Separate Blue-Green and Red Light: Design a 45° Dichroic Beamsplitter

Express blue-green reflection and red transmission as two objectives, then model, optimize, and validate polarization performance at 45°

This tutorial extends the learning target from normal incidence to a 45° splitter: reflect blue-green light, transmit red light, and check s and p polarization separately. You will set oblique-incidence optics, define two spectral objectives, and judge the splitter from the optimization results.

Prerequisite: complete A Mirror Made of Transparent Materials: Design a 99% DBR first. You should know periodic stacks and R/T spectra.

If you already know Snell's law, s/p polarization, and oblique-incidence thin films, skip the basic derivation and go straight to two-band optimization.

Band	Target	Validation quantity
450–520 nm	Reflect blue-green	Maximize average reflectance
521–619 nm	Allow the spectral transition	No objective
620–700 nm	Transmit red	Maximize average transmittance

Oblique Incidence Changes Optical Thickness

The propagation angle inside layer $i$ follows Snell's law:

n_0\sin\theta_0=n_i\sin\theta_i .

Here, $n_0$ and $n_i$ are the refractive indices of the incident medium and layer $i$ ; $\theta_0$ is the external incidence angle; and $\theta_i$ is the refracted angle inside the layer. All angles are measured from the surface normal. The one-way phase thickness of that layer is

\delta_i=\frac{2\pi n_i d_i\cos\theta_i}{\lambda} .

Here, $\delta_i$ is the phase thickness, $d_i$ is physical thickness, $\lambda$ is the vacuum wavelength, and the other symbols retain their definitions above. As incidence angle increases, the change in $\cos\theta_i$ shifts a normal-incidence quarter-wave condition. The s and p polarizations also have different interface reflection coefficients, so both must be checked for a 45° design.

Start from a Six-Pair Periodic Stack

The initial structure is $(HL)^6$ : 56 nm TiO₂ followed by 100 nm MgF₂ and repeated six times, above a 1 mm incoherent glass substrate. Using only the two shared periodic thicknesses as variables exposes the trade-off between objectives without requiring twelve independent variables at once.

Shared variable	Initial value	Allowed range	Optimized value
TiO₂ high-index layer	56 nm	25–80 nm	25.000 nm
MgF₂ low-index layer	100 nm	70–140 nm	137.618 nm

After optimization, select Apply to Structure, return to Structure, and confirm that the two group thicknesses were updated while Repeat Count remains 6.

Structure page for the 45-degree dichroic beamsplitter showing six TiO2 MgF2 pairs and optimized shared thicknesses — Figure 1 | Six-pair dichroic stack on the `Structure` page

Click Edit Group. Confirm Repeat Count is 6, with 25.000 nm TiO₂ first and 137.618 nm MgF₂ second. The thickness fields use the displayed precision, while the preview uses the fuller internal value when calculating total thickness.

Edit Layer Group dialog for the dichroic beamsplitter showing six repeats of the optimized TiO2 MgF2 unit with thicknesses and refractive indices — Figure 2 | Optimized TiO₂/MgF₂ unit in `Edit Group`

In Optics, set 400–750 nm with a 1 nm step, 45° incidence, and unpolarized light. The software represents the p-polarized fraction as pRatio: 0 is pure s, 1 is pure p, and 0.5 is an equal s/p average. Begin with 0.5 for unpolarized performance and enable Reflectance and Transmittance.

Optics page for the dichroic beamsplitter showing 400 to 750 nm, 45-degree incidence, and unpolarized light — Figure 3 | `Optics` settings for the 45° dichroic beamsplitter

Translate the Specification into Two Objectives

Add two equally weighted objectives in Optimizer:

maximize average Reflectance from 450 to 520 nm with a 5 nm step, at 45° with unpolarized light;
maximize average Transmittance from 620 to 700 nm with a 5 nm step, at 45° with unpolarized light.

Both objectives must use the same angle and polarization definition. Otherwise, the optimizer would balance two different operating conditions, and the compromise would not correspond to one physical device use case.

Dichroic Optimizer page showing blue-green reflection and red transmission objectives with two shared thickness variables — Figure 4 | Two-band objectives and shared thickness variables in `Optimizer`

Choose Nelder–Mead, set 280 maximum evaluations, and use an initial simplex scale of 0.08. The real optimization converged after 102 objective evaluations. Its 5 nm sampled report gives 91.508% blue-green reflection and 90.678% red transmission. Apply the optimized thicknesses, then perform final validation with a finer 1 nm step.

Nelder-Mead algorithm settings and evaluation budget for the dichroic optimization — Figure 5 | Nelder–Mead settings for the dichroic optimization

Unpolarized Result

The 1 nm forward run gives 91.824% average reflectance over 450–520 nm, with a band minimum of 75.787% at 520 nm. Average transmittance over 620–700 nm is 90.915%, with a band minimum of 76.209% at 620 nm. Both minima occur at the edges nearest the transition region, consistent with a spectrum changing from reflection to transmission.

Unpolarized reflectance result for the 45-degree dichroic showing a high-reflectance blue-green band — Figure 6 | `Reflectance` at 45° for unpolarized light

Unpolarized transmittance result for the 45-degree dichroic showing a high-transmission red band — Figure 7 | `Transmittance` at 45° for unpolarized light

The initial stack is already a strong blue-green reflector, but its red transmission is inadequate. Optimization gives up some reflection margin to obtain much higher red transmission.

Comparison of average blue-green reflectance and red transmittance before and after dichroic optimization — Figure 8 | Two-band average performance before and after optimization

Design	Average $R$ , 450–520 nm	Average $T$ , 620–700 nm	$R(500\ \mathrm{nm})$	$T(650\ \mathrm{nm})$
Initial 56/100 nm	99.084%	59.676%	—	—
Optimized 25.000/137.618 nm	91.824%	90.915%	90.052%	89.245%

Validate s and p Separately

An unpolarized average can hide large differences between the polarizations. Keep the structure, wavelength range, and 45° incidence fixed; set pRatio to 0 and 1 in turn and run each case.

Polarization	Average $R$ , 450–520 nm	Minimum in-band $R$	Average $T$ , 620–700 nm
s	98.820%	96.568%	84.763%
p	84.828%	55.005%	97.067%
Unpolarized average	91.824%	75.787%	90.915%

The design meets this tutorial's unpolarized average objectives, but it is not a non-polarizing beamsplitter. The p-polarized blue-green response weakens strongly near the band edge, while s-polarized red transmission is below the unpolarized average. If an application requires both polarizations to meet the same threshold, add separate s and p objectives and allow more independent thickness variables.

The optimized TiO₂ thickness is exactly at its 25 nm lower bound, showing that the search still prefers a thinner layer. A next design iteration can widen that bound. If the deposition process imposes a 25 nm minimum, keep the bound and progressively unlock the twelve layers into independent or grouped variables. Reaching a bound is not a calculation failure; it identifies where structural freedom conflicts with a manufacturing constraint.

Common Errors and Recovery Order

Symptom	Likely cause	Action
The spectrum resembles the normal-incidence DBR	The objective or forward run still uses 0°	Check 45° in both `Optics` and both optimization objectives
The average is good but one polarization fails	Only `pRatio=0.5` was validated	Run pure s and pure p, and record the minimum within each band
The transition region is too broad	Only two shared thickness variables are available	Add independent variables while constraining minimum thickness and total layer count
An optimum lies on a bound	Bounds or manufacturing constraints dominate	Decide whether the bound can move; otherwise add other structural freedom

For engineering work, repeat every acceptance calculation with real n and k dispersion, the intended substrate, deposition errors, and the actual angular distribution of the beam.

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