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

45° dichroic beamsplitter: dual-band optimization and polarization validation

Fluorescence imaging, projection, and multispectral measurements often need to turn blue-green light arriving at 45° while allowing red light to continue forward. A dichroic beamsplitter is a stack of transparent thin films that uses interference to reflect one wavelength band and transmit another, rather than absorbing and discarding either beam.

This tutorial designs a dichroic beamsplitter that reflects blue-green light from 450–520 nm and transmits red light from 620–700 nm. You will build a periodic stack, set 45° incidence and dual-band optimization objectives, and evaluate the unpolarized, s-polarized, and p-polarized results separately.

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

At 45° incidence, the propagation direction changes inside every layer. Reflections from adjacent interfaces travel different distances before recombining, so a thickness that works at normal incidence cannot simply be reused for an oblique-incidence design.

Ray path through a 45-degree dichroic filter that reflects blue light and transmits red light — Figure 1 | A 45° dichroic filter sends blue and red light in different directionsEric Magnan / Wikimedia Commons CC BY-SA 3.0

The left surface carries the dichroic coating and the right surface carries an antireflection coating. Blue light is reflected while red light passes through the substrate.

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 written as $(HL)^6$ : the high-index layer H is 56 nm TiO₂, the low-index layer L is 100 nm MgF₂, and the complete period is 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
TiO₂ high-index layer	56 nm	25–80 nm
MgF₂ low-index layer	100 nm	70–140 nm

Build the group with the initial 56/100 nm thicknesses and set Repeat Count to 6. Open Edit Group and verify the order, thicknesses, and repeat count.

Initial Structure page for the 45-degree dichroic beamsplitter showing six TiO2 MgF2 pairs and a glass substrate — Figure 2 | Initial six-pair dichroic structure

Edit Layer Group dialog for the dichroic beamsplitter showing six repeats of the initial TiO2 MgF2 unit with thicknesses and refractive indices — Figure 3 | Initial TiO₂/MgF₂ periodic unit

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 4 | Optics settings for the 45° dichroic beamsplitter

Predict before running: the initial quarter-wave stack should strongly reflect blue-green light, but its red transmission may be inadequate at 45°. The two-band optimization must redistribute performance margin between the two bands.

Run the Initial Stack First

The initial 56/100 nm stack gives 99.084% average reflectance over 450–520 nm but only 59.676% average transmittance over 620–700 nm. Blue-green reflection is already strong; red transmission is the main design gap.

Real Reflectance result for the initial six-pair dichroic, showing a high-reflectance blue-green band and residual reflection in the red region — Figure 5 | The initial structure strongly reflects blue-green light but still reflects too much red light

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 6 | 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 7 | Nelder–Mead settings for the dichroic optimization

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 8 | Six-pair dichroic structure after applying the optimum

Open Edit Group and confirm Repeat Count is 6, with 25.000 nm TiO₂ first and 137.618 nm MgF₂ second.

Edit Layer Group dialog for the dichroic beamsplitter showing six repeats of the optimized TiO2 MgF2 unit with thicknesses and refractive indices — Figure 9 | Optimized TiO₂/MgF₂ periodic unit

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 10 | Reflectance at 45° for unpolarized light

Unpolarized transmittance result for the 45-degree dichroic showing a high-transmission red band — Figure 11 | Transmittance at 45° for unpolarized light

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 12 | 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.

First set pRatio to 0, confirm that the interface shows 100% s polarization, and run the reflectance calculation.

Optics page for the dichroic showing 45-degree incidence and pRatio equal to 0 for pure s polarization — Figure 13 | Optics settings for pure s polarization

Real Reflectance result for the optimized dichroic at 45 degrees with pure s polarization — Figure 14 | Pure s polarization maintains strong blue-green reflection

Then set pRatio to 1, confirm that the interface shows 100% p polarization, and rerun the same structure.

Optics page for the dichroic showing 45-degree incidence and pRatio equal to 1 for pure p polarization — Figure 15 | Optics settings for pure p polarization

Real Reflectance result for the optimized dichroic at 45 degrees with pure p polarization — Figure 16 | Pure p polarization weakens markedly at the edge of the blue-green band

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 reaches its 25 nm lower bound. This tutorial only records that result; see Are More Layers Better? for the trade-off among variable bounds, pair count, and total thickness.

Variation Exercise

Keep the optimized structure and change only the incidence angle from 45° to 55°. Predict the change in the transition band and s/p separation, then run unpolarized, pure-s, and pure-p cases. Record which acceptance metric fails first.

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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