Abstract

Compact folded imaging systems often require freeform surfaces to correct astigmatic and other off-axis aberrations. However, aberration theory for non-rotational symmetric systems is quite complex and it is especially hard to quantify individual surface aberration contributions. In this paper we develop a matrix method based on the propagation of a differential ray pair, which allows determining the aberration contribution of each individual surface for any ray. We can mathematically prove that the sum of the aberrations is identical to the exact ray-tracing result at the image plane. A head-mounted display lens is employed for testing and verification of this method. As will be shown, the method proves to be a universal tool for aberration calculations within freeform system.

© 2016 Optical Society of America

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References

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  1. K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
    [Crossref]
  2. O. Cakmakci and J. Rolland, “Head-Worn Displays: A Review,” J. Disp. Technol. 2(3), 199–216 (2006).
    [Crossref]
  3. T Gissibl, S Thiele, A Herkommer, and H Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
    [Crossref]
  4. K. Takahashi, “Head or face mounted image display apparatus,” U.S. Patent No. 5,701,202. 23 Dec. 1997.
  5. Q. Meng, W. Wang, H. Ma, and J. Dong, “Easy-aligned off-axis three-mirror system with wide field of view using freeform surface based on integration of primary and tertiary mirror,” Appl. Opt. 53(14), 3028–3034 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  15. 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]
  16. A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).
  17. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space Optics (McGraw-Hill, 2010).
  18. H. Gross, ed., Handbook of Optical Systems, Volume 1, Fundamentals of Technical Optics (Wiley-VCH, 2005)
  19. D. Rausch and A. M. Herkommer, “Phase space approach to the use of integrator rods and optical arrays in illumination systems,” Adv. Opt. Technol. 1(1–2), 69–78 (2012).
  20. J. Koshel, Illumination Engineering: Design with Nonimaging Optics (Wiley-IEEE, 2013).
  21. A. M. Herkommer, “Phase space optics: an alternate approach to freeform optical systems,” Opt. Eng. 53(3), 031304 (2013).
    [Crossref]
  22. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69(12), 1710–1716 (1979).
    [Crossref]
  23. B. Chen and A. M. Herkommer, “High order surface aberration contributions from phase space analysis of differential rays,” Opt. Express 24(6), 5934–5945 (2016).
    [Crossref] [PubMed]
  24. Synopsis CodeV10, 8 News, https://optics.synopsys.com/codev/codev-whatsnew.html

2016 (2)

T Gissibl, S Thiele, A Herkommer, and H Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

B. Chen and A. M. Herkommer, “High order surface aberration contributions from phase space analysis of differential rays,” Opt. Express 24(6), 5934–5945 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (2)

2013 (2)

2012 (2)

D. Rausch and A. M. Herkommer, “Phase space approach to the use of integrator rods and optical arrays in illumination systems,” Adv. Opt. Technol. 1(1–2), 69–78 (2012).

K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
[Crossref]

2006 (1)

O. Cakmakci and J. Rolland, “Head-Worn Displays: A Review,” J. Disp. Technol. 2(3), 199–216 (2006).
[Crossref]

2005 (1)

1979 (1)

1976 (1)

Bastiaans, M. J.

Bauer, A.

Brewer, S. H.

Cakmakci, O.

O. Cakmakci and J. Rolland, “Head-Worn Displays: A Review,” J. Disp. Technol. 2(3), 199–216 (2006).
[Crossref]

Chen, B.

Dong, J.

Duerr, F.

Fuerschbach, K.

Giessen, H

T Gissibl, S Thiele, A Herkommer, and H Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

Gissibl, T

T Gissibl, S Thiele, A Herkommer, and H Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

Herkommer, A

T Gissibl, S Thiele, A Herkommer, and H Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

Herkommer, A. M.

B. Chen and A. M. Herkommer, “High order surface aberration contributions from phase space analysis of differential rays,” Opt. Express 24(6), 5934–5945 (2016).
[Crossref] [PubMed]

A. M. Herkommer, “Phase space optics: an alternate approach to freeform optical systems,” Opt. Eng. 53(3), 031304 (2013).
[Crossref]

D. Rausch and A. M. Herkommer, “Phase space approach to the use of integrator rods and optical arrays in illumination systems,” Adv. Opt. Technol. 1(1–2), 69–78 (2012).

Ma, H.

Meng, Q.

Meuret, Y.

Rausch, D.

