Abstract

We address, in detail, the system of differential equations determining a freeform aplanatic system with illustrative examples. We also demonstrate how two optical surfaces, in general, are insufficient in achieving freeform aplanatism through the use of integrability condition for a given reflective freeform aplanatic configuration. This result also alludes to the fact that a freeform aplanatic system fulfills a broader set of conditions than its rotationally symmetric counterpart. We also elaborate on the above results with two illustrative examples (1) A semi aplanatic system which satisfies the generalized sine condition in only one direction and (2) A fully freeform aplanatic reflective system.

© 2017 Optical Society of America

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References

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  1. J. Rolland and K. Thompson, “Freeform optics: Evolution? no, revolution!” SPIE Newsroom (2012).
  2. F. Duerr, Y. Meuret, and H. Thienpont, “Potential benefits of free-form optics in on-axis imaging applications with high aspect ratio,” Opt. Express 21(25), 31072–31081 (2013).
    [Crossref] [PubMed]
  3. H. Ries, N. E. Shatz, J. C. Bortz, and W. Spirkl, “Consequences of skewness conservation for rotationally symmetric nonimaging devices,” Proc. SPIE 3139, 47–58 (1997).
    [Crossref]
  4. J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
    [Crossref]
  5. E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9, pp. 413–468, (1873).
  6. T. T. Elazhary, P. Zhou, C. Zhao, and J. H. Burge, “Generalized sine condition,” Appl. Opt. 54(16), 5037–5049 (2015).
    [Crossref] [PubMed]
  7. J. J. Braat and P. F. Greve, “Aplanatic optical system containing two aspheric surfaces,” Appl. Opt. 18(13), 2187–2191 (1979).
    [Crossref] [PubMed]
  8. K. Schwarzschild, “Untersuchungen zur geometrischen Optik II,” Abh. Konigl. Ges. Wis. Gottingen Mathphys. Kl. 4, 1–3 (1905).
  9. J. C. Miñano, P. Benítez, and B. Narasimhan, “Freeform aplanatic systems as a limiting case of SMS,” Opt. Express 24(12), 13173–13178 (2016).
    [Crossref] [PubMed]
  10. J. H. Burge, C. Zhao, and S. H. Lu, “Use of the Abbe sine condition to quantify alignment aberrations in optical imaging systems,” Proc. SPIE 7652, 765219 (2010).
    [Crossref]
  11. B. D. Stone and G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9(3), 478 (1992).
    [Crossref]
  12. R. K. Luneburg, Mathematical Theory of Optics (University of California, Los Angeles, 1964).
  13. P. Benitez, M. Nikolic, and J. C. Miñano, “Analytical solution of an afocal two freeform mirror design problem,” Opt. Express 25(4), 4155–4161 (2017).
    [Crossref] [PubMed]
  14. G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
    [Crossref]

2017 (1)

2016 (1)

2015 (1)

2013 (1)

2010 (1)

J. H. Burge, C. Zhao, and S. H. Lu, “Use of the Abbe sine condition to quantify alignment aberrations in optical imaging systems,” Proc. SPIE 7652, 765219 (2010).
[Crossref]

2009 (1)

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[Crossref]

1997 (1)

H. Ries, N. E. Shatz, J. C. Bortz, and W. Spirkl, “Consequences of skewness conservation for rotationally symmetric nonimaging devices,” Proc. SPIE 3139, 47–58 (1997).
[Crossref]

1992 (1)

1979 (1)

1949 (1)

G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

1905 (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik II,” Abh. Konigl. Ges. Wis. Gottingen Mathphys. Kl. 4, 1–3 (1905).

Benitez, P.

Benítez, P.

J. C. Miñano, P. Benítez, and B. Narasimhan, “Freeform aplanatic systems as a limiting case of SMS,” Opt. Express 24(12), 13173–13178 (2016).
[Crossref] [PubMed]

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[Crossref]

Bortz, J. C.

H. Ries, N. E. Shatz, J. C. Bortz, and W. Spirkl, “Consequences of skewness conservation for rotationally symmetric nonimaging devices,” Proc. SPIE 3139, 47–58 (1997).
[Crossref]

Braat, J. J.

Burge, J. H.

T. T. Elazhary, P. Zhou, C. Zhao, and J. H. Burge, “Generalized sine condition,” Appl. Opt. 54(16), 5037–5049 (2015).
[Crossref] [PubMed]

J. H. Burge, C. Zhao, and S. H. Lu, “Use of the Abbe sine condition to quantify alignment aberrations in optical imaging systems,” Proc. SPIE 7652, 765219 (2010).
[Crossref]

Duerr, F.

