Abstract

The field of optical fabrication has progressed to a point where manufacturing optical quality freeform surfaces is no longer prohibitive. However, to stimulate the development of freeform systems, optical designers must be provided with the necessary tools. Full-field displays are an example of such a tool. Identifying the field dependence of the dominant aberrations of a freeform system is critical for a controlled optimization and with the help of full-field displays, this can be accomplished. Of specific interest is coma, an often system-limiting aberration and an aberration that has recently been directly addressed with freeform surfaces. In this research, we utilize nodal aberration theory to develop a ray-based method to generate a coma full-field display that circumvents wavefront fitting errors that can affect Zernike polynomial-based full-field displays for highly aberrated freeform starting designs.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics

Aaron Bauer and Jannick P. Rolland
Opt. Express 23(22) 28141-28153 (2015)

Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces

Kyle Fuerschbach, Jannick P. Rolland, and Kevin P. Thompson
Opt. Express 20(18) 20139-20155 (2012)

Theory of aberration fields for general optical systems with freeform surfaces

Kyle Fuerschbach, Jannick P. Rolland, and Kevin P. Thompson
Opt. Express 22(22) 26585-26606 (2014)

References

  • View by:
  • |
  • |
  • |

  1. Y. Tohme, “Trends in Ultra-Precision Machining of Freeform Optical Surfaces,” Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing (2008).
  2. F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technology. 62(2), 823–846 (2013).
    [Crossref]
  3. J. D. Owen, M. A. Davies, D. Schmidt, and E. H. Urruti, “On the ultra-precision diamond machining of chalcogenide glass,” CIRP Annals - Manufacturing Technology. 64(1), 113–116 (2015).
    [Crossref]
  4. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
    [Crossref] [PubMed]
  5. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014).
    [Crossref] [PubMed]
  6. A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23(22), 28141–28153 (2015).
    [Crossref] [PubMed]
  7. K. P. Thompson, “Beyond Optical Design Interaction Between The Lens Designer And The Real World,” in International Optical Design Conference, (Proc. SPIE, 1985), 426–438.
  8. R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express 20(15), 16436–16449 (2012).
    [Crossref]
  9. K. P. Thompson, “Astigmatic focal surfaces in general optical systems: a new look for an old aberration,” J. Opt. Soc. Am. 72, 1726 (1982).
  10. H. Coddington, A Treatise on the Reflexion and Refraction of Light being Part 1 of A System of Optics (Cambridge, 1829).
  11. A. Bauer, K. P. Thompson, and J. P. Rolland, “Coma full-field display for freeform imaging systems,” Proc. SPIE Optifab 2015 9633, 963316 (2015).
  12. D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007), Vol. 59.
  13. H. D. Taylor, A System of Applied Optics: being a complete system of formulæ of the second order, and the foundation of a complete system of the third order, with examples of their practical application (Macmillan and Co., limited, New York; London; 1906).
  14. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
    [Crossref] [PubMed]
  15. W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, New York, 2007).
  16. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
    [Crossref] [PubMed]
  17. H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, 1950).
  18. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
    [Crossref] [PubMed]

2015 (2)

J. D. Owen, M. A. Davies, D. Schmidt, and E. H. Urruti, “On the ultra-precision diamond machining of chalcogenide glass,” CIRP Annals - Manufacturing Technology. 64(1), 113–116 (2015).
[Crossref]

A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23(22), 28141–28153 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (1)

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technology. 62(2), 823–846 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (1)

2009 (1)

2005 (1)

1982 (1)

K. P. Thompson, “Astigmatic focal surfaces in general optical systems: a new look for an old aberration,” J. Opt. Soc. Am. 72, 1726 (1982).

Bauer, A.

Cakmakci, O.

Davies, M. A.

J. D. Owen, M. A. Davies, D. Schmidt, and E. H. Urruti, “On the ultra-precision diamond machining of chalcogenide glass,” CIRP Annals - Manufacturing Technology. 64(1), 113–116 (2015).
[Crossref]

Dunn, C.

Evans, C.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technology. 62(2), 823–846 (2013).
[Crossref]

Fang, F. Z.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technology. 62(2), 823–846 (2013).
[Crossref]

Fuerschbach, K.

Gray, R. W.

Owen, J. D.

