Are More Layers Better? Comparing Dichroic Performance and Structural Cost

4/6/8-pair dichroic stacks: spectral, polarization, and thickness trade-offs

This tutorial continues Separate Blue-Green and Red Light: Design a 45° Dichroic Beamsplitter and reuses the same materials, target bands, and 45° incidence condition.

The most direct way to increase blue-green reflection from a 45° dichroic beamsplitter is to add more TiO₂/MgF₂ pairs. More layers also increase total thickness, however, and may reduce red transmission or polarization performance, so pair count alone cannot identify the better design.

This tutorial optimizes 4-, 6-, and 8-pair stacks using the same materials, objectives, thickness ranges, and algorithm. You will compare blue-green reflection, red transmission, polarization, total thickness, and variable bounds, then select the simplest structure that meets the specification.

Three interference filters showing different transmitted and reflected colors — Different spectral selections give interference filters different transmitted and reflected colorsNASA / JPL / Wikimedia Commons Public Domain (NASA)

Fix the comparison rules first

Only Repeat Count changes among the three candidates. All other settings remain identical:

Item	Common setting
Period materials	TiO₂ / MgF₂
Initial period thicknesses	56 nm / 100 nm
Pair count	4, 6, 8
Incidence condition	45°, unpolarized light, `pRatio=0.5`
Blue-green objective	Maximize mean reflectance from 450–520 nm
Red objective	Maximize mean transmittance from 620–700 nm
TiO₂ range	25–80 nm
MgF₂ range	70–140 nm
Teaching specification	Unpolarized mean values in both bands must be at least 90%

The two optical metrics are

\overline{R}_B=\frac{1}{M_B}\sum_{j=1}^{M_B}R(\lambda_j), \qquad \overline{T}_R=\frac{1}{M_R}\sum_{j=1}^{M_R}T(\lambda_j).

Here, $\overline{R}_B$ is the mean reflectance in the blue-green band, $\overline{T}_R$ is the mean transmittance in the red band, $R(\lambda_j)$ and $T(\lambda_j)$ are the reflectance and transmittance at wavelength $\lambda_j$ , $j$ is the sample index, and $M_B$ and $M_R$ are the numbers of samples in the two bands.

If each period contains one high-index layer H and one low-index layer L, and the pair count is $N$ , the number of coating layers and the total coating thickness are

L_N=2N, \qquad D_N=N(d_H+d_L).

Here, $L_N$ is the number of coating layers, $D_N$ is the total physical coating thickness excluding the substrate, and $d_H$ and $d_L$ are the thicknesses of one TiO₂ layer and one MgF₂ layer, respectively, in nm.

Make a prediction before running the models: the 8-pair structure may have the highest blue-green reflectance, but its red transmittance, polarization performance, and total thickness may not be better.

Build three candidates from a common starting point

The incident medium is air. Below the coating is a 1 mm incoherent glass substrate, and the bottom-medium refractive index is 1.52. All three candidates start from 56 nm TiO₂ and 100 nm MgF₂; only Repeat Count changes.

Candidate	`Repeat Count`	Initial TiO₂ thickness	Initial MgF₂ thickness
4 pairs	4	56 nm	100 nm
6 pairs	6	56 nm	100 nm
8 pairs	8	56 nm	100 nm

Restore this common pair of initial thicknesses before every optimization. Otherwise, a later candidate inherits the optimum of an earlier one and the candidates no longer share the same starting point.

Optimize each candidate instead of comparing initial structures

In Optics, set 400–750 nm with a 1 nm step, 45° incidence, and unpolarized light. Add two equally weighted objectives in Optimizer:

Maximize mean Reflectance from 450–520 nm with a 5 nm step;
Maximize mean Transmittance from 620–700 nm with a 5 nm step.

For each candidate, use only the two shared thickness variables for TiO₂ and MgF₂ within the period. Use Nelder–Mead for every optimization, with at most 280 evaluations and an initial simplex scale of 0.08.

