Merit Function Editor

The Merit Function Editor is where you tell the optimizer what “good” looks like. Every target the design should hit and every constraint on the layer stack lives here, expressed as a list of operands. Each operand is a single number the optimizer tries to drive toward a target value, and the merit function (MF) is the weighted root-mean-square of how far each operand misses:

MF = √( Σ_i  w_i · residual_i²  /  Σ_i w_i )

The weight w_i enters linearly. The residual is value − target for an equality target, or a one-sided max(0, …) term for an inequality or constraint (so a satisfied constraint drops out of the sum entirely). Wavelength-valued residuals are rescaled to optical scale first so they don’t dominate. A smaller MF is better; the optimizers (Refinement, Needle and Gradual Evolution) move layer thicknesses to reduce it.

Operand types

The table below is a quick orientation. The full catalog (every type, its arguments, its output value and unit, and how it forms a residual) is on the Operand Reference page.

Group	Types	Output
Single-λ optical	T R A	T/R/A at one λ
Band average	TAV RAV AAV	mean T/R/A over a band
Spectral target	TGT RGT AGT	deviation from a flat/ramp line
Weighted integral	TIW RIW AIW	source × detector weighted mean
Worst-case	TMN RMN AMN TMX RMX AMX	band extremum of T/R/A
In dB or optical density	TDB TDBMN TDBMX RDBMX ODMN PPEF	T or R in dB, optical density, peak-to-peak error against a curve
Phase / field	PSI DEL TANPSI COSDEL PR PT DPR DPT GD* GDD* TOD* EFMX	phase, ellipsometry, dispersion, peak |E|²
Argmax/min λ	MXWT MXWR MXWA MNWT MNWR MNWA	wavelength of the extremum
Math (reference rows)	OPGT OPLT OPVA ABSO ABGT ABLT DIFF SUMM PROD	derived from other rows
Thickness and stress	TT STR MNT MXT	total / per-layer thickness, film force on the substrate
Comment	BLNK DMFS	inert

Reflection and transmission targets are typically generated in paired rows by the filter-type wizard so the optimizer can’t trade absorption for an easy win.

The wizard

The wizard sits above the table. Its bar carries the Optimize and Eval badges and the current MF and OMF; the chevron at the left folds the form away and brings it back. Under the bar are three boxes:

• Preset: the coating category (AR, mirror, beamsplitter, edge filter, bandpass or notch, gradient, integral or worst-case, custom target, curve target), the type within it, and the type’s own values, the wavelength range first. The Custom target type generates a single target of your own: a channel (T/R/A), a unit (%, dB or OD), a comparison (=, ≤, ≥), a value and a range.
• Angle and target: the angle of incidence, or a range of angles with the number of steps, the polarization, and whether the target is a continuous line or discrete points. A type that sets polarization itself, such as the polarizing beamsplitter, shows no polarization control.
• Thickness limits: minimum and maximum layer thickness (MNT/MXT) and a total thickness cap, each behind a checkbox.

A custom target in dB or OD writes one of the dB and OD rows: T ≥ in dB a TDBMN row, T ≤ in dB a TDBMX row, R ≤ in dB an RDBMX row, T ≥ in OD an ODMN row, and T = in dB one TDB row per wavelength step, with a step field in place of the line-or-points choice. Combinations with no row behind them are not offered: A has only %, R in dB only ≤, OD only ≥. Switching the unit converts the typed value to the same level, so 80 % becomes −0.97 dB. A value in dB is held at or below 0 dB, a density at or above 0.

Curve targets

The Curve target category takes a curve instead of numbers. Its rows read the curve’s own angle and polarization, so the Angle and target box has no angle of its own to set. For gain flattening it holds the two values the filter is specified by, Insertion loss and Peak-to-peak error, and for a gain also the angle and polarization its target is put at. On a design with no curve to take, the line under the boxes says so and Open Measured Spectra beside it opens the window where a curve is imported or typed.

• Curve picks one of the design’s curves from Measured Spectra and writes the block Fit… writes for it, in % or in dB. When the design has only one curve the type can take, it is taken without a pick.
• Gain flattening builds the merit function a gain-flattening filter is specified by. Its input is either a target loss curve already on the design, or an amplifier gain in dB, imported from a text file or typed into the curve editor with Type…. A gain is turned into the loss that brings every wavelength down to the lowest gain, T(λ)[dB] = G_min − G(λ): 0 dB at the gain minimum and negative everywhere else, the least loss a passive filter can flatten with. That target goes on the design as a T curve named after the gain, at the angle and polarization you give, so it draws on the plot; the gain itself is not kept.

