Abstract

It has been shown in [ J. A. Hoffnagle and C. M. Jefferson, Opt. Eng. 42, 3090, (2003)] that a pair of plano-aspheric lenses can be used to transform a radially symmetric, Gaussian beam to a radially symmetric flat-top beam. In this paper it is shown that a pair of plano-freeform lenses can be used to transform a collimated light beam of arbitrary (including, non-radially symmetric) intensity profile to a collimated output beam of constant phase and a priori specified intensity pattern over a given flat region. The curved surfaces of both lenses can be chosen strictly convex which should facilitate fabrication. The required pair of plano-freeform lenses is designed using the supporting quadric method (SQM) [ V. I. Oliker in Trends in Nonlinear Analysis, (Springer-Verlag, 2003)] combined with ideas from optimal mass transport [ V. I. Oliker, Arch. Rational Mechanics and Analysis 201, 1013 (2011)]. Such approach provides a rigorous methodology for designing freeform optics for irradiance redistribution. In this paper, this approach is applied to design of a laser beam shaping system with two lenses.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  4. J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U. S. Patent No. 3,476,463 (1969).
  5. P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19(20), 3545–3553 (1980).
    [Crossref] [PubMed]
  6. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000).
    [Crossref]
  7. V. I. Oliker, “Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport,” Arch. Rational Mechanics and Analysis 201, 1013–1045 (2011).
    [Crossref]
  8. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2-nd ed. (Springer-Verlag, 1983).
    [Crossref]
  9. 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-Verlag, 2003).
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    [Crossref] [PubMed]
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    [Crossref]
  13. J. A. Hoffnagle and M. Jefferson, “Transformation of the transverse intensity profile of a laser beam by aspheric lenses,” presented at SLAC, Oct. 2012.
  14. M. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge University, 1999).
    [Crossref]
  15. V. I. Oliker, “On design of freeform refractive beam shapers, sensitivity to figure error and convexity of lenses,” J. Opt. Soc. Am. A 25(12), 3067–3076 (2008).
    [Crossref]
  16. D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R. Driggers, ed. (Taylor & Francis, 2006).
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    [Crossref]
  18. S. V. Kravchenko, E. V. Byzov, M. C. Moiseev, and L. L. Doskolovich, “Development of multiple-surface optical elements for road lighting,” Opt. Express 25(4), A23–A35 (2017).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  23. L. L. Doskolovich, M. A. Moiseev, E. A. Bezus, and V. 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]
  24. V. I. Oliker, A rigorous method for synthesis of offset shaped reflector antennas, Computing Lett. 2(1–2), 29–49 (2006).
    [Crossref]
  25. R. Luneburg, Mathematical Theory of Optics (University of California, 1964).
  26. R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2-nd ed., (Cambridge University, 2014).
  27. V. I. Oliker, J. Rubinstein, and G. Wolansky, “Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light,” Advances in Appl. Math. 62, 160–183 (2015).
    [Crossref]
  28. P. R. Thie and G. E. Keough, An Introduction to Linear Programming and Game Theory, 3-rd ed. (J. Wiley & Sons, 2008).
    [Crossref]
  29. L. Rúshendorf and L. Uckelmann, Numerical and analytical results for the transportation problem of Monge-Kantorovich, Metrika 51(3), 245–258 (2000).
    [Crossref]
  30. Q. Mérigot and É. Oudset, “Discrete optimal transport: complexity, geometry and applications,” Discrete & Computational Geometry 55(2), 263–283 (2016).
    [Crossref]
  31. Opto-mechanical software TracePro. https://www.lambdares.com/tracepro/
  32. Rhinoceros. http://www.rhino3d.com
  33. M. A. Moiseev, E. V. Byzov, S. V. Kravchenko, and L. L. Doskolovich, “Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces,” Opt. Express 23, A1140–A1148 (2015).
    [Crossref] [PubMed]

2018 (1)

2017 (2)

2016 (1)

Q. Mérigot and É. Oudset, “Discrete optimal transport: complexity, geometry and applications,” Discrete & Computational Geometry 55(2), 263–283 (2016).
[Crossref]

2015 (4)

2013 (2)

2011 (2)

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

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]

2008 (1)

2006 (1)

V. I. Oliker, A rigorous method for synthesis of offset shaped reflector antennas, Computing Lett. 2(1–2), 29–49 (2006).
[Crossref]

2003 (1)

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng.,  42(11), 3090–3099 (2003).
[Crossref]

2000 (2)

J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000).
[Crossref]

L. Rúshendorf and L. Uckelmann, Numerical and analytical results for the transportation problem of Monge-Kantorovich, Metrika 51(3), 245–258 (2000).
[Crossref]

1980 (1)

1965 (1)

Bezus, E. A.

