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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 22 — Nov. 4, 2013
  • pp: 26080–26092
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Design method of surface contour for a freeform lens with wide linear field-of-view

Jun Zhu, Tong Yang, and Guofan Jin  »View Author Affiliations


Optics Express, Vol. 21, Issue 22, pp. 26080-26092 (2013)
http://dx.doi.org/10.1364/OE.21.026080


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Abstract

In this paper, a design method of surface contour for a freeform imaging lens with a wide linear field-of-view (FOV) is developed. During the calculation of the data points on the unknown freeform surfaces, the aperture size and different field angles of the system are both considered. Meanwhile, two special constraints are employed to find the appropriate points that can generate a smooth and accurate surface contour. The surfaces obtained can be taken as the starting point for further optimization. An f-θ single lens with a ± 60° linear FOV has been designed as an example of the proposed method. After optimization with optical design software, the MTF of the lens 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.

© 2013 Optical Society of America

1. Introduction

Compared with conventional rotationally symmetric surfaces, freeform optical surfaces have more degrees of freedom, therefore reduce the aberrations and simplify the structure of the system in optical design. In recent years, with the development of the advancing manufacture technologies, freeform surfaces have been successfully used in the imaging field, such as head-mounted-display [1

1. D. Cheng, Y. Wang, H. Hua, and M. M. Talha, “Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism,” Appl. Opt. 48(14), 2655–2668 (2009). [CrossRef] [PubMed]

3

3. Z. Zheng, X. Liu, H. Li, and L. Xu, “Design and fabrication of an off-axis see-through head-mounted display with an x-y polynomial surface,” Appl. Opt. 49(19), 3661–3668 (2010). [CrossRef] [PubMed]

], reflective systems [4

4. K. Garrard, T. Bruegge, J. Hoffman, T. Dow, and A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A, 58740A-11 (2005). [CrossRef]

9

9. T. Hisada, K. Hirata, and M. Yatsu, “Projection type image display apparatus,” U.S. Patent, 7,701,639 (April 20, 2010).

], varifocal panoramic optical system [10

10. 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]

] and microlens array [11

11. 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]

].

For freeform imaging system design, a wide FOV is difficult to be achieved. Furthermore, the real size of aperture is expected to be considered while designing a wide FOV system. So, the light beams of different fields possibly have overlap area on the unknown surface during the design. The position and shape of the overlap area should meet the imaging requirement of different fields. It is both an interest and a challenge to directly design freeform surfaces under these conditions.

In this paper, a novel method to design a freeform imaging lens with a wide FOV has been developed. The proposed method has two key contents. Firstly, the aperture size and different field angles in a wide FOV system are both considered during the calculation of the data points on the freeform surfaces. Secondly, two special constraints are employed during the calculation. With these two key contents, a series of data points can be obtained, and a smooth and accurate surface contour can be generated after curve fitting with these data points. More importantly, the coordinates and normal vectors of these original data points can be approximately maintained, and the expected imaging relationship is ensured. The surfaces obtained can be taken as the starting point for further optimization in optical design software. Here, only the design method of the two-dimensional surface contour for tangential rays is covered in this paper. An f-θ single lens with a ± 60° linear FOV is designed as an example. A good starting point is obtained with the proposed method, and it is then optimized in optical design software to achieve good image quality and scanning linearity. The design method of three-dimensional freeform surfaces will be discussed in the future study.

