## Design of freeform imaging systems with linear field-of-view using a construction and iteration process |

Optics Express, Vol. 22, Issue 3, pp. 3362-3374 (2014)

http://dx.doi.org/10.1364/OE.22.003362

Acrobat PDF (2109 KB)

### 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

## 1. Introduction

1. 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]

6. 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]

7. 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]

10. 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]

11. 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]

12. 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]

13. H. Zhang, L. Li, D. L. McCray, S. Scheiding, N. J. Naples, A. Gebhardt, S. Risse, R. Eberhardt, A. Tünnermann, and A. Y. Yi, “Development of a low cost high precision three-layer 3D artificial compound eye,” Opt. Express **21**(19), 22232–22245 (2013). [CrossRef] [PubMed]

7. 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]

9. 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]

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

19. 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]

20. 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]

## 2. Method

### 2.1 Basic idea to design the freeform surface contour

### 2.2 Construction of the freeform surface contours

#### 2.2.1 The method to construct a single freeform contour

*K*= M × N feature rays in the tangential plane are used. The intersections of the feature rays with the unknown surface are taken as the data points on the surface. The surface contour is then constructed with these points.

**(**

*P*_{i}*i*= 0,1,2…

*K*−1) on the contour of unknown surface

**, the intersections of the feature rays with surface**

*Ω*

*Ω**'*and

*Ω**”*, which are the two neighboring surfaces of

**, are employed, as shown in Fig. 2.The intersection of the ray with**

*Ω*

*Ω**'*is defined as the start point

**of a feature ray, and the intersection with**

*S*_{i}

*Ω**”*is defined as the end point

**. When the initial system and the feature rays have been decided, the start points**

*E*_{i}**(**

*S*_{i}*i*= 0,1,2…

*K*−1) of the feature rays and the directions of the rays after

*Ω**'*are known, but they are generally irregular. The end points

**(**

*E*_{i}*i*= 0,1,2…

*K*−1) are also determinate. The unit normal vector

**at each data point**

*N*_{i}**can be calculated based on the vector form of the Snell’s Law. For a refractive surface,where**

*P*_{i}*n*and

*n'*are the refractive indices of the two media in Fig. 2. Similarly, for a reflective surface

**(**

*P*_{i}*i*= 0,1,2…

*K*−1), an initial data point

**is firstly fixed which is the intersection of the marginal feature ray with the initial surface of the current iteration, as shown in Fig. 3. As the start point**

*P*_{0}**and end point**

*S*_{0}**of the initial feature ray can be easily obtained, the surface normal**

*E*_{0}**at**

*N*_{0}**can be calculated by Eq. (1) or Eq. (2). Then the tangent vector**

*P*_{0}**at**

*T*_{0}**can be obtained. To find the next data point on the unknown surface, we need to find the associated feature ray among the remaining**

*P*_{0}*K*−1 feature rays corresponding to different fields and different pupil coordinates. Here, the ray nearest to

**is taken as the feature ray corresponding to the next data point**

*P*_{i}**. As a realization of this principle,**

*P*_{i}_{+1}**is obtained by finding the point nearest to**

*P*_{1}**among the**

*P*_{0}*K*−1 intersections

**(**

*G*_{0}_{i}*i*= 1,2…

*K*−1) where the tangent vector

**intersects with the remaining**

*T*_{0}*K*−1 feature rays coming from

*Ω**'*, as shown in Fig. 3. Next, calculate the surface normal

**at**

*N*_{1}**with the start point**

*P*_{1}**and end point**

*S*_{1}**of its associated feature ray. Then find**

*E*_{1}**, which is nearest to**

*P*_{2}**among the**

*P*_{1}*K*−2 intersections of the tangent vector

**at**

*T*_{1}**with the remaining**

*P*_{1}*K*−2 feature rays. Repeat this process until all the

*K*data points on the unknown surface are obtained, as shown in Fig. 4.Finally, the contour of the surface is generated by curve fitting.

