Abstract

The newly formulated theory of aberration fields of freeform surfaces describes the aberrations that freeform Zernike polynomial surfaces can correct within folded powered optical systems. This theory has guided the design of an OLED-based reflective freeform electronic viewfinder covering a 25° full field-of-view with a 12 mm eyebox, which is reported together with a detailed methodology that begins with developing an unobscured starting point and ends with an optimized freeform design, analyzed both in display and visual spaces. In addition, tolerancing of the system points to the potential low sensitivity of these systems to manufacturing tilt (10 arcmin), decenter and despace (100 µm), and figure errors (λ/2 @ 0.632 µm).

© 2015 Optical Society of America

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References

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    [Crossref]
  4. A. Bauer and J. P. Rolland, “Visual space assessment of two all-reflective, freeform, optical see-through head-worn displays,” Opt. Express 22(11), 13155–13163 (2014).
    [Crossref] [PubMed]
  5. 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]
  6. J. Han, J. Liu, X. Yao, and Y. Wang, “Portable waveguide display system with a large field of view by integrating freeform elements and volume holograms,” Opt. Express 23(3), 3534–3549 (2015).
    [Crossref] [PubMed]
  7. Y. Tohme, “Trends in ultra-precision machining of freeform optical surfaces,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing (OSA, 2008), paper OThC6.
  8. K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, φ-polynomial, Zernike mirror,” Opt. Lett. 39(1), 18–21 (2014).
    [Crossref] [PubMed]
  9. Y.-S. Ghim, H.-G. Rhee, A. Davies, H.-S. Yang, and Y.-W. Lee, “3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry,” Opt. Express 22(5), 5098–5105 (2014).
    [Crossref] [PubMed]
  10. E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
    [Crossref]
  11. K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors,” Opt. Lett. 39(10), 2896–2899 (2014).
    [Crossref] [PubMed]
  12. 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]
  13. 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]
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  15. J. Barbur and A. Stockman, “Photopic, mesopic, and scotopic vision and changes in visual performance,” in Encyclopedia of the Eye, D. A. Dartt, J. C. Besharse, and R. Dana, eds. (Academic, 2010), pp. 323–331.
  16. 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]
  17. R. W. Gray and J. P. Rolland, “Wavefront aberration function in terms of R. V. Shack’s vector product and Zernike polynomial vectors,” J. Opt. Soc. Am. A 32(10), 1836–1847 (2015).
    [Crossref]
  18. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20(3), 2483–2499 (2012).
    [Crossref] [PubMed]
  19. I. Kaya, K. P. Thompson, and J. P. Rolland, “Comparative assessment of freeform polynomials as optical surface descriptions,” Opt. Express 20(20), 22683–22691 (2012).
    [Crossref] [PubMed]
  20. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008).
    [Crossref] [PubMed]
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  22. A. Bauer, S. Vo, K. Parkins, F. Rodriguez, O. Cakmakci, and J. P. Rolland, “Computational optical distortion correction using a radial basis function-based mapping method,” Opt. Express 20(14), 14906–14920 (2012).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2015 (2)

2014 (5)

2013 (1)

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

2012 (4)

2011 (1)

2008 (1)

2007 (1)

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

2005 (1)

2002 (1)

1999 (1)

Baker, J. G.

Bauer, A.

Cakmakci, O.

Davies, A.

Davis, G. E.

De Chiffre, L.

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

Dunn, C.

Evans, C.

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

Fang, F.

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

Forbes, G. W.

Foroosh, H.

Fuerschbach, K.

Ghim, Y.-S.

Gray, R. W.

Ha, Y.

Han, J.

Kaya, I.

Lee, Y.-W.

Liu, J.

Moore, B.

Parkins, K.

Plummer, W. T.

Rhee, H.-G.

Rodriguez, F.

Rolland, J.

Rolland, J. P.

