Fabry–Pérot Cavity Simulation Using the Transfer Matrix Method (TMM)

1. Introduction

A Fabry–Pérot cavity is one of the most important resonant structures in optics. It consists of two reflecting interfaces separated by a finite distance, allowing light to bounce repeatedly between the mirrors and form standing-wave resonances. Only wavelengths satisfying the cavity phase condition are strongly transmitted through the structure.

Fabry–Pérot resonators are widely used in optical filters, dielectric mirrors, interferometers, LEDs, and laser cavities. In this tutorial, we model a simple one-dimensional Fabry–Pérot cavity using the transfer matrix method (TMM).

Unlike FDTD, which solves Maxwell’s equations directly in the time domain, the transfer matrix method solves the optical problem in the frequency domain by propagating forward and backward travelling waves through a multilayer stack. For one-dimensional layered structures, TMM is extremely fast and computationally efficient, making it ideal for analysing cavity resonances and multilayer interference effects.

At normal incidence, the Fabry–Pérot resonance condition is

\(2nL = m\lambda\)

where \(n\) is the refractive index inside the cavity, \(L\) is the cavity length, \(\lambda\) is the wavelength, and \(m\) is an integer mode number.

Rearranging gives the allowed cavity wavelengths:

\(\lambda_m = \frac{2nL}{m}\)

This means that only discrete resonant wavelengths are efficiently transmitted through the cavity. In this tutorial, we will directly observe these resonances in the transmission and reflection spectra, and also visualise the standing-wave photon density inside the cavity.

2. Making a new simulation

Open the New simulation window and select the Transfer matrix method category (??). Then select the Fabry–Pérot cavity example (??).

After opening the example, the main interface appears as shown in ??. The simulation consists of a simple optical cavity formed by two partially reflecting layers separated by an air gap.

The cavity consists of two thin dielectric layers with refractive index \(n=3\), separated by an air region. These two high-index layers act as partially reflecting mirrors.

When broadband light enters the structure, multiple reflections occur between the mirrors. At wavelengths satisfying the cavity resonance condition, the reflected waves interfere constructively, producing enhanced field buildup inside the cavity and increased transmission through the structure.

3. Running the optical simulation

Open the Optical ribbon (??) and click the Transfer Matrix button. This launches the optical transfer matrix solver.

The transfer matrix method computes the optical field distribution, reflection spectrum, transmission spectrum, and absorbed photon density throughout the multilayer structure. Because this method works directly in the frequency domain, simulations typically complete in only a few seconds.

After the simulation finishes, the photon-density map shown in ?? appears. The horizontal axis corresponds to position inside the cavity, while the vertical axis corresponds to wavelength.

Bright regions indicate wavelengths where the optical field builds up strongly inside the structure. The most prominent feature occurs near 450–500 nm, corresponding to the fundamental Fabry–Pérot resonance of the cavity.

The oscillatory spatial structure visible in the plot is the standing-wave pattern formed by interference between forward and backward travelling waves inside the cavity.

4. Reflection and transmission spectra

The transfer matrix solver also calculates the reflected and transmitted optical power. These results are shown in ?? and ??.

The cavity resonance appears as a pronounced dip in the reflection spectrum and a corresponding peak in the transmission spectrum. Physically, this occurs because the cavity stores optical energy efficiently at the resonant wavelength, allowing light to pass through the structure rather than being reflected.

In this structure, the cavity spacing is approximately 250 nm and the cavity refractive index is close to 1. The resonance condition therefore predicts a fundamental mode near

\(\lambda \approx 2nL \approx 2 \times 1 \times 250 \approx 500\ \text{nm}\)

which is consistent with the spectra observed above.

5. Changing the cavity length

Open the Layer editor shown in ??. The cavity is formed by two thin dielectric layers separated by an air gap.

In the original structure, the cavity thickness is approximately 260 nm. Increase this to 300 nm, as shown in ??, then rerun the optical simulation.

After rerunning the simulation, the spectra shown in ?? and ?? are obtained.

Increasing the cavity length shifts the resonance toward longer wavelengths. This follows directly from the Fabry–Pérot condition:

\(\lambda = \frac{2nL}{m}\)

Since the resonance wavelength is proportional to cavity length, increasing \(L\) causes the transmission peak to move systematically toward larger wavelengths.

You should also observe additional weaker resonances corresponding to higher-order cavity modes (\(m=2,3,\dots\)).

6. Viewing photon-density snapshots

Return to the Output tab shown in ??. Open the optical_snapshots directory to inspect the photon-density distribution at individual wavelengths.

Example photon-density distributions are shown in ??– ??.

These plots show the standing-wave distribution inside the cavity at different wavelengths. Strong resonances produce enhanced photon density and pronounced standing-wave structure between the mirrors.

The spatial oscillations visible in these figures correspond to optical standing waves formed by interference between forward and backward travelling fields. Different resonant modes produce different spatial field distributions inside the cavity.

7. Changing the refractive index contrast

The strength of the cavity resonance depends strongly on the reflectivity of the mirrors. In dielectric Fabry–Pérot cavities, this reflectivity is controlled primarily by the refractive-index contrast.

Open the layer editor and reduce the mirror refractive index from \(n=3\) to \(n=1.5\), as shown in ??.

After rerunning the simulation, the reflection and transmission spectra become those shown in ?? and ??.

When the refractive-index contrast is reduced, the mirrors become weaker reflectors. This reduces the amount of light trapped inside the cavity and broadens the resonant features.

Physically, this corresponds to a reduction in cavity quality factor (Q-factor). Strong mirrors produce narrow resonances with strong field buildup, while weak mirrors produce broader and less pronounced resonances.