How Do Thickness Errors Shift a Filter? Fabry–Pérot Sensitivity Analysis

This tutorial continues Select One Color Between Two Mirrors: A Fabry–Pérot Narrowband Filter and reuses its symmetric 550 nm DBR–cavity–DBR structure.

A cavity error of only a few nanometers can shift a narrowband filter away from its target wavelength. This tutorial sweeps the Fabry–Pérot cavity thickness, quantifies changes in peak wavelength, full width at half maximum, and Q factor, and converts a spectral tolerance into a thickness tolerance.

Only the central MgF₂ cavity thickness changes in this tutorial. Both DBRs, all refractive indices, the incidence angle, and the wavelength sampling remain fixed, so any peak shift can be attributed directly to the cavity thickness.

Changing the cavity thickness changes the phase accumulated on every round trip and therefore shifts the transmission peak (Source: Krishnavedala / Wikimedia Commons · License: CC0 1.0)

The Cavity and Mirrors Jointly Determine the Peak

At normal incidence, the round-trip phase condition inside the cavity is

$$ \Phi(\lambda,d_c) =\frac{4\pi n_cd_c}{\lambda} +\phi_f(\lambda)+\phi_b(\lambda) =2\pi m. $$

Here, $\Phi$ is the total round-trip phase, $\lambda$ is the vacuum wavelength, $n_c$ and $d_c$ are the cavity refractive index and physical thickness, $\phi_f$ and $\phi_b$ are the reflection phases of the front and back DBRs, and $m$ is a positive integer resonance order. Phase is expressed in radians, and $\lambda$ and $d_c$ use the same length unit.

If the reflection phases of the two DBRs are neglected, the first-order resonance simplifies to

$$ \lambda_{\mathrm{simple}}=2n_cd_c. $$

The half-wave relation is useful for estimating the initial cavity thickness and correctly predicts that a thicker cavity shifts the peak toward longer wavelengths. Because DBR reflection phase varies with wavelength, however, the actual sensitivity must be calculated from the complete stack.

Fix the Mirrors and Optical Conditions

Reuse the 550 nm Fabry–Pérot filter:

Region	Layer sequence	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	1 mm glass, incoherent	1

Symmetric Fabry–Pérot structure used for the sensitivity analysis

The periodic sequences of the front and back mirrors are reversed. Open Edit Group for each mirror and confirm that both Repeat Count values are 4.

MgF₂/TiO₂ unit cell of the front mirror

TiO₂/MgF₂ unit cell of the back mirror

Caution

Do not change the DBR layer sequence, thicknesses, or repeat counts during the sweep. Changing the mirrors also changes their reflection phase, so the result would no longer represent cavity-thickness sensitivity alone.

In Optics, set 450–700 nm with a 0.25 nm step, 0° incidence, and unpolarized light. Enable Reflectance, Transmittance, and Absorptance.

A 0.25 nm step resolves the narrow transmission peak

At the nominal cavity thickness of 199.28 nm, the transmission peak is at 550.00 nm, peak transmittance is 95.742%, and the full width at half maximum is 3.623 nm. A 1 nm wavelength step would leave only three or four samples across the peak and cannot locate the half-maximum crossings reliably.

The full width at half maximum (FWHM) and quality factor are defined as

$$ \Delta\lambda=\lambda_R-\lambda_L, \qquad Q=\frac{\lambda_{\mathrm{peak}}}{\Delta\lambda}. $$

Here, $\lambda_L$ and $\lambda_R$ are the wavelengths of the left and right crossings where transmittance falls to half its peak value, $\Delta\lambda$ is the FWHM, $\lambda_{\mathrm{peak}}$ is the peak wavelength, and $Q$ is the dimensionless quality factor.

Sweep the Cavity Thickness

In Sweep, select MgF2 Cavity → Thickness and sweep from 190 nm to 210 nm in 5 nm steps.