J. D. Owen, M. A. Davies, D. Schmidt, and E. H. Urruti, “On the ultra-precision diamond machining of chalcogenide glass,” CIRP Annals - Manufacturing Technology. 64(1), 113–116 (2015).
[Crossref]

Rolland, J. P.

Schmid, T.

Schmidt, D.

J. D. Owen, M. A. Davies, D. Schmidt, and E. H. Urruti, “On the ultra-precision diamond machining of chalcogenide glass,” CIRP Annals - Manufacturing Technology. 64(1), 113–116 (2015).
[Crossref]

Thompson, K.

Thompson, K. P.

Urruti, E. H.

J. D. Owen, M. A. Davies, D. Schmidt, and E. H. Urruti, “On the ultra-precision diamond machining of chalcogenide glass,” CIRP Annals - Manufacturing Technology. 64(1), 113–116 (2015).
[Crossref]

Weckenmann, A.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technology. 62(2), 823–846 (2013).
[Crossref]

Zhang, G. X.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technology. 62(2), 823–846 (2013).
[Crossref]

Zhang, X. D.

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technology. 62(2), 823–846 (2013).
[Crossref]

CIRP Annals - Manufacturing Technology. (2)

F. Z. Fang, X. D. Zhang, A. Weckenmann, G. X. Zhang, and C. Evans, “Manufacturing and measurement of freeform optics,” CIRP Annals - Manufacturing Technology. 62(2), 823–846 (2013).
[Crossref]

J. D. Owen, M. A. Davies, D. Schmidt, and E. H. Urruti, “On the ultra-precision diamond machining of chalcogenide glass,” CIRP Annals - Manufacturing Technology. 64(1), 113–116 (2015).
[Crossref]

J. Opt. Soc. Am. (1)

K. P. Thompson, “Astigmatic focal surfaces in general optical systems: a new look for an old aberration,” J. Opt. Soc. Am. 72, 1726 (1982).

J. Opt. Soc. Am. A (3)

Opt. Express (4)

Other (8)

Y. Tohme, “Trends in Ultra-Precision Machining of Freeform Optical Surfaces,” Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing (2008).

H. Coddington, A Treatise on the Reflexion and Refraction of Light being Part 1 of A System of Optics (Cambridge, 1829).

A. Bauer, K. P. Thompson, and J. P. Rolland, “Coma full-field display for freeform imaging systems,” Proc. SPIE Optifab 2015 9633, 963316 (2015).

D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007), Vol. 59.

H. D. Taylor, A System of Applied Optics: being a complete system of formulæ of the second order, and the foundation of a complete system of the third order, with examples of their practical application (Macmillan and Co., limited, New York; London; 1906).

K. P. Thompson, “Beyond Optical Design Interaction Between The Lens Designer And The Real World,” in International Optical Design Conference, (Proc. SPIE, 1985), 426–438.

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, New York, 2007).

H. H. Hopkins, Wave Theory of Aberrations (Clarendon Press, 1950).

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 (9)

