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

Build a 550 nm transmission peak with two mirror-symmetric DBRs and a half-wave cavity, then validate peak value, FWHM, and Q factor

This tutorial places a transparent cavity between two DBRs to create a narrow transmission peak near 550 nm. You will set mirror-symmetric layer order, calculate the cavity thickness, and check the filter with peak transmission, FWHM, and Q.

Prerequisite: complete A Mirror Made of Transparent Materials: Design a 99% DBR first. You should know periodic stacks, Repeat Count, and reflectance spectra.

If you already know Fabry–Pérot cavities and FWHM / Q, skip the basic derivation and go straight to layer-order setup and result validation.

The Half-Wave Cavity 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 1 | 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 2 | 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 3 | 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 4 | `Optics` settings for the narrowband filter

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 5 | `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 6 | `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 7 | 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

As an independent numerical check, set cavity thickness as the only variable from 170 to 230 nm and maximize transmittance at 550 nm. The optimizer returns 199.28 nm after 26 evaluations. With materials, layer order, and target wavelength fixed, the analytical starting point is already optimal for this one-variable problem.

Common Errors and Recovery Order

Symptom	Likely cause	Action
No narrow peak near 550 nm	Both DBRs use the same direction	Check outward from the cavity and make the two layer orders mirror images
The peak is shifted from 550 nm	The cavity thickness or index is wrong	Recalculate $2n_cd_c=\lambda_0$ first
Only one or two samples define the peak	The wavelength step is too large	Reduce the step from 1 nm to 0.25 nm or finer
Peak position is correct but linewidth is not	The DBR pair count is unsuitable	Change the same number of pairs on both sides and preserve symmetry

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

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