Your First Thin-Film Design: From Bare Glass to Broadband AR

Start with reflections on a lens, then learn quarter-wave AR, broadband design, and optimization

Why does the left half of this lens show bright reflections while the right half almost disappears? The antireflection coating on the right does not block the light. It makes two reflected waves weaken each other, allowing more light to enter the lens.

Comparison of an ordinary lens and an antireflection-coated lens over the same image, with strong lamp reflections on the ordinary side and weak reflections on the coated side — Figure 1 | Ordinary lens on the left and antireflection-coated lens on the rightMaximilian Schönherr / Wikimedia Commons CC0 1.0

This tutorial is for readers making their first thin-film design. You will calculate the reflectance of bare glass, reduce the 550 nm reflection with one MgF₂ layer, and then extend the low-reflectance region to 450–650 nm by optimizing the thicknesses of a four-layer coating.

Calculate the Reflection from Bare Glass

Glass reflects even without a coating because air and glass have different refractive indices. At normal incidence from air, the reflectance of the glass surface is

R=\left(\frac{n_g-n_0}{n_g+n_0}\right)^2 .

Here, $R$ is power reflectance, $n_0$ is the refractive index of air, and $n_g$ is the refractive index of glass. Substituting $n_0=1.00$ and $n_g=1.52$ gives

R=4.258\% .

In Structure, use air as the top medium, place a 1 mm glass substrate below it, and use glass again as the bottom medium.

Structure page for the bare-glass baseline, showing air, a 1 mm incoherent glass substrate, and a glass bottom medium — Figure 2 | Structure settings for the bare-glass baseline

Mark the 1 mm glass substrate as incoherent. The task is to calculate the surface reflection of a lens, not to treat millimeter-scale glass as a coherent thin film with stable interference fringes.

In Optics, set 400–700 nm with a 1 nm step, 0° incidence, unpolarized light, and enable Reflectance and Transmittance. The same settings will be used for the single-layer coating.

Optics page shared by the bare-glass and single-layer models, showing 400 to 700 nm, normal incidence, and reflectance and transmittance detectors — Figure 3 | Optics settings shared by the bare-glass and single-layer models

Select Run. Bare-glass reflectance remains at 4.258% across the band, while transmittance is 95.742%.

Reflectance result for bare glass, showing a horizontal curve near 4.258 percent from 400 to 700 nm — Figure 4 | Reflectance of bare glass

Make Two Reflected Waves Weaken Each Other

Adding a transparent film creates two main reflected waves. One returns from the air–film boundary. The other enters the film, reflects from the film–glass boundary, and then returns to air.

Ray animation through air, an antireflection coating, and glass, showing the reflected waves from the upper and lower film boundaries — Figure 5 | Reflected waves from the two boundaries of an antireflection coatingChanli44 / Wikimedia Commons Public Domain

If the round-trip optical path in the film equals half the design wavelength, the second reflected wave returns half a cycle behind the first, so the two reflections weaken each other. The corresponding quarter-wave thickness is

d=\frac{\lambda_0}{4n_1} .

Here, $d$ is the physical film thickness, $\lambda_0$ is the vacuum design wavelength, and $n_1$ is the film refractive index. For MgF₂ with $\lambda_0=550\ \mathrm{nm}$ and $n_1=1.38$ ,

d=99.64\ \mathrm{nm} .

Add 99.64 nm of MgF₂ above the glass substrate and keep the other settings unchanged.

Structure page for the single-layer coating, showing 99.64 nm of MgF2 and a 1 mm incoherent glass substrate — Figure 6 | Single-layer AR structure with 99.64 nm of MgF₂

Select Run again. The spectrum forms a minimum near 550 nm, where reflectance falls from 4.258% for bare glass to 1.260%.

Reflectance result for the single MgF2 layer, showing a reflectance minimum near 550 nm — Figure 7 | Reflectance of the single-layer MgF₂ coating

Design	Reflectance at 550 nm
Bare glass	4.258%
99.64 nm MgF₂ on glass	1.260%

Why does it not reach zero? The quarter-wave thickness sets the phase relationship, but complete cancellation also requires equal reflected-wave amplitudes. The ideal index of a single AR layer is

n_1=\sqrt{n_0n_g}=1.233 .

MgF₂ has an index of 1.38 rather than 1.233. It therefore reduces the 550 nm reflection substantially but cannot produce exact zero reflectance.

Extend Low Reflectance across a Band

A single layer controls the phase around 550 nm. Camera lenses, display cover glass, and photovoltaic glass usually need low reflection over a visible band, so the next design uses four MgF₂/TiO₂ layers and expands the target to 450–650 nm.

For transparent hand-calculation examples, this tutorial uses constant indices: air 1.00, glass 1.52, MgF₂ 1.38, and TiO₂ 2.45. A production coating requires wavelength-dependent n and k data for the deposition materials.

