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

MgF₂/TiO₂ quarter-wave DBR: stopband, reflectance, and pair count

Transparent materials can form a mirror with more than 99% reflectance. A distributed Bragg reflector (DBR) alternates transparent high- and low-index layers. Reflections from their interfaces return in phase near the design wavelength, creating a high-reflectance stopband. DBRs are widely used in laser mirrors, narrowband filters, and optical cavities.

This tutorial builds a five-pair MgF₂/TiO₂ DBR. You will calculate quarter-wave thicknesses, create the periodic stack with Layer Group, and verify the design through its reflectance spectrum and pair-count dependence.

Dielectric high-reflecting mirror mounted in an optical experiment — A dielectric mirror shows vivid color because its reflectance depends on wavelengthEric Magnan / Wikimedia Commons CC BY-SA 4.0

DBR Operating Principle

The diagram represents a dielectric mirror made from alternating layers with indices $n_1$ and $n_2$ . Each interface returns a weak reflection. With the right layer thicknesses, these reflected waves return in phase and reinforce one another.

Diagram of incident light and multiple reflected waves in alternating high- and low-index dielectric layers — Figure 1 | Constructive interference of interface reflections in a dielectric mirrorHankwang / Wikimedia Commons CC BY 3.0

Original diagram and license

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 2 | 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 3 | TiO₂/MgF₂ periodic unit in the layer-group dialog

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 4 | DBR wavelength, incident-light, and detector settings

Before running, make a prediction: the five-pair stack should form a continuous high-reflectance band near 550 nm. A fifth pair should still raise center reflectance, but by less than the first few added pairs.

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 5 | 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 6 | 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.

The animation shows the transmission stopband and internal electric field emerging as dielectric layers are added. The upper panel is transmittance and the lower panel is field magnitude; the low-transmission band becomes more pronounced as the stack grows.

Animation showing the transmission stopband and internal electric field emerging as alternating dielectric layers are added — Figure 7 | Formation of the DBR stopband as the number of layers increasesJacopo Bertolotti / Wikimedia Commons CC0 1.0

Original animation and license

DBR center reflectance as the number of TiO2 MgF2 pairs increases — Figure 8 | Growth of 550 nm reflectance with DBR pair count

Pair count $N$	Reflectance at 550 nm
1	42.854%
2	76.699%
3	91.938%
4	97.369%
5	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.

Variation Exercise

Change only Repeat Count from 5 to 6. Estimate the remaining increase at 550 nm before running, then record the new center reflectance and the $R\ge99\%$ band. Decide whether two more layers deliver a comparable gain.

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

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