Readme¶

Overview¶

The purpose of this package is to characterize network of optical components when time delays with feedback are present. The core of the package identifies ‘trapped’ optical modes that result from feedback in the system. These modes are found by identifying the roots/poles of the transfer function containing only time delays and passive linear components. Further analysis of the network yields a linear Hamiltonian written in terms of the identified modes, linear operators coupling the system modes to the environment, and an overall scattering matrix. Nonlinear elements are added as additional Hamiltonian terms.

Our manuscript describing this method can be found on http://arxiv.org/abs/1510.08942 or http://www.epjquantumtechnology.com/content/3/1/3.

Installation¶

Simply clone the repository, open a terminal window, type:

git clone https://github.com/tabakg/potapov_interpolation
cd potapov_interpolation
python setup.py install

Files¶

Time_Delay_Network.py¶

This module includes a class to contain the information of a passive linear network with time delays. Several examples are included in this module. Each example includes matrices that yield a transfer function. This module contains methods to re-construct finite-dimensional approximations of a transfer function of passive systems.

–make_commensurate_roots(): A method to determine the roots utilizing the commensurate structure of the roots. The algorithm determines a polynomial based on the gcd of the time delays to describe where poles of the transfer function occur. The roots of the polynomial are found. The periodicity of the exponential \(e^{-zT}\) for the gcd delay is then used to find the roots within desired frequency ranges.

–make_roots(): A method that find the poles of the transfer function within a contour. This is a very general method in that the function needs only to be meromorphic with poles of order 1. In particular this yields the poles even when the delays are not commensurate.

–run_Potapov(): This function will run the Potapov procedure. The instance of Time_Delay_Network will update its roots, vecs, and T_testing. These are respectively the poles of the transfer function, a list of vectors (complex-valued column matrices) that contain the information to reconstruct the matrix-valued projectors in the Potapov representation, and an approximating transfer function that has been generated using the Potapov interpolation procedure. This method uses either make_roots() or make_commensurate_roots(), which the user can specify.

Potapov.py¶

We implement the procedure for finding Blaschke-Potapov products to approximate given functions near poles. Please see section 6.2 in our manuscript for details. This procedure is used to generate the modes of the passive linear network.

Given a rational matrix-valued function, we also construct the matrices ABCD that give the state-space representation of the system (see get_Potapov_ABCD()).

Roots.py¶

A module for identifying the zeros of a complex-valued function.

functions.py¶

Miscellaneous functions. Includes:

–Pade() Generate a Pade approximation for delays that do NOT feed back.

–spatial_modes() Finding the spatial location of modes. This is necessary to generate nonlinear terms. The required inputs to spatial_modes() are roots,M1,and E. These are respectively the poles of the transfer function, the directed connectivity matrix of the internal nodes of a network, and a diagonal matrix-valued function whose diagonal values correspond to the delays in the Fourier domain.

–make_nonlinear_interaction() Generate the weight of an interaction term due to phase-matching.

Hamiltonian.py¶

Includes a class Hamiltonian() to contain the information needed to construct the Hamiltonian of the system, including nonlinear terms. This class also includes a function make_eq_motion() to generate the classical equations of motion from the nonlinear Hamiltonian.

Also includes a class Chi_nonlin() which contains the information for a particular chi nonlinearity.

Time_Sims.py¶

Integrate the dynamics of a passive in time using the ABCD matrices.

Time_Sims_nonlin.py¶

–make_f_lin() generates outputs from ABCD model.

–make_f() generates outputs from nonlinear Hamiltonian model.

–run_ODE() integrates the equations of motion in time.

–double_up() prepares a doubled-up system which can be used for non-classical simulations.

Sample Usage – Time Domain Simulation¶

See [Simple code example](https://github.com/tabakg/potapov_interpolation/blob/master/Sample_Code_Usage.ipynb)