Abstract

A method is proposed for designing refractive optical elements focusing a collimated incident beam into a curve with specified shape. A general relationship for the freeform surface of the optical element is derived as an envelope of a parametric family of hyperboloids of revolution that focuses the incident beam into the points on the curve. Using the thin optical element approximation, the calculation of the hyperboloid parameters providing required irradiance distribution along the curve is reduced to the solution of an explicit first order differential equation. Optical elements generating line segment focus and circular arc focus are designed. The simulation results demonstrate generation of high-quality curves.

© 2013 Optical Society of America

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References

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    [CrossRef]
  4. L. L. Doskolovich, N. L. Kazanskiy, and M. A. Moiseev, “Design of high-efficient freeform LED lens for illumination of elongated rectangular regions,” Opt. Express 19, A225–A233 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. R. Hu, X. Luo, H. Zheng, Z. Qin, Z. Gan, B. Wu, and S. Liu, “Design of a novel freeform lens for LED uniform illumination and conformal phosphor coating,” Opt. Express 20, 13727–13737 (2012).
    [CrossRef]
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    [CrossRef]
  11. M. A. Moiseev and L. L. Doskolovich, “Design of TIR optics generating the prescribed irradiance distribution in the circle region,” J. Opt. Soc. Am. A 29, 1758–1763 (2012).
    [CrossRef]
  12. Y. Ding, X. Liu, Z. Zheng, and P. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
    [CrossRef]
  13. L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005).
    [CrossRef]
  14. L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, S. I. Kharitonov, and P. Perlo, “A DOE to form a line-shaped directivity diagram,” J. Mod. Opt. 51, 1999–2005 (2004).
    [CrossRef]
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    [CrossRef]
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2012 (3)

2011 (2)

2010 (5)

2008 (1)

2007 (1)

L. L. Doskolovich, N. L. Kazanskiy, and S. Bernard, “Designing a mirror to form a line-shaped directivity diagram,” J. Mod. Opt. 54, 589–597 (2007).
[CrossRef]

2005 (1)

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005).
[CrossRef]

2004 (1)

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, S. I. Kharitonov, and P. Perlo, “A DOE to form a line-shaped directivity diagram,” J. Mod. Opt. 51, 1999–2005 (2004).
[CrossRef]

Bäuer, A.

Benitez, P.

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

Bernard, S.

L. L. Doskolovich, N. L. Kazanskiy, and S. Bernard, “Designing a mirror to form a line-shaped directivity diagram,” J. Mod. Opt. 54, 589–597 (2007).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005).
[CrossRef]

Cassarly, W. J.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freefrom reflector generation using source-target maps,” Opt. Express 18, 5204–5295 (2010).
[CrossRef]

Chen, F.

Chen, J.-J.

Ding, Y.

Doskolovich, L. L.

M. A. Moiseev and L. L. Doskolovich, “Design of TIR optics generating the prescribed irradiance distribution in the circle region,” J. Opt. Soc. Am. A 29, 1758–1763 (2012).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, and M. A. Moiseev, “Design of high-efficient freeform LED lens for illumination of elongated rectangular regions,” Opt. Express 19, A225–A233 (2011).
[CrossRef]

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

L. L. Doskolovich, N. L. Kazanskiy, and S. Bernard, “Designing a mirror to form a line-shaped directivity diagram,” J. Mod. Opt. 54, 589–597 (2007).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, S. I. Kharitonov, and P. Perlo, “A DOE to form a line-shaped directivity diagram,” J. Mod. Opt. 51, 1999–2005 (2004).
[CrossRef]

Eisenhart, L. P.

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

Feng, Z.

Fournier, F. R.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freefrom reflector generation using source-target maps,” Opt. Express 18, 5204–5295 (2010).
[CrossRef]

Gan, Z.

Gu, P.

Han, Y.

Hu, R.

Huang, K.-L.

Kazanskiy, N. L.

L. L. Doskolovich, N. L. Kazanskiy, and M. A. Moiseev, “Design of high-efficient freeform LED lens for illumination of elongated rectangular regions,” Opt. Express 19, A225–A233 (2011).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, and S. Bernard, “Designing a mirror to form a line-shaped directivity diagram,” J. Mod. Opt. 54, 589–597 (2007).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, S. I. Kharitonov, and P. Perlo, “A DOE to form a line-shaped directivity diagram,” J. Mod. Opt. 51, 1999–2005 (2004).
[CrossRef]

Kharitonov, S. I.

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, S. I. Kharitonov, and P. Perlo, “A DOE to form a line-shaped directivity diagram,” J. Mod. Opt. 51, 1999–2005 (2004).
[CrossRef]

Li, H.

Lin, C.-T.

Liu, S.

Liu, T.-S.

Liu, X.

Liu, Z.

Luo, X.

Luo, Y.

Michaelis, D.

Minano, J. C.

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

Moiseev, M. A.

Oliker, V. I.

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. 197–224.

Perlo, P.

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, S. I. Kharitonov, and P. Perlo, “A DOE to form a line-shaped directivity diagram,” J. Mod. Opt. 51, 1999–2005 (2004).
[CrossRef]

Qin, Z.

