<reponame>nunoedgarhubsoftphotoflow/py-fmas
r"""
Nonlinear Schrödinger equation with loss
========================================

This example demonstrates how to perform simulations for the nonlinear
Schrödinger equation including loss.

.. codeauthor:: <NAME> <<EMAIL>>
"""

###############################################################################
# We first import the functionality needed to perform the sequence of numerical
# experiments:

import sys
import numpy as np
from fmas.models import ModelBaseClass
from fmas.config import FTFREQ, FT, IFT, C0
from fmas.grid import Grid
from fmas.solver import SySSM

###############################################################################
# Next, we implement a model for the nonlinear Schrödinger equation.  In
# particular, we here consider the standard nonlinear Schrödinger equation,
# given by
#
# .. math::
#    \partial_z u = -\frac{\alpha}{2} u -i \frac{\beta_2}{2}\partial_t^2 u + i\gamma |u|^2 u,
#
# wherein :math:`u = u(z, t)` represents the slowly varying pulse envelope,
# where :math:`alpha` is the power attenuation constant accounting for fiber
# loss, :math:`\beta_2=-1` is the second order dispersion parameter, and
# :math:`\gamma=1` is the nonlinear parameter:


class NSE(ModelBaseClass):
    def __init__(self, w, beta, alpha, gamma):
        super().__init__(w, beta_w=beta, alpha_w=alpha)
        self.gamma = gamma

    @property
    def Lw(self):
        return 1j * self.beta_w - self.alpha_w

    def N(self, uw):
        ut = IFT(uw)
        return 1j * self.gamma * FT(np.abs(ut) ** 2 * ut)


###############################################################################
# Next, we set up the computational domain, the model, an instance of a
# symmetric split-step Fourier solver and prepare an initial condition given by
# a fundamental soliton.
#
# To construct the initial condition, we use the exact single-soliton solution
# of the nonlinar Schrödinger equation, given by
#
# .. math::
#    u_{\rm{exact}}(z,t) = \sqrt{P_0} {\rm{sech}}(t/t_0)\,e^{-i\gamma P_0 z/2},
#
# with :math:`P_0=|\beta_2|/(\gamma t_0^2)`. We here consider a fundamental
# soliton of duration :math:`t_0=1` and use :math:`u_{\rm{exact}}(0,t)` as
# initial condition.  The propagation is performed up to
# :math:`z_{\rm{max}}=\pi/2`, i.e.  for one soliton period.

# -- SET MODEL PARAMETERS
t_max = -50.0
Nt = 2 ** 12
# ... PROPAGATION CONSTANT (POLYNOMIAL MODEL)
b2 = -1.0
beta = lambda w: 0.5 * b2 * w * w
# ... NONLINEAR PARAMETER
gamma = 1.0
# ... POWER ATTENUATION PARAMETER
alpha = 1.0
# ... SOLITON PARAMTERS
t0 = 1.0  # duration
P0 = np.abs(b2) / t0 / t0 / gamma  # peak-intensity
LD = t0 * t0 / np.abs(b2)  # dispersion length
# ... EXACT SOLUTION
u_exact = lambda z, t: np.sqrt(P0) * np.exp(0.5j * gamma * P0 * z) / np.cosh(t / t0)


###############################################################################
# We here measure the effect of fiber loss by monitoring the energy
#
# .. math::
#       E(z) = \sum_w |u_\omega(z)|^2
#
# which is expected to decay as
#
# .. math::
#       E(z) = E(0)\,e^{-2 \alpha z}
#


def Ce(i, zi, w, uw):
    return np.sum(np.abs(uw) ** 2)


# -- INITIALIZATION STAGE
# ... COMPUTATIONAL DOMAIN
grid = Grid(t_max=t_max, t_num=Nt)
t, w = grid.t, grid.w

model = NSE(w, beta(w), alpha, gamma)
# ... PROPAGATION ALGORITHM
solver = SySSM(model.Lw, model.N, user_action=Ce)
# ... INITIAL CONDITION
solver.set_initial_condition(w, FT(u_exact(0.0, t)))

# -- RUN SOLVER
solver.propagate(z_range=0.5 * np.pi * LD, n_steps=512, n_skip=2)  # propagation range

###############################################################################
# In the figure below, the top subfigure shows the time-domain propagation
# dynamics of a fundamental soliton for the nonlinear Schrödinger equation in
# the presence of fiber loss.
# The subfigure at the bottom show the resulting decay of the energy in the
# numerical experiment (solid line), along with the theoretical prediction
# (dashed line).

