## Automatic clocking optimization for compensating two-dimensional tolerances |

Optics Express, Vol. 21, Issue 19, pp. 22145-22152 (2013)

http://dx.doi.org/10.1364/OE.21.022145

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### Abstract

Clocking of lens elements is frequently used as an effective method of compensating for two-dimensional tolerances such as material inhomogeneity and surface figure errors. Typically, the lens designer has to determine the optimum angles of rotation by manually modeling lens element clocking in the commercial optical design software because the nature of errors resolved by lens clocking does not lead to good convergences for clocking optimization. In this paper, a method of automatic clocking optimization is developed. The method is implemented using a combination of particle swarm optimization algorithm and commercial optical design software. The optimum angles of rotation and predicted imaging performance are automatically calculated using this method. Methods of implementation and optimization examples are also given.

© 2013 OSA

## 1. Introduction

2. A. Engel, E. Mörsen, A. Jordanov, and K. Knapp, “Present and future industrial metrology needs for Qualification of High Quality Optical Micro Lithography Materials,” Proc. SPIE **4449**, 1–6 (2001). [CrossRef]

3. J. R. Rogers, “Homogeneity tolerances for Optical Elements,” (CODE V User Group Meeting, 2011). http://www.opticalres.com/

5. D. M. Williamson, “Compensator selection in the tolerancing of a microlithographic lens,” Proc. SPIE **1049**, 178–186 (1989). [CrossRef]

6. T. Matsuyama, I. Tanaka, T. Ozawa, K. Nomura, and T. Koyama, “Improving lens performance through the most recent lens manufacturing process,” Proc. SPIE **5040**, 801–810 (2003). [CrossRef]

7. T. Yoshihara, R. Koizumi, K. Takahashi, S. Suda, and A. Suzuki, “Realization of very-small aberration projection lenses,” Proc. SPIE **4000**, 559–566 (2000). [CrossRef]

## 2. Clocking Optimization

*M*, and the number of lens elements is denoted by

*N*.

*θ*

_{1},

*θ*

_{2},…

*θ*are the rotation angles for each lens element, and

_{N}*N*interferometric material inhomogeneity errors are denoted by

*s*

_{1},

*s*

_{2},…

*s*. If interferometric material inhomogeneity errors associated with the rotation angle are denoted by

_{N}*s*

_{1}(

*θ*

_{1}),

*s*

_{2}(

*θ*

_{2}),…

*s*

_{N}(

*θ*

_{N}), an expression for the merit function can be written as Eq. (1):where

*f*is the relationship between the figure of merit and lens rotation angles. Clocking optimization aims to find an optimum set of rotation angles of the lens elements with the best lens performance. However, getting an analytical expression for the merit function is impossible, which makes clocking optimization similar to the black-box optimization problem. Thus, the damped least-squares optimization algorithm used in commercial optical design software does not work well in clocking optimization. No sophisticated and automatic clocking optimization algorithms have been reported. In this paper, we propose to use the PSO algorithm to automatically calculate the optimum rotation angles of lens elements.

### 2.1 Particle swarm optimization (PSO)

### 2.2 Two methods of clocking optimization

- Step 1: The as-built imaging performance is determined and evaluated based on specifications. For lithographic lens, the two most important metrics are wavefront error and centroid-based distortion.
- Step 2: The lens is loaded and the interferograms of material inhomogeneity error from material suppliers or the inhomogeneity error generator is attached.
- Step 3: All lens elements (interferograms are actually rotated) with random clocking angles and the imaging performance are evaluated. Results are recorded and the process cycle is repeated through the macro. The exit condition is checked after each cycle.
- Step 4: The optimum rotation angles are obtained according to the best as-built imaging performances.

- Step 1: Matlab and PSOt are initialized, and the merit function and variable dimensions of the problem are defined. A single value of the merit function is calculated based on imaging performance such as RMS wavefront error and centroid-based distortion, and the variable dimension is equal to the number of clocking lens elements.
- Step 2: The PSOt sets a number of particles that fly through the hyperspace of the problem. Each particle represents a candidate solution for clocking optimization. After each cycle of optimization, the position of a particle is updated both by the previous best position of the particle and the best overall position of the entire group. The updated particle representing a new set of clocking angles is automatically passed to CodeV.
- Step 3: CodeV is commanded using the COM interface. The lens is loaded and the interferograms of material inhomogeneity error from material suppliers or the inhomogeneity error generator are attached.
- Step 4: CodeV is commanded to rotate the lens elements with certain clocking angles provided from the PSOt, and the merit function is calculated based on the imaging performance evaluation. The merit function value is returned to the PSO.
- Step 5: PSO records the relationship between clocking angels and merit function values that are returned from CodeV. The best position of the particle and the best overall position of the entire group are updated. A new set of clocking angles is updated according to the PSO algorithm and automatically passed to CodeV.
- Step 6: Steps 4 and 5 are repeated cycle by cycle. The exit condition is checked after each cycle.
- Step 7: The optimum rotation angles of lens elements are automatically listed at the end of the Matlab implementation.

