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

550 nm Fabry–Pérot filter: cavity mode, FWHM, and Q factor

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.

Blue concentric interference rings captured by a CCD after cadmium-lamp light passes through a Fabry-Pérot etalon — Real interference rings formed by cadmium-lamp light passing through a Fabry–Pérot etalonSai2020 / Wikimedia Commons 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

Incident light undergoing multiple reflections between the two interfaces of a Fabry–Pérot interferometer and producing several transmitted beams — Figure 1 | Multiple reflections and transmitted beams inside a Fabry–Pérot cavityKrishnavedala / Wikimedia Commons 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.

Structure page for the Fabry-Perot narrowband filter showing two mirror-symmetric four-pair DBRs and a central MgF2 cavity — Figure 2 | 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.

Edit Layer Group dialog for the Fabry-Perot front mirror showing four repeats of the MgF2 TiO2 unit — Figure 3 | 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.

Edit Layer Group dialog for the Fabry-Perot back mirror showing four repeats of the TiO2 MgF2 unit — Figure 4 | TiO₂/MgF₂ unit for the back mirror in Edit Group

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 page for the Fabry-Perot filter showing 450 to 700 nm and a 0.25 nm wavelength step — Figure 5 | Optics settings for the narrowband filter

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 result for the Fabry-Perot filter showing a narrow dip inside the DBR stopband — Figure 6 | 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 result for the Fabry-Perot filter showing a narrow peak at 550 nm — Figure 7 | 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.

The 550 nm Fabry-Perot transmission peak with left and right half-maximum crossings — Figure 8 | 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.

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