import matplotlib as mpl
import matplotlib.pyplot as plt
import matplotlib.colors as col

f, (ax1, ax2) = plt.subplots(2, 1, figsize=(5, 3))
plt.subplots_adjust(left=0.15, right=0.8, bottom=0.15, top=0.96, hspace=0.2)
cmap = mpl.cm.get_cmap("jet")


def _setColorbar(im, refPos):
    """colorbar helper"""
    x0, y0, w, h = refPos.x0, refPos.y0, refPos.width, refPos.height
    cax = f.add_axes([x0 + 1.02 * w, y0, 0.025 * w, h])
    cbar = f.colorbar(im, cax=cax, orientation="vertical")
    cbar.ax.tick_params(
        color="k",
        labelcolor="k",
        bottom=False,
        direction="out",
        labelbottom=False,
        labeltop=True,
        top=True,
        size=4,
        pad=0,
    )
    cbar.ax.tick_params(which="minor", bottom=False, top=False)
    return cbar


# -- TOP SUB-FIGURE: TIME-DOMAIN PROPAGATION CHARACTERISTICS
It = np.abs(solver.utz) ** 2
It /= np.max(It)

im1 = ax1.pcolorfast(
    solver.z,
    grid.t,
    np.swapaxes(It[:-1, :-1], 0, 1),
    norm=col.Normalize(vmin=0, vmax=1),
    cmap=cmap,
)
cbar1 = _setColorbar(im1, ax1.get_position())
cbar1.ax.set_ylabel(r"$|u|^2/{\rm{max}}\left(|u|^2\right)$")
ax1.xaxis.set_ticks_position("bottom")
ax1.yaxis.set_ticks_position("left")
ax1.set_ylim(-5, 5)
ax1.set_xlim([0.0, solver.z.max()])
ax1.set_ylabel(r"$t~\mathrm{(fs)}$")
ax1.ticklabel_format(useOffset=False, style="plain")
ax1.tick_params(axis="x", labelbottom=False, length=4)

# -- BOTTOM SUB-FIGURE: ENERGY DECAY
Ez = lambda z: solver.ua_vals[0] * np.exp(-2*alpha * z)
ax2.plot(solver.z, solver.ua_vals / Ez(0), lw=1.5, label="simulation")
ax2.plot(
    solver.z, Ez(solver.z) / Ez(0), color="k", lw=2.5, dashes=[2, 2], label="theory"
)
ax2.xaxis.set_ticks_position("bottom")
ax2.yaxis.set_ticks_position("left")
ax2.set_xlim([0.0, solver.z.max()])
ax2.ticklabel_format(useOffset=False, style="sci")
ax2.set_xlabel(r"Propagation distance $z~\mathrm{(\mu m)}$")
ax2.set_ylabel(r"$E(z)/E(0)$")
ax2.legend()

plt.show()

r"""
Nonlinear Schrödinger equation with loss
========================================

This example demonstrates how to perform simulations for the nonlinear
Schrödinger equation including loss.

.. codeauthor:: <NAME> <<EMAIL>>
"""

###############################################################################
# We first import the functionality needed to perform the sequence of numerical
# experiments:

import sys
import numpy as np
from fmas.models import ModelBaseClass
from fmas.config import FTFREQ, FT, IFT, C0
from fmas.grid import Grid
from fmas.solver import SySSM

###############################################################################
# Next, we implement a model for the nonlinear Schrödinger equation.  In
# particular, we here consider the standard nonlinear Schrödinger equation,
# given by
#
# .. math::
#    \partial_z u = -\frac{\alpha}{2} u -i \frac{\beta_2}{2}\partial_t^2 u + i\gamma |u|^2 u,
#
# wherein :math:`u = u(z, t)` represents the slowly varying pulse envelope,
# where :math:`alpha` is the power attenuation constant accounting for fiber
# loss, :math:`\beta_2=-1` is the second order dispersion parameter, and
# :math:`\gamma=1` is the nonlinear parameter:


class NSE(ModelBaseClass):
    def __init__(self, w, beta, alpha, gamma):
        super().__init__(w, beta_w=beta, alpha_w=alpha)
        self.gamma = gamma

    @property
    def Lw(self):
        return 1j * self.beta_w - self.alpha_w

    def N(self, uw):
        ut = IFT(uw)
        return 1j * self.gamma * FT(np.abs(ut) ** 2 * ut)


