Ψ add ( u ) , 0.5 Ψ add ( u ) , 0.25 Ψ add ( u ) in the central regions of size 3×3 mm 2 ; (d)–(f) Illuminance patterns generated in the Fresnel-Kirchhoff approximation by the eikonal functions in Fig. 6 (a)–(c).

Fig. 7
Fig. 7

(a)–(c) Fragments of the eikonal functions after 30 iterations of the GS algorithm; (d)–(f) Illuminance patterns generated using the Fresnel-Kirchhoff approximation for the eikonal functions in Fig. 7 (a)–(c).

Equations (18)

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

p( u )=( Ψ( u ), 1 [ Ψ( u ) ] 2 ).
Ψ l ( u;x )= ( ux ) 2 + f 2 + Ψ f ( x ).
{ Ψ l ( u;x )= ( ux ) 2 + f 2 + Ψ f ( x ), Ψ f ( x ) x xu ( ux ) 2 + f 2 =0, Ψ f ( x ) y yv ( ux ) 2 + f 2 =0.
Ψ l ( u )= [ ux( u ) ] 2 + f 2 + Ψ f [ x( u ) ],
x( u )= argmin xD Ψ l ( u;x )
x( u )= argmax xD Ψ l ( u;x ),
Ψ l ( u )= ( u x m ) 2 + f 2 + Ψ f,m ,
m= argmin 1iN Ψ l ( u; x i )
m= argmax 1iN Ψ l ( u; x i )
Ψ l,i ( u )= ( u x i ) 2 + f 2 + Ψ f,i ,i=1,...,N,
Ψ f,i := Ψ f,i ±Δ( I calc,i I i ),i=1,...,N,
h(u)= Ψ( u ) / ( n1 ) ,
h(u)= 1 n1 mod λ [ Ψ( u ) ].
E( x )= | 1 λf G w 0 ( u ) exp[ ik 2f ( xu ) 2 ] d 2 u | 2 .
Ψ( u )= u 2 2f + Ψ add ( u ),
E( x )= | 1 λf G E 0 ( u ) exp[ ik Ψ add ( u ) ] exp( ik f xu ) d 2 u | 2 .
Δ sub = 2λf a 2 / N im =2 N im Δ,
Ψ p ( u )= u 2 2f +p Ψ add ( u )

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