This page is the complete catalog of merit-function operands. Each row in the Merit Function Editor table is one operand: a single number the optimizer reads, compared against a target.

How an operand contributes to the merit function

The merit function (MF) is the weighted root-mean-square of the residuals:

MF = √( Σ_i  wᵢ · residualᵢ²  /  Σ_i wᵢ )

Note the weight wᵢ is applied linearly (not squared). In the least-squares solver each residual enters as √wᵢ · residualᵢ, whose square is wᵢ · residualᵢ² — consistent with the formula above.

The residual depends on the operand class:

Class	Residual	Inert when satisfied?
Equality (most optical operands)	value − target	no (two-sided)
One-sided ≥ (OPGT, ABGT, MNT, TMN…)	max(0, target − value)	yes
One-sided ≤ (OPLT, ABLT, MXT, TMX…)	max(0, value − target)	yes
Spectral target (TGT/RGT/AGT)	the RMS deviation itself	no

“Inert when satisfied” means the operand drops out of the MF entirely once its inequality holds, so it never fights the equality targets.

Mixed-unit normalization

Most operands are fractions (T/R/A ∈ [0,1]), but argmax/argmin-λ operands produce a residual in nanometres. In a single weighted RMS a 10 nm wavelength miss (residual 10) would swamp a 1 % optical miss (residual 0.01) no matter how the weights were set. To keep weight meaning importance rather than units, each residual is divided by a per-type characteristic scale σ before the RMS (a dimensionless, χ²-style sum):

MF = √( Σ wᵢ · (residualᵢ / σᵢ)²  /  Σ wᵢ )

Operand class	σ	Effect
All fraction-unit (T/R/A, averages, integrals, worst-case, spectral-target RMS, math)	1	unchanged — pure-optical MFs are identical to before
Argwave (MXW* / MNW*, nm)	500 nm	5 nm wavelength miss ≈ 1 % optical miss
Thickness (TT, MNT, MXT, nm)	1 (raw nm)	kept “hard” — a violated manufacturing bound still dominates and is fixed first

A purely optical merit function is therefore numerically unchanged; only merit functions that mix wavelength-valued and optical operands rebalance.

Common columns

Every operand row exposes the same columns; their meaning changes with the operand type (the column header updates to match the focused row):

Column	Optical / band / integral / worst-case	Argwave (MXW*/MNW*)	Constraints (MNT/MXT)	Total thickness (TT)	Math (OPGT…PROD)
λ / Start	start wavelength (nm)	band start (nm)	first layer index	comparison (≤ ≥ =)	referenced Op #
End	end wavelength (nm), band types only	band end (nm)	last layer index	—	second Op # (pair ops)
AOI (°)	angle of incidence	AOI	—	—	inherited from ref
Pol	avg / s / p	pol	—	—	inherited from ref
Target	desired value (see units below)	desired λ (nm)	bound (nm)	total (nm)	desired value (ref units)
Weight	relative importance (linear)	weight	weight	weight	weight
Current	live computed value	computed λ (nm)	min/max layer (nm)	Σ thickness (nm)	computed value
Δ	current − target	Δλ (nm)	violation (nm)	Δ (nm)	residual

Units: T/R/A-valued operands store the target as a fraction in [0,1] and display it as a percentage. Wavelength, layer-index, and thickness operands use raw numbers (nm or count). Math operands inherit the unit of the row they reference.

Polarization (avg/s/p) is chosen by the Pol column, not baked into the type code. avg is the unweighted mean of s and p, (Cs + Cp) / 2.

AOI / Snell: the angle is the angle of incidence in the incident medium; the internal substrate angle is derived from the real part of the refractive index.

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Optical — single wavelength

Evaluated at exactly one wavelength (λ / Start).

Type	Computes	Target unit	Output
T	Transmittance at λ	%	T ∈ [0,1]
R	Reflectance at λ	%	R ∈ [0,1]
A	Absorptance at λ	%	A ∈ [0,1]

Residual: value − target (two-sided). Legacy files may contain the polarization-suffixed forms TS/TP/RS/RP/AS/AP; they still evaluate (the suffix sets the polarization) but are no longer offered in the dropdown — use the Pol column instead.

Optical — band average (single target)

Sampled on a uniform grid across [λStart, λEnd] (~2 nm spacing, clamped to 13…201 points), then averaged to one number.

Type	Computes	Target unit	Output
TAV	Mean T over the band	%	mean T ∈ [0,1]
RAV	Mean R over the band	%	mean R ∈ [0,1]
AAV	Mean A over the band	%	mean A ∈ [0,1]

Residual: mean − target (two-sided). TAV/RAV/AAV are pure averages — one target = the average level over the whole band. For a per-wavelength target line use the spectral-target operands below.

Spectral target — flat or linear ramp

A per-wavelength target line across the band. Target holds two values entered as start→end (e.g. 50→50 for a flat 50 % line, 0→100 for a ramp). Sampled on a density-based grid (~2 nm, the same density as band averages); set rampPoints to override.

Type	Computes	Target unit	Output
TGT	RMS deviation of T from line	% (start→end)	RMS deviation (≥ 0)
RGT	RMS deviation of R from line	% (start→end)	RMS deviation (≥ 0)
AGT	RMS deviation of A from line	% (start→end)	RMS deviation (≥ 0)

The Current column shows the RMS deviation directly; the residual is that value (target is already folded in), so the optimizer drives it to zero. Use these for beamsplitters (flat 50 %) and gradient / ramp filters.

