## An application of the SMS method for imaging designs

Optics Express, Vol. 17, Issue 26, pp. 24036-24044 (2009)

http://dx.doi.org/10.1364/OE.17.024036

Acrobat PDF (187 KB)

### Abstract

The Simultaneous Multiple Surface (SMS) method in planar geometry (2D) is applied to imaging designs, generating lenses that compare well with aplanatic designs. When the merit function utilizes image quality over the entire field (not just paraxial), the SMS strategy is superior. In fact, the traditional aplanatic approach is actually a particular case of the SMS strategy.

© 2009 OSA

## 1. Introduction

*ρ*is the distance from a point to the optical axis (

*z*axis),

*c*is the vertex radius of curvature. The remaining parameters (

*k*,

*a*

_{4},

*a*

_{6}, …) describe the “asphericity” of the surface. More powerful ways to describe aspheric surfaces were recently proposed by Forbes [6

6. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express **15**(8), 5218–5226 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-5218. [CrossRef] [PubMed]

*k*(with the initial value

*k*= 0 for the spherical surface) until a local maximum is found and then following with

*a*

_{4}, etc. In practice, the performance of the design does not improve when the number of optimizing parameters is big (>10) because the asphericity of one surface is just cancelling the asphericity of another [3].

**P**by means of the RMS blur radius σ(

**P**) [3] of the spot image of each ray bundle. We will use two different versions of the blur radius: σ

_{2D}(

**P**), for which only tangential rays are considered, and σ(

**P**), which considers sagittal rays as well. The function of σ

_{2D}(

**P**) is used to check the attainment of the design conditions (σ

_{2D}must be zero for the N selected points).

7. P. Benítez and J. C. Miñano, “Ultra high-numerical-aperture imaging concentrator,” J. Opt. Soc. Am. A **14**(8), 1988–1997 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-14-8-1988. [CrossRef]

8. J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamaría, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE **7429**, 74290C (2009). [CrossRef]

9. R. Winston and W. Zhang, “Novel aplanatic designs,” Opt. Lett. **34**(19), 3018–3019 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-19-3018. [CrossRef] [PubMed]

10. J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. **34**(34), 7850–7856 (1995), http://www.opticsinfobase.org/abstract.cfm?URI=ao-34-34-7850. [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. W. T. Welford, “Aplanatism and Isoplanatism,” Progress in Optics **13**, 267–293 (1976). [CrossRef]

14. L. Mertz, “Geometrical design for aspheric reflecting systems,” Appl. Opt. **18**(24), 4182–4186 (1979). [CrossRef] [PubMed]

15. L. Mertz, “Aspheric potpourri,” Appl. Opt. **20**(7), 1127–1131 (1981). [CrossRef] [PubMed]

16. D. Lynden-Bell, “Exact Optics: A Unification of Optical Telescope Design,” Mon. Not. R. Astron. Soc. **334**(4), 787–796 (2002). [CrossRef]

17. R. V. Willstrop and D. Lynden-Bell, “Exact Optics — II. Exploration of Designs On- and Off-Axis,” Mon. Not. R. Astron. Soc. **342**(1), 33–49 (2003). [CrossRef]

13. W. T. Welford, “Aplanatism and Isoplanatism,” Progress in Optics **13**, 267–293 (1976). [CrossRef]

13. W. T. Welford, “Aplanatism and Isoplanatism,” Progress in Optics **13**, 267–293 (1976). [CrossRef]

7. P. Benítez and J. C. Miñano, “Ultra high-numerical-aperture imaging concentrator,” J. Opt. Soc. Am. A **14**(8), 1988–1997 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-14-8-1988. [CrossRef]

20. G. Schulz, “Higher order aplanatism,” Opt. Commun. **41**(5), 315–319 (1982). [CrossRef]

_{M}. This merit function makes sense when the optical system is going to be applied, for instance, to a focal plane array. In this case, the pixel size establishes the maximum admissible spot size (which is in general slightly smaller than the pixel size, to allow for tolerances). Going to a spot size smaller than the upper bound is in general useless.