Tip

Predict the result before running: increasing the cavity thickness should shift the transmission peak toward longer wavelengths. If the DBR reflection phase is significant, the simulated slope will differ from the simple estimate $2n_c$.

MgF₂ cavity-thickness sweep

After running the sweep, compare the five curves in Transmittance and track the same cavity mode within 500–600 nm.

Transmission-peak drift caused by cavity-thickness changes

Cavity thickness	Half-wave prediction	Simulated peak	Peak $T$	FWHM	Q
190 nm	524.39 nm	538.50 nm	95.514%	3.602 nm	149.5
195 nm	538.19 nm	544.75 nm	95.764%	3.596 nm	151.5
200 nm	551.99 nm	551.00 nm	95.355%	3.643 nm	151.2
205 nm	565.79 nm	557.00 nm	95.764%	3.686 nm	151.1
210 nm	579.59 nm	563.25 nm	95.642%	3.769 nm	149.4

Increasing the cavity thickness by 20 nm shifts the transmission peak 24.75 nm toward longer wavelengths. Peak transmittance remains approximately 95.4%–95.8%, while FWHM stays between 3.60 nm and 3.77 nm. Over this range, cavity thickness primarily moves the peak without significantly degrading it.

The half-wave formula predicts the correct direction but overestimates the actual peak sensitivity

The maximum energy-closure error across all sweep results is below $1.0\times10^{-14}$.

Derive a Thickness Tolerance from Peak Drift

A linear fit to the five simulated points gives the local sensitivity around this design:

$$ S=\frac{\mathrm{d}\lambda_{\mathrm{peak}}}{\mathrm{d}d_c} \approx1.235\ \frac{\mathrm{nm}}{\mathrm{nm}}. $$

Here, $S$ is the local sensitivity of peak wavelength to cavity thickness; $\mathrm{d}\lambda_{\mathrm{peak}}$ and $\mathrm{d}d_c$ are small changes in peak wavelength and cavity thickness, respectively. Near this design, a 1 nm change in cavity thickness moves the peak by approximately 1.24 nm.

The half-wave formula gives $S_{\mathrm{simple}}=2n_c=2.760\ \mathrm{nm}/\mathrm{nm}$, more than twice the simulated value, because it neglects DBR reflection phase and effective penetration depth.

For an allowable peak-wavelength error $\Delta\lambda_{\mathrm{allow}}$, the cavity-thickness error can be estimated as

$$ |\Delta d_c|\leq\frac{\Delta\lambda_{\mathrm{allow}}}{S}. $$

Here, $\Delta d_c$ is the cavity-thickness error, $\Delta\lambda_{\mathrm{allow}}$ is the allowable peak-wavelength error, and $S$ is the local sensitivity defined above.

Allowable peak error	Maximum cavity-thickness error
±1 nm	approximately ±0.81 nm
±2 nm	approximately ±1.62 nm
±5 nm	approximately ±4.05 nm

This is a deterministic, single-parameter sensitivity analysis, not a random-error or production-yield calculation. If the DBR pair count, materials, incidence angle, or operating wavelength changes, repeat the sweep and fit a new value of $S$.

Design Decisions

Design action	Main effect	Revalidate
Adjust cavity thickness	Moves the transmission peak	Peak wavelength and thickness tolerance
Add DBR pairs	Usually narrows the peak	Peak value, FWHM, Q, and absorption loss
Change incidence angle	Produces angular peak drift	Use angle and polarization
Use dispersive materials	Makes optical thickness wavelength dependent	Peak position and linewidth across the full band

Hitting 550 nm in the nominal design is only the first step. A manufacturable filter must also convert the spectral specification into a thickness specification and confirm that the deposition process can meet it.

Variation Exercise

Change the cavity-thickness sweep to 195–203 nm with a 1 nm step and leave all other settings unchanged. Fit $S$ again near the nominal thickness and compare the fine-sweep result with the five-point value of 1.235 nm/nm obtained above.

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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