## SMS-based optimization strategy for ultra-compact SWIR telephoto lens design |

Optics Express, Vol. 20, Issue 9, pp. 9726-9735 (2012)

http://dx.doi.org/10.1364/OE.20.009726

Acrobat PDF (2422 KB)

### Abstract

A new optical design strategy for rotational aspheres using very few parameters is presented. It consists of using the SMS method to design the aspheres embedded in a system with additional simpler surfaces (such as spheres, parabolas or other conics) and optimizing the free-parameters. Although the SMS surfaces are designed using only meridian rays, skew rays have proven to be well controlled within the optimization. In the end, the SMS surfaces are expanded using Forbes series and then a second optimization process is carried out with these SMS surfaces as a starting point. The method has been applied to a telephoto lens design in the SWIR band, achieving ultra-compact designs with an excellent performance.

© 2012 OSA

## 1. Introduction

5. P. Benítez and J. C. Miñano, “Ultrahigh-numerical-aperture imaging concentrator,” J. Opt. Soc. Am. A **14**(8), 1988–1997 (1997). [CrossRef]

6. 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]

8. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express **15**(8), 5218–5226 (2007). [CrossRef] [PubMed]

6. 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]

## 2. Improved SMS-3M design algorithm

6. 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]

9. W. Lin, P. Benítez, J. C. Minano, J. Infante, and G. Biot “Progress in the SMS design method for imaging optics,” Proc. SPIE **8128**, 81280F, 81280F-8 (2011). [CrossRef]

10. L. Wang, P. Benítez, J. C. Minano, J. Infante, M. de la Fuente, and G. Biot, “Ultracompact SWIR telephoto lens design with SMS method,” Proc. SPIE **8129**, 81290I, 81290I-10 (2011). [CrossRef]

^{2}smoothness at the connection points between the central curves and grown segments is not guaranteed. The lack of C

^{2}smoothness will finally result either in instabilities of the algorithm or poor final fitting of the curves (represented by NURBS in the SMS calculation process) with overall polynomial aspheres, which is of interest for Phase 2 of the design.

_{1}, V

_{2}and V

_{3}are the central vertices of the initial curves that can be defined beforehand by the designer as a degree of freedom. Consider the rays R

_{1}, R

_{2}and R

_{3}shown in Fig. 1, which all pass through the vertex of Curve 2: V

_{2}. R

_{2}is the on-axis ray emitted from the on-axis point Obj

_{2}to reach its on-axis image point Img

_{2}. R

_{1}is emitted from an off-axis point Obj

_{1}to reach its prescribed off-axis image point Img

_{1}and R

_{3}is its symmetrical counterpart.

_{2}from Obj

_{2}to Img

_{2}is guaranteed by normal incidence on all surfaces. The condition that R

_{1}emitted from Obj

_{1}reaches Img

_{1}sets up a relationship between the slopes of Curve 1 and Curve 2 on their rims. Furthermore, the conditions of C

^{2}smoothness at these edge points set up a second constraint on the curvatures of Curve 1 and Curve 2 on their rims. This second constraint is obtained by computing the propagation using a generalized ray-tracing [11] of the meridian curvature of the wave-fronts around R

_{1}and forcing the center of curvature of that wave-front to coincide with Img

_{1}. Equation (1) [11] establishes the relationship between the curvature of the incoming wave-front

*k*along a meridian ray, the curvature of the outgoing wave-front

_{i}*k*and the curvature of the intersected curve

_{r}*k*. Angles

_{c}*α*and

_{i}*α*are incident and refracted angles with respect to the normal. A similar law for reflective surface can be obtained by changing

_{r}*n*to −

_{r}*n*in Eq. (1). Since there are two conditions to be fulfilled, two degrees of freedom are needed, which means that Curves 1 and 3 can be represented, for instance, as spheres, the two radii being calculated with these conditions. The solution is in general not unique. Different solutions usually represent initial curves with concave or convex shapes, which lead to different families of optical designs.