Note

Spectral targets sample at ~2 nm (matching band averages), clamped to 13…201 points. For an exceptionally steep edge, raise rampPoints for an even finer fit.

Weighted integral (source × detector)

A band average weighted by w(λ) = Source(λ) · Detector(λ): C̄ = Σ wᵢ·Cᵢ / Σ wᵢ. The λ / Start cell is a preset picker (e.g. photopic-weighted Tvis, solar-weighted Tsol); the band end is read-only and driven by the preset.

Type	Computes	Target unit	Output
TIW	Source×detector weighted mean T	%	C̄ ∈ [0,1]
RIW	Source×detector weighted mean R	%	C̄ ∈ [0,1]
AIW	Source×detector weighted mean A	%	C̄ ∈ [0,1]

Residual: C̄ − target (two-sided). Source/detector specs are stored on the operand (default D65 × photopic).

Worst-case (band extremum)

Returns the true extremum of the spectrum over the band, sampled on a dense ~1 nm grid. The residual is one-sided — inert until the worst case violates the spec.

Type	Computes	Spec it enforces	Residual
TMN	Minimum T over band	min T ≥ target	max(0, target − minT)
RMN	Minimum R over band	min R ≥ target	max(0, target − minR)
AMN	Minimum A over band	min A ≥ target	max(0, target − minA)
TMX	Maximum T over band	max T ≤ target	max(0, maxT − target)
RMX	Maximum R over band	max R ≤ target	max(0, maxR − target)
AMX	Maximum A over band	max A ≤ target	max(0, maxA − target)

Output is a real T/R/A value (0…100 %), never exceeding physical bounds. The optimizer uses the single argmin/argmax wavelength as the subgradient.

Argmax / argmin wavelength

Sample C(λ) over the band, find the extremum, and refine it with a 3-point parabolic fit. Output is the wavelength (nm) at that extremum — not the T/R/A value.

Type	Computes	Target unit	Output
MXWT	λ of maximum T over band	nm	λ (nm)
MXWR	λ of maximum R over band	nm	λ (nm)
MXWA	λ of maximum A over band	nm	λ (nm)
MNWT	λ of minimum T over band	nm	λ (nm)
MNWR	λ of minimum R over band	nm	λ (nm)
MNWA	λ of minimum A over band	nm	λ (nm)

Residual: λ_extremum − target (two-sided, in nm). Use to pin a peak / notch to a desired wavelength. The default seed target is the band midpoint.

Caution

The residual is in nanometres, so a 10 nm miss is a much larger residual than a 0.01 optical miss. When mixing argwave operands with optical operands in one merit function, tune weights so the units are commensurate.

Math operands (reference another row)

Math operands do not evaluate a TMM characteristic directly. They reference one or two other rows by their stable Op # (via the λ / Start and End picker cells) and compute a derived value. Target units are inherited from the referenced row.

Type	Refs	Value	Residual	Spec it enforces
OPGT	1	ref	max(0, target − ref)	ref ≥ target
OPLT	1	ref	max(0, ref − target)	ref ≤ target
OPVA	1	ref	ref − target	ref = target
ABSO	1	`	ref	`
ABGT	1	`	ref	`
ABLT	1	`	ref	`
DIFF	2	ref1 − ref2	value − target	ref1 − ref2 = target
SUMM	2	ref1 + ref2	value − target	ref1 + ref2 = target
PROD	2	ref1 · ref2	value − target	ref1 · ref2 = target

The reference is by stable id, so inserting, deleting or reordering rows keeps the link. A reference to a deleted row renders red (“stale”). Cyclic references evaluate to a neutral (zero-residual) value. This is the familiar pattern where a target row references a measurement row by its operand number.

The Specification window’s “Generate MF” emits, for each ≥/≤ spec, a zero-weight measurement row (TAV, TMN, …) plus an OPGT/OPLT row that references it — so the table reads “spec = 99 %, value = 99.5 %”.

Thickness operands

Act on layer thicknesses, not the spectrum.

Type	λ / Start	End	Computes	Target unit	Residual
TT	comparison	—	Σ of all active layer thicknesses	nm	≤/≥ one-sided, or = two-sided
MNT	layer 1	layer 2	min thickness in layer range	nm	max(0, target − minThk) (≥ bound)
MXT	layer 1	layer 2	max thickness in layer range	nm	max(0, maxThk − target) (≤ bound)

MNT/MXT layer ranges are 1-based layer indices, clamped to the current stack — a generator can emit End = 9999 to mean “every current and future layer”. During Needle / Gradual Evolution synthesis the thickness penalties are suppressed (the dMin floor + post-refine + Cleaner enforce bounds instead); they are active during Refinement.

Comment / sentinel

Type	Effect
BLNK	Inert annotation row carrying free text — contributes nothing.
DMFS	“Default merit function” sentinel marking a generated block start.

A freshly added row is a BLNK placeholder so it can’t silently inject a stray target; pick the real type from the dropdown. Build and edit the operand table in the Merit Function Editor; the source/detector presets used by TIW/RIW/AIW come from the Integral Values tool.

References

• B. T. Sullivan, J. A. Dobrowolski, “Implementation of a numerical needle method for thin-film design,” Appl. Opt. 35, 5484 (1996).
• H. A. Macleod, Thin-Film Optical Filters, 5th ed., §2.6.4 (two-sided system), Ch. 13 (merit functions and tolerancing).