Gain flattening writes three rows. The target as a curve block in dB with its level free: the constant the design sits away from it by is taken out, so the block scores the shape and not the level. A PPEF row against that block, at the peak-to-peak value you set: least squares brings the shape in, and this row works on the worst point, the number the filter is sold against. And a TDBMN row at the wavelength where the target is highest, held at minus the insertion loss you set: if T is that high there, the peak insertion loss meets it. Both values start at 0 dB, which asks for the flattest shape and the least loss the stack can reach. Choosing Gain flattening unticks the layer maximum in the Thickness limits box, since these filters often need layers several micrometres thick; tick it again if the coating process limits layer thickness.

The synthesis tools leave the PPEF row out: its gradient comes from only the two worst wavelengths, and the needle search stalls on it. They bring the shape in with the block, and a Refinement run afterwards works the peak-to-peak error down.

The line under the boxes says how many rows the wizard will add and of which types. Start at row is where the block goes; Generate adds it. The form keeps its values while the window is closed and reopened.

The table

One row per operand. The table works like a spreadsheet:

• Click a cell to focus it, type to replace its value, or press Enter or double-click to edit it. Enter commits and moves down, Tab moves right, Escape cancels.
• The Type cell holds the operand code. Typing a letter opens the operand picker searching for it; Enter or a double-click opens it on the current type. The Pol cell shows a chevron while it is focused; the chevron, Enter or a double-click opens its three values as a list, and a, s or p typed straight in sets it without the list.
• Drag across cells, or hold Shift, to select a rectangle; hold Ctrl to add single cells. Ctrl+C copies the selection as tab-separated text, Ctrl+V pastes text over it: one value fills every selected cell, a block of values is laid out from the focused cell. Text that came from copying rows is inserted as rows.
• Rows are selected from the row-number column at the left: click one, drag down the column for a run, Shift for a run, Ctrl to add. A comment or header row is selected by a click anywhere on it. Delete, Ctrl+X and Ctrl+D act on selected rows; Insert adds a row above the focused one, Shift+Insert below.
• Right-click opens the same actions as a menu: copy and paste of the cell or cells, cut, copy and paste of operands, insert, duplicate and delete.

Load MF and Save MF in the table’s bar load and save the whole table as a named merit function you can reuse in another design.

Constraints: minimum and maximum layer-thickness bounds (MNT/MXT) per layer or per material. A bound can be written to cover layers that synthesis will add later. An MXT row is the only upper limit on a layer’s thickness; without one the optimizers let a layer grow as thick as the merit asks.

MF vs OMF

The header shows two numbers:

• MF: the full merit function, including the manufacturability rows (MNT, MXT, TT, STR).
• OMF: the optical merit function: the same RMS but counting only the optical operands (T/R/A targets, bands, and so on), with the manufacturability rows dropped.

They separate two questions the plain MF blurs together: how good is the spectrum? (OMF) versus how good is the spectrum while honoring what I can build? (MF). When MF is high but OMF is low, the optical performance is fine and it is a constraint costing you, not the optics: a too-thin or too-thick layer, a thickness budget, or a coating that bends its substrate too far.

How to read it

Each operand row shows its current value and its residual, so you can see at a glance which targets are met and which are dragging the merit up. Bigger weights make an operand count for more; bumping the stopband weight on an HR design is the most common tweak.

The operand list is saved with the design and is read by every optimizer: Refinement, Needle, Gradual Evolution and the Structural Optimizer. The synthesis tools optimize against the optical operands only while they build the stack, less any PPEF row and the math rows that refer to it; the thickness constraints and the peak-to-peak error are then enforced during Refinement. A good final sequence is Refinement, then Cleaner, then Refinement again.

References

• A. V. Tikhonravov, M. K. Trubetskov, G. W. DeBell, “Application of the needle optimization technique to the design of optical coatings,” Appl. Opt. 35, 5493 (1996).
• J. A. Dobrowolski, R. A. Kemp, “Refinement of optical multilayer systems with different optimization procedures,” Appl. Opt. 29, 2876 (1990).