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

Start from Fresnel reflection and a quarter-wave estimate, then model, optimize, and validate a four-layer visible antireflection coating

This tutorial starts from the reflectance baseline of bare glass, estimates a single 550 nm antireflection layer, and then expands the target to a 450–650 nm broadband coating. You will build a four-layer stack and check it with reflectance and energy conservation.

Prerequisite: complete the Quick Start first. You should be able to build a structure, set the optics, and run a calculation.

If you already know Fresnel reflection and quarter-wave estimates, skip the basic derivation and go straight to the broadband stack and optimization.

Design stage	Stack size	Validation quantity
Bare glass	0 layers	Establish the surface-reflection baseline
Single-layer AR	1 layer	Substantially reduce reflectance at 550 nm
Broadband AR	4 layers	Keep average reflectance below 0.1% over 450–650 nm

This tutorial uses transparent constant-index materials: air 1.00, glass 1.52, MgF₂ 1.38, and TiO₂ 2.45. The model is suitable for learning the design logic, but it does not represent the final performance of a specific deposition process.

The Bare-Glass Baseline

For normal incidence from air onto glass, the single-interface reflectance 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 a lossless model, all remaining energy is transmitted, so $T=95.742\%$ and $R+T=1$ .

This baseline matters: every later improvement must be compared with bare glass under the same incident medium, substrate, and wavelength range.

A Single Quarter-Wave AR Layer

At the design wavelength, one round trip through the film introduces a half-wave phase difference, allowing the two main reflected waves to cancel. The physical 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. With $\lambda_0=550\ \mathrm{nm}$ and $n_1=1.38$ , the MgF₂ thickness is $d=99.64\ \mathrm{nm}$ .

Perfect single-layer cancellation also requires $n_1=\sqrt{n_0n_g}=1.233$ . MgF₂ has a higher index, so the simulation gives 1.260% reflectance at 550 nm instead of exactly zero. The quarter-wave thickness matches phase; the material index determines whether the amplitudes of the two reflected waves can also match.

Extend the Target to 450–650 nm

Broadband AR must suppress a range, not just one wavelength. Start from a simple four-layer structure and let the optimizer adjust its thicknesses within finite bounds. The stack is listed from the incident side toward the glass substrate.

Layer	Material	Initial thickness	Allowed range	Optimized thickness
1	MgF₂	99.64 nm	85–115 nm	97.373 nm
2	TiO₂	112.24 nm	95–135 nm	116.149 nm
3	MgF₂	24.91 nm	15–40 nm	37.769 nm
4	TiO₂	14.03 nm	5–25 nm	11.108 nm
Substrate	Glass	1 mm, incoherent	Fixed	Fixed

Enter the optimized thicknesses in the Structure page. Set the glass substrate to incoherent and use the same index of 1.52 for the bottom medium so that the millimeter-scale substrate is not treated as a coherent thin film.

Structure page for the four-layer broadband AR coating, showing optimized MgF2 and TiO2 thicknesses and the incoherent glass substrate — Figure 1 | Four-layer broadband AR stack on the `Structure` page

In Optics, set 400–700 nm with a 1 nm step, 0° incidence, unpolarized light, and enable Reflectance and Transmittance. The wider calculation range shows behavior inside and outside the design band; the optimization objective itself uses only 450–650 nm.

Optics page for the broadband AR coating showing 400 to 700 nm, normal incidence, and reflectance and transmittance detectors — Figure 2 | Wavelength, incident-light, and detector settings in `Optics`

Set the Optimization Objective and Variables

Add one objective in Optimizer: minimize average Reflectance from 450 to 650 nm with a 5 nm step, at 0° with unpolarized light. Make all four thicknesses variables using the bounds in the table. Bounds limit the search and prevent the optimizer from hiding a poor starting structure behind extreme thicknesses.

Broadband AR Optimizer page showing the average-reflectance objective and four thickness variables — Figure 3 | Broadband reflectance objective and four thickness variables in `Optimizer`

Choose Nelder–Mead with 400 maximum evaluations and an initial simplex scale of 0.08. This four-variable continuous problem is a good fit for a derivative-free local search.

Broadband AR optimizer algorithm settings showing Nelder-Mead and the evaluation budget — Figure 4 | Nelder–Mead settings in `Optimizer`

The real optimization reached the thicknesses in the table after 174 objective evaluations and 101 iterations. After selecting 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. The optimization report samples every 5 nm, whereas final validation uses 1 nm, so their average values differ slightly.

Validate the Result

The optimized design has 0.0865% average reflectance over 450–650 nm, 0.00110% reflectance at 550 nm, and a band maximum of 0.5005% at 450 nm. The 400–700 nm average is 0.6508%; reflectance reaches 6.924% at 400 nm, demonstrating that the design does not promise performance beyond its optimization band.

Reflectance result for the four-layer broadband AR coating showing a low-reflectance region across the middle of the visible spectrum — Figure 5 | `Reflectance` result for the optimized four-layer stack

Transmittance result for the four-layer broadband AR coating showing near-unity transmission in the design band — Figure 6 | `Transmittance` result for the optimized four-layer stack

The four design stages can now be compared with the same metrics.

Comparison of average reflectance for bare glass, single-layer AR, the initial four-layer stack, and the optimized four-layer design — Figure 7 | Performance progression from bare glass to the optimized four-layer design

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%

All four lossless structures satisfy $R+T=1$ , with a maximum numerical conservation error of about $2\times10^{-15}$ . Here, $T$ is power transmittance. This check can expose missing absorption, incorrect boundaries, or result-reading errors, but it does not prove that the material model accurately represents a real sample.

Common Errors and Recovery Order

Symptom	Likely cause	Action
No reflectance minimum near 550 nm	Incorrect quarter-wave thickness or layer order	Recalculate $d=\lambda_0/(4n)$ , then check the stack from the air side
Dense fringes across the spectrum	The millimeter glass substrate is coherent	Mark the glass substrate as incoherent
Excellent performance at only one point	A single wavelength was used instead of a band average	Use average reflectance over 450–650 nm as the objective
An optimized value touches a bound	The search range limits the improving direction	Check manufacturability, then widen that variable's bound if appropriate

For an engineering design, replace the constant indices with process-specific n and k spectra, then repeat angle, polarization, thickness-error, and dispersion checks.

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