Abstract

In this paper, an analytical closed-form formula for the design of freeform lenses free of spherical aberration and astigmatism is presented. Given the equation of the freeform input surface, the formula gives the equation of the second surface to correct the spherical aberration. The derivation is based on the formal application of the variational Fermat principle under the standard geometrical optics approximation.

© 2019 Optical Society of America

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References

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  1. A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
    [Crossref]
  2. T. Yang, G.-F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6, e17081 (2017).
    [Crossref]
  3. G. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20, 2483–2499 (2012).
    [Crossref]
  4. G. Forbes, “Fitting freeform shapes with orthogonal bases,” Opt. Express 21, 19061–19081 (2013).
    [Crossref]
  5. G. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18, 19700–19712(2010).
    [Crossref]
  6. G. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15, 5218–5226 (2007).
    [Crossref]
  7. E. Muslimov, E. Hugot, W. Jahn, S. Vives, M. Ferrari, B. Chambion, D. Henry, and C. Gaschet, “Combining freeform optics and curved detectors for wide field imaging: a polynomial approach over squared aperture,” Opt. Express 25, 14598–14610 (2017).
    [Crossref]
  8. A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23, 28141–28153 (2015).
    [Crossref]
  9. D. Ochse, “Aberration fields of anamorphic systems,” Proc. SPIE 10690, 1069018 (2018).
    [Crossref]
  10. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22, 26585–26606 (2014).
    [Crossref]
  11. Y. Zhong and H. Gross, “Vectorial aberrations of biconic surfaces,” J. Opt. Soc. Am. A 35, 1385–1392 (2018).
    [Crossref]
  12. Y. Zhong and H. Gross, “Initial system design method for non-rotationally symmetric systems based on gaussian brackets and nodal aberration theory,” Opt. Express 25, 10016–10030 (2017).
    [Crossref]
  13. R. Winston, J. C. Miñano, P. G. Benitez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier, 2005).
  14. J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2016).
  15. W. Lin, P. Benítez, J. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
    [Crossref]
  16. F. Duerr, P. Benítez, J. C. Minano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012).
    [Crossref]
  17. J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
    [Crossref]
  18. J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Design of diffractive optical surfaces within the SMS design method,” in Freeform Optics (Optical Society of America, 2015), paper FTh3B-3.
  19. T. Levi-Civita, Complementi al teorema di Malus-Dupin: nota (Tipografia della R. Accademia dei Lincei, 1900).
  20. R. G. González-Acuña and H. A. Chaparro-Romo, “General formula for bi-aspheric singlet lens design free of spherical aberration,” Appl. Opt. 57, 9341–9345 (2018).
    [Crossref]
  21. R. G. González-Acuña and J. C. Guitiérrez-Vega, “Generalization of the axicon shape: the gaxicon,” J. Opt. Soc. Am. A 35, 1915–1918 (2018).
    [Crossref]
  22. A. S. Glassner, An Introduction to Ray Tracing (Elsevier, 1989).
  23. R. G. González-Acuña, “Freeform singlet lens free of spherical aberration and astigmatism,” figshare (2018), https://doi.org/10.6084/m9.figshare.7476488 .

2018 (5)

2017 (3)

2015 (1)

2014 (1)

2013 (1)

2012 (2)

2011 (1)

W. Lin, P. Benítez, J. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[Crossref]

2010 (1)

2009 (1)

J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
[Crossref]

2007 (1)

Bauer, A.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23, 28141–28153 (2015).
[Crossref]

Benitez, P. G.

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

Benítez, P.

F. Duerr, P. Benítez, J. C. Minano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012).
[Crossref]

W. Lin, P. Benítez, J. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[Crossref]

J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
[Crossref]

J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Design of diffractive optical surfaces within the SMS design method,” in Freeform Optics (Optical Society of America, 2015), paper FTh3B-3.

Biot, G.

W. Lin, P. Benítez, J. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[Crossref]

Bortz, J. C.

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

Chambion, B.

Chaparro-Romo, H. A.

Chaves, J.

J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2016).

Duerr, F.

Ferrari, M.

Forbes, G.

Fuerschbach, K.

Gaschet, C.

Glassner, A. S.

A. S. Glassner, An Introduction to Ray Tracing (Elsevier, 1989).

González-Acuña, R. G.

Gross, H.

Guitiérrez-Vega, J. C.

Henry, D.

Hugot, E.

Infante, J.

W. Lin, P. Benítez, J. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[Crossref]

J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
[Crossref]

Jahn, W.

Jin, G.-F.

T. Yang, G.-F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6, e17081 (2017).
[Crossref]

Levi-Civita, T.

T. Levi-Civita, Complementi al teorema di Malus-Dupin: nota (Tipografia della R. Accademia dei Lincei, 1900).

Lin, W.

W. Lin, P. Benítez, J. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[Crossref]

J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
[Crossref]

Mendes-Lopes, J.

J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Design of diffractive optical surfaces within the SMS design method,” in Freeform Optics (Optical Society of America, 2015), paper FTh3B-3.

Meuret, Y.

Minano, J. C.

Miñano, J.

W. Lin, P. Benítez, J. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[Crossref]

Miñano, J. C.

J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
[Crossref]

J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Design of diffractive optical surfaces within the SMS design method,” in Freeform Optics (Optical Society of America, 2015), paper FTh3B-3.

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

Muñoz, F.

J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
[Crossref]

Muslimov, E.

Ochse, D.

D. Ochse, “Aberration fields of anamorphic systems,” Proc. SPIE 10690, 1069018 (2018).
[Crossref]

Rolland, J. P.

Santamara, A.

J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
[Crossref]

Schiesser, E. M.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Shatz, N.

