In this example, we will demonstrate how INTERCONNECT can be used to design and simulate a Fabry-Perot resonator.

## Problem definition: More details

The Fabry-Perot etalon is the most basic optical resonator. The varying transmission function of an etalon is caused by interference between the multiple reflections of light between two reflecting surfaces:

There are two basic ways to simulate a Fabry-Perot resonator in INTERCONNECT. We can start by adding two optical mirrors and a straight waveguide from the element library to the schematic editor to construct the Fabry-Perot cavity.

Alternatively, we can also use the Fabry-Perot Resonator element from the element library to model the resonator directly:

## Discussion

The Optical Network Analyzer (Element Library\Analyzers\Optical) from INTERCONNECT allows you to characterize the complex transmission of the resonator, requiring minimum effort to analyze its response:

The normalized transmission for a Fabry-Perot resonator is:

$$

\frac{E_{n}}{E_{0}}=\frac{T e^{i \delta / 2}}{1-R e^{i \delta}}

$$

Where T and R are the mirror power transmission and reflection respectively. The round-trip phase shift is:

$$

\delta=-\frac{2 \pi f}{c} n_{g} .2 l

$$

Where n_{g} and l are the group index and waveguide length respectively, and c is the speed of light in a vacuum. The power transmitted is:

$$

\frac{I_{n}}{I_{0}}=\frac{(1-R)^{2}}{(1-R)^{2}+4 R \sin ^{2}(\delta / 2)}

$$

The free spectral range (FSR) is the spacing between adjacent modes of the filter:

$$

\Delta v=\frac{c}{2 n_{g} l}

$$

The straight waveguide model allows for the definition of the loss, effective index, group index and dispersion at a given center frequency f_{0}. These propagation parameters define the transfer function of the waveguide. Without including losses or dispersion, the transfer function of the straight waveguide is:

$$

\mathrm{H}(\mathrm{f})=e^{-i\left(\frac{2 \pi \mathrm{f}_{0}}{\mathrm{c}} \mathrm{n}_{\mathrm{eff}}+\frac{2 \pi}{\mathrm{c}} \mathrm{n}_{\mathrm{g}}\left(\mathrm{f}-\mathrm{f}_{0}\right)\right) \cdot l}

$$

By setting the group index and effective index at a given center frequency to the same value, the transfer function is:

$$

\mathrm{H}(\mathrm{f})=e^{-i\left(\frac{2 \pi f}{\mathrm{c}} \mathrm{ng}\right) \cdot l}=e^{i \delta / 2}

$$

For instance, for a waveguide with group index of 2.8 and length of 10 microns, from (4) the expected FSR is 5.35 THz. If the power reflection coefficient is 0.5, the plot of the power transmitted (3) versus frequency is:

## Results

The Fabry-Perot resonator can be simulated using two different configurations. The first one used the circuit of the resonator and the second one used the element directly. For the first case, the transfer function is calculated by the network analyzer, taking into account the transfer functions of the mirrors and the straight waveguide. For the second case the transfer function of the resonator is given by equation (1). In either case, the results should be identical, demonstrating the validity of equation (1) for the overall transfer function of the device. The following plot shows identical transmission for the two resonators, where the first one is labeled ‘Circuit’ and the second one ‘Element’.

The bandwidth or linewidth of the resonator changes with the reflection coefficient, a larger reflection leads to a narrower bandwidth:

Additionally, depending on the dimensions of the device, the Fabry-Perot resonator can be simulated both on the device/element level with a physics based simulation tool such as MODE or on the system/circuit level with INTERCONNECT:

### Related publications

[1] A. R. M. Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, "Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI)," Opt. Express 16, 12084-12089 (2008).