Abstract

In this paper, we proposed a general direct design method for three-dimensional freeform surfaces and freeform imaging systems based on a construction-iteration process. In the preliminary surfaces-construction process, the coordinates as well as the surface normals of the data points on the multiple freeform surfaces can be calculated directly considering the rays of multiple fields and different pupil coordinates. Then, an iterative process is employed to significantly improve the image quality or achieve a better mapping relationship of the light rays. Three iteration types which are normal iteration, negative feedback and successive approximation are given. The proposed construction-iteration method is applied in the design of an easy aligned, low F-number off-axis three-mirror system. The primary and tertiary mirrors can be fabricated on a single substrate and form a single element in the final system. The secondary mirror is simply a plane mirror. With this configuration, the alignment difficulty of a freeform system can be greatly reduced. After the preliminary surfaces-construction stage, the freeform surfaces in the optical system can be generated directly from an initial planar system. Then, with the iterative process, the average RMS spot diameter decreased by 75.4% compared with the system before iterations, and the maximum absolute distortion decreased by 94.2%. After further optimization with optical design software, good image quality which is closed to diffraction-limited is achieved.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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  8. Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  27. J. C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express 17(26), 24036–24044 (2009).
    [PubMed]
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    [Crossref] [PubMed]
  29. T. Yang, J. Zhu, and G. Jin, “Design of freeform imaging systems with linear field-of-view using a construction and iteration process,” Opt. Express 22(3), 3362–3374 (2014).
    [Crossref] [PubMed]
  30. J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A 31(11), 2401–2408 (2014).
    [Crossref] [PubMed]
  31. Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010).
    [Crossref] [PubMed]
  32. 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]

2015 (1)

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

2014 (9)

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

T. Yang, J. Zhu, and G. Jin, “Design of freeform imaging systems with linear field-of-view using a construction and iteration process,” Opt. Express 22(3), 3362–3374 (2014).
[Crossref] [PubMed]

T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
[Crossref] [PubMed]

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]

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]

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]

D. Cheng, Y. Wang, C. Xu, W. Song, and G. Jin, “Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics,” Opt. Express 22(17), 20705–20719 (2014).
[Crossref] [PubMed]

J. Zhu, X. Wu, T. Yang, and G. Jin, “Generating optical freeform surfaces considering both coordinates and normals of discrete data points,” J. Opt. Soc. Am. A 31(11), 2401–2408 (2014).
[Crossref] [PubMed]

J. Liu, P. Benítez, and J. C. Miñano, “Single freeform surface imaging design with unconstrained object to image mapping,” Opt. Express 22(25), 30538–30546 (2014).
[Crossref] [PubMed]

2013 (5)

2012 (4)

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

J. Hou, H. Li, Z. Zheng, and X. Liu, “Distortion correction for imaging on non-planar surface using freeform lens,” Opt. Commun. 285(6), 986–991 (2012).
[Crossref]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20(5), 5576–5585 (2012).
[Crossref] [PubMed]

L. Li and A. Y. Yi, “Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera,” Appl. Opt. 51(12), 1843–1852 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (4)

2009 (3)

2008 (1)

2001 (1)

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
[Crossref]

1949 (1)

G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Bauer, A.

Bäuerle, A.

Benítez, P.

Bruneton, A.

Chen, F.

Cheng, D.

Davis, G. E.

Dong, J.

Duerr, F.

Feng, Z.

Fuerschbach, K.

Gao, C.

Gong, M.

Han, Y.

He, X.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Hicks, R. A.

Hou, J.

J. Hou, H. Li, Z. Zheng, and X. Liu, “Distortion correction for imaging on non-planar surface using freeform lens,” Opt. Commun. 285(6), 986–991 (2012).
[Crossref]

Hou, W.

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
[Crossref] [PubMed]

Hsieh, C. C.

Hu, X.

Hua, H.

Huang, L.

Hung, C. C.

Infante, J.

Jin, G.

Li, H.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

J. Hou, H. Li, Z. Zheng, and X. Liu, “Distortion correction for imaging on non-planar surface using freeform lens,” Opt. Commun. 285(6), 986–991 (2012).
[Crossref]

Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010).
[Crossref] [PubMed]

Li, L.

Li, Y. H.

Liang, P.

Lin, W.

Liu, J.

Liu, P.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

Liu, Q.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Liu, S.

