Abstract

In this paper, a design method based on a construction and iteration process is proposed for designing freeform imaging systems with linear field-of-view (FOV). The surface contours of the desired freeform surfaces in the tangential plane are firstly designed to control the tangential rays of multiple field angles and different pupil coordinates. Then, the image quality is improved with an iterative process. The design result can be taken as a good starting point for further optimization. A freeform off-axis scanning system is designed as an example of the proposed method. The convergence ability of the construction and iteration process to design a freeform system from initial planes is validated. The MTF of the design result is close to the diffraction limit and the scanning error is less than 1μm. This result proves that good image quality and scanning linearity were achieved.

© 2014 Optical Society of America

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References

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2013 (2)

2012 (3)

2011 (2)

2010 (2)

2009 (3)

2008 (1)

2007 (2)

2005 (1)

K. Garrard, T. Bruegge, J. Hoffman, T. Dow, A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A (2005).
[CrossRef]

2001 (1)

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

1949 (1)

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

Benítez, P.

Bruegge, T.

K. Garrard, T. Bruegge, J. Hoffman, T. Dow, A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A (2005).
[CrossRef]

Cakmakci, O.

Chen, K.

Cheng, D.

Dow, T.

K. Garrard, T. Bruegge, J. Hoffman, T. Dow, A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A (2005).
[CrossRef]

Duerr, F.

Eberhardt, R.

Foroosh, H.

Garrard, K.

K. Garrard, T. Bruegge, J. Hoffman, T. Dow, A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A (2005).
[CrossRef]

Gebhardt, A.

He, Q.

He, X.

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

Hicks, R. A.

R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE 6668, 666802 (2007).
[CrossRef]

Hoffman, J.

K. Garrard, T. Bruegge, J. Hoffman, T. Dow, A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A (2005).
[CrossRef]

Hua, H.

Infante, J.

Jin, G.

Li, H.

Li, L.

Liang, P.

Lin, W.

Liu, Q.

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

Liu, X.

Ma, T.

McCray, D. L.

Meuret, Y.

Miñano, J. C.

Muñoz, F.

Naples, N. J.

Risse, S.

Rolland, J.

Rubinstein, J.

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

Santamaría, A.

Sasian, J.

Scheiding, S.

Shi, G.

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

Sohn, A.

K. Garrard, T. Bruegge, J. Hoffman, T. Dow, A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A (2005).
[CrossRef]

Talha, M. M.

Thienpont, H.

Tünnermann, A.

Vo, S.

Wang, C.

Wang, L.

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

Wang, Q.

Wang, T.

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

Wang, Y.

Wassermann, G. D.

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

Wolansky, G.

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

Wolf, E.

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

Xu, L.

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, 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, 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, T. Wang, “Design and fabrication of imaging optical systems with freeform surfaces,” Proc. SPIE 8486, 848607 (2012).
[CrossRef]

Zhang, H.

Zhang, X.

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

Zheng, L.

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

Zheng, Z.

Appl. Opt. (5)

Opt. Express (4)

Opt. Lett. (3)

Opt. Rev. (1)

J. Rubinstein, 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, E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[CrossRef]

Proc. SPIE (4)

R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE 6668, 666802 (2007).
[CrossRef]

K. Garrard, T. Bruegge, J. Hoffman, T. Dow, A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A (2005).
[CrossRef]

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

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

Other (3)

Code V Reference Manual, Synopsys Inc. (2012).

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

O. N. Stavroudis, “The Mathematics of Geometrical and Physical Optics” (Wiley-VCH, 2006).

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Figures (15)

Fig. 1
Fig. 1

An off-axis two-mirror system. (a) The initial system with planes. The design of surface contours starts from this initial system. (b) The system with much improved image quality for the tangential rays after design.

Fig. 2
Fig. 2

The two neighboring surfaces Ω' and Ω of the unknown surface Ω. The red rays stand for the feature rays used in the design process. The intersection of the ray with surface Ω' is defined as the start point Si of a feature ray, and the intersection with the surface Ω is defined as the end point Ei of a feature ray. Pi (i = 0,1,2…K−1) are the data points on surface Ω.

Fig. 3
Fig. 3

The method to find P1. The red rays stand for the incident feature rays used in the design process. Si (i = 0,1…K-1) are the start points of the feature rays. P1 is the point nearest to P0 among the K−1 intersections G0i (i = 1,2…K−1) where the tangent vector T0 at P0 intersects with the remaining K−1 feature rays.

Fig. 4
Fig. 4

The calculation of all the data points Pi (i = 0,1…K−1) on the unknown surface. Ni and Ti are the surface normal vector and tangent vector at each data point respectively.

Fig. 5
Fig. 5

The method to calculate the coordinate of Pi' (the end point Ei of each feature ray on surface #2) when constructing surface #1. According to Fermat’s principle, Pi' is the point on surface #2 which minimizes the optical path length between Pi-Pi'-Ii. The solid ray which passes Pi' (the point painted in green) represents the optical path which has the shortest optical path distance. The dotted rays represent optical paths with longer optical path distance.

Fig. 6
Fig. 6

The flow chart of the design process of 2D surface contours.

Fig. 7
Fig. 7

The layout of the initial system with two planes. The two planes have a 45° and 47° tilt about the z-axis in the global coordinates respectively. The distance between the entrance pupil and surface #1, between surface #1 and surface #2 and between surface #2 and the image plane are 45mm, 70mm and 135mm, respectively.

Fig. 8
Fig. 8

The layouts of the scanning system after 1, 2, 4, 8 and 12 iterations from the initial system with planes. It can be seen that the image quality is improved fast with iterations.

Fig. 9
Fig. 9

The convergence behavior of the spot diameter versus the number of iteration steps. (a) The convergence of the maximum, minimum and average spot diameter of 5 fields (0°, 2°, 4°, 6°, 8°). (b) The convergence of the standard deviation of spot diameter of these five fields.

Fig. 10
Fig. 10

The convergence behavior of the absolute distortion versus the number of iteration steps. (a) The convergence of the maximum absolute distortion of 5 fields (0°, 2°, 4°, 6°, 8°). (b) The convergence of the standard deviation of the absolute distortion of these five fields.

Fig. 11
Fig. 11

The layout of the final scanning system after optimization in Code V.

Fig. 12
Fig. 12

The image quality analysis of the final system. (a) Spot diagram. (b) The MTF curve.

Fig. 13
Fig. 13

The RMS spot diameter as a function of field in a curve.

Fig. 14
Fig. 14

Scanning errors of different fields.

Fig. 15
Fig. 15

Relative distortion of different fields.

Tables (3)

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Table 1 Parameters of the Freeform Scanning System

Tables Icon

Table 2 Changes of the Freeform Surface Profiles with Iterations

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Table 3 Profiles of the Surfaces in the Final System

Equations (5)

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N i = n ' r i ' n r i | n ' r i ' n r i | ,
N i = r i ' r i | r i ' r i | .
L = n 1 2 | P i P i ' | + n 2 i m a g e | P i ' I i | ,
Δ h = | h h | ,
z ( x , y ) = c ( x 2 + y 2 ) 1 + 1 ( 1 + k ) c 2 ( x 2 + y 2 ) + A 2 y + A 3 x 2 + A 5 y 2 + A 7 x 2 y + A 9 y 3 + A 10 x 4 + A 12 x 2 y 2 + A 14 y 4 ,

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