{"id":6564,"date":"2026-05-04T13:39:10","date_gmt":"2026-05-04T18:39:10","guid":{"rendered":"https:\/\/ykim.synology.me\/wordpress\/?p=6564"},"modified":"2026-05-06T01:03:48","modified_gmt":"2026-05-06T06:03:48","slug":"modeling-thickness-variation-in-semiconductor-thin-film-processes-a-spatial-decomposition-approach-to-machine-learning-ml","status":"publish","type":"post","link":"https:\/\/ykim.synology.me\/wordpress\/modeling-thickness-variation-in-semiconductor-thin-film-processes-a-spatial-decomposition-approach-to-machine-learning-ml-6564\/","title":{"rendered":"Modeling Thickness Variation in Semiconductor Thin-Film Processes \u2014 A Spatial Decomposition Approach to Machine Learning (ML)"},"content":{"rendered":"\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"800\" height=\"800\" src=\"https:\/\/ykim.synology.me\/wordpress\/wp-content\/uploads\/2026\/05\/Zernike-polynomials-9-orders-800px.png\" alt=\"\" class=\"wp-image-6591\" style=\"width:600px\" srcset=\"https:\/\/ykim.synology.me\/wordpress\/wp-content\/uploads\/2026\/05\/Zernike-polynomials-9-orders-800px.png 800w, https:\/\/ykim.synology.me\/wordpress\/wp-content\/uploads\/2026\/05\/Zernike-polynomials-9-orders-800px-300x300.png 300w, https:\/\/ykim.synology.me\/wordpress\/wp-content\/uploads\/2026\/05\/Zernike-polynomials-9-orders-800px-150x150.png 150w, https:\/\/ykim.synology.me\/wordpress\/wp-content\/uploads\/2026\/05\/Zernike-polynomials-9-orders-800px-768x768.png 768w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><\/figure>\n\n\n<style>.kadence-column6564_5675ce-6c > .kt-inside-inner-col,.kadence-column6564_5675ce-6c > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_5675ce-6c > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_5675ce-6c > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_5675ce-6c > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_5675ce-6c > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_5675ce-6c{position:relative;}.kadence-column6564_5675ce-6c, .kt-inside-inner-col > .kadence-column6564_5675ce-6c:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_5675ce-6c > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_5675ce-6c > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_5675ce-6c\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">\u2003Thickness uniformity in thin-film deposition determines downstream yield and device performance. Variation arises along two distinct axes \u2014 within a single wafer (Within-Wafer, <strong>WiW<\/strong>) and across wafers over time (Wafer-to-Wafer, <strong>W2W<\/strong>). These two axes have different physical origins and demand different diagnostic treatments. Mixing them into a single ML target forces the model to learn two unrelated physics simultaneously, hurting both.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003This post presents an ML framework that incorporates <strong>Spatial Decomposition<\/strong> based on <strong>Zernike Polynomials<\/strong> (Zernike 1934; Noll 1976) into<strong> label engineering<\/strong>. It compresses 13-point wafer thickness measurements into 9 physically meaningful coefficients, which then serve as targets for a Two-Head ML architecture. <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003The framework is organized along three axes: <strong>Domain<\/strong> (facts), <strong>Design<\/strong> (technical choices), and <strong>Delivery<\/strong> (user value).<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">1. Domain<\/h2>\n\n\n<style>.kadence-column6564_10b329-67 > .kt-inside-inner-col,.kadence-column6564_10b329-67 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_10b329-67 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_10b329-67 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_10b329-67 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_10b329-67 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_10b329-67{position:relative;}.kadence-column6564_10b329-67, .kt-inside-inner-col > .kadence-column6564_10b329-67:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_10b329-67 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_10b329-67 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_10b329-67\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\"><em>Domain defines the set of facts the model operates within: the environment, the data, and the mathematical tools available.<\/em><\/p>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_41e75f-85 > .kt-inside-inner-col,.kadence-column6564_41e75f-85 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_41e75f-85 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_41e75f-85 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_41e75f-85 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_41e75f-85 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_41e75f-85{position:relative;}.kadence-column6564_41e75f-85, .kt-inside-inner-col > .kadence-column6564_41e75f-85:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_41e75f-85 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_41e75f-85 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_41e75f-85\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">1.1 WiW \/ W2W Variation \u2014 Industrial Background<\/h3>\n\n\n<style>.kadence-column6564_6e36d7-5a > .kt-inside-inner-col,.kadence-column6564_6e36d7-5a > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_6e36d7-5a > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_6e36d7-5a > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_6e36d7-5a > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_6e36d7-5a > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_6e36d7-5a{position:relative;}.kadence-column6564_6e36d7-5a, .kt-inside-inner-col > .kadence-column6564_6e36d7-5a:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_6e36d7-5a > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_6e36d7-5a > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_6e36d7-5a\"><div class=\"kt-inside-inner-col\">\n<ul class=\"wp-block-list\">\n<li><strong>W2W variation<\/strong>: temporal change in mean thickness per wafer. Driven by Run-to-Run drift, source\/target depletion, recipe shifts.<\/li>\n\n\n\n<li><strong>WiW variation<\/strong>: spatial thickness distribution within a single wafer. Driven by chamber hardware asymmetry, gas flow imbalance, temperature non-uniformity.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The two axes have different physical origins, so they require different diagnostic and control approaches. Treating them as a single ML target conflates two physics regimes and degrades both.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">1.2 Measurement Setup \u2014 13-Point Wafer Location Pattern<\/h3>\n\n\n<style>.kadence-column6564_ed4ebd-f4 > .kt-inside-inner-col,.kadence-column6564_ed4ebd-f4 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_ed4ebd-f4 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_ed4ebd-f4 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_ed4ebd-f4 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_ed4ebd-f4 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_ed4ebd-f4{position:relative;}.kadence-column6564_ed4ebd-f4, .kt-inside-inner-col > .kadence-column6564_ed4ebd-f4:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_ed4ebd-f4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_ed4ebd-f4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_ed4ebd-f4\"><div class=\"kt-inside-inner-col\">\n<figure class=\"wp-block-table\"><table><thead><tr><th>Item<\/th><th>Value<\/th><\/tr><\/thead><tbody><tr><td>Wafer size<\/td><td>300 mm<\/td><\/tr><tr><td>Edge Exclusion (EE)<\/td><td>5 mm<\/td><\/tr><tr><td>Number of points<\/td><td>13<\/td><\/tr><tr><td>Layout<\/td><td>Center 1 + Middle ring (r=75 mm, 4 cardinal) + Edge ring (r=145 mm, 8 directions)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Three radial levels (0 \/ 75 \/ 145 mm) \u2014 enables radial-order decomposition.<\/li>\n\n\n\n<li>Cardinal-aligned middle ring (0\u00b0\/90\u00b0\/180\u00b0\/270\u00b0) \u2014 stabilizes astigmatism extraction.<\/li>\n\n\n\n<li>Edge ring at 8 directions \u2014 enables higher-order asymmetry (coma, trefoil) representation.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">1.3 Dataset under Study<\/h3>\n\n\n<style>.kadence-column6564_6930bd-08 > .kt-inside-inner-col,.kadence-column6564_6930bd-08 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_6930bd-08 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_6930bd-08 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_6930bd-08 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_6930bd-08 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_6930bd-08{position:relative;}.kadence-column6564_6930bd-08, .kt-inside-inner-col > .kadence-column6564_6930bd-08:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_6930bd-08 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_6930bd-08 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_6930bd-08\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\"><strong>Output side (target source)<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Data<\/th><th>Form<\/th><th>Frequency<\/th><\/tr><\/thead><tbody><tr><td>13-point thickness measurements<\/td><td>13 scalars per wafer (\u00c5 or nm)<\/td><td>per wafer or lot<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Input side (feature source)<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Data<\/th><th>Form<\/th><th>Use<\/th><\/tr><\/thead><tbody><tr><td>Equipment sensor time-series \u2014 Fault Detection and Classification (FDC)<\/td><td>RF Power, Pressure, Gas Flow, Temperature, etc.<\/td><td>step-wise statistics for feature engineering<\/td><\/tr><tr><td>Process metadata<\/td><td>Recipe ID, Chamber ID, Timestamp<\/td><td>grouping variables, context<\/td><\/tr><tr><td>Preventive Maintenance (PM) \/ maintenance history<\/td><td>event log<\/td><td>baseline definition for drift analysis<\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">1.4 Zernike Polynomials \u2014 Mathematical Foundation<\/h3>\n\n\n<style>.kadence-column6564_8d9d12-45 > .kt-inside-inner-col,.kadence-column6564_8d9d12-45 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_8d9d12-45 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_8d9d12-45 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_8d9d12-45 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_8d9d12-45 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_8d9d12-45{position:relative;}.kadence-column6564_8d9d12-45, .kt-inside-inner-col > .kadence-column6564_8d9d12-45:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_8d9d12-45 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_8d9d12-45 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_8d9d12-45\"><div class=\"kt-inside-inner-col\">\n<ul class=\"wp-block-list\">\n<li>Orthogonal basis functions defined on the unit disk (Zernike 1934).<\/li>\n\n\n\n<li>Each term is a function of normalized radius $\\rho \\in [0,1]$ and angle $\\theta \\in [0, 2\\pi]$.<\/li>\n\n\n\n<li>Standard tool in optics and metrology for wavefront decomposition and surface form analysis (Born &amp; Wolf 1999).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Decomposition formula:<\/strong><\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$T(\\rho, \\theta) = \\sum_{k=1}^{N} a_k \\cdot Z_k(\\rho, \\theta) + \\varepsilon$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">where $T(\\rho, \\theta)$ is the thickness distribution, $Z_k$ is the $k$-th Zernike basis function, $a_k$ is the corresponding scalar coefficient, $\\varepsilon$ is the residual, and $N$ is the number of terms used.