###############################################################################
# Next, we set up the computational domain, the model, an instance of a
# symmetric split-step Fourier solver and prepare an initial condition given by
# a fundamental soliton.
#
# To construct the initial condition, we use the exact single-soliton solution
# of the nonlinar Schrödinger equation, given by
#
# .. math::
#    u_{\rm{exact}}(z,t) = \sqrt{P_0} {\rm{sech}}(t/t_0)\,e^{-i\gamma P_0 z/2},
#
# with :math:`P_0=|\beta_2|/(\gamma t_0^2)`. We here consider a fundamental
# soliton of duration :math:`t_0=1` and use :math:`u_{\rm{exact}}(0,t)` as
# initial condition.  The propagation is performed up to
# :math:`z_{\rm{max}}=\pi/2`, i.e.  for one soliton period.

# -- SET MODEL PARAMETERS
t_max = -50.0
Nt = 2 ** 12
# ... PROPAGATION CONSTANT (POLYNOMIAL MODEL)
b2 = -1.0
beta = lambda w: 0.5 * b2 * w * w
# ... NONLINEAR PARAMETER
gamma = 1.0
# ... POWER ATTENUATION PARAMETER
alpha = 1.0
# ... SOLITON PARAMTERS
t0 = 1.0  # duration
P0 = np.abs(b2) / t0 / t0 / gamma  # peak-intensity
LD = t0 * t0 / np.abs(b2)  # dispersion length
# ... EXACT SOLUTION
u_exact = lambda z, t: np.sqrt(P0) * np.exp(0.5j * gamma * P0 * z) / np.cosh(t / t0)


###############################################################################
# We here measure the effect of fiber loss by monitoring the energy
#
# .. math::
#       E(z) = \sum_w |u_\omega(z)|^2
#
# which is expected to decay as
#
# .. math::
#       E(z) = E(0)\,e^{-2 \alpha z}
#


def Ce(i, zi, w, uw):
    return np.sum(np.abs(uw) ** 2)


# -- INITIALIZATION STAGE
# ... COMPUTATIONAL DOMAIN
grid = Grid(t_max=t_max, t_num=Nt)
t, w = grid.t, grid.w

model = NSE(w, beta(w), alpha, gamma)
# ... PROPAGATION ALGORITHM
solver = SySSM(model.Lw, model.N, user_action=Ce)
# ... INITIAL CONDITION
solver.set_initial_condition(w, FT(u_exact(0.0, t)))

# -- RUN SOLVER
solver.propagate(z_range=0.5 * np.pi * LD, n_steps=512, n_skip=2)  # propagation range

###############################################################################
# In the figure below, the top subfigure shows the time-domain propagation
# dynamics of a fundamental soliton for the nonlinear Schrödinger equation in
# the presence of fiber loss.
# The subfigure at the bottom show the resulting decay of the energy in the
# numerical experiment (solid line), along with the theoretical prediction
# (dashed line).

import matplotlib as mpl
import matplotlib.pyplot as plt
import matplotlib.colors as col

f, (ax1, ax2) = plt.subplots(2, 1, figsize=(5, 3))
plt.subplots_adjust(left=0.15, right=0.8, bottom=0.15, top=0.96, hspace=0.2)
cmap = mpl.cm.get_cmap("jet")


def _setColorbar(im, refPos):
    """colorbar helper"""
    x0, y0, w, h = refPos.x0, refPos.y0, refPos.width, refPos.height
    cax = f.add_axes([x0 + 1.02 * w, y0, 0.025 * w, h])
    cbar = f.colorbar(im, cax=cax, orientation="vertical")
    cbar.ax.tick_params(
        color="k",
        labelcolor="k",
        bottom=False,
        direction="out",
        labelbottom=False,
        labeltop=True,
        top=True,
        size=4,
        pad=0,
    )
    cbar.ax.tick_params(which="minor", bottom=False, top=False)
    return cbar


