This example demonstrates how to model the quantum state produced by spontaneous four-wave mixing (SFWM) in a microring resonator based on parameters that can be extracted from classical waveguide analysis. The photon pair generation rate and biphoton wavefunction are calculated for degenerate SFWM in a silicon nitride microring resonator with a non-zero 3rd order nonlinear electric susceptibility, where two pump photons are converted to pairs of signal and idler photons such that energy conservation and phase matching conditions are satisfied [1,2]. We consider a waveguide geometry such that group velocity dispersion is minimal, and consequently only work with a linearized dispersion model [1].

Simulation of the ring or racetrack resonator is best achieved by breaking the device into sub-components consisting of a directional coupler and straight and bent waveguides. These building blocks are then combined into a circuit in INTERCONNECT to simulate the complete device. This approach provides accurate models while minimizing the overall simulation time. Direct simulation of large rings using 3D FDTD is not practical due to the large memory and time requirements, although it may be possible for some small rings.

## Overview

Understand the simulation workflow and key results

The simulations in this example are performed with the MODE and qINTERCONNECT solvers. The simulation is set up and run through a Lumerical script file in MODE. Next, a python script is run to generate the state that is used as an input to qINTERCONNECT. The qINTERCONNECT simulation is then run through the Python API, and can be either run directly in INTERCONNECT or in an external Python environment. Since qINTERCONNECT requires an INTERCONNECT GUI license to run, at least two GUI licenses are required to run qINTERCONNECT directly from the INTERCONNECT environment. Documentation for the Python API is available ** here ** .

### Step 1: Coupling Coefficient

The basic characterization of the waveguide coupler can be done with a single 3D FDTD simulation by calculating the power coupling coefficient as a function of frequency for the mode of interest. In this example, we focus on the fundamental TE mode, and we neglect back reflections and backward coupling which would require multiple FDTD simulations.

### Step 2: Straight and Bent Waveguide Propagation Loss

Straight and bent waveguide sections are best characterized using the FDE solver in MODE. Effective index, group index, and dispersion as a function of frequency for the band of interest are the key results needed to describe these sub-components. Two frequency sweeps are required for this data; one for the straight waveguide and one for the bent waveguide.

### Step 3: Quality Factor

After each sub-component is characterized, the racetrack resonator can be assembled with primitive elements in INTERCONNECT. The results from the component-level simulations must be loaded into the corresponding elements in the INTERCONNECT circuit, and optical network analyzer can be used to calculate the frequency-domain response. We extract the quality factor of each resonance peak and its corresponding wavelength.

### Step 4: Effective Mode Area and Group Velocity

The effective mode area of the straight waveguide is calculated from the signal, pump and idler mode profiles. We also extract the group velocity for each mode.

### Step 5: Pair Generation Rate and Bi-photon Wave function

The quality factors, effective mode area and group velocities calculated in the previous sections are used to determine the photon pair generation rate and bi-photon wave function.

## Run and results

Instructions for running the model and discussion of key results

### Step 1: Coupling Coefficient

- Open the [[coupler_region.fsp]] simulation using FDTD.
- Run the simulation.
- Run the [[coupling_coefficient_step1.lsf]] script file to calculate and export the coupling coefficients as a function of frequency.

The coefficients will be saved to the [[ coupling_coefficient.txt]] file. The coupling coefficient is the fraction of power that couples from the fundamental TE-mode in the bus waveguide to the fundamental TE-mode of the curved waveguide in the racetrack resonator. Its value can be controlled by the gap distance between the bus waveguide and the ring resonator and the length of the straight segment of the racetrack resonator. The quality factor is inversely proportional to the coupling coefficient.

### Step 2: Straight and Bent Waveguide Propagation Loss

- Open the [[waveguide.lms]] simulation using MODE.
- Press 'Run', and from 'Modal analysis' tab ensure that the 'bent waveguide' setting is disabled.
- Press Calculate Modes.
- Select the fundamental TE mode and click 'Frequency Sweep' from the 'Frequency analysis' tab. Once the sweep is done, click the 'Export for INTERCONNECT' button. Save the waveguide parameters into [[straight_wg.ldf]] .
- Enable the bent waveguide setting. Specify the bend radius of your device. Here we assume a bend radius of 35 μm.
- Re-run the mode solver.
- Select the fundamental mode and click 'Frequency Sweep' from the 'Frequency analysis' tab. Once the sweep is done, click the 'Export for INTERCONNECT' button. Save the waveguide parameters into [[bent_wg.ldf]] .

The quality factor is inversely proportional to the propagation losses in the straight and bent waveguide sections of the racetrack resonator.

### Step 3: Quality Factor

- Open the [[ring_resonator.icp]] simulation using INTERCONNECT.
- Import FDTD and FDE results into corresponding elements. Setting the ring resonator parameters from the root element will automatically update the length or components accordingly.
- Run the [[ring_resonator_step3.lsf]] script file to obtain.

The quality factors for each resonant frequency and the free spectral range are calculated from the transmission, which is plotted and shown below.

A quality factor of is obtained for the pump resonance, which is consistent with experimentally achieved values such as those in [3].