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Physical meaning of low-order terms (Noll 1976 convention):<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Index<\/th><th>Name<\/th><th>Physical meaning<\/th><\/tr><\/thead><tbody><tr><td>$Z_1$<\/td><td>Piston<\/td><td>Mean thickness<\/td><\/tr><tr><td>$Z_2$<\/td><td>Tilt X<\/td><td>Slope along X<\/td><\/tr><tr><td>$Z_3$<\/td><td>Tilt Y<\/td><td>Slope along Y<\/td><\/tr><tr><td>$Z_4$<\/td><td>Defocus<\/td><td>Bowl \/ Dome (center\u2013edge contrast)<\/td><\/tr><tr><td>$Z_5$<\/td><td>Astigmatism 45\u00b0<\/td><td>45\u00b0\/135\u00b0 asymmetry<\/td><\/tr><tr><td>$Z_6$<\/td><td>Astigmatism 0\u00b0<\/td><td>0\u00b0\/90\u00b0 asymmetry<\/td><\/tr><tr><td>$Z_7$<\/td><td>Coma Y<\/td><td>Y-direction asymmetric variation<\/td><\/tr><tr><td>$Z_8$<\/td><td>Coma X<\/td><td>X-direction asymmetric variation<\/td><\/tr><tr><td>$Z_9$<\/td><td>Trefoil<\/td><td>3-fold pattern<\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">2. Design<\/h2>\n\n\n<style>.kadence-column6564_8e6e79-37 > .kt-inside-inner-col,.kadence-column6564_8e6e79-37 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_8e6e79-37 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_8e6e79-37 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_8e6e79-37 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_8e6e79-37 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_8e6e79-37{position:relative;}.kadence-column6564_8e6e79-37, .kt-inside-inner-col > .kadence-column6564_8e6e79-37:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_8e6e79-37 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_8e6e79-37 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_8e6e79-37\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\"><em>Design covers the technical choices and optimization strategies built on top of the Domain \u2014 the realm of decisions and trade-offs.<\/em><\/p>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_59780b-d4 > .kt-inside-inner-col,.kadence-column6564_59780b-d4 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_59780b-d4 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_59780b-d4 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_59780b-d4 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_59780b-d4 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_59780b-d4{position:relative;}.kadence-column6564_59780b-d4, .kt-inside-inner-col > .kadence-column6564_59780b-d4:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_59780b-d4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_59780b-d4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_59780b-d4\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">2.1 Why Zernike Polynomials<\/h3>\n\n\n<style>.kadence-column6564_831585-4f > .kt-inside-inner-col,.kadence-column6564_831585-4f > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_831585-4f > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_831585-4f > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_831585-4f > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_831585-4f > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_831585-4f{position:relative;}.kadence-column6564_831585-4f, .kt-inside-inner-col > .kadence-column6564_831585-4f:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_831585-4f > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_831585-4f > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_831585-4f\"><div class=\"kt-inside-inner-col\">\n<figure class=\"wp-block-table\"><table><thead><tr><th>Candidate<\/th><th>Pros<\/th><th>Cons<\/th><th>Fit<\/th><\/tr><\/thead><tbody><tr><td><strong>Zernike Polynomials<\/strong><\/td><td>Natural fit to circular domain, orthogonal, <br>physically interpretable<\/td><td>Order must be capped for 13 points<\/td><td>\u25ce<\/td><\/tr><tr><td>Polynomial ($x^n, y^n$)<\/td><td>Simple to implement<\/td><td>Non-orthogonal, unstable near circular edge<\/td><td>\u25b3<\/td><\/tr><tr><td>Fourier series (polar)<\/td><td>Orthogonal in angle<\/td><td>Poor radial expressiveness<\/td><td>\u25b3<\/td><\/tr><tr><td>Spline interpolation<\/td><td>Passes through measured points exactly<\/td><td>No physical meaning, noise-sensitive<\/td><td>\u2717<\/td><\/tr><tr><td>Principal Component Analysis <br>(PCA) \/ Autoencoder<\/td><td>Data-driven compression<\/td><td>Not interpretable, requires large data<\/td><td>\u2717<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Five reasons Zernike wins for this problem: (1) the wafer is a disk and Zernike is defined on a disk \u2014 the coordinate systems align naturally; (2) orthogonality means each coefficient represents an independent pattern; (3) low-order terms map directly to known process drivers (tilt, bowl, astigmatism); (4) Zernike is the de-facto standard in optics and semiconductor metrology (Wang &amp; Silva 1980); (5) compression from 13 points to 9 coefficients improves ML learning efficiency.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">2.2 Spatial Decomposition Structure \u2014 W2W \/ WiW Separation<\/h3>\n\n\n<style>.kadence-column6564_981f53-ec > .kt-inside-inner-col,.kadence-column6564_981f53-ec > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_981f53-ec > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_981f53-ec > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_981f53-ec > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_981f53-ec > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_981f53-ec{position:relative;}.kadence-column6564_981f53-ec, .kt-inside-inner-col > .kadence-column6564_981f53-ec:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_981f53-ec > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_981f53-ec > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_981f53-ec\"><div class=\"kt-inside-inner-col\">\n<pre style=\"font-family: consolas,monospace; font-size: 1.2rem; white-space: pre; line-height:1.2; background-color: #fff; border: none\">\n[Measurement space]            [Zernike space]\n\n13 point values         \u2500\u2500\u25ba   9 coefficients\n(T1, T2, ..., T13)             (a1, a2, ..., a9)\n\n13-D                           9-D\n(location-dependent)           (meaning-dependent)\n<\/pre>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>W2W \/ WiW group definition:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Group<\/th><th>Component<\/th><th>Count<\/th><th>Zernike terms<\/th><th>Meaning<\/th><\/tr><\/thead><tbody><tr><td><strong>W2W<\/strong><\/td><td>Mean component<\/td><td>1<\/td><td>$Z_1$ (Piston)<\/td><td>Wafer-wide mean thickness<\/td><\/tr><tr><td><strong>WiW<\/strong><\/td><td>Shape components<\/td><td>8<\/td><td>$Z_2 \\sim Z_9$<\/td><td>Spatial variation (tilt \/ bowl \/ astigmatism \/ coma \/ trefoil)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Three reasons for separation:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Higher ML accuracy<\/strong> \u2014 W2W and WiW are different physics; separating them lets each head focus on its own signal, improving accuracy and convergence speed.<\/li>\n\n\n\n<li><strong>Equipment fingerprint generation<\/strong> \u2014 the 8-element WiW vector forms a unique chamber signature that enables matching and outlier detection.<\/li>\n\n\n\n<li><strong>Continuous thickness inference<\/strong> \u2014 13 measurements reconstruct the full wafer thickness as a continuous function, allowing inference at unmeasured locations.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Fitting in matrix form:<\/strong><\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$T = A \\cdot a + \\varepsilon$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">where $T \\in \\mathbb{R}^{13 \\times 1}$ is the measurement vector, $A \\in \\mathbb{R}^{13 \\times 9}$ is the Zernike basis matrix, $a \\in \\mathbb{R}^{9 \\times 1}$ is the coefficient vector to be estimated, and $\\varepsilon \\in \\mathbb{R}^{13 \\times 1}$ is the residual. Standard solution is Least Squares (LSQ); for noise robustness, Ridge Regression (Hoerl &amp; Kennard 1970) is recommended. Detailed derivations appear in Appendices E and F.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">With 13 measurements and 9 unknowns, the residual carries 4 degrees of freedom \u2014 sufficient for stable fitting and residual diagnostics. Higher-order terms ($Z_{10}$ and above) are under-determined and flow into the residual instead.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">2.3 Model Architecture \u2014 Two-Head Design<\/h3>\n\n\n<style>.kadence-column6564_61897a-d3 > .kt-inside-inner-col,.kadence-column6564_61897a-d3 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_61897a-d3 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_61897a-d3 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_61897a-d3 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_61897a-d3 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_61897a-d3{position:relative;}.kadence-column6564_61897a-d3, .kt-inside-inner-col > .kadence-column6564_61897a-d3:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_61897a-d3 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_61897a-d3 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_61897a-d3\"><div class=\"kt-inside-inner-col\">\n<pre style=\"font-family: consolas,monospace; font-size: 1.2rem; white-space: pre; line-height:1.2; background-color: #fff; border: none\">\n                          \u250c\u2500\u25ba W2W Head \u2500\u25ba a1   (mean, 1 output)\nSensor data X \u2500\u25ba [Model] \u2500\u2524\n                          \u2514\u2500\u25ba WiW Head \u2500\u25ba a2..a9 (shape, 8 outputs)\n                                            \u2502\n                                            \u25bc\n                                    Equipment fingerprint\n<\/pre>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Input<\/strong>: feature vector X derived from sensor time-series.<\/li>\n\n\n\n<li><strong>Output<\/strong>: 9 coefficients (1 W2W + 8 WiW).<\/li>\n\n\n\n<li><strong>Reconstruction<\/strong>: 9 coefficients combined with Zernike basis yield $\\hat{T}(\\rho, \\theta)$ at any location.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Two-Head benefits: separates loss between very different output scales (W2W large, WiW small); allows feature subset specialization per head since the driving factors differ; enables independent monitoring and retraining per head in operation.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">2.4 Recommended Algorithms per Head<\/h3>\n\n\n<style>.kadence-column6564_4d93fb-84 > .kt-inside-inner-col,.kadence-column6564_4d93fb-84 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_4d93fb-84 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_4d93fb-84 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_4d93fb-84 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_4d93fb-84 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_4d93fb-84{position:relative;}.kadence-column6564_4d93fb-84, .kt-inside-inner-col > .kadence-column6564_4d93fb-84:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_4d93fb-84 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_4d93fb-84 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_4d93fb-84\"><div class=\"kt-inside-inner-col\">\n<figure class=\"wp-block-table\"><table><thead><tr><th>Head \/ Group<\/th><th>1st choice<\/th><th>2nd choice<\/th><th>Rationale<\/th><\/tr><\/thead><tbody><tr><td>W2W Head ($Z_1$)<\/td><td>Ridge Regression<\/td><td>XGBoost (shallow)<\/td><td>Strongly linear, interpretability priority, stable on small data<\/td><\/tr><tr><td>WiW low-order ($Z_2 \\sim Z_4$)<\/td><td>LightGBM \/ XGBoost<\/td><td>Random Forest<\/td><td>Mild non-linearity, multivariate interactions<\/td><\/tr><tr><td>WiW high-order ($Z_5 \\sim Z_9$)<\/td><td>XGBoost (heavy regularization)<\/td><td>1D-CNN, Stacking<\/td><td>Small signal, noise robustness needed<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">With sufficient data (&gt; 5,000 wafers), a Multi-task Learning structure with a shared backbone and group-specific heads is effective.