Born, M.

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

Bösel, C.

Bräuer, A.

Byzov, E. V.

Canavesi, C.

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

Doskolovich, L. L.

Feng, Z.

Fournier, F.

F. Fournier, “Freeform reflector design with extended sources,” Ph. D. Dissertation, (University of Central Florida, 2010).

Frieden, B. R.

Froese, B. D.

Gilbarg, D.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2-nd ed. (Springer-Verlag, 1983).
[Crossref]

Gross, H.

Hoffnagle, J.

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng.,  42(11), 3090–3099 (2003).
[Crossref]

Hoffnagle, J. A.

J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000).
[Crossref]

J. A. Hoffnagle and M. Jefferson, “Transformation of the transverse intensity profile of a laser beam by aspheric lenses,” presented at SLAC, Oct. 2012.

D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R. Driggers, ed. (Taylor & Francis, 2006).

Huang, C.-Y.

Jefferson, C.

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng.,  42(11), 3090–3099 (2003).
[Crossref]

Jefferson, C. M.

Jefferson, M.

J. A. Hoffnagle and M. Jefferson, “Transformation of the transverse intensity profile of a laser beam by aspheric lenses,” presented at SLAC, Oct. 2012.

Keough, G. E.

P. R. Thie and G. E. Keough, An Introduction to Linear Programming and Game Theory, 3-rd ed. (J. Wiley & Sons, 2008).
[Crossref]

Kravchenko, S. V.

Kreuzer, J. L.

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U. S. Patent No. 3,476,463 (1969).

Li, H.

Liang, R.

Liu, P.

Liu, X.

Luneburg, R.

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

Ma, D.

Mérigot, Q.

Q. Mérigot and É. Oudset, “Discrete optimal transport: complexity, geometry and applications,” Discrete & Computational Geometry 55(2), 263–283 (2016).
[Crossref]

Michaelis, D.

Moiseev, M. A.

Moiseev, M. C.

Oliker, V.

Oliker, V. I.

V. I. Oliker, “Controlling light with freeform multifocal lens designed with supporting quadric method (SQM),” Opt. Express 25(4), A58–A72 (2017).
[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,” Advances in Appl. Math. 62, 160–183 (2015).
[Crossref]

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. of Photonics for Energy 3, 035599 (2013).
[Crossref]

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

V. I. Oliker, “On design of freeform refractive beam shapers, sensitivity to figure error and convexity of lenses,” J. Opt. Soc. Am. A 25(12), 3067–3076 (2008).
[Crossref]

V. I. Oliker, A rigorous method for synthesis of offset shaped reflector antennas, Computing Lett. 2(1–2), 29–49 (2006).
[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-Verlag, 2003).

Oudset, É.

Q. Mérigot and É. Oudset, “Discrete optimal transport: complexity, geometry and applications,” Discrete & Computational Geometry 55(2), 263–283 (2016).
[Crossref]

Rhodes, P. W.

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,” Advances in Appl. Math. 62, 160–183 (2015).
[Crossref]

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. of Photonics for Energy 3, 035599 (2013).
[Crossref]

Rúshendorf, L.

L. Rúshendorf and L. Uckelmann, Numerical and analytical results for the transportation problem of Monge-Kantorovich, Metrika 51(3), 245–258 (2000).
[Crossref]

Schneider, R.

R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2-nd ed., (Cambridge University, 2014).

Schreiber, P.

Shealy, D. L.

P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19(20), 3545–3553 (1980).
[Crossref] [PubMed]

D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R. Driggers, ed. (Taylor & Francis, 2006).

Thie, P. R.

P. R. Thie and G. E. Keough, An Introduction to Linear Programming and Game Theory, 3-rd ed. (J. Wiley & Sons, 2008).
[Crossref]

Trudinger, N. S.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2-nd ed. (Springer-Verlag, 1983).
[Crossref]

Uckelmann, L.

L. Rúshendorf and L. Uckelmann, Numerical and analytical results for the transportation problem of Monge-Kantorovich, Metrika 51(3), 245–258 (2000).
[Crossref]

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,” Advances in Appl. Math. 62, 160–183 (2015).
[Crossref]

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. of Photonics for Energy 3, 035599 (2013).
[Crossref]

Wolf, E.