Note : Additional material losses can be added under the ‘loss’ parameter of each waveguide element, which will further decrease the quality factor. In many cases, a lower quality factor makes it easier to achieve phase matching constraints, and it can be difficult to achieve phase matching with a high-Q cavity. |

### Step 4: Effective Mode Area and Group Velocity

- Open the waveguide.lms simulation using MODE.
- Run the mode_area_step4.lsf script file.

The mode profiles at the signal, pump, and idler wavelengths are simulated and recorded, in addition to the group velocity for each mode. The effective mode area is calculated according to the formula [1]

$$

A_{\mathrm{eff}}=\frac{1}{\iint \mathrm{d} x \mathrm{~d} y u_p(x, y) u_p(x, y) u_s^*(x, y) u_i^*(x, y)}

$$

where the transverse spatial distribution of the modes are normalized according to \(\int|u(x, y)|^2 \mathrm{~d} x \mathrm{~d} y=1\). A value of \(A_{\text {eff }}=0.59 \mu \mathrm{m}^2\) is obtained, compared to the value of \(A_{\text {eff }}=0.5 \mu \mathrm{m}^2\) quoted in [1].

### Step 5: Pair Generation Rate and Biphoton Wavefunction

- Open and run the script generate_biphoton_wavefunction.py in the MODE environment.

The photon pair generation rate, squeezing level and bi-photon wave-function are calculated using the parameters from the previous steps. By default, a Gaussian pump envelope is used. The bi-photon wave-function is plotted below for a \(5 * 10^{-12}\) s pulse:

The number of photon pairs generated per second and per pump pulse is calculated, as well as the squeezing level.

## Updating the model with your parameters

When updating the model to match your component parameters, it is important to remember that multiple solvers and simulation files are involved. Changes must be made consistently in all the files. For example, changes to the waveguide width must be made in the FDTD, FDE, and CHARGE simulation files. Updated results from the component simulations must then be reloaded into INTERCONNECT.

Frequently changed component parameters:

- Ring radius (FDTD, FDE, INTERCONNECT): When updating the ring radius in FDTD, simulation objects such as port angle, FDTD simulation region, mesh position, and monitors must be updated accordingly. Particularly, the port angle must be set such that it collects only the power inside the ring section. If a larger port angle is required, increase the simulation span.
- Source bandwidth (FDTD, FDE, INTERCONNECT)
- Waveguide geometry cross section (FDTD, FDE): Note that that the assumption of insignificant group velocity dispersion may not be satisfied if these parameters are changed.
- Coupling gap distance (FDTD)
- Coupling length (FDTD, INTERCONNECT): Length of racetrack straight section, zero for circular rings.
- Material properties (FDTD, FDE)
- Frequency range (INTERCONNECT): The frequency range of the optical network analyzer can be increased to simulate additional resonance peaks
- Number of resonances (INTERCONNECT): The number of resonances in the transmission spectrum to calculate the quality factor for

The final calculation of the biphoton wavefunction and pair generation rate will depend on the results of the component simulations and the values in the script file must be updated accordingly. Frequently changed parameters include:

- Effective area
- Wavelengths of signal, pump, and idler resonances
- Group velocities at the signal, pump, and idler wavelengths
- Quality factors of signal, pump, and idler resonances
- Pulse shape of the pump laser: By default, a Gaussian pulse envelope is used, which is parametrized in terms of pulse duration. However, an arbitrary pulse envelope can be used by using a custom
**pump_envelope**lambda function - Repetition rate of the pump laser
- Average power of the pump laser
- Interaction length, corresponding to the total length of the racetrack resonator
- Second-order nonlinear refractive index of the nonlinear material
- Plot points: The resolution of the final biphoton wavefunction
- Frequency points: The number of frequency points for the discretized biphoton wavefunction for use in qINTERCONNECT

## Taking the model further

** Adding additional bus waveguides to the ring resonator. **

To simulate the quality factor of a ring resonator coupled to more than one waveguide, additional waveguide couplers can be added to the ring resonator and connected to the input port of the optical network analyzer.

** Using the biphoton wavefunction as an input to a circuit in qINTERCONNECT. **

The biphoton wavefunction produced from this model can be used as an input to a circuit in qINTERCONNECT. See for example Step 3 in Spontaneous Parametric Down-Conversion (SPDC) Photon Source – Ansys Optics .

## Additional resources

### Related publications

- I. N. Chuprina, A. A. Kalachev, P. P. An, E. G. Zubkova, V. V. Kovalyuk, and G. N. Gol’tsman, “Optimisation of spontaneous four-wave mixing in a ring microcavity,”
*Quantum Electronics (Woodbury, N.Y.)*, vol. 47, no. 10, Oct. 2017, doi: https://doi.org/10.1070/QEL16511. - L. G. Helt, Z. Yang, M. Liscidini, and J. E. Sipe, “Spontaneous four-wave mixing in microring resonators,”
*Optics Letters*, vol. 35, no. 18, pp. 3006–3008, Sep. 2010, doi: https://doi.org/10.1364/OL.35.003006. - V. D. Vaidya
*et al.*, “Broadband quadrature-squeezed vacuum and nonclassical photon number correlations from a nanophotonic device,”*Science Advances*, vol. 6, no. 39, Sep. 2020, doi: https://doi.org/10.1126/sciadv.aba9186.