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">2.5 Drift Tracking \u2014 Spatial \u00d7 Temporal<\/h3>\n\n\n<style>.kadence-column6564_b84d8f-d4 > .kt-inside-inner-col,.kadence-column6564_b84d8f-d4 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_b84d8f-d4 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_b84d8f-d4 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_b84d8f-d4 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_b84d8f-d4 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_b84d8f-d4{position:relative;}.kadence-column6564_b84d8f-d4, .kt-inside-inner-col > .kadence-column6564_b84d8f-d4:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_b84d8f-d4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_b84d8f-d4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_b84d8f-d4\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">Spatial decomposition compresses one wafer&#8217;s spatial pattern into 9 coefficients; temporal drift is then tracked on the time-series of those coefficients using Statistical Process Control (SPC) charts, Exponentially Weighted Moving Average (EWMA), or Cumulative Sum (CUSUM) \u2014 see Montgomery (2013).<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; font-size: 1.2rem; white-space: pre; line-height:1.2; background-color: #fff; border: none\">\n[Spatial: Zernike]                   [Temporal: SPC \/ time-series]\n\nT(\u03c1,\u03b8; t)  \u2192  (a1(t), ..., a9(t))  \u2192  EWMA \/ CUSUM \/ ARIMA\n(one wafer)    (coefficient series)    (drift detection)\n<\/pre>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Spatial drift<\/strong>: captured by Zernike coefficients (e.g., gradual rise in $a_4$ = bowl deepening).<\/li>\n\n\n\n<li><strong>Temporal drift<\/strong>: tracked on the 9 coefficient time-series using SPC, EWMA, or CUSUM.<\/li>\n\n\n\n<li><strong>Non-Zernike patterns<\/strong>: point defects and local hot-spots are caught by residual monitoring instead.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">2.6 Residual Interpretation and Use<\/h3>\n\n\n<style>.kadence-column6564_90281b-67 > .kt-inside-inner-col,.kadence-column6564_90281b-67 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_90281b-67 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_90281b-67 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_90281b-67 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_90281b-67 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_90281b-67{position:relative;}.kadence-column6564_90281b-67, .kt-inside-inner-col > .kadence-column6564_90281b-67:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_90281b-67 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_90281b-67 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_90281b-67\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">The 13-D measurement decomposes into a 9-D Zernike fit plus a residual carrying 4 Degrees of Freedom (DOF):<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; font-size: 1.2rem; white-space: pre; line-height:1.2; background-color: #fff; border: none\">\n13-D measurement\n   \u2502\n   \u251c\u2500\u2500 9-D (Zernike fit)  \u2500\u2500\u25ba ML training target\n   \u2502\n   \u2514\u2500\u2500 Residual (DOF 4)   \u2500\u2500\u25ba diagnostic information\n<\/pre>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Residual component<\/th><th>Origin<\/th><th>Use<\/th><\/tr><\/thead><tbody><tr><td>High-order spatial pattern<\/td><td>Process systematic missed by 9 terms<\/td><td>Signals model capacity insufficiency \u2192 consider order extension<\/td><\/tr><tr><td>Measurement system bias<\/td><td>Sensor calibration issue<\/td><td>Per-point reliability check<\/td><\/tr><tr><td>Local defect<\/td><td>Particle, scratch, etc.<\/td><td>Anomaly detection<\/td><\/tr><tr><td>Random noise<\/td><td>Measurement repeatability limit<\/td><td>Noise-floor estimation<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Two-layer monitoring strategy:<\/strong> Layer 1 (Zernike coefficients) tracks &#8220;expected variation&#8221; \u2014 drift detection, run-to-run control. Layer 2 (residual statistics) catches &#8220;unexpected anomalies&#8221; \u2014 alarm triggers, defect inspection.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">2.7 Dimensionality Reduction Strategy<\/h3>\n\n\n<style>.kadence-column6564_95a034-a5 > .kt-inside-inner-col,.kadence-column6564_95a034-a5 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_95a034-a5 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_95a034-a5 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_95a034-a5 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_95a034-a5 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_95a034-a5{position:relative;}.kadence-column6564_95a034-a5, .kt-inside-inner-col > .kadence-column6564_95a034-a5:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_95a034-a5 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_95a034-a5 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_95a034-a5\"><div class=\"kt-inside-inner-col\">\n<figure class=\"wp-block-table\"><table><thead><tr><th>Method<\/th><th>Characteristic<\/th><th>When to apply<\/th><\/tr><\/thead><tbody><tr><td>Variance filter<\/td><td>Drop coefficients with low variability<\/td><td>After initial baseline analysis<\/td><\/tr><tr><td>Domain knowledge<\/td><td>Pick dominant terms per process type \u2014 Chemical Vapor Deposition (CVD) \/ Physical Vapor Deposition (PVD) \/ Atomic Layer Deposition (ALD) \/ Etch<\/td><td>When process priors are clear<\/td><\/tr><tr><td>Target correlation<\/td><td>Select terms most correlated with yield\/quality<\/td><td>When outcome data is available<\/td><\/tr><tr><td>PCA on coefficients<\/td><td>Automatic compression (loses interpretability)<\/td><td>For ML input features only<\/td><\/tr><tr><td>Sparse Regression (Least Absolute Shrinkage and Selection Operator, LASSO)<\/td><td>Auto-selection during ML training<\/td><td>Integrated learning step<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Recommendation:<\/strong> split coefficients into &#8220;Active&#8221; (used for learning) and &#8220;Passive&#8221; (monitored only) groups. Don&#8217;t discard \u2014 keep computing and watching all coefficients.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">3. Delivery<\/h2>\n\n\n<style>.kadence-column6564_5a5088-44 > .kt-inside-inner-col,.kadence-column6564_5a5088-44 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_5a5088-44 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_5a5088-44 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_5a5088-44 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_5a5088-44 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_5a5088-44{position:relative;}.kadence-column6564_5a5088-44, .kt-inside-inner-col > .kadence-column6564_5a5088-44:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_5a5088-44 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_5a5088-44 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_5a5088-44\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\"><em>Delivery defines what the user gains by adopting this framework \u2014 operational and business value, not technical structure.<\/em><\/p>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_9ca504-79 > .kt-inside-inner-col,.kadence-column6564_9ca504-79 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_9ca504-79 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_9ca504-79 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_9ca504-79 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_9ca504-79 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_9ca504-79{position:relative;}.kadence-column6564_9ca504-79, .kt-inside-inner-col > .kadence-column6564_9ca504-79:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_9ca504-79 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_9ca504-79 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_9ca504-79\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">3.1 Application \u2014 Four Outcomes by W2W \/ WiW Group<\/h3>\n\n\n<style>.kadence-column6564_f312b8-1a > .kt-inside-inner-col,.kadence-column6564_f312b8-1a > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_f312b8-1a > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_f312b8-1a > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_f312b8-1a > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_f312b8-1a > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_f312b8-1a{position:relative;}.kadence-column6564_f312b8-1a, .kt-inside-inner-col > .kadence-column6564_f312b8-1a:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_f312b8-1a > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_f312b8-1a > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_f312b8-1a\"><div class=\"kt-inside-inner-col\">\n<figure class=\"wp-block-table\"><table><thead><tr><th>#<\/th><th>Group<\/th><th>Outcome<\/th><th>How<\/th><\/tr><\/thead><tbody><tr><td>1<\/td><td>W2W<\/td><td>Run-to-Run thickness control<\/td><td>Mean trend tracking<\/td><\/tr><tr><td>2<\/td><td>WiW<\/td><td>Early outlier-tool detection<\/td><td>Fingerprint deviation<\/td><\/tr><tr><td>3<\/td><td>WiW<\/td><td>Preventive Maintenance (PM) timing optimization<\/td><td>Fingerprint trend<\/td><\/tr><tr><td>4<\/td><td>WiW<\/td><td>Hardware root-cause diagnosis<\/td><td>Pattern-to-factor mapping<\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">3.2 W2W Outcome \u2014 Run-to-Run Thickness Control<\/h3>\n\n\n<style>.kadence-column6564_a9b1ed-f8 > .kt-inside-inner-col,.kadence-column6564_a9b1ed-f8 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_a9b1ed-f8 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_a9b1ed-f8 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_a9b1ed-f8 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_a9b1ed-f8 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_a9b1ed-f8{position:relative;}.kadence-column6564_a9b1ed-f8, .kt-inside-inner-col > .kadence-column6564_a9b1ed-f8:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_a9b1ed-f8 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_a9b1ed-f8 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_a9b1ed-f8\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\"><strong>How:<\/strong> mean trend tracking. The W2W head&#8217;s predicted mean thickness ($\\hat{a}_1$) drives recipe correction for the next wafer or lot, minimizing per-lot deviation and absorbing source-depletion or recipe-drift effects before they hit spec.