In this example, we simulate the supercontinuum generation in a photonic crystal fiber. The pitch of the photonic crystals is swept to find the condition for zero dispersion and positive dispersion slope around 835 nm. The resulting supercontinuum spectrum of the device is verified through a circuit simulation.
Overview
Understand the simulation workflow and key results
Optical fibers are widely used for supercontinuum generation. In fibers the bandwidth of the nonlinearly generated spectrum is largely controlled by the chromatic dispersion properties. These properties can be tailored by the fiber design and material properties. To promote supercontinuum generation, the dispersion length needs to be higher than nonlinear length so that nonlinear effects act before the peak power, P0, drops due to dispersion:
$$L_{Nl} = \frac{1}{γP_0}$$
where \(τ_{0}\) is the pulse width, \(D\) is the dispersion, \(γ=(n^2 ω_{0})/(cA_{eff})\), \(ω_{0}\) is the reference angular frequency, \(n^2\) is the nonlinear index and \(A_{eff}\) is the mode effective area. The wavelength of the laser used for the excitation (reference wavelength) must be close to the zero-dispersion wavelength and in the anomalous regime (positive D values) so that the nonlinear effects counterbalance the chromatic dispersion.
In this example we study the supercontinuum generation in a PCF. PCFs provide high nonlinearity due to strong mode confinement as well as good degree of control over the chromatic dispersion by proper rearrangement of the air holes. The lattice constant (pitch) of the PCF will be used to control the PCF dispersion. The reference wavelength is 835 nm.
Step 1: Calculate dispersion for different lattice constant values of PCF
Initially we run a parameter sweep to calculate the dispersion for different lattice constant values of the PCF. From the dispersion plot we choose the desired design, i.e. the lattice constant for which the reference wavelength is close to the zero-dispersion wavelength and has a positive value (anomalous regime).
Step 2: Calculate and export \(A_{eff}\), group index, dispersion and dispersion slope for the reference wavelength
The second step is to calculate the \(A_{eff}\), group index, dispersion and dispersion slope for the reference wavelength for the PCF design that was chosen in Step 1. These results are exported for the INTERCONNECT simulation.
Step 3: Calculate supercontinuum generation
In this step, we use the parameters calculated in Step 2 to calculate the supercontinuum spectrum from a circuit model of the device.
Run and Results
Instructions for running the model and discussion of key results
Step 1: Calculate dispersion for different lattice constant values of PCF
- Open the simulation file PCF.lms.
- Open and run the script file Design.lsf.
The Design.lsf script includes a parameter sweep over different pitch values of the PCF lattice.
For each pitch value, the dispersion of the fundamental mode, shown in the image below, is calculated.
After running the parameter sweep, the calculated dispersion D versus wavelength for all pitch values is plotted.
From the plot we choose 1.8 μm pitch for which the zero-dispersion wavelength is close to 835 nm and has a positive value at this wavelength.
Step 2: Calculate and export \(A_{eff}\), group index, dispersion and dispersion slope for the reference wavelength
- Open the simulation file PCF.lms.
- Open and run the script file Dispersion_export.lsf.
The Dispersion_export.lsf script runs a mode analysis and calculates the effective area \(A_{eff}\) and the group index ng of the fundamental mode. Then it runs a frequency sweep analysis around 835 nm to calculate the dispersion as well as the dispersion slope at the reference wavelength.
All the calculated parameters, \(A_{eff}\), \(n_{g}\), \(D\) and \(D\) \(slope\), are exported to be used in INTERCONNECT simulation.
Step 3: Calculate supercontinuum generation
- Open the simulation file nonlinear_NLSE.icp .
- Open and run the script file nonlinear_NLSE.lsf .
- Run the simulation file nonlinear_NLSE.icp .
For the circuit simulation, the NLSE Waveguide element is used, which models linear and nonlinear effects in waveguides. The nonlinear_NLSE.lsf script loads the calculated parameters of Step 2 to the simulation file. After the simulation is completed, choose the OSA_1 element and in the Result View right click the signal result to visualize the generated supercontinuum spectrum. Since only one mode has been defined in the NLSE element, sum signal and Mode 1 signal results of the OSA_1 will be the same. As shown in the image below, the generated supercontinuum spectrum covers a wide wavelength range from approximately 580 nm to over 1150 nm.
Important model settings
Description of important objects and settings used in this model
Step 1 & 2: MODE Simulations
Boundary conditions: Anti-symmetric and symmetric boundaries are chosen for x min and y min, respectively, to reduce the simulation time and memory, and to ensure that only TE mode is calculated for all lattice constant values.
Search near n: If you have a rough idea about the effective index of the fundamental mode, you can use the “Search near n” option as was done in this example. Otherwise, the more time-consuming “search in range” (with the default setting of “use max index”) might have to be used.
Frequency range: For Step 1 a large enough frequency range should be used in order to get the full dispersion curves for each lattice constant. In Step 2 the frequency range must be narrowed down to a small window around the reference wavelength to accurately calculate the dispersion and dispersion slope at the reference wavelength. A wavelength window of ±5nm is used.
Step 3: INTERCONNECT Simulations
Nonlinear index: For the example simulation the nonlinear index of silica (2.7e-20 m2/W) is used.
Time window: The simulation time window must be large enough to ensure that the pulse traverses the entire length of the NLSE element.
Bitrate: In order to have only one pulse, the product of the time window and the bitrate must be equal to 1. In this case the pulse will be generated at t equal to half of the time window, i.e t=(Time Window)/2.
Pulse power: The pulse peak power was set to 10000 W.
Pulse width: The pulse width is the pulse duration of the pulse full width at half maximum, defined as ratio of bit period. For a hyperbolic secant pulse used in the simulation, the half-width at the 1/e intensity point τ0 is related to the full width at half maximum, \(T_{FWHM}\) by:
$$T_{FWHM} = τ_0 2 ln(1+\sqrt{2})$$
The \(τ_{0}\) of the used pulse is 28.4 fs, so from the above equation we get that \(T_{FWHM}\) is 50.1 fs. Since the bitrate is set to 1e9 bits/s, the pulse width is 5e-5 (\(bitrate\)* \(T_{FWHM}\)).
Updating the Model With Your Parameters
Instructions for updating the model based on your device parameters
Step 1 & 2: MODE Simulations
Alternative waveguide designs can be used for supercontinuum generation. By changing their geometrical parameters or material indices the chromatic dispersion can be tuned so that the zero-dispersion wavelength approaches the reference wavelength.
Step 3: INTERCONNECT Simulations
For the INTERCONNECT simulation the following parameters can be updated:
- Use of Gaussian pulse instead of hyperbolic secant pulse.
- Use the pulse width and pulse peak power of the available laser.
- Use the desired length for the NLSE element.
- Update the nonlinear refractive index of the waveguide material.
- Enable other nonlinear effects such as “Two-photon absorption and Free carriers” and “Raman scattering”.
Additional Resources
Additional documentation, examples and training material
Related Publications
- Dudley, John & Genty, G. & Coen, Stéphane. (2006). Dudley, J. M., Genty, G. & Coen, S. Supercontinuum generation is photonic crystal fiber. Rev. Mod. Phys. 78, 1135-1184. Rev. Mod. Phys.. 78. 1135-1184. 10.1103/RevModPhys.78.1135.