Monte Carlo

Monte Carlo answers a practical question: if your coater deposits every layer with a realistic random error, how much does the spectrum move, and how often does the result still pass your Specification? You set the size and shape of the per-layer thickness and index errors, choose how many trials to run, and the tool builds a fresh randomly-perturbed version of your design for each trial. It evaluates T, R or A for every trial, keeps a running mean and standard deviation across them, and tracks the realized extremes. If your design has a Specification, every trial is re-checked against it so you get a yield figure.

The analysis runs for the surface mode set in the Design Editor (front, back, or total), shown as a badge on the window.

Settings

T / R / A: the spectral characteristic to study.

λ range / step: the wavelength grid, in nanometres.

AOI / pol: angle of incidence and polarization (s, p, or averaged).

Errors

What a trial is drawn from lives in the Errors strip below the plot, beside the trial count it was drawn with.

N trials: how many randomly-perturbed designs to simulate. More trials give a smoother corridor and a steadier yield estimate; the corridor noise shrinks roughly as 1/√N. Default is 200.

seed: the number the random draws start from. Leave it empty and Run fills it with a new seed and keeps it there, so running again with the same seed and settings gives exactly the same trials. Clear the field to draw afresh. The seed belongs to the open session and is not saved with the design; the trials dialog shows the seed each result was drawn from.

Distribution: how each layer’s error is drawn. This controls the shape of the random draw, not its size:

• Gaussian: the value you enter is one standard deviation σ. About 68 % of layers stay within ±σ and the rest exceed it; the tails are unbounded, so σ is not a hard maximum.
• Uniform: the value you enter is a hard ± bound B. Errors are spread evenly over [−B, +B], so none exceeds B, and the realized RMS works out to B/√3 ≈ 0.58·B.
• Truncated: a Gaussian bell clipped so no error exceeds the entered bound B (taken as 3σ). The realized RMS is about B/3.

Choosing Uniform or Truncated relabels the error fields from σ to ± and turns on the min/max envelope automatically, since those distributions have a true hard bound worth seeing.

σ abs / σ rel (nm / %): the per-layer thickness error. The absolute part is a fixed amount in nanometres; the relative part scales with each layer’s own thickness. They add together.

σ Re(n) / σ Im(n): the per-layer error on the real part of the refractive index (n) and on the extinction (k). k cannot go below zero, so on a layer with k = 0 the Im(n) error only ever adds absorption and its mean effect is not zero.

Per-material errors: draw one shared error per material instead of an independent error for every layer, modelling a material-chemistry drift rather than monitoring scatter.

keep n·d: links the thickness error to the index error so that the optical thickness n·d at the design reference wavelength stays constant. The plotted wavelength range does not enter it, so widening the plot does not change the draws. Only meaningful when an index error is set; with thickness errors alone it would cancel the perturbation.

k σ corridor: the width of the shaded band, in standard deviations. This is display-only: changing k redraws the band instantly without re-running the trials and never affects the yield.

min/max envelope: overlays the extreme spectra realized across all trials. For Uniform and Truncated this is the true hard bound; for Gaussian it has no fixed limit and widens as you add trials.

How to read it

The chart is in percent. The solid line is the theoretical (unperturbed) spectrum, the dotted line is the mean across all trials, and the shaded band is mean ± k·σ. A wide band means the design is sensitive to manufacturing error; a tight band means it is robust. Note that the mean can sit slightly off the theoretical curve where the spectrum is curved (for example, the mean reflectance of an antireflection minimum drifts upward). That is real physics, not noise.

If the design carries a Specification, the Results strip shows the yield (the fraction of trials that pass every requirement) with its 95 % interval in brackets, and a red chip for any requirement that fails, with its fail rate. The interval is the range the true yield lies in with 95 % confidence for the number of trials run (the Wilson score interval). Two runs whose intervals overlap may differ only by chance; more trials narrow it. Open View trials… for a deeper look: a statistics tab shows the interval and the seed, ranks the worst requirements by fail rate and the worst layers (those whose thickness deviates more in failing trials than passing ones, or, when nothing fails, those with the largest typical deviation). The trials tab lists every trial with a pass/fail mark and shows the exact per-layer Δd, Δn and Δk applied. Load thicknesses into design copies a chosen trial’s perturbed thicknesses onto the active design so you can inspect it directly; the change is undoable.

A quick sanity check: set every error to zero and every trial collapses onto the theoretical curve with zero corridor width.

References

• H. A. Macleod, Thin-Film Optical Filters, 5th ed., §13.7.
• E. B. Wilson, “Probable inference, the law of succession, and statistical inference,” Journal of the American Statistical Association 22, 209-212 (1927).