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">3.3 WiW Outcomes<\/h3>\n\n\n<style>.kadence-column6564_b45079-ea > .kt-inside-inner-col,.kadence-column6564_b45079-ea > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_b45079-ea > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_b45079-ea > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_b45079-ea > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_b45079-ea > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_b45079-ea{position:relative;}.kadence-column6564_b45079-ea, .kt-inside-inner-col > .kadence-column6564_b45079-ea:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_b45079-ea > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_b45079-ea > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_b45079-ea\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\"><strong>Early outlier-tool detection.<\/strong> Monitor distance from a normal-baseline fingerprint (e.g., Mahalanobis distance) over the 8-element WiW vector. Detects deviating tools before yield impact, maintains chamber-to-chamber matching, ensures fleet-level consistency.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>PM timing optimization.<\/strong> Move from periodic PM to condition-based PM driven by fingerprint drift trends. Improves uptime and reduces maintenance cost simultaneously, avoiding both unnecessary PMs and delayed-PM excursions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Hardware root-cause diagnosis.<\/strong> Each shape coefficient maps to specific hardware factors:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Tilt \u2192 chuck levelness, gas inlet position<\/li>\n\n\n\n<li>Bowl \u2192 center-edge temperature delta, RF coupling<\/li>\n\n\n\n<li>Astigmatism \u2192 showerhead directionality, magnetic-field asymmetry<\/li>\n\n\n\n<li>Coma \/ Trefoil \u2192 pump location, 3-zone heater non-uniformity<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Result: faster root-cause identification on excursions, better maintenance efficiency, standardized troubleshooting playbooks.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" style=\"margin-top:var(--wp--preset--spacing--60);margin-bottom:var(--wp--preset--spacing--60)\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix A. 13-Point JSON Coordinate Definition<\/h2>\n\n\n<style>.kadence-column6564_1ed300-ee > .kt-inside-inner-col,.kadence-column6564_1ed300-ee > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_1ed300-ee > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_1ed300-ee > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_1ed300-ee > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_1ed300-ee > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_1ed300-ee{position:relative;}.kadence-column6564_1ed300-ee, .kt-inside-inner-col > .kadence-column6564_1ed300-ee:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_1ed300-ee > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_1ed300-ee > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_1ed300-ee\"><div class=\"kt-inside-inner-col\">\n<div class=\"wp-block-kevinbatdorf-code-block-pro cbp-has-line-numbers\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:1rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;--cbp-line-number-color:#24292e;--cbp-line-number-width:calc(2 * 0.6 * 1rem);line-height:1.625rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span role=\"button\" tabindex=\"0\" style=\"color:#24292e;display:none\" aria-label=\"Copy\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>{\n  \"wafer_size_mm\": 300,\n  \"edge_exclusion_mm\": 5,\n  \"pattern\": \"13-points\",\n  \"points\": &#091;\n    {\"id\": \"P1\",  \"x\":    0.0, \"y\":    0.0, \"r\":   0, \"theta\":   0, \"zone\": \"Center\"},\n    {\"id\": \"P2\",  \"x\":   75.0, \"y\":    0.0, \"r\":  75, \"theta\":   0, \"zone\": \"Mid_E\"},\n    {\"id\": \"P3\",  \"x\":    0.0, \"y\":   75.0, \"r\":  75, \"theta\":  90, \"zone\": \"Mid_N\"},\n    {\"id\": \"P4\",  \"x\":  -75.0, \"y\":    0.0, \"r\":  75, \"theta\": 180, \"zone\": \"Mid_W\"},\n    {\"id\": \"P5\",  \"x\":    0.0, \"y\":  -75.0, \"r\":  75, \"theta\": 270, \"zone\": \"Mid_S\"},\n    {\"id\": \"P6\",  \"x\":  145.0, \"y\":    0.0, \"r\": 145, \"theta\":   0, \"zone\": \"Edge_E\"},\n    {\"id\": \"P7\",  \"x\":  102.5, \"y\":  102.5, \"r\": 145, \"theta\":  45, \"zone\": \"Edge_NE\"},\n    {\"id\": \"P8\",  \"x\":    0.0, \"y\":  145.0, \"r\": 145, \"theta\":  90, \"zone\": \"Edge_N\"},\n    {\"id\": \"P9\",  \"x\": -102.5, \"y\":  102.5, \"r\": 145, \"theta\": 135, \"zone\": \"Edge_NW\"},\n    {\"id\": \"P10\", \"x\": -145.0, \"y\":    0.0, \"r\": 145, \"theta\": 180, \"zone\": \"Edge_W\"},\n    {\"id\": \"P11\", \"x\": -102.5, \"y\": -102.5, \"r\": 145, \"theta\": 225, \"zone\": \"Edge_SW\"},\n    {\"id\": \"P12\", \"x\":    0.0, \"y\": -145.0, \"r\": 145, \"theta\": 270, \"zone\": \"Edge_S\"},\n    {\"id\": \"P13\", \"x\":  102.5, \"y\": -102.5, \"r\": 145, \"theta\": 315, \"zone\": \"Edge_SE\"}\n  &#093;\n}<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki github-light\" style=\"background-color: #fff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #24292E\">{<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">  <\/span><span style=\"color: #005CC5\">&quot;wafer_size_mm&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">300<\/span><span style=\"color: #24292E\">,<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">  <\/span><span style=\"color: #005CC5\">&quot;edge_exclusion_mm&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">5<\/span><span style=\"color: #24292E\">,<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">  <\/span><span style=\"color: #005CC5\">&quot;pattern&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;13-points&quot;<\/span><span style=\"color: #24292E\">,<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">  <\/span><span style=\"color: #005CC5\">&quot;points&quot;<\/span><span style=\"color: #24292E\">: &#091;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P1&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">:   <\/span><span style=\"color: #005CC5\">0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">:   <\/span><span style=\"color: #005CC5\">0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Center&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P2&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:   <\/span><span style=\"color: #005CC5\">75.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">75<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">:   <\/span><span style=\"color: #005CC5\">0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Mid_E&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P3&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:   <\/span><span style=\"color: #005CC5\">75.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">75<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">90<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Mid_N&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P4&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">-75.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">75<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">180<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Mid_W&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P5&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">-75.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">75<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">270<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Mid_S&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P6&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">145.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">145<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">:   <\/span><span style=\"color: #005CC5\">0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Edge_E&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P7&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">102.5<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">102.5<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">145<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">45<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Edge_NE&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P8&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">145.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">145<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">90<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Edge_N&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P9&quot;<\/span><span style=\"color: #24292E\">,  <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">-102.5<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">102.5<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">145<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">135<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Edge_NW&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P10&quot;<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">-145.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">145<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">180<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Edge_W&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P11&quot;<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">-102.5<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">-102.5<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">145<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">225<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Edge_SW&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P12&quot;<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:    <\/span><span style=\"color: #005CC5\">0.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">-145.0<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">145<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">270<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Edge_S&quot;<\/span><span style=\"color: #24292E\">},<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">    {<\/span><span style=\"color: #005CC5\">&quot;id&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;P13&quot;<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;x&quot;<\/span><span style=\"color: #24292E\">:  <\/span><span style=\"color: #005CC5\">102.5<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;y&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">-102.5<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;r&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">145<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;theta&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #005CC5\">315<\/span><span style=\"color: #24292E\">, <\/span><span style=\"color: #005CC5\">&quot;zone&quot;<\/span><span style=\"color: #24292E\">: <\/span><span style=\"color: #032F62\">&quot;Edge_SE&quot;<\/span><span style=\"color: #24292E\">}<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">  &#093;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #24292E\">}<\/span><\/span><\/code><\/pre><\/div>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix B. 13-Point Wafer Location Map<\/h2>\n\n\n<style>.kadence-column6564_1dc0c2-b5 > .kt-inside-inner-col,.kadence-column6564_1dc0c2-b5 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_1dc0c2-b5 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_1dc0c2-b5 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_1dc0c2-b5 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_1dc0c2-b5 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_1dc0c2-b5{position:relative;}.kadence-column6564_1dc0c2-b5, .kt-inside-inner-col > .kadence-column6564_1dc0c2-b5:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_1dc0c2-b5 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_1dc0c2-b5 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_1dc0c2-b5\"><div class=\"kt-inside-inner-col\">\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:1.2; background-color: #fff; border: none\">\n                          N (+Y)\n                           \u2502\n                    . . . P8 . . .