## Well-chosen method for an optimal design of doublet lens design

Optics Express, Vol. 17, Issue 3, pp. 1414-1428 (2009)

http://dx.doi.org/10.1364/OE.17.001414

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### Abstract

This paper presents a method for choosing a doublet design for the correction of longitudinal chromatic, spherical and coma aberrations. A secondary dispersion formula is utilized to sort out minimal longitudinal chromatic aberrations for the doublet. The program is developed with the Matlab software. An optimal doublet design to efficiently reduce both spherical aberration and coma will incorporate glass combination with a sufficiently large difference in the V-numbers and small powers. We succeed in obtaining an optimal doublet design with the proposed method.

© 2009 Optical Society of America

## 1. Introduction

1. K. D. Sharma and S. V. Rama Gopal, “Design of achromatic doublets: evaluation of the double-graph technique,” Appl. Opt. **22**, 497–500 (1983). [CrossRef] [PubMed]

2. S. Baberjee and L. Hazra, “Experiments with a genetic algorithm for structural design of cemented doublets with prespecified aberration targets,” Appl. Opt. **40**, 6265–6273 (2001). [CrossRef]

3. P. N. Robb, “Selection of optical glasses. 1: Two materials,” Appl. Opt. **24**, 1864–1877 (1985). [CrossRef] [PubMed]

5. R. E. Stephens, “Selection of glasses for three-color achromats,” J. Opt. Soc. Am. **49**, 398–401 (1959). [CrossRef]

## 2. Theory

### 2.1 Achromatic glass combinations

*K*

_{555}) of a thin lens is the sum of the powers of its component surfaces. Hence for a single lens,

*C*

_{1}and

*C*

_{2}represent the curvatures of the front and rear surfaces, respectively.

*δ*) is the difference in the power between the 460 nm light and 647 nm light [8,9]

*P*

_{555,647}of a single lens is defined by

*ε*of a single lens is given by

*K*

_{555}of the doublet can be calculated by

*K*

_{555})

_{1}and (

*K*

_{555})

_{2}are the powers of the first and the second lens, respectively.

*δ*) of the doublet is given by

*δ*

_{1}and

*δ*

_{2}are the primary color of the first and the second lens, respectively; (

*V*

_{555})

_{1}is the Abbe number of the first lens; (

*V*

_{555})

_{2}is the Abbe number of the second lens.

*δ*=0, we get

*ε*is expressed as

*ε*

_{1}is the secondary spectrum of the first lens;

*ε*

_{2}is the secondary spectrum of the second lens; (

*P*

_{555,647})

_{1}is defined as the partial dispersion of the first lens; (

*P*

_{555,647})

_{2}is the partial dispersion of the second lens.

### 2.2 Area of longitudinal chromatic aberration

### 2-3 Shape factor of the third-order spherical aberration and coma for a thin lens

*h*(

*C*

_{1}+

*C*

_{2}) =

*α*

_{1}+

*α*

_{2}, where

*h*is the marginal ray height;

*α*is the curvature factor the defined by

*α*;

_{i}=h_{i}C_{i}*U*is defined as the conjugate factor

*U*=

*u*

_{1}+

*u*

_{2};

*u*

_{1}and

*u*

_{2}represent the angles between the marginal ray and optical axis of a thin lens in object space and image space, respectively. The Lagrange invariance

*H*=

*n*

_{555}(

*h̄u*-

*hū*), where

*h̄*and

*ū*are the height and angle on the surface of the chief ray, respectively. For a simple expression of the third-order aberrations, the formula can be written as a function of shape factors, while

*S*

_{I}and

*S*

_{II}denote the values of the third-order spherical aberration and the coma. These terms are to be summed over all surfaces as usual. After some mathematical manipulation, we obtain the following aberrations [10]:

*S*

_{I1}and

*S*

_{I2}are the third-order spherical aberrations of the first lens and the second lens, respectively;

*S*

_{II1}and

*S*

_{II2}indicate the coma aberrations for the first lens and the second lens, respectively.

*U*

_{1}and

*U*

_{2}, respectively, which gives

_{1}and Λ

_{2}are the shape factors for the first lens and the second lens, we substitute Eqs. (11), (12), (15) and (16) into Eqs. (13) and (14) to find the solution for the shape factor.

*a*to

*e*are defined as

### 2.4 Thickening design of a doublet lens

*h*

_{1},

*h*

_{2},

*h*

_{3}and

*h*

_{4}are the marginal ray heights for each surface indicating the thicknesses of the doublet. Consider the stop arrangement at the first surface of the front lens. If the curvature factor is kept constant, then the power and shape factor do not change during the lens thickening procedure. We use the paraxial ray formula to find the marginal ray height and the curvature for each surface. Finally, We obtain the finished optical doublet design.