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

Thienpont, H.

Thompson, K. P.

Vives, S.

Winston, R.

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

Yang, T.

T. Yang, G.-F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6, e17081 (2017).
[Crossref]

Zhong, Y.

Zhu, J.

T. Yang, G.-F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6, e17081 (2017).
[Crossref]

Appl. Opt. (1)

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

Light: Sci. Appl. (1)

T. Yang, G.-F. Jin, and J. Zhu, “Automated design of freeform imaging systems,” Light: Sci. Appl. 6, e17081 (2017).
[Crossref]

Nat. Commun. (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Opt. Express (9)

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22, 26585–26606 (2014).
[Crossref]

G. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20, 2483–2499 (2012).
[Crossref]

G. Forbes, “Fitting freeform shapes with orthogonal bases,” Opt. Express 21, 19061–19081 (2013).
[Crossref]

G. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18, 19700–19712(2010).
[Crossref]

G. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15, 5218–5226 (2007).
[Crossref]

E. Muslimov, E. Hugot, W. Jahn, S. Vives, M. Ferrari, B. Chambion, D. Henry, and C. Gaschet, “Combining freeform optics and curved detectors for wide field imaging: a polynomial approach over squared aperture,” Opt. Express 25, 14598–14610 (2017).
[Crossref]

A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23, 28141–28153 (2015).
[Crossref]

Y. Zhong and H. Gross, “Initial system design method for non-rotationally symmetric systems based on gaussian brackets and nodal aberration theory,” Opt. Express 25, 10016–10030 (2017).
[Crossref]

F. Duerr, P. Benítez, J. C. Minano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012).
[Crossref]

Proc. SPIE (3)

J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamara, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009).
[Crossref]

D. Ochse, “Aberration fields of anamorphic systems,” Proc. SPIE 10690, 1069018 (2018).
[Crossref]

W. Lin, P. Benítez, J. Miñano, J. Infante, and G. Biot, “Advances in the SMS design method for imaging optics,” Proc. SPIE 8167, 81670M (2011).
[Crossref]

Other (6)

A. S. Glassner, An Introduction to Ray Tracing (Elsevier, 1989).

R. G. González-Acuña, “Freeform singlet lens free of spherical aberration and astigmatism,” figshare (2018), https://doi.org/10.6084/m9.figshare.7476488 .

J. Mendes-Lopes, P. Benítez, and J. C. Miñano, “Design of diffractive optical surfaces within the SMS design method,” in Freeform Optics (Optical Society of America, 2015), paper FTh3B-3.

T. Levi-Civita, Complementi al teorema di Malus-Dupin: nota (Tipografia della R. Accademia dei Lincei, 1900).

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

J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2016).

Supplementary Material (3)

NameDescription
» Code 1       Mathematica code to plot freeform surfaces free of spherical aberration
» Visualization 1       Back view of singlet lens with sinusoidal input surfaces
» Visualization 2       Subplot (a) singlet lenses with input surfaces mixing concave/convex with oscillating behavior.

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Figures (4)

Fig. 1.
Fig. 1. (a) Geometry of the problem and notation used for the distances. The origin of the coordinate system is located at the center of the input surface za(0,0)=0. (b) Zoom showing the notation for the unit vectors.
Fig. 2.
Fig. 2. Singlet lenses with non-uniform (a) convex and (b) concave input surfaces with n=1.5, T=1cm, fa=5cm, and fb=6cm.
Fig. 3.
Fig. 3. Singlet lens with sinusoidal input surfaces with n=1.5, T=1cm, fa=5cm, and fb=6cm. Please see Visualization 1.
Fig. 4.
Fig. 4. Singlet lenses with input surfaces mixing concave/convex with oscillating behavior. Parameters n=1.5 and T=1cm. Subplots (a) and (b) with fa=5cm and fb=6cm. Subplot (c) fa and fb. See Visualization 2 for the subplot (a).

Equations (16)

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

nv^2=[n^a×(n^a×v^1)]n^an2(n^a×v^1)·(n^a×v^1),
n^a=[zax,zay,1]S,v^1=[xa,ya,zafa]D,v^2=[xbxa,ybya,zbza]L,
Szax2+zay2+1,Dxa2+ya2+(zafa)2,L(xbxa)2+(ybya)2+(zbza)2,
XxbxaL=xa(zay2+1)zax(yazay+faza)nDS2zaxϕS,
YybyaL=ya(zax2+1)zay(xazax+faza)nDS2zayϕS,
ZzbzaL=(zafa)(zax2+zay2)+xazax+yazaynDS2+ϕS,
ϕ[1(yazaxxazay)2+[zax(zafa)+xa]2+[zay(zafa)+ya]2n2D2S2]1/2,
fa+nT+fb=sgn(fa)D+nL+sgn(fb)xb2+yb2+(zbTfb)2,
xb=xa+X(zbza)Z,yb=ya+Y(zbza)Z,zb=gg2+h(n21)n21,
g(zafbT)Z2+qZ+za(n21),
h[xa2+ya2za2+(T+fb)2(pnT)2]Z22zaqZza2(n21),
psgn(fa)D+fafb,
qxaX+yaYnp+n2T.
Ψa={(xa,ya,za)R3|za<zb},Ψb={(xb,ya,zb)R3|zb>za},
{nb=±xa[xb,yb,zb]×ya[xb,yb,zb]|xa[xb,yb,zb]×ya[xb,yb,zb]|,v3=[xb,yb,zbTfb]xb2+yb2+(zbTfb)2,v3=n[nb×(nb×v2)]nb1n2(nb×v2)·(nb×v2).
E=100%|v3v3v3|×100%.

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