\n                .       (0,145)      .\n            P9 .                      . P7\n        (-102,102).                  .(102,102)\n           .          P3 (0,75)         .\n          .               \u2502              .\n         .                \u2502               .\n        .                 \u2502                .\n       .                  \u2502                 .\n      P10\u2500\u2500\u2500\u2500\u2500P4\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500P1\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500P2\u2500\u2500\u2500\u2500\u2500P6   \u2500\u2500 E (+X)\n    (-145,0)(-75,0)    (0,0)     (75,0) (145,0)\n       .                  \u2502                 .\n        .                 \u2502                .\n         .                \u2502               .\n          .          P5 (0,-75)          .\n           .              \u2502             .\n       P11 .                            . P13\n       (-102,-102) .                . (102,-102)\n                .       (0,-145)      .\n                    . . . P12 . . .\n                           \u2502\n                          S (-Y)\n<\/pre>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix C. Visualization of Low-Order Zernike Terms<\/h2>\n\n\n<style>.kadence-column6564_1edce9-27 > .kt-inside-inner-col,.kadence-column6564_1edce9-27 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_1edce9-27 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_1edce9-27 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_1edce9-27 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_1edce9-27 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_1edce9-27{position:relative;}.kadence-column6564_1edce9-27, .kt-inside-inner-col > .kadence-column6564_1edce9-27:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_1edce9-27 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_1edce9-27 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_1edce9-27\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">Each Zernike term is rendered as a 17\u00d717 ASCII grid normalized to its own peak, so shape patterns are directly comparable across terms.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Shading legend (negative \u2190 zero \u2192 positive):<\/strong><\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:1.2; background-color: #fff; border: none\">\n   #   @   *   +   -   ' '   .   :   o   O   0\nstrong                                       strong\nnegative              zero                   positive\n<\/pre>\n\n\n\n<div class=\"wp-block-group is-layout-grid wp-container-core-group-is-layout-50eac3a5 wp-block-group-is-layout-grid\"><style>.kadence-column6564_796883-95 > .kt-inside-inner-col,.kadence-column6564_796883-95 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_796883-95 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_796883-95 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_796883-95 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_796883-95 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_796883-95{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_796883-95 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_796883-95 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_796883-95\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.1 $Z_1$ \u2014 Piston (mean)<\/h3>\n\n\n<style>.kadence-column6564_083301-55 > .kt-inside-inner-col,.kadence-column6564_083301-55 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_083301-55 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_083301-55 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_083301-55 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_083301-55 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_083301-55{position:relative;}.kadence-column6564_083301-55, .kt-inside-inner-col > .kadence-column6564_083301-55:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_083301-55 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_083301-55 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_083301-55\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=0, m=0)$ \u2014 wafer-wide constant. Captures the mean thickness; the W2W variation lives here.<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n        0        \n     0000000     \n   00000000000   \n  0000000000000  \n  0000000000000  \n 000000000000000 \n 000000000000000 \n 000000000000000 \n00000000000000000\n 000000000000000 \n 000000000000000 \n 000000000000000 \n  0000000000000  \n  0000000000000  \n   00000000000   \n     0000000     \n        0        \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_c40309-2d > .kt-inside-inner-col,.kadence-column6564_c40309-2d > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_c40309-2d > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_c40309-2d > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_c40309-2d > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_c40309-2d > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_c40309-2d{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_c40309-2d > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_c40309-2d > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_c40309-2d\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.2 $Z_2$ \u2014 Tilt X<\/h3>\n\n\n<style>.kadence-column6564_33d485-d8 > .kt-inside-inner-col,.kadence-column6564_33d485-d8 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_33d485-d8 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_33d485-d8 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_33d485-d8 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_33d485-d8 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_33d485-d8{position:relative;}.kadence-column6564_33d485-d8, .kt-inside-inner-col > .kadence-column6564_33d485-d8:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_33d485-d8 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_33d485-d8 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_33d485-d8\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=1, m=1)$ \u2014 linear slope along X. Diagnoses chuck levelness or asymmetric gas inlet position.<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n                 \n     +-- ..:     \n   **+-- ..:oo   \n  @**+-- ..:ooO  \n  @**+-- ..:ooO  \n @@**+-- ..:ooOO \n @@**+-- ..:ooOO \n @@**+-- ..:ooOO \n#@@**+-- ..:ooOO0\n @@**+-- ..:ooOO \n @@**+-- ..:ooOO \n @@**+-- ..:ooOO \n  @**+-- ..:ooO  \n  @**+-- ..:ooO  \n   **+-- ..:oo   \n     +-- ..:     \n                 \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_20350d-76 > .kt-inside-inner-col,.kadence-column6564_20350d-76 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_20350d-76 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_20350d-76 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_20350d-76 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_20350d-76 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_20350d-76{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_20350d-76 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_20350d-76 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_20350d-76\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.3 $Z_3$ \u2014 Tilt Y<\/h3>\n\n\n<style>.kadence-column6564_39433f-20 > .kt-inside-inner-col,.kadence-column6564_39433f-20 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_39433f-20 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_39433f-20 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_39433f-20 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_39433f-20 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_39433f-20{position:relative;}.kadence-column6564_39433f-20, .kt-inside-inner-col > .kadence-column6564_39433f-20:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_39433f-20 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_39433f-20 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_39433f-20\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=1, m=-1)$ \u2014 linear slope along Y. Diagnoses front-back chuck levelness or top-bottom flow asymmetry.<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n        #        \n     @@@@@@@     \n   @@@@@@@@@@@   \n  *************  \n  ************+  \n +++++++++++++++ \n --------------- \n --------------- \n                 \n ............... \n ............... \n ::::::::::::::: \n  ooooooooooooo  \n  ooooooooooooo  \n   OOOOOOOOOOO   \n     OOOOOOO     \n        0        \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_997967-d2 > .kt-inside-inner-col,.kadence-column6564_997967-d2 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_997967-d2 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_997967-d2 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_997967-d2 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_997967-d2 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_997967-d2{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_997967-d2 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_997967-d2 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_997967-d2\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.4 $Z_4$ \u2014 Defocus (Bowl \/ Dome)<\/h3>\n\n\n<style>.kadence-column6564_995525-7e > .kt-inside-inner-col,.kadence-column6564_995525-7e > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_995525-7e > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_995525-7e > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_995525-7e > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_995525-7e > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_995525-7e{position:relative;}.kadence-column6564_995525-7e, .kt-inside-inner-col > .kadence-column6564_995525-7e:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_995525-7e > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_995525-7e > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_995525-7e\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=2, m=0)$ \u2014 radially symmetric center-vs-edge contrast. Key diagnostic for center-edge temperature delta, RF coupling, and showerhead-to-wafer gap.<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n        0        \n     OoooooO     \n   0o:.....:o0   \n  0o.  ---  .o0  \n  o. -++*++- .o  \n O: -+**@**+- :O \n o. +*@@@@@*+ .o \n o.-+*@###@*+-.o \n0o.-*@@###@@*-.o0\n o.-+*@###@*+-.o \n o. +*@@@@@*+ .o \n O: -+**@**+- :O \n  o. -++*++- .o  \n  0o.  ---  .o0  \n   0o:.....:o0   \n     OoooooO     \n        0        \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_b0c822-72 > .kt-inside-inner-col,.kadence-column6564_b0c822-72 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_b0c822-72 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_b0c822-72 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_b0c822-72 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_b0c822-72 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_b0c822-72{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_b0c822-72 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_b0c822-72 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_b0c822-72\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.5 $Z_5$ \u2014 Astigmatism 45\u00b0<\/h3>\n\n\n<style>.kadence-column6564_8cad15-f1 > .kt-inside-inner-col,.kadence-column6564_8cad15-f1 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_8cad15-f1 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_8cad15-f1 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_8cad15-f1 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_8cad15-f1 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_8cad15-f1{position:relative;}.kadence-column6564_8cad15-f1, .kt-inside-inner-col > .kadence-column6564_8cad15-f1:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_8cad15-f1 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_8cad15-f1 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_8cad15-f1\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=2, m=-2)$ \u2014 4-fold asymmetry along diagonals. Diagnoses 45\u00b0\/135\u00b0-direction flow asymmetry or magnetic-field bias.