Abstract

The design of an axisymmetrical refractive optical element transforming a given incident beam into an output beam with prescribed illuminance distribution and wavefront is considered. The wavefront of the output beam is represented by the eikonal function defined in a certain plane behind the optical element. The design of the optical element is reduced to the solution of two explicit ordinary differential equations of the first order. These equations can be easily integrated using conventional numerical methods. As examples, we consider the design of two optical elements transforming a spherical beam from a point Lambertian light source into the uniform-illuminance beams with a plane wavefront and with a complex wavefront providing the subsequent focusing into a line segment on the optical axis.

© 2018 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems

Leonid L. Doskolovich, Dmitry A. Bykov, Evgeniy S. Andreev, Evgeni A. Bezus, and Vladimir Oliker
Opt. Express 26(19) 24602-24613 (2018)

On the use of the supporting quadric method in the problem of the light field eikonal calculation

Leonid L. Doskolovich, Mikhail A. Moiseev, Evgeni A. Bezus, and Vladimir Oliker
Opt. Express 23(15) 19605-19617 (2015)

Analogy between generalized Coddington equations and thin optical element approximation

Michael A. Golub
J. Opt. Soc. Am. A 26(5) 1235-1239 (2009)

References

  • View by:
  • |
  • |
  • |

  1. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
    [Crossref]
  2. P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
    [Crossref]
  3. 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, 5488–5499 (2000).
    [Crossref]
  4. S. Zhang, G. Neil, and M. Shinn, “Single-element laser beam shaper for uniform flat-top profiles,” Opt. Express 11, 1942–1948 (2003).
    [Crossref]
  5. H. Ma, Z. Liu, P. Jiang, X. Xu, and S. Du, “Improvement of Galilean refractive beam shaping system for accurately generating near diffraction-limited flattop beam with arbitrary beam size,” Opt. Express 19, 13105–13117 (2011).
    [Crossref]
  6. F. Duerr and H. Thienpont, “Refractive laser beam shaping by means of a functional differential equation based design approach,” Opt. Express 22, 8001–8011 (2014).
    [Crossref]
  7. X. Hui, J. Liu, Y. Wan, and H. Lin, “Realization of uniform and collimated light distribution in a single freeform-Fresnel double surface LED lens,” Appl. Opt. 56, 4561–4565 (2017).
    [Crossref]
  8. A. G. Poleshchyk and R. K. Nasyrov, “Aspherical wavefront shaping with combined computer generated holograms,” Opt. Eng. 52, 091709 (2013).
    [Crossref]
  9. J. C. Wyant and V. P. Bennett, “Using computer-generated holograms to test aspheric wavefronts,” Appl. Opt. 11, 2833–2839 (1972).
    [Crossref]
  10. S. Chen, C. Zhao, Y. Dai, and S. Li, “Reconfigurable optical null based on counter-rotating Zernike plates for test of aspheres,” Opt. Express 22, 1381–1386 (2014).
    [Crossref]
  11. Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).
  12. L. L. Doskolovich and M. A. Moiseev, “Design of radially-symmetrical refractive surface taking into account Fresnel loss,” Comput. Opt. 32, 201–203 (2008).
  13. S. V. Kravchenko, M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of axis-symmetrical optical element with two aspherical surfaces for generation of prescribed irradiance distribution,” Comput. Opt. 35, 467–472 (2011).
  14. 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]
  15. http://www.lambdares.com/ .
  16. V. Oliker, L. L. Doskolovich, and D. A. Bykov, “Beam shaping with a plano-freeform lens pair,” Opt. Express 26, 19406–19419 (2018).
    [Crossref]
  17. J.-J. Chen, T.-Y. Wang, K.-L. Huang, T.-S. Liu, M.-D. Tsai, and C.-T. Lin, “Freeform lens design for LED collimating illumination,” Opt. Express 20, 10984–10995 (2012).
    [Crossref]
  18. L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26, 24602–24613 (2018).
    [Crossref]
  19. V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997).
  20. M. A. Golub, V. Shurman, and I. Grossinger, “Extended focus diffractive optical element for Gaussian laser beams,” Appl. Opt. 45, 144–150 (2006).
    [Crossref]
  21. L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik 101, 37–41 (1995).