R. W. Gray and J. P. Rolland, “Wavefront aberration function in terms of R. V. Shack’s vector product and Zernike polynomial vectors,” J. Opt. Soc. Am. A 32(10), 1836–1847 (2015).
[Crossref]

K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors,” Opt. Lett. 39(10), 2896–2899 (2014).
[Crossref] [PubMed]

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]

K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, φ-polynomial, Zernike mirror,” Opt. Lett. 39(1), 18–21 (2014).
[Crossref] [PubMed]

A. Bauer and J. P. Rolland, “Visual space assessment of two all-reflective, freeform, optical see-through head-worn displays,” Opt. Express 22(11), 13155–13163 (2014).
[Crossref] [PubMed]

I. Kaya, K. P. Thompson, and J. P. Rolland, “Comparative assessment of freeform polynomials as optical surface descriptions,” Opt. Express 20(20), 22683–22691 (2012).
[Crossref] [PubMed]

A. Bauer, S. Vo, K. Parkins, F. Rodriguez, O. Cakmakci, and J. P. Rolland, “Computational optical distortion correction using a radial basis function-based mapping method,” Opt. Express 20(14), 14906–14920 (2012).
[Crossref] [PubMed]

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]

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]

O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008).
[Crossref] [PubMed]

Savio, E.

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

Schmitt, R.

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

Thompson, K.

Thompson, K. P.

Van Tassell, J.

Vo, S.

Wang, Y.

Weckenmann, A.

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

Yang, H.-S.

Yao, X.

Zhang, G.

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

Zhang, X.

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

Appl. Opt. (2)

CIRP Ann. (2)

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56(2), 810–835 (2007).
[Crossref]

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

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

Opt. Express (10)

A. Bauer and J. P. Rolland, “Visual space assessment of two all-reflective, freeform, optical see-through head-worn displays,” Opt. Express 22(11), 13155–13163 (2014).
[Crossref] [PubMed]

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]

J. Han, J. Liu, X. Yao, and Y. Wang, “Portable waveguide display system with a large field of view by integrating freeform elements and volume holograms,” Opt. Express 23(3), 3534–3549 (2015).
[Crossref] [PubMed]

O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16(3), 1583–1589 (2008).
[Crossref] [PubMed]

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]

G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20(3), 2483–2499 (2012).
[Crossref] [PubMed]

A. Bauer, S. Vo, K. Parkins, F. Rodriguez, O. Cakmakci, and J. P. Rolland, “Computational optical distortion correction using a radial basis function-based mapping method,” Opt. Express 20(14), 14906–14920 (2012).
[Crossref] [PubMed]

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]

I. Kaya, K. P. Thompson, and J. P. Rolland, “Comparative assessment of freeform polynomials as optical surface descriptions,” Opt. Express 20(20), 22683–22691 (2012).
[Crossref] [PubMed]

Y.-S. Ghim, H.-G. Rhee, A. Davies, H.-S. Yang, and Y.-W. Lee, “3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry,” Opt. Express 22(5), 5098–5105 (2014).
[Crossref] [PubMed]

Opt. Lett. (2)

Other (5)

Y. Tohme, “Trends in ultra-precision machining of freeform optical surfaces,” in Frontiers in Optics 2008/Laser Science XXIV/Plasmonics and Metamaterials/Optical Fabrication and Testing (OSA, 2008), paper OThC6.

A. M. Bauer and J. P. Rolland, “Design process for an all-reflective freeform electronic viewfinder,” in Imaging and Applied Optics 2015, OSA Technical Digest (online) (OSA, 2015), FW3B.2.

J. Barbur and A. Stockman, “Photopic, mesopic, and scotopic vision and changes in visual performance,” in Encyclopedia of the Eye, D. A. Dartt, J. C. Besharse, and R. Dana, eds. (Academic, 2010), pp. 323–331.

M. Chrisp and B. Primeau, “Imaging with NURBS Freeform Surfaces,” in Imaging and Applied Optics 2015, OSA Technical Digest (online) (OSA, 2015), FW2B.1.

E. Abbe, “Lens System,” United States Patent US697959 (1902).

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Figures (14)