## 3. Design procedure

*Len Design Fundamentals*by Rudolf Kingslake [9]. The design parameters are effective focal length of 10mm; F-number of 3.333; and semi-field angle of 1°. The longitudinal chromatic aberration is first analyzed by looking at primary color and the secondary spectrum formulas. The glass combination needed to obtain smaller longitudinal chromatic aberrations and the power of each lens are found. Next, the shape factor of each surface can be calculated using the formula for third-order spherical aberration and coma for a thin lens. The curvature of each surface is found by the lens thickening process. Both the third-order spherical aberration and coma are controlled. The on-axis RMS spot size in the spot diagram should be less than the diffraction-limited value so that observed area of the aberration curves in the 1.0 field meets the requirement of the spot diagram. If this condition is not satisfied, it is necessary to select an alternative glass combination. Finally, we obtain an optimal doublet design for which the on-axis RMS spot size is less than the value of the diffraction-limited, and ray aberration in the 1.0 field and longitudinal chromatic aberration are smaller. The sensitivity of the human eyes varies with the light wavelengths. The weighting factors for the three different wavelengths are defined in table 3. The area of the aberration curves from the ray-fan diagram is determined from the average area of the sum of the aberration curves at the above three wavelengths multiplied by the weighting factor.

## 4. Design examples

^{-3}mm. The on-axis RMS in the spot diagram for the doublet design is 2.2481×10

^{-3}mm, which is less than the radius of the Airy disk. However the area of the longitudinal chromatic aberration curve is the largest of all of the glass combinations. A look at table 6 shows that it has the largest difference in the area of the rayfan curve for the above three wavelengths. The ray-fan diagram and on-axis spot diagram for the doublet are shown in Fig. 6(a). The modulation transfer function (MTF) is shown in Fig. 6(b). The MTF plot reveals the image quality. The on axis RMS of the 555 nm spot diagram at is less than the radius of the Airy disk for this glass combination. It is obvious that the poor on-axis aberration at other wavelengths, caused by the on-axis MTF of the white light cannot reach the diffraction-limited.

^{2}. The glass combination which is ranked only lower than K3 and F4 is shown in table 4. Table 10 shows the areas of the ray-fan curves at three wavelengths. They have a large difference. The MTF plot is shown in Fig. 8. The on-axis MTF of a white light is less than the value of the diffraction-limited (for the large longitudinal chromatic aberration). The off-axis MTF is affected by the on-axis MTF and cannot reach the diffraction-limited.

## 5. Conclusion

## Acknowledgment

## References and links

1. | K. D. Sharma and S. V. Rama Gopal, “Design of achromatic doublets: evaluation of the double-graph technique,” Appl. Opt. |

2. | S. Baberjee and L. Hazra, “Experiments with a genetic algorithm for structural design of cemented doublets with prespecified aberration targets,” Appl. Opt. |

3. | P. N. Robb, “Selection of optical glasses. 1: Two materials,” Appl. Opt. |

4. | C. L. Tien, W. S. Sun, C. C. Sun, and C. H. Lin, “Optimization design of the split doublet using the shape factors of the third-order aberrations for a thick lens,” J. Mod. Opt. |

5. | R. E. Stephens, “Selection of glasses for three-color achromats,” J. Opt. Soc. Am. |

6. | W. S. Sun and C. H. Chu, “The best doublet design,” 6 |

7. | SCHOTT, http://www.schott.com/optics_devices/english/download/. |

8. | J. M. Geary, |

9. | R. Kingslake, |

10. | W. T. Welford, |

11. | J. W. Goodman, |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(080.2740) Geometric optics : Geometric optical design

(220.1000) Optical design and fabrication : Aberration compensation

**ToC Category:**

Geometric optics

**History**

Original Manuscript: September 12, 2008

Revised Manuscript: January 10, 2009

Manuscript Accepted: January 16, 2009

Published: January 22, 2009

**Citation**

Wen-Shing Sun, Chien-Hsun Chu, and Chuen-Lin Tien, "Well-chosen method for an optimal design of doublet lens design," Opt. Express **17**, 1414-1428 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-3-1414

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### References

- K. D. Sharma and S. V. Rama Gopal, "Design of achromatic doublets: evaluation of the double-graph technique," Appl. Opt. 22, 497-500 (1983). [CrossRef] [PubMed]
- S. Baberjee and L. Hazra, "Experiments with a genetic algorithm for structural design of cemented doublets with prespecified aberration targets," Appl. Opt. 40, 6265-6273 (2001). [CrossRef]
- P. N. Robb, "Selection of optical glasses. 1: Two materials," Appl. Opt. 24, 1864-1877 (1985). [CrossRef] [PubMed]
- C. L. Tien, W. S. Sun, C. C. Sun and C. H. Lin, "Optimization design of the split doublet using the shape factors of the third-order aberrations for a thick lens," J. Mod. Opt. 51, 31-47 (2004).
- R. E. Stephens, "Selection of glasses for three-color achromats," J. Opt. Soc. Am. 49, 398-401 (1959). [CrossRef]
- W. S. Sun and C. H. Chu, "The best doublet design," 6th ODF’08, 10PS-018 (2008).
- SCHOTT, http://www.schott.com/optics_devices/english/download/.
- J. M. Geary, Introduction to Lens Design: with Practical ZEMAX (Willmann-Bell, 2002), Chap. 18.
- R. Kingslake, Lens Design Fundamentals (Academic Press, New York, 1978), Chap. 4.
- W. T. Welford, Aberrations of the Symmetrical Optical System (Academic Press, New York, 1974).
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill), Chap. 4.

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