Select One Color Between Two Mirrors: A Fabry–Pérot Narrowband Filter

Two high-reflecting mirrors placed together seem as though they should block even more light. Yet when the round-trip phase between them is just right, the transmitted waves interfere constructively and open a very narrow transmission peak inside the high-reflectance band. A Fabry–Pérot narrowband filter uses this effect to select a specific color.

Real interference rings formed by cadmium-lamp light passing through a Fabry–Pérot etalon (Source: Sai2020 / Wikimedia Commons · License: Public Domain)

This tutorial builds that transmission peak near 550 nm. You will calculate the cavity thickness, configure two mirror-symmetric DBRs, and evaluate the filter using peak transmittance, full width at half maximum (FWHM), and Q factor.

The Half-Wave Cavity Condition

Multiple reflections and transmitted beams inside a Fabry–Pérot cavity (Source: Krishnavedala / Wikimedia Commons · License: CC0 1.0)

The red rays show the transmitted beams produced after repeated round trips between the two reflecting interfaces; $\theta$ is the propagation angle inside the cavity. The derivation below begins with the normal-incidence resonance condition.

At normal incidence, the basic cavity resonance condition is

$$ 2n_cd_c=m\lambda_0 . $$

Here, $n_c$ is the cavity refractive index, $d_c$ is its physical thickness, $m$ is a positive integer resonance order, and $\lambda_0$ is the vacuum resonance wavelength. For an MgF₂ cavity with $n_c=1.38$, $m=1$, and $\lambda_0=550\ \mathrm{nm}$,

$$ d_c=\frac{550\ \mathrm{nm}}{2\times1.38}=199.28\ \mathrm{nm}. $$

This is a half-wave optical thickness. The DBRs add reflection phase, but for this symmetric quarter-wave construction, 199.28 nm places the transmission peak exactly at 550 nm.

Mirror-Symmetric Layer Order

Build the following structure from air to the glass substrate:

Region	Layer order	Repeats
Front mirror	MgF₂ 99.64 nm / TiO₂ 56.12 nm	4
Cavity	MgF₂ 199.28 nm	1
Back mirror	TiO₂ 56.12 nm / MgF₂ 99.64 nm	4
Substrate	Glass, 1 mm and incoherent	1

The front mirror starts with low index and ends with high index. The back mirror starts with high index and ends with low index. Both cavity-facing layers are TiO₂, making the complete stack mirror-symmetric about the cavity center.

Symmetric DBR–cavity–DBR stack on the Structure page

Click Edit Group for the front mirror. Confirm MgF₂ 99.64 nm → TiO₂ 56.12 nm with Repeat Count set to 4.

MgF₂/TiO₂ unit for the front mirror in Edit Group

Open Edit Group for the back mirror. Confirm the reversed order, TiO₂ 56.12 nm → MgF₂ 99.64 nm, with Repeat Count also set to 4.

TiO₂/MgF₂ unit for the back mirror in Edit Group

Caution

Do not copy the same “MgF₂ / TiO₂” group unchanged to both sides of the cavity. The two mirror directions must be reversed; otherwise, their reflection phases do not match at the target wavelength and the 550 nm transmission peak can be strongly suppressed.

Use a Fine Wavelength Step

In Optics, set 450–700 nm with a 0.25 nm step, 0° incidence, unpolarized light, and enable Reflectance and Transmittance. The 1 nm step used for the DBR can only outline this approximately 3.6 nm-wide peak. A 0.25 nm step provides better estimates of the peak and full width at half maximum.

Optics settings for the narrowband filter

Tip

Predict before running: the broad DBR stopband should remain, but a narrow transmission window should open near 550 nm and appear as a complementary minimum in reflectance.

Run and Identify the Cavity Mode

The broad DBR stopband remains in the reflectance result, but a narrow reflectance minimum appears near 550 nm. It is not a mirror failure: cavity resonance transfers the energy to the transmitted side.

Reflectance resonance minimum inside the DBR stopband

Transmittance reaches 95.742% at 550.00 nm. With zero absorption, the reflectance minimum and transmission maximum are complementary and still satisfy $R+T=1$.

Transmittance resonance peak near 550 nm

Validate with FWHM and Q Factor

The full width at half maximum (FWHM) is the wavelength separation between the two points where transmittance falls to half its peak value. Linear interpolation between adjacent samples gives a left crossing at 548.194 nm and a right crossing at 551.817 nm, so

$$ \Delta\lambda=\lambda_R-\lambda_L=3.623\ \mathrm{nm}. $$

Here, $\Delta\lambda$ is the FWHM, while $\lambda_L$ and $\lambda_R$ are the left and right half-maximum wavelengths. The quality factor is

$$ Q=\frac{\lambda_{\mathrm{peak}}}{\Delta\lambda}=151.8, $$

where $Q$ is dimensionless, $\lambda_{\mathrm{peak}}=550.00\ \mathrm{nm}$ is the peak wavelength, and $\Delta\lambda$ is the FWHM above. A higher $Q$ means a narrower peak relative to its center wavelength.

Transmission peak, half-maximum crossings, and FWHM

Metric	Result	Tutorial criterion
Peak wavelength	550.00 nm	Matches the design wavelength
Peak transmittance	95.742%	High transmission of the target color
FWHM	3.623 nm	Narrowband selection is present
Q factor	151.8	Consistent with the measured linewidth

Variation Exercise

Change only the MgF₂ cavity thickness from 199.28 nm to 205 nm. Predict whether the transmission peak moves to a shorter or longer wavelength, then run and record its position. The next tutorial, Fabry–Pérot Sensitivity Analysis, converts this shift into a thickness tolerance.

A practical narrowband filter also requires checks of angular peak shift, material absorption, and mirror asymmetry.

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← Back to Tutorial Catalog · Next: How Do Thickness Errors Shift a Filter?

Going further

These tutorials cover the operating steps only. The physics behind them, the full parameter reference for each feature, how to read the results, and the algorithm validation all live on the documentation site:

• Transfer-Matrix Method — the physics and formulas behind these tutorials
• Feature Guide — complete reference for every screen and parameter
• Results — how to read each kind of output
• Open Dreapex TMM — build and simulate in the browser