<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n                 \n     o:. -+@     \n   0Oo:. -+*@#   \n  0Ooo:. -+**@#  \n  Ooo:.. --+**@  \n ooo:..  ---+**@ \n :::...   ---+++ \n .....     ----- \n                 \n -----     ..... \n +++---   ...::: \n @**+--- ...:ooo \n  @**+-- ..:ooO  \n  #@**+- .:ooO0  \n   #@*+- .:oO0   \n     *+- .:o     \n                 \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_98e222-d4 > .kt-inside-inner-col,.kadence-column6564_98e222-d4 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_98e222-d4 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_98e222-d4 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_98e222-d4 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_98e222-d4 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_98e222-d4{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_98e222-d4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_98e222-d4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_98e222-d4\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.6 $Z_6$ \u2014 Astigmatism 0\u00b0<\/h3>\n\n\n<style>.kadence-column6564_c8d387-4b > .kt-inside-inner-col,.kadence-column6564_c8d387-4b > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_c8d387-4b > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_c8d387-4b > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_c8d387-4b > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_c8d387-4b > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_c8d387-4b{position:relative;}.kadence-column6564_c8d387-4b, .kt-inside-inner-col > .kadence-column6564_c8d387-4b:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_c8d387-4b > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_c8d387-4b > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_c8d387-4b\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=2, m=2)$ \u2014 4-fold asymmetry along horizontal\/vertical. Diagnoses showerhead directionality and 0\u00b0\/90\u00b0 pump-position effects.<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n        #        \n     *@@@@@*     \n   -++*****++-   \n  . --+++++-- .  \n  :. ------- .:  \n o:..  ---  ..:o \n Oo:.       .:oO \n Oo:..     ..:oO \n0Oo:..     ..:oO0\n Oo:..     ..:oO \n Oo:.       .:oO \n o:..  ---  ..:o \n  :. ------- .:  \n  . --+++++-- .  \n   -++*****++-   \n     *@@@@@*     \n        #        \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_41b473-bb > .kt-inside-inner-col,.kadence-column6564_41b473-bb > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_41b473-bb > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_41b473-bb > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_41b473-bb > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_41b473-bb > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_41b473-bb{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_41b473-bb > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_41b473-bb > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_41b473-bb\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.7 $Z_7$ \u2014 Coma Y<\/h3>\n\n\n<style>.kadence-column6564_4b5cc0-f0 > .kt-inside-inner-col,.kadence-column6564_4b5cc0-f0 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_4b5cc0-f0 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_4b5cc0-f0 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_4b5cc0-f0 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_4b5cc0-f0 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_4b5cc0-f0{position:relative;}.kadence-column6564_4b5cc0-f0, .kt-inside-inner-col > .kadence-column6564_4b5cc0-f0:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_4b5cc0-f0 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_4b5cc0-f0 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_4b5cc0-f0\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=3, m=-1)$ \u2014 Y-direction asymmetric tilt with stronger curvature on one side. Diagnoses asymmetric Y flow and pump-position bias.<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n        #        \n     *++-++*     \n   *+  ...  +*   \n  *- .::o::. -*  \n  - .:ooooo:. -  \n - .::ooooo::. - \n - ..:::::::.. - \n    .........    \n                 \n    ---------    \n . --+++++++-- . \n . -++*****++- . \n  . -+*****+- .  \n  o. -++*++- .o  \n   o:  ---  :o   \n     o::.::o     \n        0        \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_cfa9d2-98 > .kt-inside-inner-col,.kadence-column6564_cfa9d2-98 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_cfa9d2-98 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_cfa9d2-98 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_cfa9d2-98 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_cfa9d2-98 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_cfa9d2-98{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_cfa9d2-98 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_cfa9d2-98 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_cfa9d2-98\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.8 $Z_8$ \u2014 Coma X<\/h3>\n\n\n<style>.kadence-column6564_2d9bc3-9a > .kt-inside-inner-col,.kadence-column6564_2d9bc3-9a > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_2d9bc3-9a > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_2d9bc3-9a > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_2d9bc3-9a > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_2d9bc3-9a > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_2d9bc3-9a{position:relative;}.kadence-column6564_2d9bc3-9a, .kt-inside-inner-col > .kadence-column6564_2d9bc3-9a:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_2d9bc3-9a > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_2d9bc3-9a > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_2d9bc3-9a\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=3, m=1)$ \u2014 X-direction asymmetric tilt. Diagnoses asymmetric X gas inlet\/outlet or one-sided chamber hardware bias.<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n                 \n     --   ..     \n   *-       .o   \n  *- ..   -- .o  \n  + .:.. --+- :  \n * .:::. -+++- o \n + :oo:. -+**+ : \n +.:oo:. -+**+-: \n#-.ooo:. -+***-.0\n +.:oo:. -+**+-: \n + :oo:. -+**+ : \n * .:::. -+++- o \n  + .:.. --+- :  \n  *- ..   -- .o  \n   *-       .o   \n     --   ..     \n                 \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n\n\n<style>.kadence-column6564_405579-d3 > .kt-inside-inner-col,.kadence-column6564_405579-d3 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_405579-d3 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_405579-d3 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_405579-d3 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_405579-d3 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_405579-d3{position:relative;}@media all and (max-width: 1024px){.kadence-column6564_405579-d3 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_405579-d3 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_405579-d3\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">C.9 $Z_9$ \u2014 Trefoil (3-fold)<\/h3>\n\n\n<style>.kadence-column6564_873d50-bb > .kt-inside-inner-col,.kadence-column6564_873d50-bb > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_873d50-bb > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_873d50-bb > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_873d50-bb > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_873d50-bb > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_873d50-bb{position:relative;}.kadence-column6564_873d50-bb, .kt-inside-inner-col > .kadence-column6564_873d50-bb:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_873d50-bb > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_873d50-bb > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_873d50-bb\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">$(n=3, m=-3)$ \u2014 3-fold pattern repeating every 120\u00b0. Marks 3-zone heater non-uniformity or 3-fold chamber hardware effects (3-leg lift pins, 3-port gas).<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:0.7; background-color: #fff; border: none\">\n        0        \n     :ooooo:     \n   +-..:::..-+   \n  @+- ..... -+@  \n  @+-  ...  -+@  \n @*+--     --+*@ \n *+--       --+* \n ---         --- \n                 \n ...         ... \n o:..       ..:o \n Oo:..     ..:oO \n  O:.  ---  .:O  \n  O:. ----- .:O  \n   :.--+++--.:   \n     +*****+     \n        #        \n<\/pre>\n<\/div><\/div>\n<\/div><\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">C.10 Drift Diagnostic Guide<\/h3>\n\n\n<style>.kadence-column6564_c03e10-37 > .kt-inside-inner-col,.kadence-column6564_c03e10-37 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_c03e10-37 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_c03e10-37 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_c03e10-37 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_c03e10-37 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_c03e10-37{position:relative;}.kadence-column6564_c03e10-37, .kt-inside-inner-col > .kadence-column6564_c03e10-37:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_c03e10-37 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_c03e10-37 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_c03e10-37\"><div class=\"kt-inside-inner-col\">\n<figure class=\"wp-block-table\"><table><thead><tr><th>Coefficient that suddenly grows<\/th><th>Hypothesized cause<\/th><\/tr><\/thead><tbody><tr><td>$a_1$ (Piston)<\/td><td>Source depletion, deposition time\/power shift<\/td><\/tr><tr><td>$a_2, a_3$ (Tilt)<\/td><td>Chuck levelness change, gas inlet position<\/td><\/tr><tr><td>$a_4$ (Defocus)<\/td><td>Center-edge temperature change, RF coupling, showerhead distance<\/td><\/tr><tr><td>$a_5, a_6$ (Astigmatism)<\/td><td>Showerhead directionality, magnetic-field asymmetry, pump position<\/td><\/tr><tr><td>$a_7, a_8$ (Coma)<\/td><td>Asymmetric gas flow, one-sided hardware bias<\/td><\/tr><tr><td>$a_9$ (Trefoil)<\/td><td>3-zone heater or 3-fold hardware issues<\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix D. Why 13 Measurements Map to 9 Coefficients<\/h2>\n\n\n<style>.kadence-column6564_562a6e-11 > .kt-inside-inner-col,.kadence-column6564_562a6e-11 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_562a6e-11 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_562a6e-11 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_562a6e-11 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_562a6e-11 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_562a6e-11{position:relative;}.kadence-column6564_562a6e-11, .kt-inside-inner-col > .kadence-column6564_562a6e-11:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_562a6e-11 