pyLLE：Python中的Lugiato Lefever方程求解器

# pyLLE  [](https://gregmoille.github.io/pyLLE/)  [](https://doi.org/10.6028/jres.124.012) pyLLE is a tool to solve the Lugiato Lefever Equations (LLE)^[1](#ref1)^,^[2](#ref2)^,^[3](#ref3)in a fast and easy way. Thanks to a user-friendly front-end in python and an efficient back-end in Julia, solving this problem becomes easy and fast. For a complete documentation of the package, please visit the [github page](https://gregmoille.github.io/pyLLE/) ## Instalation As pyLLE relies on a Julia back-end, please prior to installing this package be sure that Julia is installed on your machine or visit the julia [package download page](https://julialang.org/downloads/) to install it by selecting &#9888; **v1.1.1** &#9888;. **Windows users**: Please, keep julia in the default directory during the installation (i.e. ~\AppData\Local\Julia-1.1.1\ for windows). **Mac Os User**: You would need to add the julia binary to the path. The easiest way to do it is to create a simlink in the terminal ```bash ln -s /Applications/Julia-1.1.app/Contents/Resources/julia/bin/julia /usr/local/bin/julia ``` Once Julia installed, the different packages needed to run pyLLE, either python or julia related, will be automatically downloaded and installed. Just a heads up, the installation of the package can vary in time, especially because of Julia that might rebuild the cache. For a automatic install, just pip it : ```bash pip install pyLLE ``` For a manual install, download the .zip of the repository or clone it and install with the setup.py script ```bash git clone https://github.com/gregmoille/pyLLE.git cd pyLLE python setup.py install ``` If the julia location is custom, please before installing change in the setup.py, line 18 to the correct location, as in pyLLE/llesolver.py line 430 to point to the correct location. Thanks ## Checking that everything works correctly Launch a julia console and within type the commands: ```julia using HDF5 using FFTW using ProgressMeter ``` if any of the previous command throw an issue, mostly it is because it is not installed. One way to fix it is to remove the installed packaged to remove the cache - for linux and mac os user: remove everything in ~/.julia/ - for windows users: remove everything in C:\Users\<your user name>\.julia\ Then enter the pacakge manager for julia by typing in the julia console: ```julia julia>] ``` then ```julia (v1.1) pkg>add HDF5 (v1.1) pkg>add FFTW ``` ## Example A complete example is available in the example directory [notebook](https://github.com/gregmoille/pyLLE/tree/master/example/NotebookExample.ipynb) with the corresponding file needed in the folder. Here we will only go through the main aspect of pyLLE - First import the package: ```python import pyLLE ``` - Define a resonator and a simulation dictionary such as (parameters from Li et _al._^[4](#ref4)): ```python res = {'R': 23e-6, # ring radius 'Qi': 1e6, # intrinsic quality factor 'Qc': 1e6, # coupling quality factor 'γ': 2, # non-linear coefficient 'dispfile': 'TestDispersion.csv' # name of the dispersion file } sim = {'Pin': 100e-3, #input power in W 'Tscan': 1e6, # Total time for the simulation in unit of round trip 'f_pmp': 191e12, # frequency of the pump in Hz 'δω_init': 4e9*2*np.pi, # start frequency of detuning ramp in Hz 'δω_end': -10e9*2*np.pi, # stop frequency of detuning ramp in Hz 'μ_sim': [-70,170], # limit of the mode on the left and right side of the pump to simulate 'μ_fit': [-60, 160], # limit of the mode on the left and right side of the pump to fit the dispersion with } ``` It is important to note the format of the dispersion file *TestDispersion.csv*. It must be formatted such that each line represents a resonance, with first the azimuthal mode order listed and then the frequency of the resonance, separated by a comma ',' <br> We can now set up the pyLLE class: ```python solver = pyLLE.LLEsolver(sim=sim, res=res, debug=False) ``` The debug input allows the script to generate a log file in the working directory with useful information on the simulation. The authors highly encourage to keep this key to True, unless some loops are run, which could create an issue with the read/write access to the file. <br> To analyze the dispersion just the *Analyze* ```python solver.Analyze(plot=True, plottype='all') ``` To start the simulation, first we need to set up an hdf5 file which makes the bridge between python and julia ```python solver.Setup() ``` <br> For the sake of simplicity and to retrieve the different parameters, all keys of the sim and res dictionary are translated with the previously described translator dictionary and cannot be accessed anymore with the greek alphabet. Hence, in an ipython console, one can retrieve the parameter with, for example: ```python IN [1]: solver.sim['mu_sim'] OUT [1]: [-70,170] ``` <br> Then we can start the simulation ```python solver.SolveTemporal() ``` To retrieve the data computed by julia, we call the *RetrieveData* method ```python solver.RetrieveData() ``` We can finally start to plot the result of the simulation. One can start with a complete overview of the simulation, where a spectral and a temporal map vs the LLE step is displayed in addition to the comb power Vs the LLE step ```python solver.PlotCombPower() ``` From there, we can find the step of the LLE where we want to see the spectra: ```python ind = 600 solver.PlotCombSpectra(ind) ``` One can also quickly solve the LLE through a steady-state method to find its roots ```python sim['δω'] = -5e9*2*np.pi Ering, Ewg, f, ax = solver.SolveSteadyState() ``` ## How to Cite Us? You can cite our paper published in the Journal of Research of National Institute of Standards and Technology available [here](https://doi.org/10.6028/jres.124.012): > Moille G, Li Q, Lu X, Srinivasan K (2019) pyLLE: A Fast and User Friendly Lugiato-Lefever Equation Solver. J Res Natl Inst Stan 124:124012. https://doi.org/10.6028/jres.124.012 ## Works Using pyLLE - Gregory Moille, Xiyuan Lu, Ashutosh Rao, Qing Li, Daron A. Westly, Leonardo Ranzani, Scott B. Papp, Mohammad Soltani, and Kartik Srinivasan "_Kerr-Microresonator Soliton Frequency Combs at Cryogenic Temperatures_," Phys. Rev. Applied 12, 034057 (2019) - Gregory Moille, Qing Li, Travis C. Briles, Su-Peng Yu, Tara Drake, Xiyuan Lu, Ashutosh Rao, Daron Westly, Scott B. Papp, and Kartik Srinivasan "_Broadband resonator-waveguide coupling for efficient extraction of octave-spanning microcombs_," Optics Letters Vol. 44, Issue 19, pp. 4737-4740 (2019) - Lin Chang, Weiqiang Xie, Haowen Shu, Qifan Yang, Boqiang Shen, Andreas Boes, Jon D. Peters, Warren Jin, Songtao Liu, Gregory Moille, Su-Peng Yu, Xingjun Wang, Kartik Srinivasan, Scott B. Papp, Kerry Vahala, John E. Bowers "_Ultra-efficient frequency comb generation in AlGaAs-on-insulator microresonators_," arXiv:1909.09778 (2019) ## References <a name="ref1">1</a>: Luigi A. Lugiato and René Lefever. "Spatial dissipative structures in passive optical systems." Physical review letters 58, no. 21 (1987): 2209. <a name="ref1">2</a>: Yanne K. Chembo and Curtis R. Menyuk. "Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators." Physical Review A 87, no. 5 (2013): 053852. <a name="ref1">3</a>: Stéphane Coen, Hamish G. Randle, Thibaut Sylvestre, and Miro Erkintalo. "Modeling of octave-spanning Kerr frequency combs using a generalized mean-field