Abstract

We present here aplanatic systems in 3D geometry as a limiting case of a SMS 3D design. We extend the basic formulations governing rotationally symmetric aplanatic systems to freeform aplanatic systems and provide a formal proof that a SMS 3D design in the limiting case of 3 coincident points leads to a freeform aplanatic system.

© 2016 Optical Society of America

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References

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  1. P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
    [Crossref]
  2. P. Benitez, J. C. Miñano, J. Chaves, and A. Santamaria, “SMS freeforms for illumination,” Adv. Opt. Technol. 2, 323–329 (2013).
  3. J. Chaves, Introduction to Nonimaging Optics, II nd edition (CRC, 2015).
  4. J. J. Braat and P. F. Greve, “Aplanatic optical system containing two aspheric surfaces,” Appl. Opt. 18(13), 2187–2191 (1979).
    [Crossref] [PubMed]
  5. A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70(10), 945–949 (1957).
    [Crossref]
  6. D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
    [Crossref]
  7. R. V. Willstrop and D. Lynden-Bell, “Exact optics - II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
    [Crossref]
  8. K. Schwarzschild, “Untersuchungen zur geometrischen optik I-III,” Abh. Konigl. Ges. Wis. Gottingen Mathphys.Kl. 4, Nos. 1–3 (1905–1906).
  9. 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]
  10. E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv Für Mikroskopische Anatomie, 9(1), 413–418 (1873).
    [Crossref]
  11. J. H. Burge, C. Zhao, and M. Dubin, “Use of the abbe sine condition to quantify alignment aberrations in optical imaging systems,” in International Optical Design Conference and Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuD5.
    [Crossref]
  12. T. T. Elazhary, P. Zhou, C. Zhao, and J. H. Burge, “Generalized sine condition,” Appl. Opt. 54(16), 5037–5049 (2015).
    [Crossref] [PubMed]
  13. S. A. Comastri, J. M. Simon, and R. Blendowske, “Generalized sine condition for image-forming systems with centering errors,” J. Opt. Soc. Am. A 16(3), 602–612 (1999).
    [Crossref]
  14. C. Zhao and J. H. Burge, “Conditions for correction of linear and quadratic field-dependent aberrations in plane-symmetric optical systems,” J. Opt. Soc. Am. A 19(12), 2467–2472 (2002).
    [Crossref] [PubMed]
  15. J. C. Miñano, R. Mohedano, and P. Benítez, Nonimaging Optics. The Optics Encyclopedia (Wiley, 2015).
  16. M. Herzberger, “On the fundamental optical invariant, the optical tetrality principle, and on the new development of gaussian optics based on this Law,” J. Opt. Soc. Am. 25(9), 295–304 (1935).
    [Crossref]
  17. M. Herzberger, “First-order laws in asymmetrical optical systems part I. The image of a given congruence: fundamental conceptions,” J. Opt. Soc. Am. 26(9), 354–359 (1936).
    [Crossref]
  18. M. Herzberger, “First-order laws in asymmetrical optical systems II. The image congruences belonging to the rays emerging from a point in object and image space; fundamental forms,” J. Opt. Soc. Am. 26(11), 389–406 (1936).
    [Crossref]
  19. B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

2015 (2)

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

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

2013 (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]

P. Benitez, J. C. Miñano, J. Chaves, and A. Santamaria, “SMS freeforms for illumination,” Adv. Opt. Technol. 2, 323–329 (2013).

2004 (1)

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

2003 (1)

R. V. Willstrop and D. Lynden-Bell, “Exact optics - II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[Crossref]

2002 (2)

1999 (1)

1979 (1)

1957 (1)

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70(10), 945–949 (1957).
[Crossref]

1936 (2)

1935 (1)

1873 (1)

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv Für Mikroskopische Anatomie, 9(1), 413–418 (1873).
[Crossref]

Abbe, E.

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv Für Mikroskopische Anatomie, 9(1), 413–418 (1873).
[Crossref]

Benitez, P.

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

P. Benitez, J. C. Miñano, J. Chaves, and A. Santamaria, “SMS freeforms for illumination,” Adv. Opt. Technol. 2, 323–329 (2013).

Benítez, P.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Blen, J.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Blendowske, R.

Braat, J. J.

Burge, J. H.

Chaves, J.

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

P. Benitez, J. C. Miñano, J. Chaves, and A. Santamaria, “SMS freeforms for illumination,” Adv. Opt. Technol. 2, 323–329 (2013).

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Comastri, S. A.

Dross, O.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Duerr, F.

Elazhary, T. T.

Falicoff, W.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Grabovickic, D.

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

Greve, P. F.

Head, A. K.

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70(10), 945–949 (1957).
[Crossref]

Hernández, M.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Herzberger, M.

Infante, J.

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

Lynden-Bell, D.

R. V. Willstrop and D. Lynden-Bell, “Exact optics - II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[Crossref]

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
[Crossref]

Meuret, Y.

Miñano, J. C.

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

P. Benitez, J. C. Miñano, J. Chaves, and A. Santamaria, “SMS freeforms for illumination,” Adv. Opt. Technol. 2, 323–329 (2013).

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Mohedano, R.

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Narasimhan, B.

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

Nikolic, M.

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

Santamaria, A.

P. Benitez, J. C. Miñano, J. Chaves, and A. Santamaria, “SMS freeforms for illumination,” Adv. Opt. Technol. 2, 323–329 (2013).

