Photonic circuits incorporate ring resonators and Fabry-Perot cavities to produce peaks that are periodic as a function of frequency. The spacing in frequency between adjacent peaks is called the Free Spectral Range (FSR) and may, itself, also vary as a function of frequency. The FSR is an important quantity in the design of these circuits, and therefore, its accurate calculation is highly desirable. For ring resonator structures, the FSR is inversely proportional to the group delay associated with propagation over one complete loop. Similarly, for Fabry-Perot cavities, the FSR is inversely proportional to the round-trip group delay.
In this example, we are going to demonstrate delay compensation in TSM simulations of a Fabry-Perot cavities and a ring resonators to show the importance of this option in FSR measurements and simulation accuracy.
In time-domain Transient Sample Mode simulations, for an element to process data, a valid signal sample is required at each port. In bidirectional simulations involving reflections, this requires that the INTERCONNECT dynamic data flow simulator introduce delays, and as a result each connection between elements introduces an additional delay of one sampling period. As a result, the group delay through an arm of a circuit comprised. For example, the round-trip delay in a Fabry-Perot cavity composed of a waveguide and a pair of partially reflecting mirrors will be (slightly, in most instances) longer than expected, resulting in an FSR that is smaller than expected. When the sampling rate is large such that the time step (or sampling period) is small in relation to the expected group delay, the difference will be small and can therefore be reduced to an acceptable tolerance by increasing the sampling. However, in the case of photonic circuits that are based on resonant devices, particularly in circuits with gain, the tolerances can be quite exigent. Moreover, increasing the sampling rate leads to longer and longer simulation times for a given desired spectral resolution. Fortunately, by internally reducing the number of delays or digital filter taps used by the waveguide element, the effect of the additional delays introduced by the dynamic data flow simulator can be offset. This is done by setting the delay compensation parameter in the waveguide elements in INTERCONNECT to a non-zero integer equal to the number of connections in the arm of the circuit.
In the following examples, the delay compensation parameter in the INTERCONNECT waveguide element is employed to offset additional delays introduced by the INTERCONNECT dynamic data flow simulator thus enabling accurate, efficient calculation and measurement of the FSR for a ring resonator and a Fabry-Perot cavity, without resorting to large sampling rates. The Ring Resonator Simulation is first described, followed by the Fabry-Perot Cavity Simulation. Finally the results for both simulations are discussed together in the Results and Discussion section.
Ring Resonator Simulation
The INTERCONNECT project file Ring_FSR_delay_compensation.icp contains three circuits. They are comprised of an Optical Network Analyzer (ONA) and a pair of couplers and pair of waveguides linked together to form a loop as shown in Figure 1. Referring to the figure, a highly spectrally sampled frequency-domain simulation is performed by the ONA_1/WGD_1/C_1/WGD_2/C_2 circuit and the resulting transmission spectrum from the through port recorded by ONA_1 at input 1. This spectrum serves as a reference. The ONA_2/WGD_3/C_3/WGD_4/C_4 and ONA_3/WGD_5/C_5/WGD_6/C_6 circuits perform simulations in the time-domain with the delay compensation parameter on their respective waveguides set to 0 and 2, respectively. In the latter case, note that the setting of two (2) for each waveguide is that such that the total amount of delay compensation, 4, in going around a loop, corresponds to the number of connections traversed.
- Open INTERCONNECT, load the file Ring_FSR_delay_compensation.icp, and run the simulation
- Load and run the script, plotResults.lsf.
Fabry-Perot Cavity Simulation
The INTERCONNECT project file FP_FSR_delay_compensation.icp contains three circuits, as shown in Figure 2. They are comprised of an Optical Network Analyzer (ONA) and pair of partially reflecting mirrors linked by an optical waveguide. Referring to Figure 2, a highly spectrally sampled frequency-domain simulation is performed by the ONA_1/WGD_1 circuit and the spectrum, recorded by ONA_1 at input 1, serves as a reference. The ONA_2/WGD_2 and ONA_3/WGD_3 perform similar simulations in the time-domain with the delay compensation parameter on their respective waveguides set to 0 and 2, respectively. In the latter case, note that the setting of two (2), corresponds to the two links from the waveguide, one to each mirror.
- Open INTERCONNECT, load the file FP_FSR_delay_compensation.icp, and run the simulation
- Load and run the script, plotResults.lsf.
Results and Discussion
Figure 3 depicts the power transmission spectra from the through port recorded at input 1 of the respective ONA in each ring resonator circuit shown in Figure 1. Figure 4 depicts the reflectivity spectra from the Fabry-Perot resonator in each circuit recorded by their respective ONA's.
As can be seen from the figures, the spectra from the circuits with the delay compensation set to 2 on the waveguides produced much better agreement with the reference, i.e., the highly-spectrally sampled frequency domain (scattering analysis), than the circuit with the delay compensation set to 0. Moreover, with delay compensation, the agreement obtained in terms of FSR and positions of the peaks was excellent.
It should be emphasized that all of the time domain (impulse response) simulations for a given resonator type were carried out at the same sample rate. Therefore, it is possible to increase the accuracy of group delay simulation (as demonstrated by the FSR calculations) and eliminate the effects of the addition delays introduced by the dynamic data flow modeler, without having to resort to high sample rates and long simulations.