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.. "auto_tutorials/photon_number_conservation/g_FMAS.py"
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.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_tutorials_photon_number_conservation_g_FMAS.py:


FMAS
====

This example demonstrates photon number conservation for the forward model for
the analytic signal (FMAS).

The considered propagation model provides a proper conservation law as class
method `claw`. However, for clarity, we here re-implement the conservation
law and explicitly pass this  user-defined function to the solver class upon
initialization.

As exemplary propagation scenario, the setup used in the step-by-step demo

:ref:`sphx_glr_auto_tutorials_specific_g_spectrogram.py`

is used.

.. codeauthor:: Oliver Melchert <melchert@iqo.uni-hannover.de>

.. GENERATED FROM PYTHON SOURCE LINES 21-45

.. code-block:: default

    import fmas
    import numpy as np
    from fmas.solver import IFM_RK4IP
    from fmas.analytic_signal import AS
    from fmas.grid import Grid
    from fmas.propagation_constant import PropConst, define_beta_fun_ESM
    from fmas.tools import sech, change_reference_frame, plot_claw

    beta_fun = define_beta_fun_ESM()
    pc = PropConst(beta_fun)

    grid = Grid(t_max=5500.0, t_num=2 ** 14)  # (fs)  # (-)

    Ns = 8.0  # (-)
    t0 = 7.0  # (fs)
    w0 = 1.7  # (rad/fs)
    n2 = 3.0e-8  # (micron^2/W)
    chi = 8.0 * n2 * pc.beta(w0) * pc.c0 / w0 / 3.0
    gam0 = 3 * w0 * w0 * chi / (8 * pc.c0 * pc.c0 * pc.beta(w0))
    A0 = Ns * np.sqrt(abs(pc.beta2(w0)) / gam0) / t0
    E_0t_fun = lambda t: np.real(A0 * sech(t / t0) * np.exp(1j * w0 * t))
    Eps_0w = AS(E_0t_fun(grid.t)).w_rep









.. GENERATED FROM PYTHON SOURCE LINES 46-47

As model we here consider the forward model for the analytic signal (FMAS) 

.. GENERATED FROM PYTHON SOURCE LINES 47-51

.. code-block:: default


    from fmas.models import FMAS
    model = FMAS(w=grid.w, beta_w=pc.beta(grid.w), chi=chi)








.. GENERATED FROM PYTHON SOURCE LINES 52-60

For the FMAS :math:`z`-propagation model we consider a conserved quantity
that is related to the classical analog of the photon number, see Eq. (24) of
Ref. [AD2010] below. In particular we here implement

.. math::
   C_p(z) = \sum_{\omega>0} \omega^{-2} \beta(\omega) |u_\omega(z)|^2,

which is, by default, provided as method `model.claw` .

.. GENERATED FROM PYTHON SOURCE LINES 60-74

.. code-block:: default


    beta_w = pc.beta(grid.w)  # pre-compute beta(w) for convenience


    def Cp(i, zi, w, uw):
        _a2_w = np.divide(
            np.abs(beta_w) * np.abs(uw) ** 2,
            w * w,
            out=np.zeros(w.size, dtype="float"),
            where=w > 0,
        )
        return np.sum(_a2_w)









.. GENERATED FROM PYTHON SOURCE LINES 75-83

As shown below, this conserved quantity can be provided when an instance of
the desired solver is initialized. Here, for simply monitoring the
conservation law we use the Runge-Kutta in the ineraction picture method.
However, a proper conserved quantity is especially important when the
conservation quantity error method (CQE) is used, see, e.g., demo

:ref:`sphx_glr_auto_tutorials_tests_g_performance_CQE.py`


.. GENERATED FROM PYTHON SOURCE LINES 83-89

.. code-block:: default


    solver = IFM_RK4IP(model.Lw, model.Nw, user_action=Cp)
    solver.set_initial_condition(grid.w, Eps_0w)
    solver.propagate(z_range=0.01e6, n_steps=4000, n_skip=8)  # (micron)  # (-)  # (-)









.. GENERATED FROM PYTHON SOURCE LINES 90-103

The figure below shows the dynamic evolution of the pulse in the time domain
(top subfigure) and in the frequency domain (center subfigure). The subfigure
at the bottom shows the conservation law (c-law) given by the normalized
photon number variation

.. math::
   \delta_{\rm{Ph}}(z) = \frac{ C_p(z)-C_p(0)}{C_p(0)}

as function of the proapgation coordinate :math:`z`. For the considered
discretization of the computational domain the normalized photon number
variation is of the order :math:`\delta_{\rm{Ph}}\approx 10^{-7}` and thus
very small. The value can be still decreased by decreasing the stepsize
:math:`\Delta z`.

.. GENERATED FROM PYTHON SOURCE LINES 103-110

.. code-block:: default


    utz = change_reference_frame(solver.w, solver.z, solver.uwz, pc.vg(w0))

    plot_claw(
        solver.z, grid.t, utz, solver.ua_vals, t_lim=(-25, 125), w_lim=(0.5, 4.5)
    )




.. image:: /auto_tutorials/photon_number_conservation/images/sphx_glr_g_FMAS_001.png
    :alt: g FMAS
    :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 111-116

**References:**

[AD2010] Sh. Amiranashvili, A. Demircan, Hamiltonian structure of propagation
equations for ultrashort optical pulses, Phys. Rev. E 10 (2010) 013812,
http://dx.doi.org/10.1103/PhysRevA.82.013812.


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