# -- TOP SUB-FIGURE: TIME-DOMAIN PROPAGATION CHARACTERISTICS
It = np.abs(solver.utz) ** 2
It /= np.max(It)

im1 = ax1.pcolorfast(
    solver.z,
    grid.t,
    np.swapaxes(It[:-1, :-1], 0, 1),
    norm=col.Normalize(vmin=0, vmax=1),
    cmap=cmap,
)
cbar1 = _setColorbar(im1, ax1.get_position())
cbar1.ax.set_ylabel(r"$|u|^2/{\rm{max}}\left(|u|^2\right)$")
ax1.xaxis.set_ticks_position("bottom")
ax1.yaxis.set_ticks_position("left")
ax1.set_ylim(-5, 5)
ax1.set_xlim([0.0, solver.z.max()])
ax1.set_ylabel(r"$t~\mathrm{(fs)}$")
ax1.ticklabel_format(useOffset=False, style="plain")
ax1.tick_params(axis="x", labelbottom=False, length=4)

# -- BOTTOM SUB-FIGURE: ENERGY DECAY
Ez = lambda z: solver.ua_vals[0] * np.exp(-2*alpha * z)
ax2.plot(solver.z, solver.ua_vals / Ez(0), lw=1.5, label="simulation")
ax2.plot(
    solver.z, Ez(solver.z) / Ez(0), color="k", lw=2.5, dashes=[2, 2], label="theory"
)
ax2.xaxis.set_ticks_position("bottom")
ax2.yaxis.set_ticks_position("left")
ax2.set_xlim([0.0, solver.z.max()])
ax2.ticklabel_format(useOffset=False, style="sci")
ax2.set_xlabel(r"Propagation distance $z~\mathrm{(\mu m)}$")
ax2.set_ylabel(r"$E(z)/E(0)$")
ax2.legend()

plt.show()
Nonlinear Schrödinger equation with loss ========================================  This example demonstrates how to perform simulations for the nonlinear Schrödinger equation including loss.  .. codeauthor:: <NAME> <<EMAIL>> ############################################################################### # We first import the functionality needed to perform the sequence of numerical # experiments: ############################################################################### # Next, we implement a model for the nonlinear Schrödinger equation.  In # particular, we here consider the standard nonlinear Schrödinger equation, # given by # # .. math:: #    \partial_z u = -\frac{\alpha}{2} u -i \frac{\beta_2}{2}\partial_t^2 u + i\gamma |u|^2 u, # # wherein :math:`u = u(z, t)` represents the slowly varying pulse envelope, # where :math:`alpha` is the power attenuation constant accounting for fiber # loss, :math:`\beta_2=-1` is the second order dispersion parameter, and # :math:`\gamma=1` is the nonlinear parameter: ############################################################################### # Next, we set up the computational domain, the model, an instance of a # symmetric split-step Fourier solver and prepare an initial condition given by # a fundamental soliton. # # To construct the initial condition, we use the exact single-soliton solution # of the nonlinar Schrödinger equation, given by # # .. math:: #    u_{\rm{exact}}(z,t) = \sqrt{P_0} {\rm{sech}}(t/t_0)\,e^{-i\gamma P_0 z/2}, # # with :math:`P_0=|\beta_2|/(\gamma t_0^2)`. We here consider a fundamental # soliton of duration :math:`t_0=1` and use :math:`u_{\rm{exact}}(0,t)` as # initial condition.  The propagation is performed up to # :math:`z_{\rm{max}}=\pi/2`, i.e.  for one soliton period. # -- SET MODEL PARAMETERS # ... PROPAGATION CONSTANT (POLYNOMIAL MODEL) # ... NONLINEAR PARAMETER # ... POWER ATTENUATION PARAMETER # ... SOLITON PARAMTERS # duration # peak-intensity # dispersion length # ... EXACT SOLUTION ############################################################################### # We here measure the effect of fiber loss by monitoring the energy # # .. math:: #       E(z) = \sum_w |u_\omega(z)|^2 # # which is expected to decay as # # .. math:: #       E(z) = E(0)\,e^{-2 \alpha z} # # -- INITIALIZATION STAGE # ... COMPUTATIONAL DOMAIN # ... PROPAGATION ALGORITHM # ... INITIAL CONDITION # -- RUN SOLVER # propagation range ############################################################################### # In the figure below, the top subfigure shows the time-domain propagation # dynamics of a fundamental soliton for the nonlinear Schrödinger equation in # the presence of fiber loss. # The subfigure at the bottom show the resulting decay of the energy in the # numerical experiment (solid line), along with the theoretical prediction # (dashed line). colorbar helper # -- TOP SUB-FIGURE: TIME-DOMAIN PROPAGATION CHARACTERISTICS # -- BOTTOM SUB-FIGURE: ENERGY DECAY