#### 2.2.2 Constructing the contour of surface #2

#### 2.2.3 Constructing the contour of surface #1

**in the object space,**

*S*_{i}**is redirected into**

*S*_{i}P_{i}**by surface #1. Then,**

*P*_{i}P_{i}'**is redirected to its ideal image point**

*P*_{i}P_{i}'**by surface #2. Here, Fermat's principle is used to calculate the unknown coordinates of**

*I*_{i}**. According to Fermat's principle,**

*P*_{i}'**is the point on surface #2 which minimizes the optical path length between**

*P*_{i}'

*P*_{i}*-*

*P*_{i}'*-*

**. The optical path length**

*I*_{i}*L*of

*P*_{i}*-*

*P*_{i}'*-*

**can be expressed aswhere**

*I*_{i}*n*

_{1-2}and

*n*

_{2-image}are the refractive indices of the medium between surface #1 and surface #2 and the medium between surface #2 and the image plane respectively. So

**can be obtained by minimizing**

*P*_{i}'*L*and it is taken as the end point of a feature ray on surface #2. With

**,**

*S*_{i}**and**

*P*_{i}**, the incident and outgoing directions of a feature ray can be calculated and the normal**

*P*_{i}'**at each data point on surface #1 can be obtained using Eq. (1) or Eq. (2). All the data points on surface #1 can be calculated following the procedure depicted in Section 2.2.1. The new contour of surface #1 is finally obtained by curve fitting. Then, the previous surface #1 is replaced..**

*N*_{i}### 2.3 Iterative process

## 3. Design example

### 3.1 System parameters and the iterative process

### 3.2 Image quality analysis

_{spo}of the spot diameters of these five fields. It can be seen that the spot sizes of different fields and the difference between them reduce rapidly in few iterations, which means the convergence of the design process is fast. Here, σ

_{spo}of the initial system is not plotted in Fig. 9(b) as it is generally zero due to the structure of planes. After several iterations, the average 100% spot diameter converges to a steady value around 240μm. The standard deviation of different field is less than 10μm, which indicates that the image qualities of different fields are significantly improved simultaneously.

*∆h*for each field is defined as the absolute value of the difference between the actual image height and the ideal image heightwhere

*h*is the ideal image height of each field,

*h'*is the actual image height. Figure 10(a) shows the convergence behavior of the maximum absolute distortion of the five sample fields (0°, 2°, 4°, 6°, 8°) versus the number of iterations. Figure 10(b) shows the convergence of the standard deviation σ

_{dis}of the absolute distortion of these five fields. The distortion converges rapidly to a steady value in several iterations. Although some residual distortion exists, the standard deviation of the distortion of different fields is very small, which is around 100μm after several iterations, as shown in Fig. 10(b). It indicates that the distortion actually introduces a translation of the actual image plane.

### 3.3 Further optimization with optical design software

1. 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. 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]

10. 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]

11. 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]

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

*x*in XY polynomials are kept. Moreover, the high order terms are not used as they lower the ray tracing speed and increase the difficulty in manufacture. So, an eight terms XY polynomials up to the 4th order is used:where

*c*is the curvature of the surface,

*k*is the conic constant, and

*A*is the coefficient of the

_{i}*xy*terms. This kind of freeform surface is continuous and can be fabricated. In addition, the 4th order polynomials have enough design freedom for optimization. The default transverse ray aberration error function in Code V is used in this optimization. The scanning error (distortion) is controlled by constraining the imaging coordinate of the chief ray in each field using real ray trace data. The final system with good image quality in both the tangential plane and the sagittal plane using two 3D freeform surfaces was obtained quickly by optimization, as shown in Fig. 11.The primary mirror (surface #1) has a rectangular size of 6mm × 22.84mm. The secondary mirror (surface #2) has a rectangular size of 2.7mm × 26.18mm. The exact profiles of the two surfaces are given in Table 3.The final design has an F# of 47.7, which is normal for an f-theta scanning system. The spot diagram is shown in Fig. 12(a).The MTF of each field is close to the diffraction limit, which is shown in Fig. 12(b). Figure 13 shows the RMS spot diameter as a function of field in a curve. Figure 14 shows the scanning errors (i.e. the distortion values) of different fields. For all of the sampling fields, the error is not more than ± 1μm. Figure 15shows the relative distortion of the system, which is within 0.009% over the full FOV. This result proves that good scanning linearity was achieved.