_{i}**17**(26), 24036–24044 (2009). [CrossRef] [PubMed]

## 3. Tele-photo design specifications and the RXXR architecture

12. R. H. Shepard III and S. W. Sparrold, “Material selection for color correction in the short-wave infrared,” Proc. SPIE **7060**, 70600E, 70600E-10 (2008). [CrossRef]

13. C. Olson, T. Goodman, C. Addiego, and S. Mifsud, “Design and construction of a short-wave infrared 3.3X continuous zoom lens,” Proc. SPIE **7652**, 76522A, 76522A-12 (2010). [CrossRef]

12. R. H. Shepard III and S. W. Sparrold, “Material selection for color correction in the short-wave infrared,” Proc. SPIE **7060**, 70600E, 70600E-10 (2008). [CrossRef]

_{2D}distribution curves (defined as the RMS spot diameter calculated with only meridian rays within the field) present a constant ripple over the field of view. Since the SMS method guarantees that the RMS

_{2D}is null at the design object angles, the general form of the function RMS

_{2D}(

*θ*) is [6

**17**(26), 24036–24044 (2009). [CrossRef] [PubMed]

*θ*

_{1}is a designed off-axis object angle and

*A*(

*θ*) is an arbitrary even function. The best performing SMS designs usually have

*A*(

*θ*) ≈constant [6

**17**(26), 24036–24044 (2009). [CrossRef] [PubMed]

_{2D}function is plotted in Fig. 3(a) . A 3rd order polynomial has been fitted (with

*A*= 0.208 mm/deg

^{3}). However, the convex exit surface design on Fig. 2(b) a constant

*A*(

*θ*) does not fit well, but a sixth-order approximation is needed (

*A*(

*θ*) = 1.78156−10.5

*θ*

^{2}+ 34.5628

*θ*

^{4}−36.4749

*θ*

^{6},

*θ*in degrees,

*A*in mm/deg

^{3}), as shown in Fig. 3(b). Comparing the maximum RMS

_{2D}values for both graphs in Fig. 3, the concave exit surface design has much lower value than the convex one. Therefore the concave exit surface design has a better control over all meridian rays, and thus in the following sections, only the concave exit surface type will be considered.

## 4. Phase I optimization

_{3D}spot size within the field (calculated by 3D ray-tracing over the whole pupil). We have found that for a fixed set of vertices positions, varying central curvature of frontal parabola can always give a strong reduction to the merit function.

_{2D}curves are very similar), but strongly affects the RMS

_{3D}. In this optimization process, the biggest RMS

_{3D}has dropped from 80μm to 12μm. The merit function over the front curvature, shown in Fig. 4(c), is very smooth and possesses only one minimum, which demonstrates that the SMS method can effectively avoid the problem of excessive local minimum trapping. We started from a design far from the optimum position, and a much better monochromatic design has still been found by only optimizing one parameter. Therefore, the SMS method provides an effective way of designing because: First, by controlling 3 on-axis and off-axis meridian ray-bundles, on-axis spherical aberration is removed and all meridian rays are controlled by SMS design points. Later, by optimizing the central curvature of the first parabola, skew rays are efficiently controlled.

_{3D}(on Fig. 5) in the range explored are small compared to the variation given by the front central curvature (on Fig. 4(c)), as it is common to find in conventional multi-parameter optimization of the spherical optics [14

14. F. Bociort and M. van Turnhout, “Finding new local minima in lens design landscapes by constructing saddle points,” Opt. Eng. **48**(6), 063001 (2009). [CrossRef]

## 5. Phase II optimization

8. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express **15**(8), 5218–5226 (2007). [CrossRef] [PubMed]

16. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express **18**(19), 19700–19712 (2010). [CrossRef] [PubMed]

### 5.1. Further optimization of the monochromatic design

_{3D}distributions, as shown in Fig. 6(a) . The two RMS

_{3D}curves are very well matched. Three designed object angles can be seen to correspond to 3 local minimum RMS

_{3D}values. In Fig. 6(b), the MTF performance for all fields of view is very close to the diffraction limit at 1,300 nm. However, further improvement for this monochromatic design is still possible, because the RMS

_{3D}distribution curve is not well balanced between the central field and the edge field.

_{3D}distribution, as shown in Fig. 7 . The central field now performs not exactly at the diffraction limit, but the edge and middle field performances have clearly improved.

### 5.2. Polychromatic design

## 6. Conclusion

^{2}smoothness condition. A comparison between these two types of solutions reveals that the concave structure has a better performance than the convex type.

