Full case study: /case-studies/inverse-rcwa

Observable

Broadband reflectance (300-2500 nm) from periodic GAAFET stacks, measured with inline optical tools.

Claim

With fabrication priors and restricted measurement diversity, trench depth is recoverable. Secondary parameters remain degenerate.

Load-Bearing Constraints

Axiomatic

  • RCWA forward map (geometry + materials → spectrum) is deterministic but many-to-one
  • Multiple (thickness, index, geometry) combinations produce near-identical far-field spectra

Measurement

  • Instrument: bandwidth, spot-size (~10-50 μm), spectral resolution limit detail
  • Real samples: roughness, line-edge roughness, thickness variation violate periodic assumptions
  • Single polarization used: reduces sensitivity, amplifies parameter correlations

Fabrication

  • Process windows define feasible ranges (used as hard bounds in inversion)
  • Sidewall profiles and roughness produce effective responses not captured by rectangular geometries
  • Stack composition known from process flow (reduces free parameters)

Statistical

  • Identifiability determined by covariance structure, not fit quality
  • Multi-start optimization showed multiple local minima with similar χ²
  • Parameter correlations >0.7 for secondary parameters

Computational

  • RCWA convergence verified (50-100 Fourier orders needed)
  • Dispersion models can absorb modeling error and hide geometric degeneracies

Primary Limiting Factor

Intrinsic ill-posedness of inverse problem.

Single-polarization broadband reflectance underdetermines geometry. Depth-duty-cycle trade-offs and thickness-index coupling produce high covariances for secondary parameters.

What This Ruled Out

  • Unique geometry recovery from fit quality alone (multiple solutions with χ² < 0.03 found)
  • Sub-resolution feature inference (beyond measurement sensitivity)
  • Secondary parameter claims without explicit covariance analysis

What Remains Non-Identifiable

  • Thickness-refractive-index trade-offs (both affect optical path length similarly)
  • Depth vs. duty cycle for secondary parameters (similar phase accumulation)
  • Roughness-driven scattering vs. effective index changes
  • Sidewall profile details (45° vs. 50° taper produces <1% spectral difference)

What Would Help

  • Add measurement diversity (multiple angles, polarizations, azimuthal rotations)
  • Encode fabrication knowledge as explicit priors (not just bounds)
  • Cross-validate with orthogonal metrology (CD-SEM, AFM, TEM on calibration samples)
  • Run full identifiability analysis (posterior sampling, multi-start clustering)

Methods Referenced

Analysis date: Summer 2025
My experience: First industry internship, first large-scale inverse problem

This analysis reflects what I learned during the internship about when inverse optical metrology works and when it doesn’t. The constraint analysis (ill-posed inverse problem as primary limit) was surprising to me initially—I thought more optimization would solve it.

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