D. Rausch and A. M. Herkommer, “Phase space approach to the use of integrator rods and optical arrays in illumination systems,” Adv. Opt. Technol. 1(1–2), 69–78 (2012).

Rolland, J.

O. Cakmakci and J. Rolland, “Head-Worn Displays: A Review,” J. Disp. Technol. 2(3), 199–216 (2006).
[Crossref]

Rolland, J. P.

Thiele, S

T Gissibl, S Thiele, A Herkommer, and H Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

Thienpont, H.

Thompson, K.

Thompson, K. P.

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. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
[Crossref]

Wang, W.

Adv. Opt. Technol. (1)

D. Rausch and A. M. Herkommer, “Phase space approach to the use of integrator rods and optical arrays in illumination systems,” Adv. Opt. Technol. 1(1–2), 69–78 (2012).

Appl. Opt. (1)

J. Disp. Technol. (1)

O. Cakmakci and J. Rolland, “Head-Worn Displays: A Review,” J. Disp. Technol. 2(3), 199–216 (2006).
[Crossref]

J. Opt. Soc. Am. (2)

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

Nat. Photonics (1)

T Gissibl, S Thiele, A Herkommer, and H Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016).
[Crossref]

Opt. Eng. (1)

A. M. Herkommer, “Phase space optics: an alternate approach to freeform optical systems,” Opt. Eng. 53(3), 031304 (2013).
[Crossref]

Opt. Express (4)

Opt. Photonics News (1)

K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012).
[Crossref]

Other (11)

Synopsis CodeV10, 8 News, https://optics.synopsys.com/codev/codev-whatsnew.html

A. Cox, A System of Optical Design (Focal Press, 1964).

J. Koshel, Illumination Engineering: Design with Nonimaging Optics (Wiley-IEEE, 2013).

A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space Optics (McGraw-Hill, 2010).

H. Gross, ed., Handbook of Optical Systems, Volume 1, Fundamentals of Technical Optics (Wiley-VCH, 2005)

J. M. Rodgers, “Catoptric optical system including concave and convex reflectors,” (1994). US Patent 5,309,276.

K. Takahashi, “Head or face mounted image display apparatus,” U.S. Patent No. 5,701,202. 23 Dec. 1997.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

R. Kingslake and R. B. Johnson, Lens Design Fundamental (SPIE Press, 2010).

W. Smith, Modern Optical Engineering (McGraw-Hill, 2000).

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Figures (5)

Fig. 1
Fig. 1

Ray definition and propagation in between parallel reference planes.

Fig. 2
Fig. 2

Illustration of ray propagation: a) defines the reference ray and the dummy surfaces, b) illustrates the ray propagation in the unfolded reference system of the dummy surfaces, c) illustrates the ray positions in phase space.

Fig. 3
Fig. 3

Freeform prism from reference [4].

Fig. 4
Fig. 4

Distortion analysis: In (a) – (d) the surface contributions are shown, where the red circles are the paraxial positions of the rays on image, and the blue cross represents the lateral distortion contribution of the surface, (e) illustrates the sum of the distortion contributions and is compared to the exact ray-tracing result at the image, (f) shows the surface contributions for a single ray.

Fig. 5
Fig. 5

Transverse aberrations for the (0°,0°) field, resolved for individual surface contributions.

Tables (1)

Tables Icon

Table 1 Lens data of the freeform prism of reference [4] including additional dummy surfaces

Equations (11)

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

v=n M N u=n L N .
r i+1 = T i r ' i where T i =( I dI 0 I ).
p ' i = S i r i .
Δ i = M I,i ( r ' i p ' i ).
M I,i = T n S n S i+1 T i ,
M I,i = M I,i+1 S i+1 T i .
Δ 1 = M I,1 ( r ' 1 p ' 1 ), Δ 2 = M I,2 ( r ' 2 p ' 2 ), Δ n = M I,n ( r ' n p ' n ).
M I,i r ' i = M I,i+1 S i+1 T i r ' i r i+1 = M I,i+1 p ' i+1 ,
Δ 1 = M I,1 ( r ' 1 p ' 1 )= M I,2 p ' 2 M I,1 p ' 1 , Δ 2 = M I,2 ( r ' 2 p ' 2 )= M I,3 p ' 2 M I,2 p ' 2 , Δ n = M I,n ( r ' n p ' n )= M I,n r ' n M I,n p ' n .
i Δ i = M I,n r ' n M I,1 p ' 1 .
i Δ i = r I p I .

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