n,k Characterization

n,k Characterization turns a measurement of a witness sample into a material. Give it either a transmittance and reflectance pair from a spectrophotometer, or a Ψ and Δ pair from an ellipsometer, and it returns the film’s n(λ), k(λ) and thickness, ready to save into a catalog and use in a design.

Import the measurement first: R and T in Measured Spectra, Ψ and Δ in Measured Ellipsometry. The T / R and Ψ / Δ buttons at the left of the toolbar pick which kind of measurement the run uses, and each mode offers only the curves that belong to it.

One film, one substrate

The model is a single film on a substrate. A sample with a second layer under the film, an intentional interlayer or a thick thermal oxide, is not what this window solves, and fitting one as a single film returns a film that does not exist. The residual is what tells you: a sample the model cannot describe leaves degrees of residual in Δ rather than hundredths.

Settings

Setting	What it does
Index model	Cauchy or Sellmeier for a dielectric, Drude or Drude-Lorentz for a metal. The number of terms is chosen from the data.
Sample	Whether the measurement saw the substrate’s back face (slab) or a coating on a semi-infinite substrate. Photometry only; an ellipsometer sees the coated surface alone.
Substrate and its thickness	The substrate the film sits on. It has to be right: an error here goes straight into k.
Δ convention	Ellipsometry only. Must match what the imported file carries.
λ range	The part of the measurement to fit.
Film	Solve for the thickness, or hold it.

The Film setting

Hold keeps the thickness exactly as typed and fits only n and k. Use it when the thickness is known from a profilometer or a quartz monitor.

Solve fits the thickness too, and the number in the box is still read. What it is used for depends on the measurement:

• A transmittance with interference fringes carries the thickness in the fringe spacing. That is read first, and the typed value is ignored.
• Anything else, which includes every ellipsometric fit and a reflectance on its own, has no fringe spacing to read. The typed value is then the centre of the thickness search, and the search covers half to one and a half times it.

So under Solve, a Ψ/Δ fit needs a starting thickness good to about a third. Put 500 nm in the box for a 100 nm film and the right answer is outside the searched range: the fit will still converge, on a different film, with a small residual and no complaint. A design open next to the window seeds the box from its own single layer, which is usually what you want and occasionally is not.

What you get back

The plot

Three views, chosen from the toolbar.

n and k shows the fitted model as two lines, and the per-wavelength solve as two sets of dots. The dots are the answer to a different question: at each measured wavelength, holding the fitted thickness, the two measured values are solved directly for n and k with no dispersion model imposed. For an R and T pair that is the classic pair extraction. It is the independent check on the model: where the dots sit on the lines, the smooth model is describing the measurement; where they wander off it, it is not.

Two properties of the dots are worth knowing. Only wavelengths the solve actually reached are drawn, and the results table counts them. And a wavelength’s own pair of measurements has more than one (n, k) that reproduces it at a given thickness, so the solve is started from the fitted model to pick the root beside it. That does not pull the dots toward the model: they still have to reproduce the measurement exactly, so a model that is wrong is left standing away from them.

Fit plots the measurement against what the fitted film calculates. Residual plots the difference, which is where a systematic error shows itself as structure rather than noise.

The table

Beside the thickness and the model, the table reports the residual per channel, in the units of that channel: absolute for T and R, degrees for Ψ and Δ. This is the first number to read. It says whether the model reproduces the measurement at all, and no warning substitutes for it.

The rest is there to say how much of the answer the measurement determined:

• Smallest k this measurement resolves. Single-pass absorptance is 4πkd/λ, so the instrument’s own accuracy puts a floor under k. An extracted k below this line is describing the instrument, not the film.
• Wavelengths solved. How many points the per-wavelength solve reached.
• Strongest parameter correlation. Thickness and index enter a measurement largely as the product n·d. When there is too little structure to separate them, the residual stays small, the spreads grow, and this goes to one. A correlation near 1 with a small residual means the measurement did not pin the answer, whatever the residual says.

Warnings

A warning names a condition a source calls wrong. Anything that is a matter of degree is a number in the table instead.

• The extracted k rises toward longer wavelengths. A film absorbs at its band edge, in the ultraviolet, so k should fall as wavelength rises. Macleod shows two ways to produce the opposite, an inhomogeneous film fitted as a homogeneous one and a photometric scale error, and both recalculate the input perfectly while describing the wrong film. Treat it as Macleod does: with deep suspicion.
• The fitted n rises toward longer wavelengths on a film that absorbs nowhere in the range. Usually at an end of the range, where the fringes run out and the measurement stops fixing the index.
• The fitted n or k leaves the values a real film can have. For k this is the absorption model running away rather than an unusual film; do not save that material.
• The measured R and T add to more than one, which is a calibration fault.
• k was fitted from a reflectance alone. Reflectance barely responds to absorption, so that value is not measured.

Saving the result

Save as material writes the fit into a catalog, stored the way a fitted tabular material is stored: the analytic model, with a sampled table behind it. Nothing downstream needs to know it came from a measurement.

Save and open design additionally builds a design holding this film on the substrate and evaluation mode it was characterized on, carrying the measured curves it was fitted to. That design reproduces the measurement, which makes it the natural place to check the fit against anything else in the application.

Export writes the model and the per-wavelength points side by side as CSV, with a column saying which points were solved.

Method

Four steps, in this order, because each supplies what the next needs.

1. The transmittance fringe envelopes give a first index and, from the fringe positions, a thickness. Closed form, no starting guess. With no fringes to read, a constant-index scan takes its place and the thickness comes from the Film setting.
2. At each trial thickness, n and k are solved outright at every measured wavelength.
3. The thickness whose extracted index wanders least is kept, and a dispersion model is fitted to its n and k. A film’s index is a smooth function of wavelength; extracted at the wrong thickness it is not, and picks up an oscillation at the fringe period. That is the signal being minimised.
4. Model and thickness are refined together against the measurement through the exact transfer-matrix calculation, which is the only step that sees the sample geometry, angle of incidence and polarization.

Steps 1 to 3 exist to put step 4 in the right place. A fringed spectrum has one solution per interference order and the residual cannot tell them apart, so starting anywhere is not an option.

Reference throughout: H. A. Macleod, Thin-Film Optical Filters, 5th ed., “Measurement of the Optical Properties”.