# Statistical Tool Size Study for Computer-Controlled Optical Surfacing

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## Abstract

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## 1. Introduction

## 2. Fourier Analysis of Surface Error

#### 2.1. The Fourier Transform

#### 2.2. Power Spectral Density

#### 2.3. Encircled Error

#### 2.4. Characteristic Frequency of a Surface Error Map

## 3. The Reference TIF

#### 3.1. The Reference Surface

#### 3.2. Preston’s Equation and the Line TIF

#### 3.3. The Reference TIF

#### 3.4. Integrated PSD and the Characteristic Frequency of a TIF

## 4. Extending the Calibration to Other TIF Shapes

#### 4.1. The Gaussian TIF

#### 4.2. The Spin TIF

#### 4.3. The Orbital TIF

#### 4.4. Determining the TIF Size to Match a Desired CF

## 5. The Characteristic Frequency Ratio

## 6. Computer-Assisted Study of the Optimal CFR for Tool Size Selection

#### 6.1. Simulation Specifications

#### 6.2. Simulation Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the relationship between relevant surface parameters, where “★” is the auto-correlation operator, $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ represent the forward and inverse Fourier transforms, respectively, and ${\sigma}^{2}$ is the variance of the surface profile.

**Figure 2.**An example surface error map (

**a**) and its 2D PSD map (

**b**). The EE of the PSD (

**c**) demonstrates that 80% of the RMS error was due to spatial frequencies lower than 3.41 m${}^{-1}$.

**Figure 4.**(

**a**) Reference Line TIF with characteristic frequency ${f}_{TIF}^{c}=0.05$ mm${}^{-1}$, (

**b**) 1D PSD of the reference Line TIF, and (

**c**) IP of the reference TIF.

**Figure 6.**Simulation example for one iteration of test case with CFR = 0.7 showing (

**a**) initial error map, (

**b**) first residual error map, (

**c**) second residual error map, and (

**d**) final residual error map, with a table keeping track of the RMS value and CF of each map, the FWHM of the Gaussian TIF used on each map, and the calculated run time required for each run.

**Figure 7.**Simulation results for the average total run time (blue), the standard deviation of the total run time (red), the average number of iterations (yellow), and the figure of merit (purple).

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## Share and Cite

**MDPI and ACS Style**

Pullen, W.C.; Wang, T.; Choi, H.; Ke, X.; Negi, V.S.; Huang, L.; Idir, M.; Kim, D.
Statistical Tool Size Study for Computer-Controlled Optical Surfacing. *Photonics* **2023**, *10*, 286.
https://doi.org/10.3390/photonics10030286

**AMA Style**

Pullen WC, Wang T, Choi H, Ke X, Negi VS, Huang L, Idir M, Kim D.
Statistical Tool Size Study for Computer-Controlled Optical Surfacing. *Photonics*. 2023; 10(3):286.
https://doi.org/10.3390/photonics10030286

**Chicago/Turabian Style**

Pullen, Weslin C., Tianyi Wang, Heejoo Choi, Xiaolong Ke, Vipender S. Negi, Lei Huang, Mourad Idir, and Daewook Kim.
2023. "Statistical Tool Size Study for Computer-Controlled Optical Surfacing" *Photonics* 10, no. 3: 286.
https://doi.org/10.3390/photonics10030286