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

Wu, R.

Xu, L.

Zhang, Y.

Zheng, Z.

Advances 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,” Advances in Appl. Math. 62, 160–183 (2015).
[Crossref]

Appl. Opt. (4)

Arch. Rational Mechanics and Analysis (1)

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

Computing Lett. (1)

V. I. Oliker, A rigorous method for synthesis of offset shaped reflector antennas, Computing Lett. 2(1–2), 29–49 (2006).
[Crossref]

Discrete & Computational Geometry (1)

Q. Mérigot and É. Oudset, “Discrete optimal transport: complexity, geometry and applications,” Discrete & Computational Geometry 55(2), 263–283 (2016).
[Crossref]

J. of Photonics for Energy (1)

V. I. Oliker, J. Rubinstein, and G. Wolansky, “Ray mapping and illumination control,” J. of Photonics for Energy 3, 035599 (2013).
[Crossref]

J. Opt. Soc. Am. A (2)

Metrika (1)

L. Rúshendorf and L. Uckelmann, Numerical and analytical results for the transportation problem of Monge-Kantorovich, Metrika 51(3), 245–258 (2000).
[Crossref]

Opt. Eng. (1)

J. Hoffnagle and C. Jefferson, “Beam shaping with a plano-aspheric lens pair,” Opt. Eng.,  42(11), 3090–3099 (2003).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Other (14)

J. A. Hoffnagle and M. Jefferson, “Transformation of the transverse intensity profile of a laser beam by aspheric lenses,” presented at SLAC, Oct. 2012.

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

F. Fournier, “Freeform reflector design with extended sources,” Ph. D. Dissertation, (University of Central Florida, 2010).

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

F. M. Dickey, ed., Laser Beam Shaping: Theory and Techniques, 2nd ed. (CRC Press, 2014).
[Crossref]

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U. S. Patent No. 3,476,463 (1969).

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2-nd ed. (Springer-Verlag, 1983).
[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-Verlag, 2003).

Opto-mechanical software TracePro. https://www.lambdares.com/tracepro/

Rhinoceros. http://www.rhino3d.com

D. L. Shealy and J. A. Hoffnagle, “Aspheric optics for laser beam shaping,” in Encyclopedia of Optical Engineering, R. Driggers, ed. (Taylor & Francis, 2006).

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

R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2-nd ed., (Cambridge University, 2014).

P. R. Thie and G. E. Keough, An Introduction to Linear Programming and Game Theory, 3-rd ed. (J. Wiley & Sons, 2008).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 A system of two plano-freeform lenses for beam shaping; only the active (non-planar) sides of the lenses are shown. The two lenses R1 and R2 are required to transform a collimated input beam with an arbitrary radiance I1(x) into a collimated output beam with a prescribed irradiance I2(p) at a given region on the plane α′ : z = d; n = const is the normalized refractive index of R1 and R2 (as explained at the beginning of Subsection 2.1). From Ref. [7].
Fig. 2
Fig. 2 Design of a two-lens system for transforming a non-uniform laser beam into a uniform intensity RS beam requires first circularization (a)→(b) and filtering (b)→(c) of the beam from the laser. Then a RS beam shaper can convert it into a “flat-top” (c)→(d); from Ref. [13] with permission.
Fig. 3
Fig. 3 Hyperbola Hl maps the segment [a, b] into Fr.
Fig. 4
Fig. 4 The function z in Eq. (9) defines the first lens R1. The second lens R2 is defined by Eq. (10).
Fig. 5
Fig. 5 The designed systems of two plano-freeform lenses in a Keplerian (a, b) and a Galilean (c, d) configurations to transform an incident circular beam into a rectangular outgoing beam. The dashed lines show the z axis.
Fig. 6
Fig. 6 Normalized radiance distributions generated in the planes z = 10 cm (a, c) and z = 100 cm (b, d) by the designed Keplerian (a, b) and Galilean (c, d) lens systems calculated in TracePro. The radiance cross-sections along the coordinate axes are shown at the top and at the right of the figures.
Fig. 7
Fig. 7 (a) The calculated system of two plano-freeform lenses transforming an incident circular beam into a star-shaped beam. The dashed line shows the z axis. The first plano-freeform lens with the surface z(x) (b) reshapes the input radiance, whereas the second lens with the surface w(p) (c) reshapes the phase.
Fig. 8
Fig. 8 Normalized radiance distributions generated by the two-lens system shown in Fig. 7 in the planes z = 10 cm (a) and z = 100 cm (b) calculated in TracePro. The radiance cross-sections along the coordinate axes are shown at the top and at the right of the figures.