Fig. 1
Fig. 1 Cross-sectional view of the Polaroid SX-70 camera. Released in 1972, this camera represented the first commercial product leveraging freeform optical surfaces. The freeform lenses were low-order XY-polynomials and are circled above. (Image adapted from [1])
Fig. 2
Fig. 2 The resulting aberration contributions after adding a Zernike (a) astigmatism, (b) coma, or (c) elliptical coma contribution to a surface within an optical system. The resulting aberrations are (a) field-constant astigmatism, (b) field-linear medial field curvature, field-linear field-asymmetric astigmatism, and field-constant coma, and (c) field-constant elliptical coma and field-linear field-conjugate astigmatism. (Image adapted from [12])
Fig. 3
Fig. 3 Two possible unobscured two-mirror geometries for the electronic viewfinder that may serve as a starting point. Note that the OLEDs could be tilted to mitigate the focal plane tilt in these starting points, but that degree of freedom was removed with the system telecentricity constraint.
Fig. 4
Fig. 4 Aberration FFDs for the starting point shown in Fig. 3(a) for Zernike (a) defocus/FC, (b) astigmatism, (c) astigmatism after the field-constant component is digitally removed for illustration purposes, and (d) coma. The tilt-induced field-constant astigmatism is dominating in (b). All other aberration contributions not shown here are negligible.
Fig. 5
Fig. 5 Aberration FFDs for the starting point shown in Fig. 3(b) for Zernike (a) defocus/FC, (b) astigmatism, (c) astigmatism after the field-constant component is digitally removed for illustration purposes, and (d) coma. The limiting aberrations are similar to those shown in Fig. 4, but the orientation of the coma is rotated 180° due to the alternate rotation of the secondary mirror. All other aberration contributions not shown here are negligible.
Fig. 6
Fig. 6 Aberration FFDs for the system after adding an astigmatic contribution to correct the field-constant astigmatism. The plots show Zernike (a) defocus/FC, (b) astigmatism, and (c) coma. Notice the field-constant astigmatism has essentially been optically eliminated, leaving field-linear field asymmetric astigmatism.
Fig. 7
Fig. 7 Aberration FFDs for the system after optimizing with a comatic contribution on both surfaces. The plots show Zernike (a) defocus/FC, (b) astigmatism, (c) coma, (d) elliptical coma, (e) oblique spherical, and (f) RMS wavefront error. Note the scale decrease of 6x from Fig. 6.
Fig. 8
Fig. 8 Aberration FFDs for the system after optimizing with an elliptical coma contribution on both surfaces. The plots show Zernike (a) defocus/FC, (b) astigmatism, (c) coma, (d) elliptical coma, (e) oblique spherical, and (f) RMS wavefront error. Note the ~2x scale decrease from Fig. 7.
Fig. 9
Fig. 9 Aberration FFDs for the system after adding conics and spherical aberration contributions to the mirrors. The plots show Zernike (a) defocus/FC, (b) astigmatism, (c) coma, (d) elliptical coma, (e) oblique spherical, and (f) RMS wavefront error.
Fig. 10
Fig. 10 (left) The final design layout. The two mirrors are in a slightly different orientation than the starting design. (right) The primary and secondary mirror shown with the best-fit manufacturing sphere removed. The maximum departure of the primary and secondary mirrors are 350 µm and 330 µm, respectively.
Fig. 11
Fig. 11 Aberration FFDs for the final viewfinder system. The plots show Zernike (a) defocus/FC, (b) astigmatism, (c) coma, (d) elliptical coma, (e) oblique spherical, and (f) RMS wavefront error. The average RMS wavefront error is 0.46 waves. Note the scale decrease of 2.5x from Fig. 9.
Fig. 12
Fig. 12 The black outer circle represents the full 12 mm eyebox diameter over which the 3 mm subpupils are sampled to simulate the eye as the aperture stop of the viewfinder. The system is symmetric about the vertical axis, so only fields on one half are sampled. The limiting subpupil is the filled circle.
Fig. 13
Fig. 13 (left) MTF FFD results for 0 degree and 90 degree orientations of the object in display space for the limiting case subpupil highlighted in Fig. 12. The columns show 40 lp/mm (80% Nyquist of OLED) and 50 lp/mm (Nyquist of OLED). (right) A simulated image seen through the viewfinder. The 4% distortion is best seen at the bottom corners.
Fig. 14
Fig. 14 MTF FFD results for the viewfinder in visual space for 0 degree and 90 degree object orientations for the limiting case subpupil highlighted in Fig. 12. The angular frequencies correspond to the spatial frequencies shown in the MTF FFDs of Fig. 13.

Tables (2)

Tables Icon

Table 1 Electronic Viewfinder Specifications

Tables Icon

Table 2 Percent MTF drop for each tolerance at 35 lp/mm, (λ = 632 nm).

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