From design to production: thickness-tolerance Monte Carlo

Tolerance Monte Carlo

Given the 1σ thickness error the coater actually delivers, each trial perturbs every layer by an independent Gaussian of that width and re-solves the whole stack. Sorting a few hundred results wavelength by wavelength gives the P5–P95 tolerance band — the range that holds 90 % of the parts.

• Percentiles, not mean ± σ — a bounded quantity such as R has a lopsided distribution, so mean − σ often lands at a value no part can reach.
• Metric distributions — a histogram of one figure of merit instead of a spectrum: R min/max, peak wavelength, Q, FWHM and L*a*b* colour for coatings; outcoupling efficiency, Purcell factor and channel fractions for OLEDs; J_sc, η, V_oc and FF for solar cells (including current matching in tandems).
• Per-layer σ — a sputtered oxide, an evaporated metal and an unprocessed substrate have different tolerances. Each layer can carry its own σ or be held exactly; a name filter configures hundreds of rugate layers in one step.
• Reproducible — a fixed seed gives the same band for the same design, so designs can be compared and results quoted in a report.
• Runs in the background with a progress counter and STOP.

Without the Monte Carlo — analytic sensitivity

The analytic thickness derivatives from the solver propagate the tolerance to first order. Because it updates during editing, it is the tool for steering a design; the Monte Carlo is the tool for reporting one.

• Each layer's share of the risk — its fraction of the band variance, which shows directly which layer to tighten.
• Correlated error ρ — a common mode in which every layer errs the same way (ρ > 0, rate or calibration drift) and the anti-correlation of optical monitoring (ρ < 0). A drift-coupling figure shows how immune the design is to a uniform drift.
• Spec margin and yield — one-sided (R ≤ 0.5 %) and two-sided (T = 50 ± 2 %) specifications give different pass probabilities (an upper bound set by the worst wavelength; the Monte Carlo gives the exact value).
• Thickness control for 3σ — the inverse: how tightly each thickness must be held to put three standard deviations inside the specification. That is the number to take to a process engineer.

Designs that tolerate error

Robust design optimises the merit averaged over deposition error rather than the merit as drawn. The design with the lowest merit on paper often sits in a narrow well; averaging moves toward a wider one — slightly worse on paper, better at the upper percentiles once error is applied.

Where it is used

• Production-yield prediction for AR, bandpass and dichroic filters, and deposition specifications agreed with the coater
• Passband-centre drift of narrowband filters and GDD-ripple tolerance of chirped mirrors
• How far thickness error moves the colour and outcoupling efficiency of microcavity OLEDs
• The effect of AR and window-layer thickness error on J_sc and η, and the current-matching tolerance of tandem cells

Try it

Phazic runs on Windows 10 and 11 and is available from the Microsoft Store. Run a tolerance band on any stack you design.

Related: Anisotropy (Berreman 4×4) · Mixed layers (EMA) · OLED dipole emission