Fig. 1
Fig. 1 (left) An unobscured freeform starting design. This early stage system is highly aberrated as a result of the many surface tilts. (middle) The Zernike coma FFD in commercial software fails to plot due to a wavefront fitting error. (right). The ray-based method in this paper successfully plots a FFD for the same system, illustrating the value of the algorithm.
Fig. 2
Fig. 2 (left) The arrows illustrate the effect that elliptical coma has on circular coma in a rotationally symmetric optical system. (middle) A full circular coma blur is shown sampled over the pupil (shown in black) together with the corresponding blur after the addition of elliptical coma (shown in blue). (right) The final blur is the combination of circular and elliptical coma shown alone. (Image adapted from [14])
Fig. 3
Fig. 3 The coma FFD process is shown, from using the algorithm to plotting the symbols, in the above flow chart.
Fig. 4
Fig. 4 (top) A nominally rotationally symmetric f/4 Cooke Triplet with a 20° full FOV. The middle element is decentered in both X and Y by 5 microns. (bottom, left) The effect of the decentering can be seen as the node is moved off-axis. (bottom, right) The Zernike coma FFD shows good correlation between the two types of FFDs. The 180° symbol rotation is due to the transfer from the pupil plane (Zernike FFD) to the image plane (ray-based coma FFD).
Fig. 5
Fig. 5 (left) A fully unobscured freeform viewfinder with an external pupil. This design covers a 25° full diagonal FOV with a 12 mm eyebox. (right) The resulting coma FFD shows the impact of the elliptical coma and the non-symmetric nature of the optical system.
Fig. 6
Fig. 6 (left) Ray-based coma FFD showing only the elliptical coma contribution plotted as lines for a direct comparison with the (right) Zernike trefoil FFD, which shows only elliptical coma through 5th-order.
Fig. 7
Fig. 7 (left) A fully unobscured freeform telescope from [16]. This f/1.9 system with an 8°x6° full FOV has a larger elliptical coma contribution than it does circular coma. (right) The resulting coma FFD for this system. The elliptical blur completely overtakes the classic “coma” shape.
Fig. 8
Fig. 8 (left) Ray-based coma FFD showing only the circular coma contribution. (right) Zernike coma (Z7/8) FFD. At the full aperture, the Zernike coma FFD has a significant contribution from aberrations that are higher-order in aperture, which were made negligible by generating this Zernike FFD at a reduced aperture, allowing a comparison to the ray-based coma FFD.
Fig. 9
Fig. 9 (left) Ray-based coma FFD showing only the elliptical coma contribution plotted as lines for a direct comparison with (right) the Zernike trefoil (Z10/11) FFD. At the full aperture, the Zernike trefoil FFD has a significant contribution from aberrations that are higher-order in aperture, which were made negligible by generating this Zernike FFD at a reduced aperture, allowing a comparison to the ray-based coma FFD.

Equations (45)