Elazhary, T. T.

Forbes, G. W.

Greve, P. F.

Lu, S. H.

J. H. Burge, C. Zhao, and S. H. Lu, “Use of the Abbe sine condition to quantify alignment aberrations in optical imaging systems,” Proc. SPIE 7652, 765219 (2010).
[Crossref]

Meuret, Y.

Miñano, J. C.

Narasimhan, B.

Nikolic, M.

Ries, H.

H. Ries, N. E. Shatz, J. C. Bortz, and W. Spirkl, “Consequences of skewness conservation for rotationally symmetric nonimaging devices,” Proc. SPIE 3139, 47–58 (1997).
[Crossref]

Rolland, J.

J. Rolland and K. Thompson, “Freeform optics: Evolution? no, revolution!” SPIE Newsroom (2012).

Santamaría, A.

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[Crossref]

Schwarzschild, K.

K. Schwarzschild, “Untersuchungen zur geometrischen Optik II,” Abh. Konigl. Ges. Wis. Gottingen Mathphys. Kl. 4, 1–3 (1905).

Shatz, N. E.

H. Ries, N. E. Shatz, J. C. Bortz, and W. Spirkl, “Consequences of skewness conservation for rotationally symmetric nonimaging devices,” Proc. SPIE 3139, 47–58 (1997).
[Crossref]

Spirkl, W.

H. Ries, N. E. Shatz, J. C. Bortz, and W. Spirkl, “Consequences of skewness conservation for rotationally symmetric nonimaging devices,” Proc. SPIE 3139, 47–58 (1997).
[Crossref]

Stone, B. D.

Thienpont, H.

Thompson, K.

J. Rolland and K. Thompson, “Freeform optics: Evolution? no, revolution!” SPIE Newsroom (2012).

Wassermann, G.

G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Wolf, E.

G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Zhao, C.

T. T. Elazhary, P. Zhou, C. Zhao, and J. H. Burge, “Generalized sine condition,” Appl. Opt. 54(16), 5037–5049 (2015).
[Crossref] [PubMed]

J. H. Burge, C. Zhao, and S. H. Lu, “Use of the Abbe sine condition to quantify alignment aberrations in optical imaging systems,” Proc. SPIE 7652, 765219 (2010).
[Crossref]

Zhou, P.

Abh. Konigl. Ges. Wis. Gottingen Mathphys. Kl. (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik II,” Abh. Konigl. Ges. Wis. Gottingen Mathphys. Kl. 4, 1–3 (1905).

Appl. Opt. (2)

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

Opt. Express (3)

Opt. Rev. (1)

J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[Crossref]

Proc. Phys. Soc. B (1)

G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Proc. SPIE (2)

H. Ries, N. E. Shatz, J. C. Bortz, and W. Spirkl, “Consequences of skewness conservation for rotationally symmetric nonimaging devices,” Proc. SPIE 3139, 47–58 (1997).
[Crossref]

J. H. Burge, C. Zhao, and S. H. Lu, “Use of the Abbe sine condition to quantify alignment aberrations in optical imaging systems,” Proc. SPIE 7652, 765219 (2010).
[Crossref]

Other (3)

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

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9, pp. 413–468, (1873).

J. Rolland and K. Thompson, “Freeform optics: Evolution? no, revolution!” SPIE Newsroom (2012).

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Figures (4)

Fig. 1
Fig. 1 (Left) Two mirror aplanat under freeform prescription and (Right) plot of the integrability condition of the same illustrating it is not zero.
Fig. 2
Fig. 2 Illustration of a three surface freeform aplanatic system.
Fig. 3
Fig. 3 (Left) Illustration of a semi aplanatic system formed by two freeform mirrors and (Right) Plot describing the dependence of RMS spot size with object position in x and y directions.
Fig. 4
Fig. 4 (Left) Perspective of a three mirror freeform aplanatic system and (Right) distribution of RMS spot diameter along two directions.

Equations (5)

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u + | O O + u u u u | + u = L u = f ( u , u , u , L )
N = u O O + u u u u | O O + u u u u |
u + v + v + u = L u u + v v v v u u = O O
u x = p 0 + p M X u y = q 0 + q M Y
u u p v = u p ( 1 u v ) u u q v = u q ( 1 u v ) u u p v = u p ( 1 u v ) u u q v = u q ( 1 u v )

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