2. Method

2.1 The feature rays for calculating the data points on the unknown surface

During the design of the freeform contour, each two neighboring fields are taken as a group. Several feature rays corresponding to different fields and pupil coordinates are defined in each group, and their intersections with the front surface can be calculated based on the relationships between the incident and outgoing rays. So, the data points on the unknown surface can be obtained group after group and the contour of the front surface can be then constructed with these data points. Consider an arbitrary group shown in Fig. 4
Fig. 4 The definition of the feature rays used in each field group to calculate the data points on the freeform surface. (a) When the two neighboring fields have overlap area on the unknown surface, two data points (point 1 and 2) are calculated in one field group using three feature rays (①②③). (b) When the light beams are separated, two data points (point 1 and 2) on the surface are calculated in one field group with only two feature rays (①②).
. The two neighboring fields are labeled as field #1 and field #2 and the feature rays are respectively marked with a circled number and plotted in bold. Feature ray ① is the chief ray of field #1 and it will be refracted to its ideal image point Pf1 by the front surface at data point 1. Feature ray ② is the bottom marginal ray of field #2 and it will be refracted to its ideal image point Pf2 by the front surface at data point 2. When the beams of neighboring fields have overlap area (yellow area in Fig. 4(a)) on the unknown surface, the beams from two different fields have to be simultaneously controlled at this area on the surface. It means that two rays from field #1 and #2 which hit on the same point in the overlap area should be refracted to Pf1 and Pf2 respectively. This problem generally does not have an exact solution due to the over-determined problem. So in this paper, the calculation of the data points is taken as an optimization problem. As only the starting point of the system is concerned, an approximate solution obtained by a mathematical optimization process is adequate. Here in particular, at data point 2 (the bottom of the overlap area), feature ray ② from field #2 as well as another feature ray ③ (blue bold dotted ray) from field #1 will be refracted to their ideal image point Pf2 and Pf1 respectively. In short, two data points (point 1 and 2) on the surface are calculated in one field group using three feature rays (①②③) when the light beams have overlap area on the unknown surface. When the field angle is increasing and the light beams are separated, as shown in Fig. 4(b), there is no feature ray ③ and two data points (point 1 and 2) on the surface are calculated in one field group using only two feature rays (①②).

2.2 Establishing the constraints to generate a smooth link line of the points

After the feature rays used in each group are defined, the data points on the front surface can be calculated based on the relationships between the incident and outgoing light rays. However, there are still some problems. As there are no geometric relationships between different groups, the two data points calculated in each field group are the optimum solution of a single problem specific to two fields. Therefore, the data points from different groups distribute irregularly in the tangential plane. A smooth and accurate surface contour is difficult to be obtained.

N3e23=0,
(2)
N4e34=0.
(3)

Equations (2) and (3) establish the geometric relationships between neighboring field groups. The data points no longer distribute irregularly in the tangential plane after the constraint is added. Moreover, in this constraint, the original normal vector at each data point which determines the outgoing direction of light ray is perpendicular to the line connecting the neighboring point. As a consequence, the consistency of the normal vectors after curve fitting is approximately ensured, and the light rays can be shifted in the expected directions.

2.3 Calculating the data points on the unknown freeform surface

tanθ=βγ.
(11)

3. Design example: A freeform f-θ single lens

3.1 Designing the starting point of the system

As an example of the proposed method, an f-θ single lens with a wide linear FOV has been designed. The f-θ lens is used for a scanning range of ± 210mm in y direction. The system has a linear FOV of ± 60°, and it is divided equally into 61 fields with a 2° interval during the design process. As the scanning width y (mm) has a linear relationship with the scanning angle θ (°) for an f-θ lens, the f-θ property can be written as
y=21060θ=3.5θ.
(12)
The system has a circular entrance pupil with 3mm diameter. The scanning light is 780nm infrared laser. The material of the lens is PMMA.

3.2 Optimization of the starting point

4. Conclusion

Acknowledgment

This work is supported by the National Basic Research Program of China (973, No. 2011CB706701).

References and links

1.

D. Cheng, Y. Wang, H. Hua, and M. M. Talha, “Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism,” Appl. Opt. 48(14), 2655–2668 (2009). [CrossRef] [PubMed]

2.

Q. Wang, D. Cheng, Y. Wang, H. Hua, and G. Jin, “Design, tolerance, and fabrication of an optical see-through head-mounted display with free-form surface elements,” Appl. Opt. 52(7), C88–C99 (2013). [CrossRef] [PubMed]

3.

Z. Zheng, X. Liu, H. Li, and L. Xu, “Design and fabrication of an off-axis see-through head-mounted display with an x-y polynomial surface,” Appl. Opt. 49(19), 3661–3668 (2010). [CrossRef] [PubMed]

4.

K. Garrard, T. Bruegge, J. Hoffman, T. Dow, and A. Sohn, “Design tools for free form optics,” Proc. SPIE 5874, 58740A, 58740A-11 (2005). [CrossRef]

5.

R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE 6668, 666802, 666802-10 (2007). [CrossRef]

6.