Simon, J. M.

Thienpont, H.

Willstrop, R. V.

R. V. Willstrop and D. Lynden-Bell, “Exact optics - II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[Crossref]

Zhao, C.

Zhou, P.

Adv. Opt. Technol. (1)

P. Benitez, J. C. Miñano, J. Chaves, and A. Santamaria, “SMS freeforms for illumination,” Adv. Opt. Technol. 2, 323–329 (2013).

Appl. Opt. (2)

Archiv Für Mikroskopische Anatomie, (1)

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Archiv Für Mikroskopische Anatomie, 9(1), 413–418 (1873).
[Crossref]

J. Opt. Soc. Am. (3)

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

Mon. Not. R. Astron. Soc. (2)

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
[Crossref]

R. V. Willstrop and D. Lynden-Bell, “Exact optics - II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[Crossref]

Opt. Eng. (1)

P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489 (2004).
[Crossref]

Opt. Express (1)

Proc. Phys. Soc. London Sec. B (1)

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Sec. B 70(10), 945–949 (1957).
[Crossref]

Proc. SPIE (1)

B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

Other (4)

J. C. Miñano, R. Mohedano, and P. Benítez, Nonimaging Optics. The Optics Encyclopedia (Wiley, 2015).

J. Chaves, Introduction to Nonimaging Optics, II nd edition (CRC, 2015).

K. Schwarzschild, “Untersuchungen zur geometrischen optik I-III,” Abh. Konigl. Ges. Wis. Gottingen Mathphys.Kl. 4, Nos. 1–3 (1905–1906).

J. H. Burge, C. Zhao, and M. Dubin, “Use of the abbe sine condition to quantify alignment aberrations in optical imaging systems,” in International Optical Design Conference and Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuD5.
[Crossref]

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Figures (3)

Fig. 1
Fig. 1 Illustration of the SMS design principle in 2d and 3d cases.
Fig. 2
Fig. 2 Nomenclature used in establishing the link between SMS and aplanatism.
Fig. 3
Fig. 3 A three mirror freeform aplanatic system showing rays linking the origin of the object space with the origin of the image space.

Equations (16)

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ρ = ( x , y , 0 ) t = ( p , q , r ) ρ = ( x , y , 0 ) t = ( p , q , r )
ρ x t p ρ p t x = ρ x t p ρ p t x = 1 ρ x t q ρ q t x = ρ x t q ρ q t x = 0 ρ y t q ρ q t y = ρ y t q ρ q t y = 1 ρ y t p ρ p t y = ρ y t p ρ p t y = 0
x ´ = A ( x , y , p , q ) = a 00 ( p , q ) + a 10 ( p , q ) x + a 01 ( p , q ) y + ... y ´ = B ( x , y , p , q ) = b 00 ( p , q ) + b 10 ( p , q ) x + b 01 ( p , q ) y + ... p ´ = C ( x , y , p , q ) = c 00 ( p , q ) + c 10 ( p , q ) x + c 01 ( p , q ) y + ... q ´ = D ( x , y , p , q ) = d 00 ( p , q ) + d 10 ( p , q ) x + d 01 ( p , q ) y + ...
x = A ( 0 , 0 , p , q ) = a 00 ( p , q ) = 0 y = B ( 0 , 0 , p , q ) = b 00 ( p , q ) = 0
( a 10 b 10 a 01 b 01 ) ( c 00 p c 00 q d 00 p d 00 q ) = ( 1 0 0 1 )
( c 00 p c 00 q d 00 p d 00 q ) = ( M X 1 0 0 M Y 1 )
p = p 0 + p M X q = q 0 + q M Y
x = M X x + ... y = M Y y + ... p = p 0 + p M X + ... q = q 0 + q M Y + ...
x = A 0 + A 1 x + A 2 y + F ( x , y , p , q , x α , y α , x β , y β , x γ , y γ )
( A 0 A 1 A 2 ) = ( 1 x α y α 1 x β y β 1 x γ y γ ) 1 ( x α x β x γ )
F ( x , y , p , q , x α , y α , x β , y β , x γ , y γ ) = i , j , k = { 0 , 1 } A i j k ( x x α ) i ( y y α ) 1 i ( x x β ) j ( y y β ) 1 j ( x x γ ) k ( y y γ ) 1 k
F x | x = x α y = y α = j , k = { 0 , 1 } A 1 j k ( x α x β ) j ( y α y β ) 1 j ( x α x γ ) k ( y α y γ ) 1 k F y | x = x α y = y α = j , k = { 0 , 1 } A 0 j k ( x α x β ) j ( y α y β ) 1 j ( x α x γ ) k ( y α y γ ) 1 k
x x | x = x α y = y α = A 1 x y | x = x α y = y α = A 2
( A 0 A 1 A 2 ) = ( 1 0 0 1 Δ x 0 1 0 Δ y ) 1 ( 0 Δ x ´ 0 ) = ( 1 0 0 Δ x 1 Δ x 1 0 Δ y 1 0 Δ y 1 ) ( 0 Δ x ´ 0 ) = ( 0 Δ x / Δ x 0 )
( B 0 B 1 B 2 ) = ( 0 0 Δ y / Δ y )
x = Δ x Δ x x + ... y ' = Δ y Δ y y + ...

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