## 4. Conclusion

## Acknowledgment

## References and links

1. | O. Cakmakci and J. Rolland, “Design and fabrication of a dual-element off-axis near-eye optical magnifier,” Opt. Lett. |

2. | O. Cakmakci, S. Vo, H. Foroosh, and J. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis magnifier,” Opt. Lett. |

3. | R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE |

4. | K. Garrard, T. Bruegge, J. Hoffman, T. Dow, and A. Sohn, “Design tools for free form optics,” Proc. SPIE |

5. | L. Xu, K. Chen, Q. He, and G. Jin, “Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism,” Appl. Opt. |

6. | 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 |

7. | 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. |

8. | D. Cheng, Y. Wang, H. Hua, and J. Sasian, “Design of a wide-angle, lightweight head-mounted display using free-form optics tiling,” Opt. Lett. |

9. | 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. |

10. | 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. |

11. | 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 |

12. | 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. |

13. | H. Zhang, L. Li, D. L. McCray, S. Scheiding, N. J. Naples, A. Gebhardt, S. Risse, R. Eberhardt, A. Tünnermann, and A. Y. Yi, “Development of a low cost high precision three-layer 3D artificial compound eye,” Opt. Express |

14. | G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B |

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

16. | D. Cheng, Y. Wang, and H. Hua, “Free form optical system design with differential equations,” Proc. SPIE |

17. | J. Rubinstein and G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. |

18. | O. N. Stavroudis, “ |

19. | 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 |

20. | 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 |

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

**OCIS Codes**

(080.2740) Geometric optics : Geometric optical design

(080.4035) Geometric optics : Mirror system design

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: November 12, 2013

Revised Manuscript: January 28, 2014

Manuscript Accepted: January 29, 2014

Published: February 5, 2014

**Citation**

Tong Yang, Jun Zhu, and Guofan Jin, "Design of freeform imaging systems with linear field-of-view using a construction and iteration process," Opt. Express **22**, 3362-3374 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3362

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### References

- O. Cakmakci, J. Rolland, “Design and fabrication of a dual-element off-axis near-eye optical magnifier,” Opt. Lett. 32(11), 1363–1365 (2007). [CrossRef] [PubMed]
- O. Cakmakci, S. Vo, H. Foroosh, J. Rolland, “Application of radial basis functions to shape description in a dual-element off-axis magnifier,” Opt. Lett. 33(11), 1237–1239 (2008). [CrossRef] [PubMed]
- 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]
- L. Xu, K. Chen, Q. He, G. Jin, “Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism,” Appl. Opt. 48(15), 2871–2879 (2009). [CrossRef] [PubMed]
- 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, 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]
- D. Cheng, Y. Wang, H. Hua, J. Sasian, “Design of a wide-angle, lightweight head-mounted display using free-form optics tiling,” Opt. Lett. 36(11), 2098–2100 (2011). [CrossRef] [PubMed]
- Q. Wang, D. Cheng, Y. Wang, H. Hua, 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]
- Z. Zheng, X. Liu, H. Li, 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]
- T. Ma, J. Yu, P. Liang, 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]
- L. Li, 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]
- H. Zhang, L. Li, D. L. McCray, S. Scheiding, N. J. Naples, A. Gebhardt, S. Risse, R. Eberhardt, A. Tünnermann, A. Y. Yi, “Development of a low cost high precision three-layer 3D artificial compound eye,” Opt. Express 21(19), 22232–22245 (2013). [CrossRef] [PubMed]
- G. D. Wassermann, E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949). [CrossRef]
- D. Knapp, “Conformal Optical Design,” Ph.D. Thesis, University of Arizona (2002).
- D. Cheng, Y. Wang, H. Hua, “Free form optical system design with differential equations,” Proc. SPIE 7849, 78490Q (2010). [CrossRef]
- J. Rubinstein, G. Wolansky, “Reconstruction of optical surfaces from ray data,” Opt. Rev. 8(4), 281–283 (2001). [CrossRef]
- O. N. Stavroudis, “The Mathematics of Geometrical and Physical Optics” (Wiley-VCH, 2006).
- J. C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express 17(26), 24036–24044 (2009). [CrossRef] [PubMed]
- F. Duerr, P. Benítez, J. C. Miñano, Y. Meuret, 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]
- Code V Reference Manual, Synopsys Inc. (2012).

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