Modeling Thickness Variation in Semiconductor Thin-Film Processes — A Spatial Decomposition Approach to Machine Learning (ML)

The model: $T = A \cdot a + e$

 The Zernike decomposition writes a measured wavefront $T$ as a linear combination of Zernike basis functions plus measurement noise:

$$T = A \cdot a + e$$

 This appendix walks through what each piece of the equation means and what makes $A$ — the Zernike basis matrix — useful.

Symbols

Per-point equation

 Written out for one point $i$, the model becomes:

$$T_i = a_1 Z_1(\rho_i, \theta_i) + a_2 Z_2(\rho_i, \theta_i) + \ldots + a_N Z_N(\rho_i, \theta_i) + e_i$$

 Each measurement is decomposed into a weighted sum of $N$ predefined shapes; the unknowns are the weights $a_j$. Stacking the $m$ such per-point equations gives the matrix form $T = A \cdot a + e$.

Structure of $A$ ($m \times N$)

 The entries of $A$ are Zernike-basis evaluations at the measurement points:

$$A[i, j] = Z_j(\rho_i, \theta_i)$$

 That is, $A[i, j]$ is “the value of the $j$-th basis at the $i$-th measurement point”. $A$ has $m$ rows (one per measurement point) and $N$ columns (one per Zernike basis function):

            j=1     j=2     j=3    ...   j=N
        ┌─────────────────────────────────────┐
i=1     │  Z_1    Z_2    Z_3    ...    Z_N    │ ← all basis values at point 1
i=2     │  Z_1    Z_2    Z_3    ...    Z_N    │ ← all basis values at point 2
...     │   …      …      …             …     │
i=m     │  Z_1    Z_2    Z_3    ...    Z_N    │ ← all basis values at point m
        └─────────────────────────────────────┘
          ↑
        column 1 = Z_1 evaluated at every measurement point

Row and column meaning

Noll normalization

$A^{\!\top}\!A \approx m \cdot I$ — sampling density and regression stability

 The least-squares solution is $\hat{a} = (A^{\!\top}\!A)^{-1} A^{\!\top}\, T$, so the structure of $A^{\!\top}\!A$ controls the conditioning of the fit. Its $(j, k)$ entry is the inner product of column $j$ and column $k$ of $A$:

$$(A^{\!\top}\!A)_{jk} = \sum_{i=1}^{m} Z_j(\rho_i, \theta_i)\, Z_k(\rho_i, \theta_i)$$

 Dense sampling ($m \rightarrow \infty$, with the points uniformly covering the disk): the discrete sum tends to the integral average:

$$\frac{1}{m} \sum_{i=1}^{m} Z_j(\rho_i, \theta_i)\, Z_k(\rho_i, \theta_i) \;\approx\; \frac{1}{\pi} \iint_{\text{disk}} Z_j Z_k \, dA \;=\; \delta_{jk}$$

hence $A^{\!\top}\!A \approx m \cdot I$ — a diagonal matrix with all diagonal entries equal to $m$. In this regime:

• $(A^{\!\top}\!A)^{-1} \approx I/m$ — the inverse is essentially “multiply by $1/m$”. Numerically very stable.
• $\hat{a}_j \approx \frac{1}{m} \sum_i Z_j(\rho_i, \theta_i)\, T_i$ — each coefficient is just a $Z_j$-weighted average of the measurements.
• Condition number $\approx 1$ — measurement noise propagates to the coefficients with virtually no amplification.

 Sparse sampling (e.g. the $m = 13$ measurement points used in this project): the integral approximation breaks. $A^{\!\top}\!A$ is no longer diagonal; it is a general $N \times N$ matrix with non-zero off-diagonal entries, and its condition number grows. With $m = 13 \geq N = 9$ the fit is still solvable, but if the condition number becomes too large, ridge regression (adding $\lambda I$ to $A^{\!\top}\!A$) is a standard way to stabilize the solve.