Dual-band objectives and two shared thickness variables for comparing dichroic structures — Figure 1 | Dual-band objectives and thickness variables shared by all three candidates

The two objectives constrain blue-green reflection and red transmission, while the variables control the TiO₂ and MgF₂ thicknesses in one period. Objective weights and variable ranges remain unchanged when pair count changes.

Nelder Mead algorithm settings used to compare the dichroic structures — Figure 2 | Nelder–Mead algorithm settings shared by all three candidates

All three runs use the same algorithm and evaluation limit so that differences in search settings are not mistaken for pair-count effects.

Run the three optimizations separately and check each Optimization Report. The optimization objectives use 5 nm sampling, whereas final acceptance uses 1 nm spectra, so the report values and final table will differ slightly.

Optimization Report for the four-pair dichroic stack — Figure 3 | Optimization report for the 4-pair candidate

Both optimized thicknesses for the four-pair candidate remain inside their permitted ranges, so the result can be applied directly before fine-step acceptance testing.

Optimization Report for the six-pair dichroic stack — Figure 4 | Optimization report for the 6-pair candidate

The six-pair result pushes TiO₂ to its 25 nm lower bound. Its reported objective values must therefore be judged together with the bound condition and the final 1 nm spectrum.

Optimization Report for the eight-pair dichroic stack — Figure 5 | Optimization report for the 8-pair candidate

For eight pairs, TiO₂ and MgF₂ stop at their lower and upper bounds, respectively. The objective has converged, but the permitted thickness ranges constrain further trade-offs.

Apply the optima and verify the structures

Click Apply to Structure in each report, then run the forward calculation again with a 1 nm step. The optimized period thicknesses are:

Pair count	Optimized TiO₂ thickness	Optimized MgF₂ thickness	Layer count	Total coating thickness
4	32.555 nm	130.787 nm	8	653.37 nm
6	25.000 nm	137.618 nm	12	975.71 nm
8	25.000 nm	140.000 nm	16	1320.00 nm

Each Layer Group now has a different pair count and optimized thicknesses. Check each Structure overview and open Edit Group to verify the group name, Repeat Count, layer order, and material parameters.

Four-pair candidate

Structure page for the optimized four-pair dichroic stack — Figure 6 | 4-pair candidate after applying the optimum

The structure overview contains one periodic group with Repeat Count set to 4, giving eight functional layers. At 653.37 nm, it is the thinnest of the three candidates.

Edit Group dialog for the four-pair dichroic stack — Figure 7 | Optimized TiO₂/MgF₂ period unit of the 4-pair structure

The TiO₂ and MgF₂ layers are 32.555 nm and 130.787 nm thick, respectively. Neither variable stops at a permitted bound.

Six-pair candidate

Structure page for the optimized six-pair dichroic stack — Figure 8 | 6-pair candidate after applying the optimum

Increasing the pair count to 6 gives 12 functional layers and raises total thickness to 975.71 nm. The basic alternating high- and low-index structure remains unchanged.

Edit Group dialog for the six-pair dichroic stack — Figure 9 | Optimized TiO₂/MgF₂ period unit of the 6-pair structure

The optimized TiO₂ thickness is 25.000 nm, exactly at its lower bound, while MgF₂ is 137.618 nm thick. This bound condition must be considered together with the final spectrum.

Eight-pair candidate

Structure page for the optimized eight-pair dichroic stack — Figure 10 | 8-pair candidate after applying the optimum

The eight-pair candidate contains 16 functional layers and reaches a total thickness of 1320.00 nm, about twice that of the four-pair candidate.

Edit Group dialog for the eight-pair dichroic stack — Figure 11 | Optimized TiO₂/MgF₂ period unit of the 8-pair structure

TiO₂ stops at its 25.000 nm lower bound and MgF₂ at its 140.000 nm upper bound. Both variables reaching bounds indicates that this periodic structure has little room left to balance the two target bands within the permitted thickness ranges.