2018 (2)

2017 (1)

2014 (2)

2013 (1)

A. G. Poleshchyk and R. K. Nasyrov, “Aspherical wavefront shaping with combined computer generated holograms,” Opt. Eng. 52, 091709 (2013).
[Crossref]

2012 (2)

2011 (2)

S. V. Kravchenko, M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of axis-symmetrical optical element with two aspherical surfaces for generation of prescribed irradiance distribution,” Comput. Opt. 35, 467–472 (2011).

H. Ma, Z. Liu, P. Jiang, X. Xu, and S. Du, “Improvement of Galilean refractive beam shaping system for accurately generating near diffraction-limited flattop beam with arbitrary beam size,” Opt. Express 19, 13105–13117 (2011).
[Crossref]

2008 (1)

L. L. Doskolovich and M. A. Moiseev, “Design of radially-symmetrical refractive surface taking into account Fresnel loss,” Comput. Opt. 32, 201–203 (2008).

2006 (1)

2003 (1)

2000 (1)

1995 (1)

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik 101, 37–41 (1995).

1980 (1)

1972 (1)

1965 (1)

Andreev, E. S.

Bennett, V. P.

Bezus, E. A.

Bykov, D. A.

Chen, J.-J.

Chen, S.

Dai, Y.

Doskolovich, L.

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

Doskolovich, L. L.

V. Oliker, L. L. Doskolovich, and D. A. Bykov, “Beam shaping with a plano-freeform lens pair,” Opt. Express 26, 19406–19419 (2018).
[Crossref]

L. L. Doskolovich, D. A. Bykov, E. S. Andreev, E. A. Bezus, and V. Oliker, “Designing double freeform surfaces for collimated beam shaping with optimal mass transportation and linear assignment problems,” Opt. Express 26, 24602–24613 (2018).
[Crossref]

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]

S. V. Kravchenko, M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of axis-symmetrical optical element with two aspherical surfaces for generation of prescribed irradiance distribution,” Comput. Opt. 35, 467–472 (2011).

L. L. Doskolovich and M. A. Moiseev, “Design of radially-symmetrical refractive surface taking into account Fresnel loss,” Comput. Opt. 32, 201–203 (2008).

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik 101, 37–41 (1995).

Du, S.

Duerr, F.

Frieden, B. R.

Golub, M. A.

Grossinger, I.

Hoffnagle, J. A.

Huang, K.-L.

Hui, X.

Jefferson, C. M.

Jiang, P.

Kazanskiy, N. L.

S. V. Kravchenko, M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of axis-symmetrical optical element with two aspherical surfaces for generation of prescribed irradiance distribution,” Comput. Opt. 35, 467–472 (2011).

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik 101, 37–41 (1995).

Kotlyar, V.

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

Kravchenko, S. V.

S. V. Kravchenko, M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of axis-symmetrical optical element with two aspherical surfaces for generation of prescribed irradiance distribution,” Comput. Opt. 35, 467–472 (2011).

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

Li, S.

Lin, C.-T.

Lin, H.

Liu, J.

Liu, T.-S.

Liu, Z.

Ma, H.

Moiseev, M. A.

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]

S. V. Kravchenko, M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of axis-symmetrical optical element with two aspherical surfaces for generation of prescribed irradiance distribution,” Comput. Opt. 35, 467–472 (2011).

L. L. Doskolovich and M. A. Moiseev, “Design of radially-symmetrical refractive surface taking into account Fresnel loss,” Comput. Opt. 32, 201–203 (2008).

Nasyrov, R. K.

A. G. Poleshchyk and R. K. Nasyrov, “Aspherical wavefront shaping with combined computer generated holograms,” Opt. Eng. 52, 091709 (2013).
[Crossref]

Neil, G.

Oliker, V.

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

Poleshchyk, A. G.

A. G. Poleshchyk and R. K. Nasyrov, “Aspherical wavefront shaping with combined computer generated holograms,” Opt. Eng. 52, 091709 (2013).
[Crossref]

Rhodes, P. W.

