Operand Reference

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
Ellipsometry PSI / DEL (deg)	90 / 180	~1° miss ≈ 1 % optical miss
Group delay GD* / GDD* (fs, fs²)	50	a ~0.5 fs / fs² miss ≈ 1 % optical miss
Third-order dispersion TOD* (fs³)	500	a ~5 fs³ miss ≈ 1 % optical miss
TANPSI, COSDEL, EFMX (O(1))	1	already comparable to an optical fraction

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	n/a	second Op # (pair ops)
AOI (°)	angle of incidence	AOI	n/a	n/a	inherited from ref
Pol	avg / s / p	pol	n/a	n/a	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
% of MF²	row’s share of the weighted squared residual	same	same	same	same

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.

The final column uses the same definition for every row. If the merit function is nonzero, its percentages sum to 100%. If every row is exactly met, every contribution is zero. Disabled and unevaluable rows show a dash. Hover the cell to see the row’s residual in its own unit, labelled as a difference, constraint slack, or RMS deviation as appropriate.

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.

Phase / field operands

Quantities derived from the complex amplitude coefficients or the internal electric field of the front coating, rather than an intensity T/R/A. They carry physical units (degrees, femtoseconds, or normalized field), so they use the per-type σ scales above and drive the optimizer through the finite-difference Jacobian. These match the Ellipsometry, GD & GDD and E-field analysis windows on the front surface.

Ellipsometry

Evaluated at one wavelength (λ / Start). Ψ and Δ come from the complex ratio ρ = r_p / r_s = tan Ψ · e^{iΔ}, so they use both polarizations and the Pol column does not apply.

Type	Computes	Target unit	Output
PSI	Ellipsometric Ψ at λ	deg	Ψ ∈ [0°, 90°]
DEL	Ellipsometric Δ at λ	deg	Δ ∈ [0°, 360°)
TANPSI	tan Ψ (ellipsometer-native)	none	≥ 0
COSDEL	cos Δ (ellipsometer-native)	none	[−1, 1]

Residual: value − target (two-sided). Use PSI/DEL to match a measured ellipsometric spectrum, or to force a specific reflection-phase relationship.

Group delay & dispersion

Phase, group delay, GDD, and TOD come from the complex reflection or transmission amplitude at exactly the requested wavelength. DPR and DPT are the cyclic p-minus-s phase difference. The Pol column selects s or p; avg is their mean for non-differential operands.

Type	Computes	Target unit	Output
PR, PT	Reflection or transmission phase at λ	deg	phase (deg)
DPR, DPT	p-minus-s differential phase at λ	deg	phase (deg)
GD, GDT	Reflection or transmission group delay at λ	fs	GD (fs)
GDD, GDDT	Reflection or transmission GDD at λ	fs²	GDD (fs²)
TOD, TODT	Reflection or transmission TOD at λ	fs³	TOD (fs³)
GDFLAT, GDTFLAT	RMS deviation of GD from a flat level	fs	RMS deviation (≥ 0)
GDDFLAT, GDDTFLAT	RMS deviation of GDD from a flat level	fs²	RMS deviation (≥ 0)
TODFLAT, TODTFLAT	RMS deviation of TOD from a flat level	fs³	RMS deviation (≥ 0)

Point residuals are two-sided (value - target). Phase residuals wrap to the shortest difference in the range -180° to 180°. The *FLAT operands carry their RMS deviation directly, so the optimizer drives it to zero. In the merit table, Current for a flatness row is the arithmetic mean GD, GDD, or TOD across the band, which can be read directly against the target level. The RMS deviation used by the merit function remains in the contribution-cell tooltip. Every point operand uses the same analytic evaluator as the GD / GDD window; there is no nearby sample or finite-difference wavelength grid. These operands score the front coating normally and the back coating when the design surface mode is back-only. Total-system phase-dispersion operands are not defined. In Total merit mode, ordinary R and T operands score the complete element while phase, GD, GDD, and TOD operands in the same table keep scoring that one coating. The Merit Function Editor and Refinement window show this scope beside the table.

Analytic phase derivatives are evaluated only inside every participating material model’s stated wavelength range. An operand outside that range shows Error in its Current cell; hover the row to see the material and reason. Other rows continue to display, but MF and OMF remain unavailable and Refinement will not start until every enabled target is valid.

Electric-field peak

Type	Computes	Target unit	Output
EFMX	Peak normalized |E|² anywhere in coating	none	≥ 0

Evaluated at λ / Start; the Pol column selects s or p (avg takes the larger of the two peaks, the damage-relevant one). Residual: value − target; with the default target 0 it monotonically minimizes the peak field, the usual laser-damage-threshold objective.

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	n/a	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 new constraint therefore covers layers 1 to 1000 by default: the end is deliberately past any stack you start from, so the constraint keeps covering the layers synthesis adds. 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).