## 2. Design procedure

7. P. Benítez and J. C. Miñano, “Ultra high-numerical-aperture imaging concentrator,” J. Opt. Soc. Am. A **14**(8), 1988–1997 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-14-8-1988. [CrossRef]

22. J. C. Miñano and J. C. González, “New method of design of nonimaging concentrators,” Appl. Opt. **31**(16), 3051–3060 (1992), http://www.opticsinfobase.org/abstract.cfm?URI=ao-31-16-3051. [CrossRef] [PubMed]

24. J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. **34**(34), 7850–7856 (1995), http://www.opticsinfobase.org/abstract.cfm?URI=ao-34-34-7850. [CrossRef] [PubMed]

25. P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, J. Alvarez, and W. Falicoff, “SMS Design Method in 3D Geometry: Examples and Applications,” Proc. SPIE **5185**, 18–29 (2003). [CrossRef]

**14**(8), 1988–1997 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-14-8-1988. [CrossRef]

## 3. Results

*θ*be a ray’s off-axis angle. We are going to compare aplanatic designs with equivalent two-surface SMS designs. In both cases we assume the object points to be at infinity (

*i.e.,*parallel rays). To evaluate the imaging quality of a design we will use the RMS spot radius functions σ(

*θ*) = σ(

**P**) and σ

_{2D}(θ) = σ

_{2D}(

**P**), where

**P**is the image point of the bundle of parallel rays incoming at the direction angle θ. Because all design rays are in the meridian plane, an SMS design of N surfaces will have σ

_{2D}(

*θ*

_{i}) = 0 for N design directions

*θ*

_{i}assuming that a negligible contribution of the initial parts of the four-surface SMS design, which is not theoretically perfect). Then,

27. F. Muñoz, Doctoral Thesis “Sistemas ópticos avanzados de gran compactibilidad con aplicaciones en formación de imagen y en iluminación” 2004 http://www-app.etsit.upm.es/tesis_etsit/documentos_biblioteca/masinformacion.php?sgt=TESIS-04-030

*A*(

*θ*) is an arbitrary analytic function of

*θ*which, in a first approximation can be considered a constant function of

*θ*(although the value of this constant varies with different designs),

*i.e.*, σ

_{2D}(

*θ*) is approximated by the absolute value of an N-degree polynomial. The ray tracing results of Fig. 5 show that this approximation is good in this case, and that the spot radius σ

_{2D}(

*θ*) is the absolute value of a parabolic function of

*θ*for small

*θ*. Figure 5 left plots the RMS spot radius σ(

*θ*) (dotted lines) and σ

_{2D}(

*θ*) (continuous) of three lenses with the same on-axis thickness, the same focal length

*f =*14.31mm, and the same f-number

*f/*1.576. The leftmost surface acts as stop surface. One of the lenses is designed for

*θ*

_{i}= ± 2° (this design will be called SMS ± 2°) and another for

*θ*

_{i}= ± 1° (SMS ± 1°), while the last one is the aplanatic case, which can be seen as an SMS 0° case with its two off-axis directions become on-axis (

*θ*

_{i}→0°). The value of the constant

*A*turns out to be quite similar in all three cases. The lens profiles of the three lenses considered in Fig. 5 are very similar to the one shown in Fig. 2 (the maximum deviation between curve profiles of these three lenses is less than 13 microns).

**14**(8), 1988–1997 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-14-8-1988. [CrossRef]

28. N. Shatz and J. Bortz, “Optimized image-forming cemented-doublet concentrator,” Proc. SPIE **5942**, 59420G (2005). [CrossRef]

_{M},

*i.e.*, the problem is to maximize the FOV when the maximum allowable spot size in the FOV is limited to a given value. Figure 5-left shows that no matter the value of σ

_{M}, there is always an SMS design with a wider range of incidence angles (

*i.e.*, a wider FOV) than the aplanatic design of equal focal length and

*f*-number. This best SMS design is the one having the maximum allowable spot size at both normal incidence and the edge of the field. For example, when the maximum RMS spot radius is σ

_{M}≈11.1 μm, the SMS design having the widest FOV diameter (which is ≈5.9°) is the one designed for incidence angles

*θ*

_{i}= ± 2°. Included in this comparison is the SMS ± 0° design, which is simply the aplanatic design. Figure 5-right shows the maximum RMS spot radius σ

_{M}

*vs.*FOV diameter for the aplanatic case and for these SMS designs. This result does not, however, prove that the present SMS designs reach the absolute maximum value of that merit function. In ref [28

28. N. Shatz and J. Bortz, “Optimized image-forming cemented-doublet concentrator,” Proc. SPIE **5942**, 59420G (2005). [CrossRef]

28. N. Shatz and J. Bortz, “Optimized image-forming cemented-doublet concentrator,” Proc. SPIE **5942**, 59420G (2005). [CrossRef]

*θ*) and σ

_{2D}(

*θ*) of these two-surface SMS designs are quite similar, which means that the selection of the tangential ray bundles as design rays gives good control of the function σ(

*θ*),

*i.e.*, of the RMS spot radius for all the rays.