Rolland, J. P.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Fast freefrom reflector generation using source-target maps,” Opt. Express 18, 5204–5295 (2010).
[CrossRef]

Schreiber, P.

Soifer, V. A.

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, S. I. Kharitonov, and P. Perlo, “A DOE to form a line-shaped directivity diagram,” J. Mod. Opt. 51, 1999–2005 (2004).
[CrossRef]

Tsai, M.-D.

Wang, K.

Wang, T.-Y.

Winston, R.

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

Wu, B.

Zheng, H.

Zheng, Z.

J. Mod. Opt. (4)

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

L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, S. I. Kharitonov, and P. Perlo, “A DOE to form a line-shaped directivity diagram,” J. Mod. Opt. 51, 1999–2005 (2004).
[CrossRef]

L. L. Doskolovich, N. L. Kazanskiy, and S. Bernard, “Designing a mirror to form a line-shaped directivity diagram,” J. Mod. Opt. 54, 589–597 (2007).
[CrossRef]

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

Opt. Express (8)

Opt. Lett. (1)

Other (4)

R. Winston, J. C. Minano, and P. Benitez, Nonimaging Optics (Elsevier Academic, 2005).

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

Lambda Research Corporation, “TracePro Suite of Optical and Illumination Design Software,” http://www.lambdares.com/software_products/tracepro/ .

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. 197–224.

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Figures (5)

Fig. 1.
Fig. 1.

Geometry of the considered problem.

Fig. 2.
Fig. 2.

Geometry of the rays cone.

Fig. 3.
Fig. 3.

Designed refractive optical elements producing (a) a line-segment focus and (b) a circular arc focus.

Fig. 4.
Fig. 4.

Cross-section profiles of the designed optical elements producing (a) a line-segment focus and (b) a circular arc focus.

Fig. 5.
Fig. 5.

Simulated irradiance distributions generated by the designed refractive optical elements producing (a) a line segment focus and (b) a circular arc focus.

Equations (20)

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X(ξ)=(X(ξ),Y(ξ),f),
n·h(x,y)+|X(ξ)S|=ψ(ξ),
(X(ξ)S|X(ξ)S|,dX(ξ)dξ)=dψ(ξ)dξ.
τ(ξ)=(dX(ξ)dξ,dY(ξ)dξ,0),n(ξ)=(dY(ξ)dξ,dX(ξ)dξ,0),b=(0,0,1).
p(ξ,φ)=τ(ξ)cos(α)+n(ξ)sin(α)sin(φ)+bsin(α)cos(φ),
S(ξ,φ)=(x(ξ,φ),y(ξ,φ),z(ξ,φ))=X(ξ)+p(ξ,φ)·l(ξ,φ),
l(ξ,φ)=ψ(ξ)nfnpz(ξ,φ)+1.
x(ξ,φ)=X(ξ)+(nfψ(ξ))(dX(ξ)dξdψ(ξ)dξdY(ξ)dξ1(dψ(ξ)dξ)2sin(φ))n1(dψ(ξ)dξ)2cos(φ)1,y(ξ,φ)=Y(ξ)+(nfψ(ξ))(dY(ξ)dξdψ(ξ)dξ+dX(ξ)dξ1(dψ(ξ)dξ)2sin(φ))n1(dψ(ξ)dξ)2cos(φ)1.
ΔΦ=Δξφ1(ξ)φ2(ξ)I0(ξ,φ)|J(ξ,φ;dψdξ;d2ψdξ2)|dφ,
J(ξ,φ;ψ(ξ);dψdξ;d2ψdξ2)=x(ξ,φ)ξy(ξ,φ)φx(ξ,φ)φy(ξ,φ)ξ
x2(ξ,φ)+y2(ξ,φ)=R2.
I(ξ)=limΔξ0ΔΦΔξ=φ1(ξ)φ2(ξ)I0(ξ,φ)|J(ξ,φ;ψ(ξ);dψdξ;d2ψdξ2)|dφ.
x(ξ,η)=X(ξ)+a(ξ)f2+η2dX(ξ)dξηdY(ξ)dξ,y(ξ,η)=Y(ξ)+a(ξ)f2+η2dY(ξ)dξ+ηdX(ξ)dξ,
a(ξ)=dψ(ξ)dξ/1(dψ(ξ)dξ)2,η=ftg(φ).
J(ξ,η;a(ξ);dadξ)=[1+da(ξ)dξf2+η2]+η(a2(ξ)+1)K(ξ),
I(ξ)=η1(ξ)η2(ξ)I0(ξ,η)|J(ξ,η;a(ξ);dadξ)|dη
da(ξ)dξ=I(ξ)(a2(ξ)+1)K(ξ)η1(ξ)η2(ξ)I0(ξ,η)ηdηη1(ξ)η2(ξ)I0(ξ,η)dηη1(ξ)η2(ξ)I0(ξ,η)f2+η2dη.
dψ(ξ)dξ=a(ξ)1+a2(ξ),ψ(ξ)=0ξa(t)1+a2(t)dt.
Xseg(ξ)=(ξd2,0,f),ξ[0,d],
Xarc(ξ)=(Rarcsin([ξd/2]/Rarc),RarcRarccos([ξd/2]/Rarc),f),ξ[0,d],

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