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_562a6e-11 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_562a6e-11\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">For a linear measurement model $T = A \\cdot a + \\varepsilon$ with $m$ measurements and $N$ basis terms, three regimes exist:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Condition<\/th><th>Name<\/th><th>Result<\/th><\/tr><\/thead><tbody><tr><td>$N &gt; m$<\/td><td>Under-determined<\/td><td>Infinitely many solutions \u2014 no unique answer<\/td><\/tr><tr><td>$N = m$<\/td><td>Exactly-determined<\/td><td>Residual = 0 but noise also fitted (overfitting)<\/td><\/tr><tr><td>$N &lt; m$<\/td><td>Over-determined<\/td><td>Residual-minimizing LSQ solution exists \u2014 recommended<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">For 13 points, the choice of $N$ is constrained:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>$N$<\/th><th>Patterns expressible<\/th><th>DOF<\/th><th>Verdict<\/th><\/tr><\/thead><tbody><tr><td>4<\/td><td>Piston, Tilt X\/Y, Defocus<\/td><td>9<\/td><td>Insufficient (no asymmetry)<\/td><\/tr><tr><td>6<\/td><td>+ Astigmatism 0\u00b0\/45\u00b0<\/td><td>7<\/td><td>Acceptable<\/td><\/tr><tr><td><strong>9<\/strong><\/td><td>+ Coma X\/Y, Trefoil<\/td><td><strong>4<\/strong><\/td><td><strong>Recommended balance<\/strong><\/td><\/tr><tr><td>11<\/td><td>+ Spherical, Quadrafoil<\/td><td>2<\/td><td>DOF too low<\/td><\/tr><tr><td>13<\/td><td>+ further terms<\/td><td>0<\/td><td>No residual monitoring possible<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">DOF of 4 means the residual $\\varepsilon$ moves freely in a 4-dimensional subspace, providing the information channel for residual diagnostics (Section 2.6). The 13 \u2192 9 mapping is therefore not an arbitrary choice but the simultaneous optimum of three constraints: over-determined system, sufficient expressiveness, and residual-monitoring DOF.<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix E. Zernike basis matrix A<\/h2>\n\n\n<style>.kadence-column6564_84f3c1-a7 > .kt-inside-inner-col,.kadence-column6564_84f3c1-a7 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_84f3c1-a7 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_84f3c1-a7 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_84f3c1-a7 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_84f3c1-a7 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_84f3c1-a7{position:relative;}.kadence-column6564_84f3c1-a7, .kt-inside-inner-col > .kadence-column6564_84f3c1-a7:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_84f3c1-a7 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_84f3c1-a7 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_84f3c1-a7\"><div class=\"kt-inside-inner-col\">\n<h3 class=\"wp-block-heading\">The model: $T = A \\cdot a + e$<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003The Zernike decomposition writes a measured wavefront $T$ as a linear combination of Zernike basis functions plus measurement noise:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$T = A \\cdot a + e$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003This appendix walks through what each piece of the equation means and what makes $A$ \u2014 the&nbsp;<em>Zernike basis matrix<\/em>&nbsp;\u2014 useful.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Symbols<\/h3>\n\n\n<style>.kadence-column6564_24c770-11 > .kt-inside-inner-col,.kadence-column6564_24c770-11 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_24c770-11 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_24c770-11 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_24c770-11 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_24c770-11 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_24c770-11{position:relative;}.kadence-column6564_24c770-11, .kt-inside-inner-col > .kadence-column6564_24c770-11:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_24c770-11 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_24c770-11 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_24c770-11\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">\u2003The pieces of the model carry the following meaning:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Symbol<\/th><th>Meaning<\/th><\/tr><\/thead><tbody><tr><td>$Z_j(\\rho, \\theta)$<\/td><td>The $j$-th Zernike basis function on the unit disk<\/td><\/tr><tr><td>$(\\rho_i, \\theta_i)$<\/td><td>Polar coordinates of the $i$-th measurement point<\/td><\/tr><tr><td>$a_j$<\/td><td>Coefficient (weight) of $Z_j$ \u2014 the unknown to fit<\/td><\/tr><tr><td>$T_i$<\/td><td>Measured value at the $i$-th point (e.g. wafer thickness)<\/td><\/tr><tr><td>$e_i$<\/td><td>Measurement noise at the $i$-th point<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003$Z_j$ is one element of a sequence of fixed shapes defined on the unit disk (the disk of radius $1$). Just as a Fourier series uses sines and cosines as a basis, a Zernike expansion uses $Z_1, Z_2, Z_3, \\ldots$ as its basis. The Noll convention numbers the basis functions starting from $j = 1$.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>$j$<\/th><th>Shape of $Z_j$<\/th><th>Optical name<\/th><\/tr><\/thead><tbody><tr><td>1<\/td><td>constant (flat)<\/td><td>Piston<\/td><\/tr><tr><td>2<\/td><td>tilt along $x$<\/td><td>Tilt X<\/td><\/tr><tr><td>3<\/td><td>tilt along $y$<\/td><td>Tilt Y<\/td><\/tr><tr><td>4<\/td><td>bowl (center vs. edge)<\/td><td>Defocus<\/td><\/tr><tr><td>$\\ldots$<\/td><td>$\\ldots$<\/td><td>$\\ldots$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003For the $i$-th measurement point, $\\rho_i$ is the radius from the disk center (normalized so that the edge sits at $\\rho = 1$) and $\\theta_i$ is the angle from the positive $x$-axis (in radians).<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Per-point equation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003Written out for one point $i$, the model becomes:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$T_i = a_1 Z_1(\\rho_i, \\theta_i) + a_2 Z_2(\\rho_i, \\theta_i) + \\ldots + a_N Z_N(\\rho_i, \\theta_i) + e_i$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003Each 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 $T = A \\cdot a + e$.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Structure of $A$ ($m \\times N$)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003The entries of $A$ are Zernike-basis evaluations at the measurement points:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$A[i, j] = Z_j(\\rho_i, \\theta_i)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003That is, $A[i, j]$ is &#8220;the value of the $j$-th basis at the $i$-th measurement point&#8221;. $A$ has $m$ rows (one per measurement point) and $N$ columns (one per Zernike basis function):<\/p>\n\n\n\n<pre style=\"font-family: consolas,monospace; white-space: pre; line-height:1.2; background-color: #fff; border: none\">\n            j=1     j=2     j=3    ...   j=N\n        \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\ni=1     \u2502  Z_1    Z_2    Z_3    ...    Z_N    \u2502 \u2190 all basis values at point 1\ni=2     \u2502  Z_1    Z_2    Z_3    ...    Z_N    \u2502 \u2190 all basis values at point 2\n...     \u2502   \u2026      \u2026      \u2026             \u2026     \u2502\ni=m     \u2502  Z_1    Z_2    Z_3    ...    Z_N    \u2502 \u2190 all basis values at point m\n        \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\n          \u2191\n        column 1 = Z_1 evaluated at every measurement point\n<\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Row and column meaning<\/h3>\n\n\n<style>.kadence-column6564_e15b42-04 > .kt-inside-inner-col,.kadence-column6564_e15b42-04 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_e15b42-04 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_e15b42-04 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_e15b42-04 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_e15b42-04 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_e15b42-04{position:relative;}.kadence-column6564_e15b42-04, .kt-inside-inner-col > .kadence-column6564_e15b42-04:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_e15b42-04 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_e15b42-04 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_e15b42-04\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">\u2003The rows and columns of $A$ have very different meanings, and both are useful:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Column $j$<\/strong>: the values of one fixed basis function $Z_j$ sampled at all $m$ measurement points \u2014\u00a0<em>$Z_j$&#8217;s sampling pattern at the measurement points (the $j$-th column vector of $A$)<\/em>. It is a snapshot of the continuous function $Z_j$ &#8220;photographed&#8221; at the chosen $m$ points.<\/li>\n\n\n\n<li><strong>Row $i$<\/strong>: the values of all $N$ basis functions evaluated at one measurement point \u2014 the coefficients on the right-hand side of the\u00a0<em>expansion equation (writing the wavefront as a sum of basis functions)<\/em>\u00a0at that point.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003Here\u00a0<em>expansion<\/em>\u00a0means writing a function as a sum of basis functions. The Fourier series, which expands a function as a sum of sines and cosines, is the textbook example. In Zernike land we expand the wavefront $T(\\rho, \\theta)$ as a sum of $Z_j$.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Noll normalization<\/h3>\n\n\n<style>.kadence-column6564_d7d3c9-b6 > .kt-inside-inner-col,.kadence-column6564_d7d3c9-b6 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_d7d3c9-b6 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_d7d3c9-b6 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_d7d3c9-b6 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_d7d3c9-b6 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_d7d3c9-b6{position:relative;}.kadence-column6564_d7d3c9-b6, .kt-inside-inner-col > .kadence-column6564_d7d3c9-b6:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_d7d3c9-b6 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_d7d3c9-b6 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_d7d3c9-b6\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">\u2003The Noll convention scales each $Z_j$ so that all coefficients $a_j$ are directly comparable. After normalization, on the unit disk:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\iint_{\\text{disk}} Z_j(\\rho, \\theta)^2 \\, dA = \\pi \\quad \\text{for every } j \\quad\\text{(unit norm)}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\iint_{\\text{disk}} Z_j \\, Z_k \\, dA = 0 \\quad \\text{for } j \\neq k \\quad\\text{(orthogonality)}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003The scaling constants depend on the $(n, m)$ pair (radial order $n$, azimuthal frequency $m$) that each $j$ corresponds to:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For $m = 0$ terms (purely radial: Piston, Defocus, \u2026): multiply by $\\sqrt{n+1}$.<\/li>\n\n\n\n<li>For $m \\neq 0$ terms (with $\\cos$ or $\\sin$: Tilt, Astigmatism, \u2026): multiply by $\\sqrt{2(n+1)}$.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003The $j \\rightarrow (n, m)$ conversion follows Noll&#8217;s 1976 algorithm. The benefit of normalization: $a_j^2$ is exactly &#8220;$j$&#8217;s contribution to the wavefront RMS&#8221;, so absolute coefficient values can be compared at a glance.