_{2D}(

*θ*) calculated by ray tracing of the four-surface SMS design of Fig. 4 (f/2.241, focal length of 8.59 mm). Although the design of Fig. 4 shows equally spaced incoming directions (

*θ*

_{i}= ± 2°, ± 4°), this is not necessary in an SMS design. Equally spacing

*θ*

_{i}gives approximately a constant ripple of σ

_{2D}(

*θ*), as Fig. 9 shows. When a maximum spot radius σ

_{M}is fixed, the constant ripple, constrained to σ

_{2D}(

*θ*) ≤ σ

_{M}, maximizes the angular field of view for the function σ

_{2D}(

*θ*). Again the function σ(

*θ*) (dotted line in Fig. 9) is quite close to σ

_{2D}(

*θ*) in this four-surface design, showing that the selected design ray bundles give good control of the function σ

_{2D}(

*θ*) (as expected) and also for the RMS spot radius of all the rays (σ(

*θ*)), which is the relevant function for the evaluation of the image quality. The selection of the design ray bundles for an optimum control of σ(

*θ*) is a subject of our ongoing research, because the control of the RMS spot radius for all the rays by means of designing for only a limited subsets of tangential rays is not always as good as that shown in Fig. 9.

## 4. Conclusions

*θ*

_{i}and the output bundles are their focal points. Once a maximum spot radius in the field is fixed, the SMS design has a wider FOV than that of the equivalent aplanatic designs for the two-surface design (the FOV diameter of the best SMS design is about 1.42 times that of the equivalent aplanat), which is a trivial result when we note that aplanatic designs are just particular cases of SMS designs (when the SMS design directions

*θ*

_{i}coincide with the optical axis).

## Acknowledgement

## References and Links

1. | R. Kingslake, |

2. | W. J. Smith, |

3. | R. E. Fisher, and B. Tadic-Galeb, |

4. | A. E. Conrady, |

5. | G. G. Slyusarev, in |

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

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

8. | J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamaría, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE |

9. | R. Winston and W. Zhang, “Novel aplanatic designs,” Opt. Lett. |

10. | J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. |

11. | K. Schwarzschild, “Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen |

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

13. | W. T. Welford, “Aplanatism and Isoplanatism,” Progress in Optics |

14. | L. Mertz, “Geometrical design for aspheric reflecting systems,” Appl. Opt. |

15. | L. Mertz, “Aspheric potpourri,” Appl. Opt. |

16. | D. Lynden-Bell, “Exact Optics: A Unification of Optical Telescope Design,” Mon. Not. R. Astron. Soc. |

17. | R. V. Willstrop and D. Lynden-Bell, “Exact Optics — II. Exploration of Designs On- and Off-Axis,” Mon. Not. R. Astron. Soc. |

18. | R. Winston, J. C. Miñano, and P. Benítez, |

19. | M. Mansuripur, |

20. | G. Schulz, “Higher order aplanatism,” Opt. Commun. |

21. | G. Schulz, Aspheric surfaces, E. Wolf, (Ed.), Progress in Optics, |

22. | J. C. Miñano and J. C. González, “New method of design of nonimaging concentrators,” Appl. Opt. |

23. | J. C. Miñano, P. Benítez, and J. C. González, “RX: a nonimaging concentrator,” Appl. Opt. |

24. | J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. |

25. | P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, J. Alvarez, and W. Falicoff, “SMS Design Method in 3D Geometry: Examples and Applications,” Proc. SPIE |

26. | J. Chaves, |

27. | F. Muñoz, Doctoral Thesis “Sistemas ópticos avanzados de gran compactibilidad con aplicaciones en formación de imagen y en iluminación” 2004 http://www-app.etsit.upm.es/tesis_etsit/documentos_biblioteca/masinformacion.php?sgt=TESIS-04-030 |

28. | N. Shatz and J. Bortz, “Optimized image-forming cemented-doublet concentrator,” Proc. SPIE |

**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: November 24, 2009

Manuscript Accepted: December 9, 2009

Published: December 16, 2009

**Citation**

Juan C. Miñano, Pablo Benítez, Wang Lin, José Infante, Fernando Muñoz, and Asunción Santamaría, "An application of the SMS method for imaging designs," Opt. Express **17**, 24036-24044 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-24036