Compare performance, thickness, and bounds together

Pair count	Blue-green mean $R$	Blue-green minimum $R$	Red mean $T$	Red minimum $T$	Total thickness	Variables at bounds
4	91.228%	87.100%	91.160%	78.693%	653.37 nm	None
6	91.824%	75.787%	90.915%	76.209%	975.71 nm	TiO₂ at lower bound
8	96.332%	81.602%	87.882%	81.419%	1320.00 nm	TiO₂ at lower bound; MgF₂ at upper bound

Each structure is rerun with a 1 nm step after applying its optimized thicknesses. The reflectance spectra show the trade-off directly: more pairs raise average blue-green reflectance, but they do not automatically remove transition-band ripple or guarantee lower reflection in the red band.

Four pairs: both bands meet the mean-value specification

Reflectance spectrum of the optimized four-pair dichroic stack — Figure 12 | Final reflectance spectrum of the four-pair candidate

The four-pair design reaches 91.228% mean blue-green reflectance and 91.160% mean red transmittance, so both means pass the 90% teaching specification. Its minimum blue-green reflectance is 87.100%, showing that a band mean does not represent every wavelength in the band.

Six pairs: a small gain in the mean but a lower band minimum

Reflectance spectrum of the optimized six-pair dichroic stack — Figure 13 | Final reflectance spectrum of the six-pair candidate

The six-pair design raises mean blue-green reflectance to 91.824%, but its minimum falls to 75.787%; mean red transmittance is 90.915%. Compared with four pairs, two extra pairs provide only about 0.60 percentage points of blue-green mean-reflectance gain.

Eight pairs: higher blue-green reflection but insufficient red transmission

Reflectance spectrum of the optimized eight-pair dichroic stack — Figure 14 | Final reflectance spectrum of the eight-pair candidate

The eight-pair design raises mean blue-green reflectance to 96.332%, but mean red transmittance falls to 87.882%. More periods strengthen one target band while causing the other to miss the 90% specification.

Compare optical performance with structural cost

Band performance and total thickness comparison for four-pair six-pair and eight-pair dichroic stacks — Figure 15 | Increasing the pair count does not improve both target bands simultaneously

The combined chart places the two mean optical metrics beside total thickness. Both four and six pairs pass the dual-band mean-value specification, but the six-pair coating is about 49% thicker. Eight pairs are the thickest and miss the red-transmission requirement. Increasing pair count therefore does not by itself produce the better overall design.

A variable reaching a bound is not a calculation error. It indicates that, for the current objectives and structure, the optimizer would still prefer to decrease TiO₂ or increase MgF₂. If the bounds come from a real fabrication process, the appropriate response is to change the structural degrees of freedom or objective weights, not to ignore the bounds and compare objective-function values alone.

For all three lossless structures, the maximum energy-closure error is no greater than $4.9\times10^{-15}$ .

An unpolarized average does not replace polarization acceptance

Keep the incidence angle at 45° and set pRatio to 0 and 1 in turn to evaluate pure s and pure p polarization.

Pair count	s: blue-green mean $R$	p: blue-green mean $R$	s: red mean $T$	p: red mean $T$
4	97.720%	84.737%	86.559%	95.762%
6	98.820%	84.828%	84.763%	97.067%
8	99.806%	92.857%	82.437%	93.327%

Increasing the pair count improves some polarization metrics, but it does not improve all objectives simultaneously. If the specification requires every target wavelength to exceed a threshold for either polarization, evaluate each candidate by its minimum value and add s and p as separate optimization objectives.

Select the simplest structure that meets the specification

Under the specification used in this tutorial, the 4-pair structure keeps the mean values in both bands at or above 90%, uses the fewest layers and the smallest total thickness, and has no variables at their bounds. It is therefore the preferred design.

This conclusion applies only to the current materials, bands, angle, objective weights, and thickness ranges. If the blue-green mean reflectance must exceed 95%, the 8-pair candidate is closer to that target, but its insufficient red transmittance and bound-limited variables must be addressed again.

Variation Exercise

Build a five-pair stack from the common 56/100 nm starting point, keep the objectives, bounds, and algorithm unchanged, and optimize again. Insert the result into the 4/6/8-pair comparison table and determine whether the five-pair stack can pass both 90% mean-value specifications with less total thickness than the six-pair stack.

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