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

n u ε = 2 [ ] 220 m ρ + [ ] 222 2 ρ * + 2 { ( [ ] 131 + [ ] 331 m ) ρ } ρ + ( ρ ρ ) ( [ ] 131 + [ ] 331 m ) + 3 4 [ ] 333 3 ρ 2 * + O ( ρ 3 + ) ,
ϵ a = 2 [ ] 220 m ρ a + [ ] 222 2 ρ a * ,
ϵ a 1 = 2 [ ] 220 m ρ a 1 + [ ] 222 2 ρ a 1 * ,
ϵ a 2 = 2 [ ] 220 m ρ a 2 + [ ] 222 2 ρ a 2 * ,
ϵ a c = 2 [ ] 220 m ρ c + [ ] 222 2 ρ c * + 2 { ( [ ] 131 + [ ] 331 m ) ρ c } ρ c + ( ρ c ρ c ) ( [ ] 131 + [ ] 331 m ) + 3 4 [ ] 333 3 ρ c 2 * ,
ϵ c = 2 { ( [ ] 131 + [ ] 331 m ) ρ c } ρ c + ( ρ c ρ c ) ( [ ] 131 + [ ] 331 m ) + 3 4 [ ] 333 3 ρ c 2 * ,
ϵ c = ϵ a c 2 [ ] 220 m ρ c + [ ] 222 2 ρ c * .
ϵ c 1 = 2 { ( [ ] 131 + [ ] 331 m ) ρ c 1 } ρ c 1 + ( ρ c 1 ρ c 1 ) ( [ ] 131 + [ ] 331 m ) + 3 4 [ ] 333 3 ρ c 1 2 * ,
ϵ c 2 = 2 { ( [ ] 131 + [ ] 331 m ) ρ c 2 } ρ c 2 + ( ρ c 2 ρ c 2 ) ( [ ] 131 + [ ] 331 m ) + 3 4 [ ] 333 3 ρ c 2 2 * .
ρ c 1 = 0 x ^ + ρ y ^ ,
ρ c 2 = ρ x ^ + 0 y ^ .
θ e l l = 1 2 ( 3 θ 33 θ 31 ) .
[ ] 220 m W 220 m ( H H ) 2 ( H A 220 m ) + B 220 m
[ ] 222 2 W 222 H 2 2 H A 222 + B 222 2 ,
[ ] 131 W 131 H A 131 ,
[ ] 331 m W 331 m ( H H ) H 2 ( H A 331 m ) H + 2 B 331 m H ( H H ) A 331 m + B 331 m 2 H * C 331 m
[ ] 333 3 W 333 H 3 3 H 2 A 333 + 3 H B 333 2 C 333 3 ,
A k l m j W k l m j σ j ,
B k l m j W k l m j ( σ j σ j ) ,
B k l m 2 j W k l m j σ j 2 ,
C k l m j W k l m j ( σ j σ j ) σ j ,
C k l m 3 j W k l m j σ j 3 .
ϵ a 1 = 2 [ ] 220 m ρ a 1 + [ ] 222 2 ρ a 1 * ,
ϵ a 2 = 2 [ ] 220 m ρ a 2 + [ ] 222 2 ρ a 2 * .
ϵ a 1 ρ a 1 = 2 [ ] 220 m ρ a 1 2 + [ ] 222 2 | ρ a 1 | 2 ,
ϵ a 2 ρ a 2 = 2 [ ] 220 m ρ a 2 2 + [ ] 222 2 | ρ a 2 | 2 .
ϵ a 2 ρ a 2 ϵ a 1 ρ a 1 = 2 [ ] 220 m ( ρ a 2 2 ρ a 1 2 ) .
( ϵ a 2 ρ a 2 ϵ a 1 ρ a 1 ) ( ρ a 2 2 ρ a 1 2 ) = 2 [ ] 220 m ( ρ a 2 2 ρ a 1 2 ) ( ρ a 2 2 ρ a 1 2 ) ,
[ ] 220 m = ( ϵ a 2 ρ a 2 ϵ a 1 ρ a 1 ) ( ρ a 2 2 ρ a 1 2 ) 2 ( ρ a 2 2 ρ a 1 2 ) ( ρ a 2 2 ρ a 1 2 ) .
[ ] 222 2 = ϵ a 1 ρ a 1 2 [ ] 220 m ρ a 1 2 | ρ a 1 | 2 .
ϵ c 1 = 2 { ( [ ] 131 + [ ] 331 m ) ρ c 1 } ρ c 1 + ( ρ c 1 ρ c 1 ) ( [ ] 131 + [ ] 331 m ) + 3 4 [ ] 333 3 ρ c 1 2 * ,
ϵ c 2 = 2 { ( [ ] 131 + [ ] 331 m ) ρ c 2 } ρ c 2 + ( ρ c 2 ρ c 2 ) ( [ ] 131 + [ ] 331 m ) + 3 4 [ ] 333 3 ρ c 2 2 * .
α [ ] 131 + [ ] 331 m ,
β [ ] 333 3 .
ϵ c 1 = 2 ( α ρ c 1 ) ρ c 1 + ( ρ c 1 ρ c 1 ) α + 3 4 β ρ c 1 2 * ,
ϵ c 2 = 2 ( α ρ c 2 ) ρ c 2 + ( ρ c 2 ρ c 2 ) α + 3 4 β ρ c 2 2 * .
ρ c 1 = 0 x ^ + ρ c y ^ ,
ρ c 2 = ρ c x ^ + 0 y ^ .
ϵ c 1 = 2 ( α x ρ c 1 x + α y ρ c 1 y ) ( ρ c 1 x x ^ + ρ c 1 y y ^ ) + | ρ c 1 | 2 α + 3 4 { [ β x ( ρ c 1 y 2 ρ c 1 x 2 ) 2 β y ρ c 1 x ρ c 1 y ] x ^ + [ β y ( ρ c 1 y 2 ρ c 1 x 2 ) + 2 β x ρ c 1 x ρ c 1 y ] y ^ }
ϵ c 2 = 2 ( α x ρ c 2 x + α y ρ c 2 y ) ( ρ c 2 x x ^ + ρ c 2 y y ^ ) + | ρ c 2 | 2 α + 3 4 { [ β x ( ρ c 2 y 2 ρ c 2 x 2 ) 2 β y ρ c 2 x ρ c 2 y ] x ^ + [ β y ( ρ c 2 y 2 ρ c 2 x 2 ) + 2 β x ρ c 2 x ρ c 2 y ] y ^ }
ϵ c 1 = 2 α y ρ c 2 y ^ + ρ c 2 α + 3 4 ρ c 2 ( β x x ^ + β y y ^ ) ,
ϵ c 2 = 2 α x ρ c 2 x ^ + ρ c 2 α 3 4 ρ c 2 ( β x x ^ + β y y ^ ) .
ϵ c 1 + ϵ c 2 = 4 ρ c 2 α .
α = ϵ c 1 + ϵ c 2 4 ρ c 2 .
β = 4 3 ( ϵ c 1 ρ c 2 α 2 α y y ^ ) ,

Metrics