Shealy, D. L.

Shinn, M.

Shurman, V.

Soifer, V.

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

Soifer, V. A.

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik 101, 37–41 (1995).

Thienpont, H.

Tsai, M.-D.

Tzaregorodtzev, A. Y.

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik 101, 37–41 (1995).

Wan, Y.

Wang, T.-Y.

Wyant, J. C.

Xu, X.

Zhang, S.

Zhao, C.

Appl. Opt. (6)

Comput. Opt. (2)

L. L. Doskolovich and M. A. Moiseev, “Design of radially-symmetrical refractive surface taking into account Fresnel loss,” Comput. Opt. 32, 201–203 (2008).

S. V. Kravchenko, M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of axis-symmetrical optical element with two aspherical surfaces for generation of prescribed irradiance distribution,” Comput. Opt. 35, 467–472 (2011).

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

Opt. Eng. (1)

A. G. Poleshchyk and R. K. Nasyrov, “Aspherical wavefront shaping with combined computer generated holograms,” Opt. Eng. 52, 091709 (2013).
[Crossref]

Opt. Express (7)

Optik (1)

L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik 101, 37–41 (1995).

Other (3)

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

http://www.lambdares.com/ .

Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer, 1990).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1. Geometry of the problem.
Fig. 2.
Fig. 2. (a) Profile of the optical element transforming a spherical beam generated by a Lambertian point source to a uniform-illuminance beam with a plane wavefront; (b), (c) normalized illuminance distributions generated by the element in the planes z=30  mm and z=50  mm, respectively, calculated using the ray-tracing software TracePro. The illuminance cross sections along the coordinate axes are shown at the right of the distributions.
Fig. 3.
Fig. 3. (a) Profile of the optical element transforming a spherical beam generated by a Lambertian point source to a uniform-illuminance beam with the eikonal function of Eq. (18). Normalized illuminance distributions generated by the element in the plane z=30  mm (b) and in the xz plane (c) calculated using the ray-tracing software TracePro. The illuminance cross sections along the coordinate axes are shown at the right of the distributions.

Equations (18)

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

p(r)=(px(r),pz(r))=(Ψ(r),1[Ψ(r)]2).
xwf(r)=r+px(r)l(r),zwf(r)=f+pz(r)l(r),
γ(α)=αarctanw(α)w(α).
x1(α)=w(α)sinα+sin(γ(α))l1(α),z1(α)=w(α)cosα+cos(γ(α))l1(α),
Ψ1(x1(α),z1(α))=l(α)+Ψ0,
dl1(α)dα=np1,x(α)x1(α)+np1,z(α)z1(α)=nsin(β(α))x1(α)+ncos(β(α))z1(α).
dl1(α)dα=[sin(αβ(α))w(α)cos(αβ(α))w(α)l1(α)sin(β(α)γ(α))γ(α)]·[cos(β(α)γ(α))n1]1,
dl1(α)dα=l1(α)sin(β(α)α)n1cos(β(α)α),
E0(α)dS0=E0(α)·2π[w(α)]2+[w(α)]2sinα·dα=E(r)2πrdr,
dr2(α)dα=2E0(α)E(r2)[w(α)]2+[w(α)]2sinα.
p2(α)=(p2,x(α),p2,z(α))=p(r(α)).
x2(α)=r(α)p2,x(α)l2(α),z2(α)=fp2,z(α)l2(α),
β(α)=arctanx2(α)x1(α)z2(α)z1(α).
Ψ0+l1(α)+n[x2(α)x1(α)]2+[z2(α)z1(α)]2+l2(α)=Ψ˜(α),
[r(α)p2,x(α)l2(α)w(α)sinαsin(γ(α))l1(α)]2+[fp2,z(α)l2(α)w(α)cosαcos(γ(α))l1(α)]2=1n2(Ψ˜(α)[l1(α)+l2(α)+Ψ0])2.
Ψ(r)=(n1)h0+f,r[0,r0].
r(α)=r01cos2α1cos2α0.
Ψ(r)=12aln[1+2a(f1+ar2+r2+(f1+ar2)2)]+Ψf,r[0,r0],