## Merit function regression method for efficient alignment control of two-mirror optical systems

Optics Express, Vol. 15, Issue 8, pp. 5059-5068 (2007)

http://dx.doi.org/10.1364/OE.15.005059

Acrobat PDF (248 KB)

### Abstract

The precision alignment of high-performance, wide-field optical systems is generally a difficult and often laborious process. We report a new merit function regression method that has the potential to bring to such an optical alignment process higher efficiency and accuracy than the conventional sensitivity table method. The technique uses actively damped least square algorithm to minimize the Zernike coefficient-based merit function representing the difference between the designed and misaligned optical wave fronts. The application of this method for the alignment experiment of a Cassegrain type collimator of 900mm in diameter resulted in a reduction of the mean system rms wave-front error from 0.283λ to 0.194λ, and in the field dependent wave-front error difference from ±0.2λ to ±0.014λ in just two alignment actions. These results demonstrate a much better performance than that of the conventional sensitivity table method simulated for the same steps of experimental alignment.

© 2007 Optical Society of America

## 1. Introduction

## 2. KRISS Collimator

6. SVD, “Matlab function reference,”http://www-ccs.ucsd.edu/matlab/techdoc/ref/svd.html.

## 3. Theoretical Basis of Merit Function Regression Method

### 3.1 Concept and limitations of conventional sensitivity table method

*Z*is the differences of Zernike coefficients between the measured and model WFEs. A is the Zernike sensitivity table that is calculated from the ideal configuration model. Δ

*D*represents the amount of disturbances in the alignment parameters (

*x*) such as displacement, tilt or decenter.

_{i}*m*and N are the total number of Zernike coefficients fitted and the total number of alignment parameters, respectively. This equation is commonly solved for ∆

*D*using singular value decomposition technique [6

6. SVD, “Matlab function reference,”http://www-ccs.ucsd.edu/matlab/techdoc/ref/svd.html.

### 3.2 Merit function regression method

_{i}and T

_{i}are the current and target values of chosen parameter that are the Zernike coefficients in our case. W

_{i}is the weighting factor. Many commercial optical design software use the actively damped least square method to minimize the MF value to produce the best-fit parameters [8, 9

9. J. Meiron, “Damped Least-Squares Method for Automatic Lens Design,” J. Opt. Soc. Am. **55**, 1105–1109 (1965) [CrossRef]

- 1. Read the Zernike coefficients from the interferometric measurements. They are T
_{i}which represents the misaligned system WFE. - 2. Assign the ideal model Zernike coefficients to V
_{i}, which represents the current alignment status (i.e. wave front) of the designed optical system. - 3. Run the optimization algorithm (e.g. damped least square technique [8, 9]) embedded in the software to minimize the MF. This operation varies the alignment parameters so that V
9. J. Meiron, “Damped Least-Squares Method for Automatic Lens Design,” J. Opt. Soc. Am.

**55**, 1105–1109 (1965) [CrossRef]_{i}approaches T_{i}as closely as possible. - 4. When MF is minimized, read the alignment parameters that indicate the misalignment state of the optical system.

_{5}:Astig X, Z

_{6}:Astig Y, Z

_{7}:Coma X, Z

_{8}:Coma Y, and Z

_{9}:Spherical) of Fringe Zernike polynomials at 5 different fields (on-axis and 4 extreme fields). The weighting factor for the MF regression method was set to unity for all variables. The results are summarized in Table 1. The sensitivity table method resulted in accuracy of prediction worse than that of the merit function minimization method, and the estimation error increases with the magnitude of misalignment perturbation. This can be caused either by the nonlinearity of the Zernike coefficient sensitivity to the alignment parameters and/or by cross-coupling among the alignment parameters, as explained in the earlier study [7]. In practice, therefore, the sensitivity table method tends to give slow convergence in iterative alignment processes, when the initial alignment state is far from the design tolerance range of the target optical system.

^{-5}even with Dx = Dy = 1 mm and Tx = Ty = 1 deg. This indicates that the MF regression method can provide fast convergence in iterative alignment processes, even if it starts from the misalignment state that is initially far from the design tolerance of the target optical system.

## 4. Error Sources of the MF Regression Method

### 4.1 Effects of number of measurement fields

_{5}∼ Z

_{9}) extracted from the single field measurement can be used to solve Eq. 2 for the 5 misalignment parameters. However, the interferometric measurement has, in general, several error sources including air turbulence and vibration. Such error sources can produce inaccurate Zernike coefficients, hence leading to increased error in estimating the misalignment parameters. This can be improved by taking multiple field measurements, as it tends to average the effects of the interferometric measurement error sources out, but at the expense of stretching the process overhead.

_{5}∼Z

_{9}coefficient terms were used to fit the optical path difference map from each field measurement. Figure 3 shows the field numbering that the measurement sequence followed in this simulation.

### 4.2 Effects of field positioning error

## 5. Alignment Experiment with KRISS Collimator

^{st}alignment action) was the mean rms WFE of 0.257λ and the maximum rms WFE difference of 0.18λ. The difference between the prediction and the measurement indicates some errors in the MF regression run. The main reason for such difference might be from the errors in the WFE measurement that could be much more than 10 %, as explained in section 4.

^{st}alignment action (i.e. correction of field error and misalignment of SM) was performed and produced the measured WFE (i.e. solid square symbol) of “1

^{st}alignment” in Fig. 6. The measured mean rms WFE and the maximum rms WFE difference was reduced to 0.259λ and 0.102λ, respectively. The MF regression run in this state showed 0.222λ in mean rms WFE and 0.076λ of maximum rms WFE difference, which shows some calculation errors from the MF regression run. After the 2

^{nd}alignment action, the measurement result shows 0.194λ in mean rms WFE and 0.014λ in maximum rms WFE difference. Considering that the final rms WFEs observed both from Fig. 6 and 8 come mostly from the deformation of PM and thus cannot be improved by continuing the alignment step, the KRISS collimator was successfully aligned. It is also shown in Fig. 7 that the field error was reduced from 0.05 degrees to 0.01 degrees. The field error of 0.05 degrees is about 70 % of a half field of view. Since this large error was decreased to 14% after two alignment actions, the MF regression method must be an effective tool in alignment, even with a large initial field error.

