{
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   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "The plasma dispersion function\n",
    "==============================\n",
    "\n",
    "Let's import some basics (and `PlasmaPy`!)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "import plasmapy.dispersion.dispersionfunction\n",
    "\n",
    "help(plasmapy.dispersion.dispersionfunction.plasma_dispersion_func)"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "raw_mimetype": "text/restructuredtext"
   },
   "source": [
    "Take a look at the docs to :func:`~plasmapy.dispersion.dispersionfunction.plasma_dispersion_func` for more information on this."
   ]
  },
  {
   "source": [
    "We'll now make some sample data to visualize the dispersion function:\n",
    "\n"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "x = np.linspace(-1, 1, 1000)\n",
    "X, Y = np.meshgrid(x, x)\n",
    "Z = X + 1j * Y\n",
    "print(Z.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we start plotting, let's make a visualization function first:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "def plot_complex(X, Y, Z, N=50):\n",
    "    fig, (real_axis, imag_axis) = plt.subplots(1, 2)\n",
    "    real_axis.contourf(X, Y, Z.real, N)\n",
    "    imag_axis.contourf(X, Y, Z.imag, N)\n",
    "    real_axis.set_title(\"Real values\")\n",
    "    imag_axis.set_title(\"Imaginary values\")\n",
    "    for ax in [real_axis, imag_axis]:\n",
    "        ax.set_xlabel(\"Real values\")\n",
    "        ax.set_ylabel(\"Imaginary values\")\n",
    "    fig.tight_layout()\n",
    "\n",
    "\n",
    "plot_complex(X, Y, Z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now apply our visualization function to our simple dispersion relation\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    },
    "tags": [
     "nbsphinx-thumbnail"
    ]
   },
   "outputs": [],
   "source": [
    "# sphinx_gallery_thumbnail_number = 2\n",
    "F = plasmapy.dispersion.dispersionfunction.plasma_dispersion_func(Z)\n",
    "plot_complex(X, Y, F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So this is going to be a hack and I'm not 100% sure the dispersion function\n",
    "is quite what I think it is, but let's find the area where the dispersion\n",
    "function has a lesser than zero real part because I think it may be important\n",
    "(brb reading Fried and Conte):\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "plot_complex(X, Y, F.real < 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also visualize the derivative:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "F = plasmapy.dispersion.dispersionfunction.plasma_dispersion_func_deriv(Z)\n",
    "plot_complex(X, Y, F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting the same function on a larger area:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
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   },
   "outputs": [],
   "source": [
    "x = np.linspace(-2, 2, 2000)\n",
    "X, Y = np.meshgrid(x, x)\n",
    "Z = X + 1j * Y\n",
    "print(Z.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "F = plasmapy.dispersion.dispersionfunction.plasma_dispersion_func(Z)\n",
    "plot_complex(X, Y, F, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we examine the derivative of the dispersion function as a function\n",
    "of the phase velocity of an electromagnetic wave propagating through\n",
    "the plasma. This is recreating figure 5.1 in:\n",
    "J. Sheffield, D. Froula, S. H. Glenzer, and N. C. Luhmann Jr,\n",
    "Plasma scattering of electromagnetic radiation: theory and measurement\n",
    "techniques. Chapter 5 Pg 106 (Academic press, 2010).\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "xs = np.linspace(0, 4, 100)\n",
    "ws = (-1 / 2) * plasmapy.dispersion.dispersionfunction.plasma_dispersion_func_deriv(xs)\n",
    "wRe = np.real(ws)\n",
    "wIm = np.imag(ws)\n",
    "\n",
    "plt.plot(xs, wRe, label=\"Re\")\n",
    "plt.plot(xs, wIm, label=\"Im\")\n",
    "plt.axis([0, 4, -0.3, 1])\n",
    "plt.legend(\n",
    "    loc=\"upper right\", frameon=False, labelspacing=0.001, fontsize=14, borderaxespad=0.1\n",
    ")\n",
    "plt.show()"
   ]
  }
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