The goal of this notebook is to examine the ABCD matrices for a specific example and confirm several relationships between them as generated by the Potapov code.



In [1]:

    
import Roots
import Potapov
import Time_Delay_Network
import Time_Sims
import functions
import tests

import numpy as np
import numpy.linalg as la

import matplotlib.pyplot as plt
%matplotlib inline



In [2]:

    
def contour_plot(Mat,func = abs):
    '''
    Make a simple plot to view a matrix

    Args:
        Mat (compelx-valued matrix): a matrix to view
        func (optional[function]): a function to apply to each component

    Generates a plot of the matrix.
    '''
    fig = plt.figure()
    ax = fig.add_subplot(111)
    cax = ax.matshow(func(Mat), interpolation='nearest')
    fig.colorbar(cax)
    plt.show()



In [15]:

    
Ex = Time_Delay_Network.Example3(r1 = 0.9, r3 = 0.9, max_linewidth=35.,max_freq=25.)
Ex.run_Potapov()
E = Ex.E
roots = Ex.roots
M1 = Ex.M1
delays = Ex.delays
modes = functions.spatial_modes(roots,M1,E)



In [16]:

    
roots









    Out[16]:





[(-0.36514939263783186-10.901513851604911j),
 (-0.33972600185569823+20.178467995423109j),
 (-0.36514939263783186+10.901513851604911j),
 (-0.33972600185569823-20.17846799542311j),
 (-0.3376937738186606-4.1474624048492497e-23j)]



In [17]:

    
A,B,C,D = Potapov.get_Potapov_ABCD(Ex.roots,Ex.vecs,Ex.T,z=0.)



In [18]:

    
Omega = (A-A.H)/(2j)

To obtain the SLH from the ABCD, we can use that for a passive system $\Omega = (A - A^H)/2i$.

We generate $B$ and $C$ in such a way that $B = C^H$.

Also notice that $(A+A^H)/2 = BC/2$.



In [19]:

    
contour_plot(Omega)



In [28]:

    
## Let's look at the off-diagonal elements
Omega2 = - (A.imag+A.imag.H)/2
for i in range(Omega2.shape[0]):
    Omega2[i,i]=0
    
contour_plot(Omega2)
plt.show









    












    Out[28]:





<function matplotlib.pyplot.show>



In [20]:

    
contour_plot((A+A.H)/2.)



In [21]:

    
contour_plot(B*C/2.)



In [22]:

    
contour_plot(B.H)



In [23]:

    
contour_plot(C)