A Mirror Made of Transparent Materials: Design a 99% DBR

Use an MgF₂/TiO₂ quarter-wave periodic stack to study constructive interference, stopband formation, and the effect of pair count

This tutorial turns the destructive interference of an AR coating into constructive interference, using an MgF₂/TiO₂ periodic stack to build a 99% DBR mirror. You will use Layer Group and Repeat Count to set the period, then check the center reflectance and stopband.

Prerequisite: complete Your First Thin-Film Design: From Bare Glass to Broadband AR first. You should know quarter-wave thickness, basic structure setup, and R/T spectra.

If you already know quarter-wave periodic stacks and Layer Group, skip the basic derivation and go straight to structure setup and pair-count comparison.

The Quarter-Wave Period

Keep the design wavelength at $\lambda_0=550\ \mathrm{nm}$ . The physical quarter-wave thickness of each layer is

d_i=\frac{\lambda_0}{4n_i} .

Here, $d_i$ is the physical thickness of material $i$ , $n_i$ is its refractive index, and $\lambda_0$ is the vacuum design wavelength. With $n_H=2.45$ and $n_L=1.38$ , the high-index TiO₂ layer is $d_H=56.12\ \mathrm{nm}$ and the low-index MgF₂ layer is $d_L=99.64\ \mathrm{nm}$ .

Use the order $(HL)^N$ from the air side, meaning TiO₂ followed by MgF₂ and repeated $N$ times. At the design wavelength, the main reflected components from the interfaces return with nearly the same phase. The ideal quarter-wave center reflectance is

R_0=\left[\frac{n_s(n_H/n_L)^{2N}-n_0}{n_s(n_H/n_L)^{2N}+n_0}\right]^2 .

Here, $R_0$ is the power reflectance at $\lambda_0$ ; $n_0$ and $n_s$ are the indices of air and the glass substrate; $n_H$ and $n_L$ are the high- and low-index material indices; and $N$ is the number of layer pairs. Because $n_H/n_L$ is greater than 1, the exponent $2N$ drives reflectance rapidly toward unity.

Build a Five-Pair DBR

Add a Layer Group in Structure. Put 56.12 nm TiO₂ first and 99.64 nm MgF₂ second, then set Repeat Count to 5. Place a 1 mm incoherent glass substrate under the group and set the bottom medium index to 1.52.

Structure page for a five-pair TiO2 MgF2 distributed Bragg reflector — Figure 1 | Five-pair TiO₂/MgF₂ DBR on the `Structure` page

Click Edit Group to open the periodic unit. Confirm that the first layer is 56.12 nm TiO₂, the second is 99.64 nm MgF₂, and Repeat Count is 5.

Edit Layer Group dialog for the DBR showing five repeats of the TiO2 MgF2 unit with thicknesses and refractive indices — Figure 2 | TiO₂/MgF₂ periodic unit in `Edit Group`

The Layer Group stores the period once. To compare $N=1$ through $N=5$ , edit only Repeat Count rather than manually copying and deleting ten individual layers.

In Optics, set 400–900 nm with a 1 nm step, 0° incidence, unpolarized light, and enable Reflectance and Transmittance. The broad wavelength range reveals both the central stopband and the transmission regions on its sides.

Optics page for the DBR showing 400 to 900 nm, normal incidence, and reflectance and transmittance detectors — Figure 3 | DBR wavelength, incident-light, and detector settings in `Optics`

Run and Identify the Stopband

The five-pair structure reaches 99.158% reflectance at 550 nm. Using $R\ge99\%$ as the high-reflectance criterion for this tutorial gives a continuous stopband from 523 to 580 nm. The design wavelength lies inside this band, but the shape need not be perfectly symmetric about 550 nm because the incident medium and glass substrate have different indices.

Reflectance result for the five-pair DBR showing a high-reflectance stopband near 550 nm — Figure 4 | `Reflectance` result for the five-pair DBR

Transmittance falls over the same band. All materials are lossless, so every wavelength should satisfy $R+T=1$ ; the maximum numerical error in this run is about $4.7\times10^{-15}$ .

Transmittance result for the five-pair DBR showing low transmission over the high-reflectance band — Figure 5 | `Transmittance` result for the five-pair DBR

Change the Pair Count

Set Repeat Count to 1, 2, 3, 4, and 5 in turn, running the calculation and recording reflectance at 550 nm each time. Keep all other settings fixed so that changes in the curve can be attributed to pair count alone.

DBR center reflectance as the number of TiO2 MgF2 pairs increases — Figure 6 | Growth of 550 nm reflectance with DBR pair count

Pair count $N$	Total film layers	Reflectance at 550 nm
1	2	42.854%
2	4	76.699%
3	6	91.938%
4	8	97.369%
5	10	99.158%

The results show two design rules. Additional pairs raise center reflectance substantially, but with diminishing returns. High reflectance at one wavelength also does not guarantee a sufficiently wide stopband; a device specification must state both a reflectance threshold and a wavelength range.

Common Errors and Recovery Order

Symptom	Likely cause	Action
The stopband is not near 550 nm	H or L thickness was not calculated with the stated index	Recalculate $d_H$ and $d_L$ separately; do not swap them
The curve does not change with more pairs	Ordinary layers were copied while an unused group was edited	Expand the effective stack and confirm the active `Repeat Count`
Dense fringes appear around the reflection band	The glass substrate is coherent	Mark the millimeter-scale substrate as incoherent
$R+T$ differs substantially from 1	A material absorbs or the boundaries are inconsistent	Check every layer's $k$ , then verify top and bottom media

A practical DBR also requires dispersive material data, angle and polarization checks, and thickness-error analysis.

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