<\/p>\n<\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">$A^{\\!\\top}\\!A \\approx m \\cdot I$ \u2014 sampling density and regression stability<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003The 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 $A$:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$(A^{\\!\\top}\\!A)_{jk} = \\sum_{i=1}^{m} Z_j(\\rho_i, \\theta_i)\\, Z_k(\\rho_i, \\theta_i)$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003<strong>Dense sampling<\/strong>&nbsp;($m \\rightarrow \\infty$, with the points uniformly covering the disk): the discrete sum tends to the integral average:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\frac{1}{m} \\sum_{i=1}^{m} Z_j(\\rho_i, \\theta_i)\\, Z_k(\\rho_i, \\theta_i) \\;\\approx\\; \\frac{1}{\\pi} \\iint_{\\text{disk}} Z_j Z_k \\, dA \\;=\\; \\delta_{jk}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">hence $A^{\\!\\top}\\!A \\approx m \\cdot I$ \u2014 a diagonal matrix with all diagonal entries equal to $m$. In this regime:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$(A^{\\!\\top}\\!A)^{-1} \\approx I\/m$ \u2014 the inverse is essentially &#8220;multiply by $1\/m$&#8221;. Numerically very stable.<\/li>\n\n\n\n<li>$\\hat{a}_j \\approx \\frac{1}{m} \\sum_i Z_j(\\rho_i, \\theta_i)\\, T_i$ \u2014 each coefficient is just a $Z_j$-weighted average of the measurements.<\/li>\n\n\n\n<li>Condition number $\\approx 1$ \u2014 measurement noise propagates to the coefficients with virtually no amplification.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u2003<strong>Sparse sampling<\/strong>&nbsp;(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.<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix F. Least Squares Normal-Equation Derivation<\/h2>\n\n\n<style>.kadence-column6564_487a1c-7b > .kt-inside-inner-col,.kadence-column6564_487a1c-7b > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_487a1c-7b > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_487a1c-7b > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_487a1c-7b > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_487a1c-7b > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_487a1c-7b{position:relative;}.kadence-column6564_487a1c-7b, .kt-inside-inner-col > .kadence-column6564_487a1c-7b:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_487a1c-7b > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_487a1c-7b > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_487a1c-7b\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">Find the coefficients that minimize squared residual:<\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$\\hat{a} = \\arg\\min_{a} \\| T &#8211; A \\cdot a \\|^2$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Expanding the objective $J(a) = \\| T &#8211; A \\cdot a \\|^2$:<\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$J(a) = (T &#8211; A \\cdot a)^\\top (T &#8211; A \\cdot a) = T^\\top T &#8211; 2 \\, T^\\top A \\cdot a + a^\\top A^\\top A \\cdot a$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Setting $\\partial J \/ \\partial a = 0$:<\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$-2 \\, A^\\top T + 2 \\, A^\\top A \\cdot a = 0 \\quad \\Rightarrow \\quad A^\\top A \\cdot a = A^\\top T$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">This is the <strong>normal equation<\/strong>. When $A^\\top A$ is invertible (i.e., $A$ has full column rank), the solution is:<\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$\\hat{a} = (A^\\top A)^{-1} A^\\top T$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Geometrically, $\\hat{a}$ is the orthogonal projection of $T$ onto the column space of $A$, so the residual $\\varepsilon = T &#8211; A \\cdot \\hat{a}$ satisfies $A^\\top \\varepsilon = 0$. The Hessian $2 A^\\top A$ is positive semi-definite, so this is a global minimum.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For 13-point fitting, $A$ must have full column rank \u2014 the cardinal-aligned 13-point pattern satisfies this for $N=9$. If $A^\\top A$ is ill-conditioned, use SVD-based pseudo-inverse or Ridge Regression (Appendix G).<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix G. Ridge Regression Derivation (Tikhonov Regularization)<\/h2>\n\n\n<style>.kadence-column6564_8643a8-a8 > .kt-inside-inner-col,.kadence-column6564_8643a8-a8 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_8643a8-a8 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_8643a8-a8 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_8643a8-a8 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_8643a8-a8 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_8643a8-a8{position:relative;}.kadence-column6564_8643a8-a8, .kt-inside-inner-col > .kadence-column6564_8643a8-a8:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_8643a8-a8 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_8643a8-a8 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_8643a8-a8\"><div class=\"kt-inside-inner-col\">\n<p class=\"wp-block-paragraph\">Pure LSQ becomes unstable when $A^\\top A$ is ill-conditioned or measurement noise is large. Hoerl &amp; Kennard (1970) addressed this by adding an L2 penalty to the objective:<\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$J_{\\text{ridge}}(a) = \\| T &#8211; A \\cdot a \\|^2 + \\lambda \\| a \\|^2$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">where $\\lambda \\geq 0$ is the regularization strength. With $\\lambda = 0$ this reduces to LSQ; as $\\lambda \\to \\infty$, $a \\to 0$. Expanding:<\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$J_{\\text{ridge}}(a) = T^\\top T &#8211; 2 \\, T^\\top A \\cdot a + a^\\top (A^\\top A + \\lambda I) \\cdot a$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Setting the gradient to zero:<\/p>\n\n\n\n<div style=\"background-color: #fff; border: none\">\n$$(A^\\top A + \\lambda I) \\cdot a = A^\\top T \\quad \\Rightarrow \\quad \\hat{a}_{\\text{ridge}} = (A^\\top A + \\lambda I)^{-1} A^\\top T$$\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why this is always invertible:<\/strong> if $A^\\top A$ is positive semi-definite, then for any $\\lambda &gt; 0$, $A^\\top A + \\lambda I$ is strictly positive-definite \u2014 guaranteeing a unique solution even when LSQ fails.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Aspect<\/th><th>LSQ<\/th><th>Ridge<\/th><\/tr><\/thead><tbody><tr><td>Singular $A^\\top A$<\/td><td>No solution<\/td><td>Solution exists<\/td><\/tr><tr><td>Noise sensitivity<\/td><td>High<\/td><td>Low<\/td><\/tr><tr><td>High-order coefficient stability<\/td><td>Unstable<\/td><td>Stable<\/td><\/tr><tr><td>Bias<\/td><td>None<\/td><td>Mild bias introduced<\/td><\/tr><tr><td>13-point recommendation<\/td><td>Only when noise is very small<\/td><td><strong>Recommended in general<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">To choose $\\lambda$ for 13-point fitting: collect coefficient distributions from many normal wafers via LSQ, sweep $\\lambda \\in \\{0.001, 0.01, 0.1, 1.0, 10.0\\}$, run Leave-One-Out Cross-Validation (LOOCV) by holding out one of the 13 measurements, pick the $\\lambda$ minimizing average prediction error, and re-evaluate periodically (every few months).<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">References<\/h2>\n\n\n<style>.kadence-column6564_db594f-a4 > .kt-inside-inner-col,.kadence-column6564_db594f-a4 > .kt-inside-inner-col:before{border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;}.kadence-column6564_db594f-a4 > .kt-inside-inner-col{column-gap:var(--global-kb-gap-sm, 1rem);}.kadence-column6564_db594f-a4 > .kt-inside-inner-col{flex-direction:column;}.kadence-column6564_db594f-a4 > .kt-inside-inner-col > .aligncenter{width:100%;}.kadence-column6564_db594f-a4 > .kt-inside-inner-col:before{opacity:0.3;}.kadence-column6564_db594f-a4{position:relative;}.kadence-column6564_db594f-a4, .kt-inside-inner-col > .kadence-column6564_db594f-a4:not(.specificity){margin-left:var(--global-kb-spacing-sm, 1.5rem);}@media all and (max-width: 1024px){.kadence-column6564_db594f-a4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}@media all and (max-width: 767px){.kadence-column6564_db594f-a4 > .kt-inside-inner-col{flex-direction:column;justify-content:center;}}<\/style>\n<div class=\"wp-block-kadence-column kadence-column6564_db594f-a4\"><div class=\"kt-inside-inner-col\">\n<ul class=\"wp-block-list\">\n<li>Born, M., &amp; Wolf, E. (1999). <em>Principles of Optics<\/em> (7th ed.). Cambridge University Press.<\/li>\n\n\n\n<li>Hoerl, A. E., &amp; Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. <em>Technometrics<\/em>, 12(1), 55\u201367.<\/li>\n\n\n\n<li>Montgomery, D. C. (2013). <em>Introduction to Statistical Quality Control<\/em> (7th ed.). Wiley.<\/li>\n\n\n\n<li>Noll, R. J. (1976). Zernike polynomials and atmospheric turbulence. <em>Journal of the Optical Society of America<\/em>, 66(3), 207\u2013211.<\/li>\n\n\n\n<li>Wang, J. Y., &amp; Silva, D. E. (1980). Wavefront interpretation with Zernike polynomials. <em>Applied Optics<\/em>, 19(9), 1510\u20131518.<\/li>\n\n\n\n<li>Zernike, F. (1934). Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode. <em>Physica<\/em>, 1(7\u201312), 689\u2013704.<\/li>\n<\/ul>\n<\/div><\/div>\n<div style='text-align:center' class='yasr-auto-insert-overall'><\/div><div style='text-align:center' class='yasr-auto-insert-visitor'><\/div>","protected":false},"excerpt":{"rendered":"<p>\u2003Thickness uniformity in thin-film deposition determines downstream yield and device performance. Variation arises along two distinct axes \u2014 within a single wafer (Within-Wafer, WiW) and across wafers over time (Wafer-to-Wafer, W2W). These two axes have different physical origins and demand different diagnostic treatments. Mixing them into a single ML target forces the model to learn&#8230;<\/p>\n","protected":false},"author":4,"featured_media":6591,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"_kadence_starter_templates_imported_post":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_kad_post_classname":"","yasr_overall_rating":0,"yasr_post_is_review":"","yasr_auto_insert_disabled":"","yasr_review_type":"","fifu_image_url":"","fifu_image_alt":"","iawp_total_views":6,"footnotes":""},"categories":[56,18,374,4,375],"tags":[],"class_list":["post-6564","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data-science-slug","category-ai-powered-slug","category-label-engineering-slug","category-semiconductor-slug","category-tree-based-model-slug"],"yasr_visitor_votes":{"stars_attributes":{"read_only":false,"span_bottom":false},"number_of_votes":1,"sum_votes":5},"jetpack_featured_media_url":"https:\/\/ykim.synology.me\/wordpress\/wp-content\/uploads\/2026\/05\/Zernike-polynomials-9-orders-800px.png","_links":{"self":[{"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/posts\/6564","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/comments?post=6564"}],"version-history":[{"count":20,"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/posts\/6564\/revisions"}],"predecessor-version":[{"id":6613,"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/posts\/6564\/revisions\/6613"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/media\/6591"}],"wp:attachment":[{"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/media?parent=6564"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/categories?post=6564"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ykim.synology.me\/wordpress\/wp-json\/wp\/v2\/tags?post=6564"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}