{
 "cells": [
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   "cell_type": "markdown",
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   "source": [
    "# Theory"
   ]
  },
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    "ExecuteTime": {
     "end_time": "2024-04-02T04:45:32.196685Z",
     "start_time": "2024-04-02T04:45:30.147713Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "interactive(children=(FloatSlider(value=1.0, description='E0', max=2.0, min=0.1), FloatSlider(value=1.0, descr…",
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       "version_minor": 0,
       "model_id": "7a7b5d47e29d4c0b9938d747926b281d"
      }
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": "<function __main__.plot_fields(E0=1.0, beta=1.0, m=0, k=2.0, N=5, R=0.0, Phi=0.1, d=5.0)>"
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "from scipy.special import jv as J\n",
    "from scipy.integrate import cumulative_trapezoid\n",
    "from ipywidgets import interact\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "\n",
    "def plot_fields(E0=1.0, beta=1.0, m=0, k=2.0, N=5, R=0.0, Phi=0.1, d=5.0):\n",
    "    Z = np.linspace(0, 10, 100)\n",
    "\n",
    "    def Ez(z=Z, r=R, phi=Phi):\n",
    "        return E0 * np.exp(-1j*beta*Z) * J(m, (k**2-beta**2)**0.5*R) * np.cos(m*Phi)\n",
    "        \n",
    "    mag_Ez = Ez()\n",
    "    \n",
    "    def En(n):\n",
    "        Z = np.linspace(-d/2, d/2, 100)\n",
    "        return cumulative_trapezoid(Ez(Z, 0, Phi*np.exp(1j*(beta+2*np.pi*n/d)*Z)))[-1] / d\n",
    "                                \n",
    "    Ez_sum = sum([En(n) * np.exp(-1j*(beta+2*np.pi*n/d)*Z) * J(m, (k**2-(beta+2*np.pi*n/d)**2)**0.5*R) * np.cos(m*Phi) for n in range(-N, N+1, 1)])\n",
    "    mag_Ez_sum = np.abs(Ez_sum)\n",
    "    \n",
    "    fig, axs = plt.subplots(1, 2, figsize=(12, 5))\n",
    "    \n",
    "    cax1 = axs[0].plot(Z, mag_Ez)\n",
    "    axs[0].set_title(\"Magnitude of Ez\")\n",
    "    axs[0].set_xlabel(\"z\")\n",
    "    axs[0].set_ylabel(\"Ez\")\n",
    "    \n",
    "    cax2 = axs[1].plot(Z, mag_Ez_sum)\n",
    "    axs[1].set_title(\"Magnitude of Ez_sum\")\n",
    "    axs[1].set_xlabel(\"z\")\n",
    "    axs[1].set_ylabel(\"Ez_sum\")\n",
    "    \n",
    "    plt.tight_layout()\n",
    "    fig.show()\n",
    "\n",
    "\n",
    "interact(plot_fields,\n",
    "         E0=(0.1, 2.0, 0.1),\n",
    "         beta=(0.1, 5.0, 0.1),\n",
    "         m=(0, 10),\n",
    "         k=(0.1, 5.0, 0.1),\n",
    "         N=(0, 20),\n",
    "         R=(0.0, 10.0, 0.1),\n",
    "         Phi=(0.1, 10.0, 0.1),\n",
    "         d=(1.0, 10.0, 0.1))"
   ]
  }
 ],
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