Sort: Year | Journal | Reset

### References

- R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978).
- W. J. Smith, Modern Optical Engineering, 3rd ed., (McGraw-Hill, 2000).
- R. E. Fisher, and B. Tadic-Galeb, Optical System Design (McGraw-Hill, 2000).
- A. E. Conrady, Applied Optics and Optical Design, Part 1, New edition 1992 (Oxford University Press and Dover Publications, 1929).
- G. G. Slyusarev, in Aberration and Optical Design Theory, pp. 499–502, Adam Hilger, (Techno House, Bristol, 1984).
- G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-5218 . [CrossRef] [PubMed]
- P. Benítez and J. C. Miñano, “Ultra high-numerical-aperture imaging concentrator,” J. Opt. Soc. Am. A 14(8), 1988–1997 (1997), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-14-8-1988 . [CrossRef]
- J. C. Miñano, P. Benítez, W. Lin, F. Muñoz, J. Infante, and A. Santamaría, “Overview of the SMS design method applied to imaging optics,” Proc. SPIE 7429, 74290C (2009). [CrossRef]
- R. Winston and W. Zhang, “Novel aplanatic designs,” Opt. Lett. 34(19), 3018–3019 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-19-3018 . [CrossRef] [PubMed]
- J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34(34), 7850–7856 (1995), http://www.opticsinfobase.org/abstract.cfm?URI=ao-34-34-7850 . [CrossRef] [PubMed]
- K. Schwarzschild, “Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen 10, 3 (1905), Reprinted: Selected Papers on Astronomical Optics,” SPIE Milestone Ser. 73, 3 (1993).
- G. D. Wassermann and E. Wolf, “On the Theory of Aplanatic Aspheric Systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949). [CrossRef]
- W. T. Welford, “Aplanatism and Isoplanatism,” Progress in Optics 13, 267–293 (1976). [CrossRef]
- L. Mertz, “Geometrical design for aspheric reflecting systems,” Appl. Opt. 18(24), 4182–4186 (1979). [CrossRef] [PubMed]
- L. Mertz, “Aspheric potpourri,” Appl. Opt. 20(7), 1127–1131 (1981). [CrossRef] [PubMed]
- D. Lynden-Bell, “Exact Optics: A Unification of Optical Telescope Design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002). [CrossRef]
- R. V. Willstrop and D. Lynden-Bell, “Exact Optics — II. Exploration of Designs On- and Off-Axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003). [CrossRef]
- R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Academic Press, New York, 2005)
- M. Mansuripur, Classical Optics and its Applications (Cambridge University Press, Cambridge, 2002) p. 16.
- G. Schulz, “Higher order aplanatism,” Opt. Commun. 41(5), 315–319 (1982). [CrossRef]
- G. Schulz, Aspheric surfaces, E. Wolf, (Ed.), Progress in Optics, 25, 1988, pp. 349–415.
- J. C. Miñano and J. C. González, “New method of design of nonimaging concentrators,” Appl. Opt. 31(16), 3051–3060 (1992), http://www.opticsinfobase.org/abstract.cfm?URI=ao-31-16-3051 . [CrossRef] [PubMed]
- J. C. Miñano, P. Benítez, and J. C. González, “RX: a nonimaging concentrator,” Appl. Opt. 34(13), 2226–2235 (1995), http://www.opticsinfobase.org/abstract.cfm?URI=ao-34-13-2226 . [CrossRef] [PubMed]
- J. C. Miñano, J. C. González, and P. Benítez, “A high-gain, compact, nonimaging concentrator: RXI,” Appl. Opt. 34(34), 7850–7856 (1995), http://www.opticsinfobase.org/abstract.cfm?URI=ao-34-34-7850 . [CrossRef] [PubMed]
- P. Benítez, J. C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, J. Alvarez, and W. Falicoff, “SMS Design Method in 3D Geometry: Examples and Applications,” Proc. SPIE 5185, 18–29 (2003). [CrossRef]
- J. Chaves, Introduction to Nonimaging Optics, (CRC Press, Boca Ratón, 2008).
- F. Muñoz, Doctoral Thesis “Sistemas ópticos avanzados de gran compactibilidad con aplicaciones en formación de imagen y en iluminación” 2004 http://www-app.etsit.upm.es/tesis_etsit/documentos_biblioteca/masinformacion.php?sgt=TESIS-04-030
- N. Shatz and J. Bortz, “Optimized image-forming cemented-doublet concentrator,” Proc. SPIE 5942, 59420G (2005). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.