Liu, X.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

J. Hou, H. Li, Z. Zheng, and X. Liu, “Distortion correction for imaging on non-planar surface using freeform lens,” Opt. Commun. 285(6), 986–991 (2012).
[Crossref]

Loosen, P.

Luo, X.

Luo, Y.

Ma, H.

Ma, T.

Meng, Q.

Meuret, Y.

Miñano, J. C.

Muñoz, F.

Qin, Z.

Rolland, J. P.

Rubinstein, J.

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
[Crossref]

Santamaría, A.

Shi, G.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Song, W.

Stollenwerk, J.

Talha, M. M.

Thienpont, H.

Thompson, K. P.

Wang, C.

Wang, K.

Wang, L.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Wang, S.

Wang, T.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Wang, W.

Wang, Y.

Wassermann, G. D.

G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Wester, R.

Wolansky, G.

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
[Crossref]

Wolf, E.

G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Wu, D.

Wu, R.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

Wu, X.

Xiang, H.

Xu, C.

Xu, L.

Yang, T.

Yi, A. Y.

Yu, J.

Yu, S.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zhang, B.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zhang, F.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zhang, X.

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zhang, Y.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

Zheng, L.

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

Zheng, Z.

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

R. Wu, L. Xu, P. Liu, Y. Zhang, Z. Zheng, H. Li, and X. Liu, “Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation,” Opt. Lett. 38(2), 229–231 (2013).
[Crossref] [PubMed]

J. Hou, H. Li, Z. Zheng, and X. Liu, “Distortion correction for imaging on non-planar surface using freeform lens,” Opt. Commun. 285(6), 986–991 (2012).
[Crossref]

Zhenrong, Z.

Zhu, J.

Appl. Opt. (5)

J. Opt. (1)

J. Zhu, W. Hou, X. Zhang, and G. Jin, “Design of a low F-number freeform off-axis three-mirror system with rectangular field-of-view,” J. Opt. 17(1), 015605 (2015).
[Crossref]

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

Opt. Commun. (2)

Y. Zhang, R. Wu, P. Liu, Z. Zheng, H. Li, and X. Liu, “Double freeform surfaces design for laser beam shaping with Monge-Ampère equation method,” Opt. Commun. 331, 297–305 (2014).
[Crossref]

J. Hou, H. Li, Z. Zheng, and X. Liu, “Distortion correction for imaging on non-planar surface using freeform lens,” Opt. Commun. 285(6), 986–991 (2012).
[Crossref]

Opt. Express (16)

J. C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express 17(26), 24036–24044 (2009).
[PubMed]

Y. Luo, Z. Feng, Y. Han, and H. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010).
[Crossref] [PubMed]

F. Chen, K. Wang, Z. Qin, D. Wu, X. Luo, and S. Liu, “Design method of high-efficient LED headlamp lens,” Opt. Express 18(20), 20926–20938 (2010).
[Crossref] [PubMed]

Z. Feng, Y. Luo, and Y. Han, “Design of LED freeform optical system for road lighting with high luminance/illuminance ratio,” Opt. Express 18(21), 22020–22031 (2010).
[Crossref] [PubMed]

T. Ma, J. Yu, P. Liang, and C. Wang, “Design of a freeform varifocal panoramic optical system with specified annular center of field of view,” Opt. Express 19(5), 3843–3853 (2011).
[Crossref] [PubMed]

S. Wang, K. Wang, F. Chen, and S. Liu, “Design of primary optics for LED chip array in road lighting application,” Opt. Express 19(S4Suppl 4), A716–A724 (2011).
[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]

F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20(5), 5576–5585 (2012).
[Crossref] [PubMed]

J. Liu, P. Benítez, and J. C. Miñano, “Single freeform surface imaging design with unconstrained object to image mapping,” Opt. Express 22(25), 30538–30546 (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]

D. Cheng, Y. Wang, C. Xu, W. Song, and G. Jin, “Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics,” Opt. Express 22(17), 20705–20719 (2014).
[Crossref] [PubMed]

H. Hua, X. Hu, and C. Gao, “A high-resolution optical see-through head-mounted display with eyetracking capability,” Opt. Express 21(25), 30993–30998 (2013).
[Crossref] [PubMed]

T. Yang, J. Zhu, and G. Jin, “Design of freeform imaging systems with linear field-of-view using a construction and iteration process,” Opt. Express 22(3), 3362–3374 (2014).
[Crossref] [PubMed]

