### pypho v3

#### Example 1: Pulsebroadening by chromatic dispersion

A simple example to show the effect of chromatic dispersion is to show the pulse broadening of a single pulse, which has field $E$ has a gaussian shape with a $T_0$ 1/e-pulse width:
$$E(0, t) = e^{-\frac{1}{2} \left( \frac{t^2}{T_0^2} \right)^2 }$$

According to Agrawal the pulse width increases with the transmission length $z$: $$T_1(z) = T_0 \underbrace{ \sqrt{1 + \left( \frac{z}{L_D} \right)^2} }_{\text{Broadening factor}}$$ where $L_D = T_0^2 / |\beta_2|$ represents the dispersion length. This formula can be used to calculate the FWHM-pulse width, as well. For this the input FWHM-pulse width has to be multiplicated simply by the broadening factor.

In the following a signal with $T_{FWHM}=20ps$ pulse width is transmitted over $z=100km$ fiber with a dispersion coefficent $D=17 ps/nm/km$. The coefficent $\alpha$, $\beta_{1x}$, $\beta_{1y}$, $\beta_3$ and the non linear coeffiecent $\gamma$ is set to zero, so that only the effect of first order chromatic dispersion can be studied.

From the plot can be seen, that the FWHM pulse width is $T_{FWHM}=47.1 ps$. After the transmission the FWHM pulse width is $T_{FWHM}=259.72 ps$. This is in good accordance to the theorie, as the calculated FWHM pulse width is 259.59ps. The difference gets smaller with a higher number of samples per symbol.

Fig. 1 : Input and broadened output signal

##### Good to know!
###### Definition of pulse width

The FWHM pulse width of a gaussian pulse of the electrical field as defined above can be calculated by $T_{FWHM} = 2\sqrt{\ln4} T_0 \approx 2.35482 T_0$.
If the power of the pulse $|E(0, t)|^2$ is analysed the FWHM pulse width is given by $T_{FWHM,power} = \sqrt{0.5} T_{FWHM} = 2\sqrt{\ln2} T_0 \approx 1.665109 T_0$. The 1/e width is $T_{0,power} = \sqrt{0.5} T_{0}$ accordingly.

Import functions and libraries

import sys
sys.path.append('../')
from pypho_setup import pypho_setup
from pypho_symbols import pypho_symbols
from pypho_signalsrc import pypho_signalsrc
from pypho_lasmod import pypho_lasmod
from pypho_fiber import pypho_fiber
from pypho_functions import *
import numpy as np
import copy
import matplotlib.pyplot as plt
from scipy.interpolate import UnivariateSpline

# Define network elements
gp       = pypho_setup(nos =2**4, sps = 256, symbolrate = 10.0e9)

symbolsrc   = pypho_symbols(glova = gp, nos = gp.nos, pattern = 'debruijn')
esigsrc  = pypho_signalsrc(glova = gp, pulseshape = 'rect' , fwhm = 0.85)
sig_1550 = pypho_lasmod(glova = gp, power = 0, Df = 0, teta = 0)
SSMF     = pypho_fiber(glova = gp, l = 60.0e3,  D = 17.0,   S = 0, alpha = 0.2e-12, gamma = 1.4e-12, phi_max = 10.0)

# Simulation
bits = symbolsrc()
esig = esigsrc(bitsequence = bits)

E_Tx = sig_1550(esig = esig)

T_0 = 25.0e-12
z = SSMF.l
D = 17.0
beta_2, beta_3 = DS2beta(17.0, 0, gp.lambda0)

# Create a single pulse with gaussian shape (not power!)
E_Tx[0]['E'][0] = E_Tx[0]['E'][0]*0 + np.exp(-(gp.timeax()-gp.timeax()[-1]/2)**2 / (2.0*T_0**2) )

E = copy.deepcopy(E_Tx)

# Fiber transmission
E = SSMF(E = E, D = D, l = z)
#sys.exit()

# Plot Input and Output signal
plt.figure(1)
plt.plot(gp.timeax()*1.0e12, np.abs(E_Tx[0]['E'][0]), 'r', label='$E(0, t)$')
plt.plot(gp.timeax()*1.0e12, np.abs(E[0]['E'][0]), 'g', label= '$E(z='+ str(SSMF.l/1e3)+ 'km, t)') # Get FWHM of the input signal E_Tx spline = UnivariateSpline(gp.timeax()*1.0e12, np.abs(E_Tx[0]['E'][0])-1*np.max(np.abs(E_Tx[0]['E'][0]))/2, s=0) r1, r2 = spline.roots() # find the roots plt.annotate(s='', xy=(r1,np.max(np.abs(E_Tx[0]['E'][0]))/2), xytext=(r2,np.max(np.abs(E_Tx[0]['E'][0]))/2), arrowprops=dict(arrowstyle='<->')) plt.text(r1+(r2-r1)/2.0, 0.01 +np.max(np.abs(E_Tx[0]['E'][0]))/2, '$T_{FWHM,0}= ' + str(np.round(r2-r1,2)) + ' ps', fontsize=12, horizontalalignment='center') T_FWHM_0 = (r2-r1) * 1e-12 T_0_plot = T_FWHM_0 / 2.35482 # Get FWHM of the output signal E spline = UnivariateSpline(gp.timeax()*1.0e12, np.abs(E[0]['E'][0])-1*np.max(np.abs(E[0]['E'][0]))/2, s=0) r1, r2 = spline.roots() # find the roots plt.annotate(s='', xy=(r1,np.max(np.abs(E[0]['E'][0]))/2), xytext=(r2,np.max(np.abs(E[0]['E'][0]))/2), arrowprops=dict(arrowstyle='<->')) plt.text(r1+(r2-r1)/2.0, 0.01 + np.max(np.abs(E[0]['E'][0]))/2, 'T_{FWHM,1}= ' + str(np.round(r2-r1,2)) + ' ps', fontsize=12, horizontalalignment='center') T_FWHM_1 = (r2-r1) * 1e-12 plt.ylabel('|E|$a.u.'); plt.xlabel('Time$t\$ [ps]'); legend = plt.legend(loc='upper right')
plt.show()
L_D = (T_0_plot)**2  / np.abs(beta_2)
# Print the results
print ('Input signal 1/e-pulse width by definition : T_0 = ' , T_0*1e12, ' ps')
print ('Input signal 1/e-pulse width from plot : T_0 = ' , T_0_plot*1e12, ' ps')
print ('Input signal FWHM-pulse width from plot : T_FWHM,0 = ' , T_FWHM_0*1e12, ' ps')
print ('Output signal FWHM-pulse width from plot : T_FWHM,1 = ' , T_FWHM_1*1e12, ' ps')
print( 'Calculated output FWHM-pulse width : T_FWHM,1 = ' , T_FWHM_0 * np.sqrt(1 + (z/L_D)**2)*1e12, ' ps')