## 6. Concluding Remarks

10. H. Lee, G. B. Dalton, I. A. Tosh, and S.-W. Kim, “Computer guided alignment I: Phase and amplitude modulation of alignment-influenced optical wave front,” Opt. Express **15**, 3127–3139 (2007) [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | R. N. Wilson, “Aberration Theory of Telescopes,” in |

2. | M. A. Lundgren and W. L. Wolfe, “Simultaneous Alignment and Multiple Surface Figure Testing of Optical System Components Via Wavefront Aberration Measurement and Reverse Optimization,” in |

3. | J. W. Figoski, T. E. Shrode, and G. F. Moore, “Computer-aided Alignment of a Wide-field, Three-mirror, Unobscured, High-resolution Sensor,” in |

4. | Z. Bin, Z. Xiaohui, W. Cheng, and H. Changyuan, “Investigation on Computer-aided Alignment of the Complex Optical System,” in |

5. | H. S. Yang, Y. W. Lee, E. D. Kim, Y. W. Choi, and A. A. A. Rashed, “Alignment methods for Cassegrain and RC telescope with wide field of view,” in |

6. | SVD, “Matlab function reference,”http://www-ccs.ucsd.edu/matlab/techdoc/ref/svd.html. |

7. | E. D. Kim, Y.-W. Choi, M.-S. Kang, and S. C. Choi, “Reverse-optimization alignment algorithm using Zernike sensitivity,” J. Opt. Soc. Kor. |

8. |
Zemax development corporation, “Optimization,” in |

9. | J. Meiron, “Damped Least-Squares Method for Automatic Lens Design,” J. Opt. Soc. Am. |

10. | H. Lee, G. B. Dalton, I. A. Tosh, and S.-W. Kim, “Computer guided alignment I: Phase and amplitude modulation of alignment-influenced optical wave front,” Opt. Express |

11. | H. S. Yang, S.-W. Kim, Y.-W. Lee, and S. Kim are preparing a manuscript to be called “Extending the merit-function regression method for effective alignment of multiple mirror optical systems”. |

**OCIS Codes**

(110.6770) Imaging systems : Telescopes

(120.4820) Instrumentation, measurement, and metrology : Optical systems

(220.1140) Optical design and fabrication : Alignment

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: February 12, 2007

Revised Manuscript: April 3, 2007

Manuscript Accepted: April 9, 2007

Published: April 11, 2007

**Citation**

Seonghui Kim, Ho-Soon Yang, Yun-Woo Lee, and Sug-Whan Kim, "Merit function regression method for efficient alignment control of two-mirror optical systems," Opt. Express **15**, 5059-5068 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5059

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### References

- R. N. Wilson, "Aberration Theory of Telescopes," in Reflecting Telescope Optics I (Springer, Berlin, 1996), Ch. 3.
- M. A. Lundgren, and W. L. Wolfe, "Simultaneous Alignment and Multiple Surface Figure Testing of Optical System Components Via Wavefront Aberration Measurement and Reverse Optimization," in 1990 Intl Lens Design Conference, G.N. Lawrence, ed., Proc. of SPIE 1354, 533-539 (1990).
- J. W. Figoski, T. E. Shrode, and G. F. Moore, "Computer-aided Alignment of a Wide-field, Three-mirror, Unobscured, High-resolution Sensor," in Recent Trends in Optical Systems Design and Computer Lens Design Workshop II, R. E. Fischer, and R. C. Juergens, eds., Proc. of SPIE 1049, 166-177 (1989).
- Z. Bin, Z. Xiaohui, W. Cheng, and H. Changyuan, "Investigation on Computer-aided Alignment of the Complex Optical System," in Advanced Optical Manufacturing and testing technology, L. Yang, H. M. Pollicove, Q. Xin, and J. C. Wyant, eds., Proc. of SPIE 4231, 67-72 (2000).
- H. S. Yang, Y. W. Lee, E. D. Kim, Y. W. Choi, and A. A. A. Rashed, "Alignment methods for Cassegrain and RC telescope with wide field of view," in Space systems engineering and optical alignement mechanisms, L. D. Peterson, and R. C. Guyer, eds., Proc. SPIE 5528, 334-341 (2004).
- SVD, "Matlab function reference,"http://www-ccs.ucsd.edu/matlab/techdoc/ref/svd.html.
- E. D. Kim, Y.-W. Choi, M.-S. Kang, and S. C. Choi, "Reverse-optimization alignment algorithm using Zernike sensitivity," J. Opt. Soc. Kor. 9, 67-73 (2005).
- Zemax development corporation, "Optimization," in Zemax optical design program user's guide (2004), Ch.14.
- J. Meiron, "Damped Least-Squares Method for Automatic Lens Design," J. Opt. Soc. Am. 55, 1105-1109 (1965) [CrossRef]
- H. Lee, G. B. Dalton, I. A. Tosh and S.-W. Kim, "Computer guided alignment I: Phase and amplitude modulation of alignment-influenced optical wave front," Opt. Express 15,3127-3139 (2007) [CrossRef] [PubMed]
- H. S. Yang, S.-W. Kim. Y.-W. Lee and S. Kim are preparing a manuscript to be called "Extending the merit-function regression method for effective alignment of multiple mirror optical systems".

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