## Acknowledgment

## References and links

1. | M. Born and E. Wolf, |

2. | R. K. Luneburg, |

3. | R. Winston, J. C. Miñano, and P. Benítez, with contributions of N. Shatz, J. Bortz, |

4. | J. Chaves, |

5. | P. Benítez and J. C. Miñano, “Ultrahigh-numerical-aperture imaging concentrator,” J. Opt. Soc. Am. A |

6. | 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 |

7. | J.M.I HerreroF. Muñoz, P. Benitez, J. C. Miñano, W. Lin, J. Vilaplana, G. Biot, and M. de la Fuente, “Novel fast catadioptric objective with wide field of view,” Proc. SPIE |

8. | G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express |

9. | W. Lin, P. Benítez, J. C. Minano, J. Infante, and G. Biot “Progress in the SMS design method for imaging optics,” Proc. SPIE |

10. | L. Wang, P. Benítez, J. C. Minano, J. Infante, M. de la Fuente, and G. Biot, “Ultracompact SWIR telephoto lens design with SMS method,” Proc. SPIE |

11. | N. Orestes, Stavroudis, |

12. | R. H. Shepard III and S. W. Sparrold, “Material selection for color correction in the short-wave infrared,” Proc. SPIE |

13. | C. Olson, T. Goodman, C. Addiego, and S. Mifsud, “Design and construction of a short-wave infrared 3.3X continuous zoom lens,” Proc. SPIE |

14. | F. Bociort and M. van Turnhout, “Finding new local minima in lens design landscapes by constructing saddle points,” Opt. Eng. |

15. | W. Ulrich, “Freeform Surfaces: Hype or handy Design Tool?” Opening presentation at SPIE Optical Systems Design, Marseille (2011). |

16. | G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express |

**OCIS Codes**

(080.2740) Geometric optics : Geometric optical design

(080.3620) Geometric optics : Lens system design

(080.4035) Geometric optics : Mirror system design

**History**

Original Manuscript: February 15, 2012

Revised Manuscript: March 30, 2012

Manuscript Accepted: April 1, 2012

Published: April 13, 2012

**Citation**

Wang Lin, Pablo Benítez, Juan C. Miñano, José M. Infante, Guillermo Biot, and Marta de la Fuente, "SMS-based optimization strategy for ultra-compact SWIR telephoto lens design," Opt. Express **20**, 9726-9735 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-9726

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### References

- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
- R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
- R. Winston, J. C. Miñano, and P. Benítez, with contributions of N. Shatz, J. Bortz, Nonimaging Optics, (Academic Press Elsevier, 2004), Chap. 8.
- J. Chaves, Introduction to Nonimaging Optics (CRC Press, 2008).
- P. Benítez and J. C. Miñano, “Ultrahigh-numerical-aperture imaging concentrator,” J. Opt. Soc. Am. A14(8), 1988–1997 (1997). [CrossRef]
- 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]
- J.M.I HerreroF. Muñoz, P. Benitez, J. C. Miñano, W. Lin, J. Vilaplana, G. Biot, and M. de la Fuente, “Novel fast catadioptric objective with wide field of view,” Proc. SPIE7787, 778704 (2010).
- G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express15(8), 5218–5226 (2007). [CrossRef] [PubMed]
- W. Lin, P. Benítez, J. C. Minano, J. Infante, and G. Biot “Progress in the SMS design method for imaging optics,” Proc. SPIE8128, 81280F, 81280F-8 (2011). [CrossRef]
- L. Wang, P. Benítez, J. C. Minano, J. Infante, M. de la Fuente, and G. Biot, “Ultracompact SWIR telephoto lens design with SMS method,” Proc. SPIE8129, 81290I, 81290I-10 (2011). [CrossRef]
- N. Orestes, Stavroudis, The mathematics of geometrical and physical optics (Wiley-VCH, 2006).
- R. H. Shepard and S. W. Sparrold, “Material selection for color correction in the short-wave infrared,” Proc. SPIE7060, 70600E, 70600E-10 (2008). [CrossRef]
- C. Olson, T. Goodman, C. Addiego, and S. Mifsud, “Design and construction of a short-wave infrared 3.3X continuous zoom lens,” Proc. SPIE7652, 76522A, 76522A-12 (2010). [CrossRef]
- F. Bociort and M. van Turnhout, “Finding new local minima in lens design landscapes by constructing saddle points,” Opt. Eng.48(6), 063001 (2009). [CrossRef]
- W. Ulrich, “Freeform Surfaces: Hype or handy Design Tool?” Opening presentation at SPIE Optical Systems Design, Marseille (2011).
- G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010). [CrossRef] [PubMed]

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