Equations (31)

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I 2 ( T d ( x ) ) | det J ( T d ( x ) ) | = I 1 ( x ) , x Ω ,
I 2 ( x , f ( x ) ) [ f 11 f 22 f 12 2 ] = I 1 ( x ) , x Ω . ( Here f i j = 2 f x i x j , i , j = 1 , 2 . )
P ( x ) = x β z ( x ) M ( x ) , x Ω ¯ .
w ( P ( x ) ) = z ( x ) + β 1 n 2 [ n + 1 M ( x ) ] , x Ω ¯ .
H ( x , p ) : = n β c ( x , p ) n 2 1 , where c ( x , p ) : = β 2 + ( n 2 1 ) ( x p ) 2 .
ξ p , η l ( x ) = H ( x , p ) + η , x α ,
F p , η l = ( p , η + n n 2 1 ( β | β | ) ) , F p , η r = ( p , η + n n 2 1 ( β + | β | ) ) , C p , η = ( p , η + n β n 2 1 ) .
η x , ξ r ( p ) = H ( x , p ) + ξ , p α .
F x , ξ l = ( x , ξ n n 2 1 ( β + | β | ) ) , F x , ξ r = ( x , ξ n n 2 1 ( β | β | ) ) , C x , ξ = ( x , ξ n β n 2 1 ) .
z ( x ) = min p T ¯ { H ( x , p ) + w ( p ) } , x Ω ¯ ,
w ( p ) = max x Ω ¯ { H ( x , p ) + z ( x ) } , p T ¯ .
z ( x ) = max p T ¯ { H ( x , p ) + w ( p ) } , x Ω ¯ ,
w ( p ) = min x Ω ¯ { H ( x , p ) + z ( x ) } , p T ¯ .
P z , w ( x ) = { p T ¯ | z ( x ) = H ( x , p ) + w ( p ) } , x Ω ¯ ,
P z , w 1 ( p ) = { x Ω ¯ | w ( p ) = H ( x , p ) + z ( x ) } , p T ¯ .
( i ) P z , w ( Ω ¯ ) = P z ^ , w ( α ) = T ¯ and ( i i ) P z , w 1 ( T ¯ ) = P z , w ^ 1 ( α ) = Ω ¯ .
G z , w ( τ ) = P z , w 1 ( τ ) I 1 ( x ) d x
lim k T ¯ f ( p ) G z k , w k ( d p ) = T ¯ f ( p G z , w ( d p ) ) f C ( T ¯ ) .
P z , w ( Ω ¯ ) = T ¯
G z , w ( τ ) = μ ( τ ) τ ( T ¯ ) , where μ ( τ ) : = τ I 2 ( p ) d p .
Ω ¯ I 1 ( x ) d x = μ ( T ¯ ) .
Adm A ( Ω ¯ , T ¯ ) = { ( z , w ) C ( Ω ¯ ) × C ( T ¯ ) w ( p ) z ( x ) H ( x , p ) ( x , p ) Ω ¯ × T ¯ } .
( z , w ) = T ¯ w ( p ) μ ( d p ) Ω ¯ z ( x ) I 1 ( x ) d x , ( z , w ) Adm A ( Ω ¯ , T ¯ ) .
( z , w ) min over Adm A ( Ω ¯ , T ¯ )
i = 1 M ω ¯ i M = Ω ¯ , ω t M ω k M = if t k ; j = 1 N τ ¯ j N = T ¯ , τ h N τ l N = if h l .
E i M = ω i M I 1 ( x ) d x , F i N = μ ( τ j N ) .
E M ( ω ) = x i M ω E i M and F N ( τ ) = p i N τ F i k .
i = 1 M E i M = j = 1 N F j N .
M , N ( z 1 M , , z M M , w 1 N , , w N N ) : = j = 1 N w j N F j N i = 1 M z i M E i M
Adm M , N = { w j N z i M H ( x i , p j ) , i = 1 , , M , j = 1 , , N } .
( z ¯ M , w ¯ N ) = inf Adm M , N M , N ( z M , w N ) .

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