O. Cakmakci and J. Rolland, “Design and fabrication of a dual-element off-axis near-eye optical magnifier,” Opt. Lett. 32(11), 1363–1365 (2007). [CrossRef] [PubMed]

7.

L. Xu, K. Chen, Q. He, and G. Jin, “Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism,” Appl. Opt. 48(15), 2871–2879 (2009). [CrossRef] [PubMed]

8.

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, 848607-10 (2012). [CrossRef]

9.

T. Hisada, K. Hirata, and M. Yatsu, “Projection type image display apparatus,” U.S. Patent, 7,701,639 (April 20, 2010).

10.

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]

11.

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]

12.

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

13.

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

14.

D. Cheng, Y. Wang, and H. Hua, “Free form optical system design with differential equations,” Proc. SPIE 7849, 78490Q, 78490Q-8 (2010). [CrossRef]

15.

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). [CrossRef] [PubMed]

16.

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]

17.

Y. Wang and H. H. Hopkins, “Ray-tracing and aberration formulae for a general optical system,” J. Mod. Opt. 39(9), 1897–1938 (1992). [CrossRef]

18.

1stOpt Manual, 7D-Soft High Technology Inc. (2012).

19.

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

OCIS Codes
(220.2740) Optical design and fabrication : Geometric optical design
(080.4225) Geometric optics : Nonspherical lens design

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: July 29, 2013
Revised Manuscript: October 16, 2013
Manuscript Accepted: October 16, 2013
Published: October 24, 2013

Citation
Jun Zhu, Tong Yang, and Guofan Jin, "Design method of surface contour for a freeform lens with wide linear field-of-view," Opt. Express 21, 26080-26092 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-26080


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References

  1. D. Cheng, Y. Wang, H. Hua, and M. M. Talha, “Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism,” Appl. Opt.48(14), 2655–2668 (2009). [CrossRef] [PubMed]
  2. Q. Wang, D. Cheng, Y. Wang, H. Hua, and G. Jin, “Design, tolerance, and fabrication of an optical see-through head-mounted display with free-form surface elements,” Appl. Opt.52(7), C88–C99 (2013). [CrossRef] [PubMed]
  3. Z. Zheng, X. Liu, H. Li, and L. Xu, “Design and fabrication of an off-axis see-through head-mounted display with an x-y polynomial surface,” Appl. Opt.49(19), 3661–3668 (2010). [CrossRef] [PubMed]
  4. K. Garrard, T. Bruegge, J. Hoffman, T. Dow, and A. Sohn, “Design tools for free form optics,” Proc. SPIE5874, 58740A, 58740A-11 (2005). [CrossRef]
  5. R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE6668, 666802, 666802-10 (2007). [CrossRef]
  6. O. Cakmakci and J. Rolland, “Design and fabrication of a dual-element off-axis near-eye optical magnifier,” Opt. Lett.32(11), 1363–1365 (2007). [CrossRef] [PubMed]
  7. L. Xu, K. Chen, Q. He, and G. Jin, “Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism,” Appl. Opt.48(15), 2871–2879 (2009). [CrossRef] [PubMed]
  8. 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. SPIE8486, 848607, 848607-10 (2012). [CrossRef]
  9. T. Hisada, K. Hirata, and M. Yatsu, “Projection type image display apparatus,” U.S. Patent, 7,701,639 (April 20, 2010).
  10. 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. Express19(5), 3843–3853 (2011). [CrossRef] [PubMed]
  11. 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]
  12. G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B62(1), 2–8 (1949). [CrossRef]
  13. D. Knapp, “Conformal Optical Design,” Ph.D. Thesis, University of Arizona (2002).
  14. D. Cheng, Y. Wang, and H. Hua, “Free form optical system design with differential equations,” Proc. SPIE7849, 78490Q, 78490Q-8 (2010). [CrossRef]
  15. J. C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express17(26), 24036–24044 (2009). [CrossRef] [PubMed]
  16. 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. Express20(5), 5576–5585 (2012). [CrossRef] [PubMed]
  17. Y. Wang and H. H. Hopkins, “Ray-tracing and aberration formulae for a general optical system,” J. Mod. Opt.39(9), 1897–1938 (1992). [CrossRef]
  18. 1stOpt Manual, 7D-Soft High Technology Inc. (2012).
  19. Code V Reference Manual, Synopsys Inc. (2012).

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