From the air side toward the glass, the initial stack is:

Film	Material	Initial thickness
1	MgF₂	99.64 nm
2	TiO₂	112.24 nm
3	MgF₂	24.91 nm
4	TiO₂	14.03 nm
Substrate	Glass	1 mm, incoherent

In Optics, set 400–700 nm with a 1 nm step, 0° incidence, unpolarized light, and enable Reflectance and Transmittance. The calculation range is wider than the target band so that you can see behavior both inside 450–650 nm and at its edges.

Optics page for the broadband AR coating showing 400 to 700 nm, normal incidence, and reflectance and transmittance detectors — Figure 8 | Optics settings for the broadband AR coating

Before selecting Run, make a prediction: the initial four-layer stack should have lower band-average reflectance than the single MgF₂ layer, but its spectrum should still have ripple. The calculated four-layer baseline is 0.341% at 550 nm and 0.648% on average over 450–650 nm.

Design	Reflectance at 550 nm	Average reflectance, 450–650 nm
Single MgF₂ layer	1.260%	1.348%
Initial four-layer stack	0.341%	0.648%

The four layers improve the full band, but the design has not yet reached the target of less than 0.1% average reflectance.

Let the Optimizer Try the Thicknesses

Four films give four adjustable thicknesses, so manual trial and error quickly becomes inefficient. Optimizer changes those thicknesses repeatedly within limits that you provide, recalculates the spectrum, and scores each combination by its 450–650 nm average reflectance. It does not choose the materials or add layers for you; you still define the structure, target, and adjustable parameters.

In Optimizer, add one objective: minimize average Reflectance from 450 to 650 nm with a 5 nm step, at 0° with unpolarized light. Then add all four film thicknesses as variables:

Film	Initial thickness	Range the optimizer may try
MgF₂ 1	99.64 nm	85–115 nm
TiO₂ 1	112.24 nm	95–135 nm
MgF₂ 2	24.91 nm	15–40 nm
TiO₂ 2	14.03 nm	5–25 nm

Broadband AR Optimizer page showing the average-reflectance objective and four film-thickness variables — Figure 9 | Broadband reflectance objective and four thickness variables

Select Nelder–Mead, set the maximum number of evaluations to 400, and set the initial simplex ratio to 0.08. Nelder–Mead compares objective values without calculating derivatives, which makes it suitable for this introductory problem with four continuous variables.

Broadband AR optimizer algorithm settings showing Nelder-Mead and the evaluation budget — Figure 10 | Nelder–Mead algorithm settings

Select Optimize. This run reached the following result after 174 objective evaluations and 101 iterations. The optimized values appear only now, after the optimization has been set up and run:

Film	Initial thickness	Optimized thickness
MgF₂ 1	99.64 nm	97.373 nm
TiO₂ 1	112.24 nm	116.149 nm
MgF₂ 2	24.91 nm	37.769 nm
TiO₂ 2	14.03 nm	11.108 nm

Select Apply to Structure, return to Structure, and confirm that all four values were written back. Then run a forward calculation with a finer 1 nm step. Optimization uses 5 nm sampling to reduce the calculation cost, while final validation uses 1 nm sampling, so the two pages may report slightly different averages.

Structure page for the four-layer broadband AR coating, showing the optimized MgF2 and TiO2 thicknesses and the incoherent glass substrate — Figure 11 | Four-layer broadband AR stack after applying the optimization result

Optimized Result Against the Target

The optimized average reflectance over 450–650 nm is 0.0865%, which passes the 0.1% target. Reflectance is 0.00110% at 550 nm. The maximum within the band is 0.5005% at 450 nm.

Reflectance result for the four-layer broadband AR coating showing a low-reflectance region across the middle of the visible spectrum — Figure 12 | Reflectance of the optimized four-layer stack

Transmittance result for the four-layer broadband AR coating showing nearly one hundred percent transmission within the target band — Figure 13 | Transmittance of the optimized four-layer stack

Reflectance rises to 6.924% at 400 nm. This is not a failed calculation: the objective constrains only 450–650 nm, and the optimizer does not automatically improve wavelengths outside that range.

Comparison of average reflectance for bare glass, a single AR layer, the initial four-layer stack, and the optimized four-layer design — Figure 14 | Results from bare glass to the four-layer broadband AR coating

Design	Reflectance at 550 nm	Average reflectance, 450–650 nm
Bare glass	4.258%	4.258%
Single MgF₂ layer	1.260%	1.348%
Initial four-layer stack	0.341%	0.648%
Optimized four-layer stack	0.00110%	0.0865%

Every material in this teaching model is lossless, so the result should satisfy $R+T=1$ . Energy removed from reflection becomes transmitted light; it does not disappear.

Change One Thickness Yourself

Restore the four-layer stack to its initial thicknesses, then change only the first MgF₂ layer to 90, 99.64, and 110 nm. Before each run, predict whether the reflectance minimum will move toward shorter or longer wavelengths. Then compare the 550 nm reflectance and the 450–650 nm average. This experiment shows directly that the best thickness for one wavelength is not necessarily the best thickness for a full band.

For an engineering design, replace the constant indices with wavelength-dependent n and k data for the deposition process, then repeat the checks for angle, polarization, and thickness error.

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