T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
[Crossref] [PubMed]

A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21(9), 10563–10571 (2013).
[Crossref] [PubMed]

Z. Feng, L. Huang, M. Gong, and G. Jin, “Beam shaping system design using double freeform optical surfaces,” Opt. Express 21(12), 14728–14735 (2013).
[Crossref] [PubMed]

Opt. Lett. (3)

Opt. Rev. (1)

J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001).
[Crossref]

Proc. Phys. Soc. B (1)

G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Proc. SPIE (2)

X. Zhang, L. Zheng, X. He, L. Wang, F. Zhang, S. Yu, G. Shi, B. Zhang, Q. Liu, and T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[Crossref]

D. Cheng, Y. Wang, and H. Hua, “Free form optical system design with differential equations,” Proc. SPIE 7849, 78490Q (2010).
[Crossref]

Other (1)

D. Knapp, “Conformal Optical Design,” Ph.D. Thesis, University of Arizona (2002).

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Figures (14)

Fig. 1
Fig. 1 The schematic view of the unknown surface Ω, the feature light ray Ri, the corresponding data point Pi, the target surface, and the target point Ti.
Fig. 2
Fig. 2 The procedures of fitting the data points into a freeform surface with a base conic. (a) Find the surface vertex (xo, yo, zo). (b) Fit the data points into a base sphere and transform the coordinates and surface normal of the data points into the local coordinates system. (c) Fit the data points into a base conic in the local coordinates system. (d) Fit the residual coordinates and surface normal into freeform terms and obtain the final freeform surface.
Fig. 3
Fig. 3 Three types of the iterative process. (a) Normal iterations. (b) Negative feedback. (c) Successively approximation. The solid line represents the actual feature light ray. The dotted line represents the ideal light ray.
Fig. 4
Fig. 4 The flow diagram of the construction-iteration method. M denotes the number of the freeform surfaces in the system.
Fig. 5
Fig. 5 The layout of the initial system with three planes.
Fig. 6
Fig. 6 The layouts of the systems. (a) After generating freeform M3. (b) After generating freeform M1.
Fig. 7
Fig. 7 (a) The average RMS spot diameter of the two systems after generating M3 and after generating M1. (b) The distortion grid of the system after generating freeform M3. (c) The distortion grid of the system after generating freeform M1.
Fig. 8
Fig. 8 The convergence behavior of the RMS deviation σRMS for the three iteration types versus the number of iteration steps.
Fig. 9
Fig. 9 The layout of system after iterations.
Fig. 10
Fig. 10 (a) The spot diagram of the system after iterations. (b) The comparison of the average RMS spot diameter of the systems before and after iterations.
Fig. 11
Fig. 11 The distortion grid of the system after iterations.
Fig. 12
Fig. 12 Optical layout of the final system after optimization.
Fig. 13
Fig. 13 MTF plots of the final system at LWIR.
Fig. 14
Fig. 14 (a) The RMS wavefront error of the final system. (b) The distortion grid.

Tables (3)

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Table 1 Comparisons of the three iteration types.

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Table 2 Specifications of the freeform off-axis three-mirror system.

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Table 3 Comparisons of the three iteration types for the design example (ε = 0.3, ρ = 0.7).

Equations (11)

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δS=δ P i T i nds =0
z(x,y)= c( x 2 + y 2 ) 1+ 1(1+k) c 2 ( x 2 + y 2 ) + j=1 N A j g j (x,y) .
θ=arctan( y o y c z o z c )
{ x i '= x i x o y i '=( y i y o )cosθ( z i z o )sinθ z i '=( y i y o )sinθ+( z i z o )cosθ
{ α i '= α i β i '= β i cosθ γ i sinθ γ i '= β i sinθ+ γ i cosθ
( x i '', y i '', z i '')=( x i ', y i ', z i ' z ic ')
( α i '', β i '',1)=( α i ' γ i ' + α ic ' γ ic ' , β i ' γ i ' + β ic ' γ ic ' ,1)
T i = T i, ideal
T i ={ T i, ideal +εΔif( T i, ideal T i * )>Δ T i, ideal +ε( T i, ideal T i * )ifΔ( T i, ideal T i * )Δ T i, ideal εΔif( T i, ideal T i * )<Δ
T i = T i * +ρ( T i, ideal T i * )
σ RMS = m=1 K σ m 2 K

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