Identification of Zernike-Polynomial Systematic Pattern with Spatial Variation Spectrum

Zernike多項式常於生醫及半導體製造用於描述所觀察到的圓狀表面的系統變異並建立模型。但是因為Zernike多項式包含無數個多項式，故建模時會遭遇多項式選擇的問題。統的Zernike多項式建模選擇方式僅是依照Zernike多項式的順序選擇，並不考慮Zernike多項式與所觀察的圓面間的關係，或採用bootstrap方法，須花費大量時間在計算及模擬所有可能的模型上以期找到最適合的模型。解決上述兩種方法的缺失，本研究提出考慮到圓面及Zernike多項式之關係的多項式選擇方法，目的是分析圓面的系統變異，並找出與其最相似的Zernike多項式來建立模型，而無須考慮加入與圓面系統變異不相似的Zernike多項式。研究使用空間移動變異及半徑移動變異辨識不同頻率下的系統變異特徵。彼此間變異特徵類似的Zernike多項式會歸入同組，並建立預測組內與圓面最相近之多項式的模型。此預測模型的結果會用於順向選取法選擇與圓面最相近的Zernike多項式進行建模。驗證本研究所提出之Zernike多項式項次選擇方法，透過模擬產生的資料及實際資料，與其他Zernike多項式項次選擇方法比較。我們發現所提出的項次選擇方法可以得到近似或更佳的結果。Zernike polynomials are commonly used to characterize and model systematic patterns of observed circular-shaped topography, such as wafer topography in semiconductor fabrication and corneal aberration in biomedical engineering. However, the infinite number of Zernike polynomials leads to difficulties in modeling the observed topography. Conventional Zernike polynomials selection methods either select the Zernike polynomials in sequence without considering the similarity between the observed topography and the Zernike polynomials or use the bootstrap method, which demand great simulation and computation time on all possible Zernike-polynomials combination models to find the best-fit model.n this research, we propose a Zernike polynomials selection method based on the characteristics of circular topography. The objective is to analyze observed topography and select the Zernike polynomials with similar spatial variation patterns for modeling, without attempting all possible model combinations.patial moving variance and radial moving variance are used to characterize the systematic variation over the frequency spectrum. The Zernike polynomials will be classified into groups based on their spatial variation pattern and prediction models will be established for each group. Using the prediction models, a forward selection algorithm will be proposed to select the Zernike polynomials, with systemic variation patterns similar to the observed topography, into the model.o validate, the proposed method is compared with other conventional selection methods via simulated data and real cases. It is shown that the same or better modeling results can be obtained through the proposed methods.1 Introduction............................................................................................................1.1 Background and Motivation......................................................................1.2 The Zernike Polynomials...........................................................................3.3 Conventional Zernike Polynomials Selection Methods and Literature Review...................................................................................................................5.4 Shortcomings of Traditional and Bootstrap Zernike Polynomials Selection Methods..................................................................................................7.5 Research Objective....................................................................................8.6 Thesis Organization...................................................................................8 Spatial Variation Characterization of Circular-shaped Topography....................10.1 Exampler-free Characterization of Spatial Variation...............................10.2 Spatial Sample Variation..........................................................................12.3 Spatial Variation Spectrum......................................................................13.4 Frequency-partition of SVS......................................................................15.5 Radial Sample Variance...........................................................................17.6 Radial Variation Spectrum.......................................................................19.7 Frequency-partition of RVS......................................................................20 Spatial-Variation Characterization of Zernike polynomials................................23.1 SVS-characterization of Zernike polynomials..........................................23.2 RVS-characterization of Zernike polynomials.........................................24.3 Grouping of Zernike polynomials based on SVS and RVS.......................26.3.1 Selection of Zernike-polynomial Order by SVS and RVS frequency-partitions.....................................................................................26.3.2 Selection of Zernike-polynomial azimuthal frequency by SVS and RVS frequency-partitions.............................................................................32.3.3 Selection of Zernike-polynomial difference between order and azimuthal frequency by SVS and RVS frequency-partitions........................34 Zernike Polynomials Model Selection.................................................................39.1 Model Performance evaluation: the Similarity........................................40.2 Stopping Criteria for Zernike Polynomials Selection..............................41.3 Topography-Correlation-Based Zernike Polynomials Model Selection Algorithm.............................................................................................................42.4 Zernike polynomials forward selection...................................................44.4.1 Forward Selection Principle.........................................................44.4.2 Inaccuracy tolerance of within group prediction.........................46.4.3 Forward Zernike polynomials Selection Algorithm....................47 Case Study...........................................................................................................49.1 General Cases...........................................................................................49.2 The CDU Dataset.....................................................................................52 Conclusion...........................................................................................................56EFERENCE...............................................................................................................5