## 3. Examples of clocking optimization

*W*and

*D*are the worst wavefront error and distortion, respectively, across the image field and are expressed in nanometers, and

*a*and

*b*are the imaging performance weights for the wavefront error and distortion, respectively. A balanced imaging performance between the wavefront error and distortion can be obtained by adjusting the values of

*a*and

*b*. For instance, we set

*a =*1 and

*b =*0.25 in this paper.

## 4. Conclusion

## References and links

1. | K. Becker, B. Dörband, R. Lörcher, and M. Schmidt, “Aspheric Optics at Different Quality Levels and Functional Need,” Proc. SPIE |

2. | A. Engel, E. Mörsen, A. Jordanov, and K. Knapp, “Present and future industrial metrology needs for Qualification of High Quality Optical Micro Lithography Materials,” Proc. SPIE |

3. | J. R. Rogers, “Homogeneity tolerances for Optical Elements,” (CODE V User Group Meeting, 2011). http://www.opticalres.com/ |

4. | T. I. Harris, “Overview of CODE V Development,” Proc. SPIE |

5. | D. M. Williamson, “Compensator selection in the tolerancing of a microlithographic lens,” Proc. SPIE |

6. | T. Matsuyama, I. Tanaka, T. Ozawa, K. Nomura, and T. Koyama, “Improving lens performance through the most recent lens manufacturing process,” Proc. SPIE |

7. | T. Yoshihara, R. Koizumi, K. Takahashi, S. Suda, and A. Suzuki, “Realization of very-small aberration projection lenses,” Proc. SPIE |

8. | J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” in Proceedings of IEEE International Conference on Neural Networks (Perth, Australia, 1995), pp. 1942–1948. [CrossRef] |

9. | Y. Omura, European Patent 1139138, embodiment 5. |

10. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

**OCIS Codes**

(080.3620) Geometric optics : Lens system design

(220.1000) Optical design and fabrication : Aberration compensation

**ToC Category:**

Geometric Optics

**History**

Original Manuscript: June 18, 2013

Revised Manuscript: August 26, 2013

Manuscript Accepted: August 27, 2013

Published: September 12, 2013

**Citation**

Weicai Xu, Wei Huang, Chunlai Liu, and Hongbo Shang, "Automatic clocking optimization for compensating two-dimensional tolerances," Opt. Express **21**, 22145-22152 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22145

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### References

- K. Becker, B. Dörband, R. Lörcher, and M. Schmidt, “Aspheric Optics at Different Quality Levels and Functional Need,” Proc. SPIE3739, ••• (1999).
- A. Engel, E. Mörsen, A. Jordanov, and K. Knapp, “Present and future industrial metrology needs for Qualification of High Quality Optical Micro Lithography Materials,” Proc. SPIE4449, 1–6 (2001). [CrossRef]
- J. R. Rogers, “Homogeneity tolerances for Optical Elements,” (CODE V User Group Meeting, 2011). http://www.opticalres.com/
- T. I. Harris, “Overview of CODE V Development,” Proc. SPIE1354, 104 (1990).
- D. M. Williamson, “Compensator selection in the tolerancing of a microlithographic lens,” Proc. SPIE1049, 178–186 (1989). [CrossRef]
- T. Matsuyama, I. Tanaka, T. Ozawa, K. Nomura, and T. Koyama, “Improving lens performance through the most recent lens manufacturing process,” Proc. SPIE5040, 801–810 (2003). [CrossRef]
- T. Yoshihara, R. Koizumi, K. Takahashi, S. Suda, and A. Suzuki, “Realization of very-small aberration projection lenses,” Proc. SPIE4000, 559–566 (2000). [CrossRef]
- J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” in Proceedings of IEEE International Conference on Neural Networks (Perth, Australia, 1995), pp. 1942–1948. [CrossRef]
- Y. Omura, European Patent 1139138, embodiment 5.
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge Univ. Press, 2007), Chap. 10.5 and Chap. 10.12.

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