Fresnel minima.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "plt.rcParams['figure.figsize'] = (10.0, 8.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x108d4fdd8>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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nzQohI9tdd7mfemrp2xs2LASoTL17u0+fXvw23n47fGCnA+W6deH+W28V9/oN\nG8IXihdfDPdHjnS/6qriXvvxx5tf+/TT7t27FxdC169379HD/cknw7FaKJSkDR0aQtXnPuc+dWrh\n8q+8EoLPwQe733hjceV33DG8p3wBIu2GG0J4ffbZwmUbG92vvLJw+2za5D55svvKlYW3+f774fcN\nVFO5Acw8jkWTYmRmXmt1wpZ+/etwlthNN1W7JtEWLw5DAJnrBBVr06Ywh+fxx7dcXTwOxxwT2i7f\ntSWnTQtDC3PnVrav738/1H/UqObPXXxxmDv14x+Xvt2PPgpnaa1cueVk5gsvDPOtLrustO2NGxfm\n02QOrR18cJiHU8ocuC98QfrOd8Kwyh13SH/8Y2kXEB85MgzTXXttGAKbNq3465tedlk4w+6jj0K7\n/uxnxb3u+uvD39CaNeHs0R12KPyapUvDvK8BA6SxY4sbYp0+Pczd+/73i1tX7s03w3Z32qlwWQCS\nmcndS54RTQBDyU49VfrmN8NFaGuRewhRCxeGD8RSPP10mHeS6zp6lfjXfw3r/Vx1Ve4yV14Z/r3m\nmsr2NXly2Ma8ec2f+9znpL/9LSxCWo4BA8K8lLPPDvc3bgyTgv/xj/zLNkR5770wn2b58tA2//Vf\nYQ7euHGlbWf8+BCeHnoozIu65JLSjs/XXgvznYYMCWehlnJW2Ntvh/lQ7duHeV3Fzl3btCnMizr8\n8LA6PYD6VG4AYxI+SpK+lloxE5SrJX2pm3KWo4hz/a9sxawH1thY3CTuQvr3D2Eye52fZctC6Dns\nsPK3fd550u23b74/Z07obSs1fEkhdA0bJv3P/4Qz33772zCJuVRDh4YeqEGDQiDKNXk7l913D6Fv\n223DxO5SdOsWJro/9ljx4UsKSzaccw7hC2irCGAoycMPh7N/Mq+lVovKXZC12LPfynHiiWG5g08+\niX7+jTfCyuWZ17MrV8eOoadq4sQtH7/rrnAmWL7V4QsZPjwEyZdfDvdvvFEaMaL87f3bv0ljxkhf\n+UoIccVcHDxb+/ah12/AgHBmXjlLeJx+uvT734feUwBoaQQwlGT69Nq5+HY+5fSAbdrU/Lprcdp+\n+zCvKFe9pk4NPVflrv+V7ayzwnyoTH/9q3TuuZVtt2vXcIr7FVeEQDl/vvTtb5e/vf32C0Oi++zT\nvL6l2H136Uc/yr8GFwDUCgIYSlKr639lO/TQMJl45criX/PUU+GyLnFPvs80fLh0zz3Rz02eHM/w\nY9pZZ4W382QhAAAgAElEQVS1jdKLlM6dGy4XVOkFxqUwuf/550NY/f3vm19upVSnnCL98pe137MK\nAHGJJYCZ2WAzW2Jmz5tZxHlXkpk1mNnjZva0mXEd9jq0enU4O7CYFcqrrV27UM9SziZsyeHHtOHD\nw4TxzMvQSGExy0mTSp+7lE/HjuHMt5Ejw/5++lPpJz8p7ky4QrbbLvR+vftuvHUGgLai4gBmZu0k\n3ShpkKReks4zswOzynSV9L+STnf3QySdXel+kbw5c8JK6JX2diSl1HlgxVw7sFL77htWnc4Ohg88\nEK7NGOfq+1JYauKdd8IK5+3bS+efH9+2zULIAwCULo4esN6Slrr7cndfL+kOSdnfib8s6R53XyFJ\n7v5WDPtFwupl+DGtX7/NF70tZMOGEDDjugB3PuefH872y/S730nf+Eb8+9pqqzCv7Z57wmVyOnSI\nfx8AgNLFEcB6SHol4/6rqccyHSBpRzObZWaPmNnXYtgvEjZjRn1MwE876qiw5MILLxQu+9BDYT2q\nJM6AO//80JbpuVmPPBIW4jznnJbZX8eO4czCuCb3AwAql9T34Q6SjpLUX9K2kh40swfdPfKjcfTo\n0Z/ebmhoUEMS3RLI6+23wwWujz662jUpXrt24eLD998vXXpp/rKTJkmnnZZMvbp0kf7930MQ+/Of\nw7+//GW4SC8AoLY1NTWpqdCijkWoeCV8M+sjabS7D07d/4nCdZGuyygzStJn3P3fU/f/KGmyuzc7\nH4yV8GvTvfeGYbIpU6pdk9JMnBgu+TKrwGkfhx0W3t/xxydTr02bwgT5228Pq8pfe21xl5UBANSW\nql2KyMzaS3pO0imSVkp6WNJ57r44o8yBkv5H0mBJHSUtkHSOuz8bsT0CWA364Q/D8Nzll1e7JqVZ\nty5MbF+2LCwxEWX58jABftWqeM4QBAC0HVW7FJG7b5R0saSpkp6RdIe7LzazC83sglSZJZIaJT0p\n6SFJY6PCF2rX7NnJTFCPW6dOYWmJ7BXhM915Z1hKgfAFAEgKF+NGQe++Gy5q/c479TmR+667pJtu\nyn1G5OGHh0vh1GPABABUFxfjRouZOzdcn7Aew5cULtT89NPSiy9KjY2NGjhwuAYOHK7GxkY9+WQI\nmHGsDl+K7Hq0lGuuuUbduu2nbt320zXXXNNi+wEAlIZVgVBQU1NYU6tedewoffWr0uWXL9PEiSO0\nbl04P2TevBHq3/9hjRixV0UXpy5VY2Ojzjxzy3qMHz9Og+K8DpFC+Lrqql9JukGSdNVVl0iSrrzy\nylj3AwAoHUOQKOjzn5duuEHq27faNSnfP/8pHXDAh1q/fpKk9NWo71SHDkO0YkUX7bJLcnUZOHC4\npk0bKmlE6pFxGjBggqZOzXGRyDJ167af3nnnp1vsZ8cdf6a33y5iYTQAQFEYgkSLeO+9cNHlY46p\ndk0qs/feUvfusxUu3JB2rPbcc2qi4QsAAIkAhgLmzZOOPbZ1LBJ6ww0dZbZa0jJJ/5TZWxozZrvE\n6zFy5AXq1GmUpHGSxqlTp1EaOfKC2Pdz2WXnS7rk0/1Il6QeAwBUG0OQyOtHP5K220766U+rXZN4\n3HffdP3rv66SJP3nf+6qM86ozsUtGxsbdf31YyWFQBb3/K+0a665Rr/+9S2SQiBj/hcAxKtqC7HG\njQBWW445Rvr1r6UTT6x2TQAAqD0EMMTugw+kHj2kt94KZxICAIAtMQkfsZs3L/SAEb4AAIgXAQw5\nzZ5d3+t/5ZLUIqi1Uo9qvN9aaWMAqFUMQSKnY4+VfvWr1hXCshdB7dRpVIssglor9ajG+62VNgaA\nJDAHDLH68EOpe/cw/+szn6l2beKT1CKotVKParzfWmljAEgCc8AQq/nzwwr4rSl8AQBQK7gWJCLN\nni01NFS7FvEbOfICzZs3QuvWhfthEdRxrbYe1Xi/tdLGAFDLGIJEpOOOk665Rurfv9o1iV9Si6DW\nSj2q8X5rpY0BoKUxBwyxWbNG2nVX6Y03pG22qXZtAACoXcwBQ2weekg6/HDCFwAALYUAhmbmzuXS\nQwAAtCQCGJohgAEA0LKYA4YtrF8v7bij9PLL0g47VLs2AADUNuaAIRaPPy599rOELwAAWhIBDFtg\n+BEAgJZHAMMWCGAAALQ85oDhU+7SzjtLixZJe+xR7doAAFD7mAOGii1ZInXpQvgCAKClxRLAzGyw\nmS0xs+fNbFSecseY2XozOyuO/SJeDD8CAJCMigOYmbWTdKOkQZJ6STrPzA7MUe5aSY2V7hMtY948\n6YQTql0LAABavzh6wHpLWuruy919vaQ7JA2LKPd9SXdLeiOGfaIF0AMGAEAy4ghgPSS9knH/1dRj\nnzKz3SWd4e43SSp5ohpa3quvSqtXSwc267sEAABx65DQfv5bUubcsLwhbPTo0Z/ebmhoUENDQ4tU\nCpulhx+NeAwAQE5NTU1qamqqeDsVL0NhZn0kjXb3wan7P5Hk7n5dRpmX0jcl7SRpjaQL3H1CxPZY\nhqIKvvc9aZ99pJEjq10TAADqRzWXoXhE0n5m1tPMtpZ0rqQtgpW775P6+azCPLDvRoUvVA/zvwAA\nSE7FQ5DuvtHMLpY0VSHQ3ezui83swvC0j81+SaX7RLzefVdatkw68shq1wQAgLaBlfChSZOkX/9a\nmjGj2jUBAKC+sBI+ysbwIwAAySKAgQAGAEDCGIJs49atk3baSVq1Surcudq1AQCgvjAEibI88ojU\nqxfhCwCAJBHA2jiGHwEASB4BrI3jAtwAACSPOWBt2MaNUrdu0tKl0s47V7s2AADUH+aAoWRPPil1\n7074AgAgaQSwNmzePOZ/AQBQDQSwNmzuXOZ/AQBQDQSwNsqdMyABAKgWAlgb9dJLUvv20t57V7sm\nAAC0PQSwNio9/Ggln7cBAAAqRQBroxh+BACgeghgbRQBDACA6iGAtUGrVklvvhmuAQkAAJJHAGuD\n5s2Tjj8+TMIHAADJI4C1QQw/AgBQXQSwNogV8AEAqC4uxt3GfPhhuP7j229LHTtWuzYAANQ3LsaN\nojz4oHTUUYQvAACqiQDWxjD8CABA9RHA2hguwA0AQPUxB6wN+eQTaccdpRUrpK5dq10bAADqX1Xn\ngJnZYDNbYmbPm9moiOe/bGZPpH7mmdmhcewXpXnsMWn//QlfAABUW8UBzMzaSbpR0iBJvSSdZ2YH\nZhV7SdJJ7n64pJ9L+kOl+0XpGH4EAKA2xNED1lvSUndf7u7rJd0haVhmAXd/yN3fT919SFKPGPaL\nErEAKwAAtSGOANZD0isZ919V/oD1bUmTY9gvSrBpkzR/Pj1gAADUgg5J7szMTpZ0viRiQMIWL5a2\n317affdq1wQAAMQRwFZI2ivj/h6px7ZgZodJGitpsLu/m2+Do0eP/vR2Q0ODGhoaYqhm28bwIwAA\nlWtqalJTU1PF26l4GQozay/pOUmnSFop6WFJ57n74owye0maIelr7v5Qge2xDEUL+MpXpP79pW99\nq9o1AQCg9ajaMhTuvlHSxZKmSnpG0h3uvtjMLjSzC1LFfippR0m/NbPHzezhSveL0rACPgAAtYOF\nWNuAl1+WjjlGev11yUrO6AAAIBcuxo2c0ut/Eb4AAKgNBLA2gOFHAABqCwGsDWAFfAAAagtzwFq5\nt9+WPvtZ6Z13pA6JrvoGAEDrxxwwRPrHP6Q+fQhfAADUEgJYK8fwIwAAtYcA1sqxAj4AALWHOWCt\n2Nq10s47S2++KW2zTbVrAwBA68McMDTz8MPSYYcRvgAAqDUEsFaM+V8AANQmAlgrxvwvAABqE3PA\nWqkNG6Qdd5SWLZO6dat2bQAAaJ2YA4YtPPGEtOeehC8AAGoRAayVYvgRAIDaRQBrpbgANwAAtYs5\nYK2Qu7TbbmEZip49q10bAABaL+aA4VNLl0odOxK+AACoVQSwVojhRwAAahsBrBViAVYAAGobAawV\n4gxIAABqGwGslVm5Unr3Xengg6tdEwAAkAsBrJWZN0/q21dqx28WAICaxcd0K8P8LwAAah8BrJVh\n/hcAALWPhVhbkfffl3r0kN55R9p662rXBgCA1q+qC7Ga2WAzW2Jmz5vZqBxlbjCzpWa2yMyOiGO/\n2NKDD0pHH034AgCg1lUcwMysnaQbJQ2S1EvSeWZ2YFaZIZL2dff9JV0o6XeV7hfNMfwIAEB9iKMH\nrLekpe6+3N3XS7pD0rCsMsMk3SpJ7r5AUlcz2zWGfSMDK+ADAFAf4ghgPSS9knH/1dRj+cqsiCiD\nCnz8sfToo1KfPtWuCQAAKKRDtSsQZfTo0Z/ebmhoUENDQ9XqUi+eeUY66CBpu+2qXRMAAFqvpqYm\nNTU1Vbydis+CNLM+kka7++DU/Z9Icne/LqPM7yTNcve/pe4vkdTP3VdFbI+zIMv0ySdMwAcAIEnV\nPAvyEUn7mVlPM9ta0rmSJmSVmSDp69Knge29qPCFyhC+AACoDxUPQbr7RjO7WNJUhUB3s7svNrML\nw9M+1t0fMLNTzewFSWsknV/pfgEAAOoVC7ECAACUqaoLsQIAAKB4BDAAAICEEcAAAAASRgADAABI\nGAEMAAAgYQQwAACAhBHAAAAAEkYAAwAASBgBDAAAIGEEMAAAgIQRwAAAABJGAAMAAEgYAQwAACBh\nBDAAAICEEcAAAAASRgADAABIGAEMAAAgYQQwAACAhBHAAAAAEkYAAwAASBgBDAAAIGEEMAAAgIQR\nwAAAABJGAAMAAEgYAQwAACBhFQUwM9vBzKaa2XNm1mhmXSPK7GFmM83sGTN7yswuqWSfyK2pqana\nVahrtF9laL/y0XaVof0qQ/tVR6U9YD+RNN3dPydppqTLI8pskHSZu/eSdJyk75nZgRXuFxH4I6oM\n7VcZ2q98tF1laL/K0H7VUWkAGyZpXOr2OElnZBdw99fdfVHq9mpJiyX1qHC/AAAAdavSALaLu6+S\nQtCStEu+wma2t6QjJC2ocL8AAAB1y9w9fwGzaZJ2zXxIkku6StL/ufuOGWXfdvduObbTWVKTpJ+5\n+3159pe/QgAAADXE3a3U13QoYqMDcj1nZqvMbFd3X2Vmu0l6I0e5DpLulnRbvvCV2l/JbwIAAKCe\nVDoEOUHSN1K3R0jKFa7+JOlZdx9T4f4AAADqXsEhyLwvNttR0p2S9pS0XNKX3P09M+su6Q/ufrqZ\n9ZU0R9JTCkOXLukKd59Sce0BAADqUEUBDAAAAKWruZXwzexXZrbYzBaZ2T1mtl2161QPzGywmS0x\ns+fNbFS161NPWCy4cmbWzsweM7MJ1a5LvTGzrmZ2V+r/vWfM7Nhq16memNnlqXZ70sz+YmZbV7tO\ntczMbk7N334y47GCi6ojZ9uVnVlqLoBJmiqpl7sfIWmpohd3RQYzayfpRkmDJPWSdB6L3ZaExYIr\n9wNJz1a7EnVqjKQH3P0gSYcrrJWIIphZT0nfkXSkux+mcGLZudWtVc27ReGzIlMxi6ojuu3Kziw1\nF8Dcfbq7b0rdfUjSHtWsT53oLWmpuy939/WS7lBYJBdFYLHgypjZHpJOlfTHatel3qS+LZ/o7rdI\nkrtvcPcPqlytevKBpE8kbZs6234bSa9Vt0q1zd3nSXo36+GCi6ojuu0qySw1F8CyfFPS5GpXog70\nkPRKxv1XRYAoC4sFl+U3kn6kcIINSvNZSW+Z2S2pIdyxZtap2pWqF+7+rqTrJb0saYWk99x9enVr\nVZdKWlQdOZWUWaoSwMxsWmq8Pv3zVOrfL2aUuVLSene/vRp1RNuTWiz4bkk/SPWEoQAzO03SqlQP\noqV+ULwOko6S9L/ufpSktQrDQSiCme0j6VJJPSXtLqmzmX25urVqFfgyVaJyMkvBhVhbQr7FXSXJ\nzL6hMKTRP5EK1b8VkvbKuL9H6jEUqZTFgrGFvpKGmtmpkjpJ6mJmt7r716tcr3rxqqRX3H1h6v7d\nkjiJpnhHS5rv7u9Ikpn9XdLxkvjiXpqiFlVHtHIzS80NQZrZYIXhjKHu/nG161MnHpG0n5n1TJ0B\ndK7CIrkoHosFl8Hdr3D3vdx9H4Xjbibhq3ipYZ9XzOyA1EOniJMZSvGcpD5m9hkzM4X24ySGwrJ7\nq4tdVB1ZbVdJZqm5dcDMbKmkrSW9nXroIXf/bhWrVBdSB8EYhVB9s7tfW+Uq1Q0WC46HmfWTNNLd\nh1a7LvXEzA5XOIFhK0kvSTrf3d+vbq3qh5n9SCE8bJT0uKRvp05GQgQzu11Sg6RuklZJulrSvZLu\nUtai6tWqY63K0XZXqMzMUnMBDAAAoLWruSFIAACA1o4ABgAAkDACGAAAQMIIYAAAAAkjgAEAACSM\nAAYAAJAwAhgAAEDCCGAAAAAJI4ABAAAkjAAGAACQMAIYAABAwghgAAAACSOAAQAAJIwABgAAkDAC\nGAAAQMIIYAAAAAkjgAEAACSMAAYAAJAwAhgAAEDCCGAAAAAJI4ABAAAkjAAGAACQMAIYAABAwioO\nYGbW0cwWmNnjZvaMmf0iokw/M3vPzB5L/VxV6X4BAADqVYdKN+DuH5vZye6+1szaS5pvZn3dfX5W\n0TnuPrTS/QEAANS7WIYg3X1t6mbH1DbfjShmcewLAACg3sUSwMysnZk9Lul1SU3u/mxEsePMbJGZ\nTTKzg+PYLwAAQD0yd49vY2bbSZoqaZS7z854vLOkTalhyiGSxrj7ATm2EV+FAAAAWpi7lzzKF+tZ\nkO7+gaRJko7Oenx1epjS3SdL2srMdsyzHX4yfq6++uqq16EWf2gX2oV2oU1oF9ql2j/liuMsyJ3M\nrGvqdidJAyQtyiqza8bt3go9b+9Uum8AAIB6VPFZkJK6SxpnZqYQ6G5z9xlmdqEkd/exkv7FzC6S\ntF7SOknnxLBfAACAuhTHMhRPSToq4vHfZ9z+X0n/W+m+2qqGhoZqV6Em0S7RaJdotEtztEk02iUa\n7RKvWCfhx8HMvNbqBAAAEMXM5NWehA8AAIDCCGAAAAAJI4ABQBvT2NiogQOHa+DA4WpsbKx2dVDj\nOF5aBnPAAKANaWxs1JlnjtC6dddJkjp1GqXx48dp0KBBVa4ZahHHS2HlzgEjgAFAGzJw4HBNmzZU\n0ojUI+M0YMAETZ16TzWrhRrF8VIYk/ABAADqRBwLsQIA6sTIkRdo3rwRWrcu3O/UaZRGjhxX3Uqh\nZnG8tByGIAGgjWlsbNT114+VFD5gmc+DfDhe8mMOGAAAQMKYAwYAAFAnCGAAAAAJI4ABAAAkjAAG\nAACQMAIYAABAwghgAAAACSOAAQAAJIwABgAAkDACGAAAQMIIYAAAAAkjgAEAACSMAAYAAJAwAhgA\nAEDCCGAAAAAJI4ABAAAkjAAGAACQMAIYAABAwghgAAAACSOAAQAAJIwABgAAkDACGAAAQMIIYAAA\nAAkjgAEAACSMAAYAAJAwAhgAAEDCCGAAAAAJI4ABAAAkjAAGAACQMAIYAABAwghgAAAACSOAAQAA\nJKziAGZmHc1sgZk9bmbPmNkvcpS7wcyWmtkiMzui0v0CAADUqw6VbsDdPzazk919rZm1lzTfzPq6\n+/x0GTMbImlfd9/fzI6V9DtJfSrdNwAAQD2KZQjS3dembnZMbfPdrCLDJN2aKrtAUlcz2zWOfQMA\nANSbWAKYmbUzs8clvS6pyd2fzSrSQ9IrGfdXpB4DAABoc+LqAdvk7kdK2kPSSWbWL47tAgAAtEYV\nzwHL5O4fmNkkSUdLmp3x1ApJe2bc3yP1WKTRo0d/eruhoUENDQ1xVhMAAKAsTU1Nampqqng75u6V\nbcBsJ0nr3f19M+skqVHSv7v7jIwyp0r6nrufZmZ9JP23u0dOwjczr7ROAAAASTAzubuV+ro4esC6\nSxpnZqYwpHmbu88wswslubuPdfcHzOxUM3tB0hpJ58ewXwAAgLpUcQ9Y3OgBAwAA9aLcHjBWwgcA\nAEgYAQwAACBhBDAAAICEEcAAAAASRgADAABIGAEMAAAgYQQwAACAhBHAAAAAEkYAAwAASBgBDAAA\nIGEEMAAAgIQRwAAAABJGAAMAAEgYAQwAACBhBDAAAICEEcAAAAASRgADAABIGAEMAAAgYQQwAACA\nhBHAgFaisbFRAwcO18CBw9XY2Fjt6gAA8jB3r3YdtmBmXmt1AmpdY2OjzjxzhNatu06S1KnTKI0f\nP06DBg2qcs0AoHUzM7m7lfy6Wgs7BDCgdAMHDte0aUMljUg9Mk4DBkzQ1Kn3VLNaANDqlRvAGIIE\nAABIWIdqVwBA5UaOvEDz5o3QunXhfqdOozRy5LjqVgoAkBNDkEAr0djYqOuvHyspBDLmfwFAy2MO\nGAAAQMKYAwYAAFAnCGAAAAAJI4ABAAAkjAAGAACQMAIYAABAwghgAAAACSOAAQAAJIwABgAAkDAC\nGAAAQMIIYAAAAAkjgAEAACSMAAYAAJAwAhgAAEDCCGAAAAAJI4ABAAAkjAAGAACQsIoDmJntYWYz\nzewZM3vKzC6JKNPPzN4zs8dSP1dVul8AAIB61SGGbWyQdJm7LzKzzpIeNbOp7r4kq9wcdx8aw/4A\nAADqWsU9YO7+ursvSt1eLWmxpB4RRa3SfQEAALQGsc4BM7O9JR0haUHE08eZ2SIzm2RmB8e5XwAA\ngHoSxxCkJCk1/Hi3pB+kesIyPSppL3dfa2ZDJN0r6YBc2xo9evSntxsaGtTQ0BBXNQEAAMrW1NSk\npqamirdj7l75Rsw6SJooabK7jymi/DJJn3f3dyKe8zjqBAAA0NLMTO5e8jSruIYg/yTp2Vzhy8x2\nzbjdWyH4NQtfAAAAbUHFQ5Bm1lfSVyQ9ZWaPS3JJV0jqKcndfaykfzGziyStl7RO0jmV7hcAAKBe\nxTIEGSeGIAEAQL2o9hAkAAAAikQAAwAASBgBDAAAIGEEMAAAgIQRwAAAABJGAAMAAEgYAQwAACBh\nBDAAAICEEcAAAAASRgADAABIGAEMAAAgYQQwAACAhBHAAAAAEkYAAwAASBgBDAAAIGEEMAAAgIQR\nwAAAABJGAAMAAEgYAQwAACBhBDAAAICEEcAAAAASRgADAABIGAEMAAAgYQQwAACAhBHAAAAAEkYA\nAwAASBgBDAAAIGEEMAAAgIQRwAAAABJGAAMAAEgYAQwAACBhBDAAAICEEcAAAAASRgADAABIGAEM\nAAAgYQQwAACAhBHAAAAAEkYAAwAASBgBDAAAIGEEMAAAgIQRwAAAABJGAAMAAEhYxQHMzPYws5lm\n9oyZPWVml+Qod4OZLTWzRWZ2RKX7BQAAqFcdYtjGBkmXufsiM+ss6VEzm+ruS9IFzGyIpH3dfX8z\nO1bS7yT1iWHfAAAAdafiHjB3f93dF6Vur5a0WFKPrGLDJN2aKrNAUlcz27XSfQMAANSjWOeAmdne\nko6QtCDrqR6SXsm4v0LNQxoAAECbEMcQpCQpNfx4t6QfpHrCyjZ69OhPbzc0NKihoaGiugEAAMSh\nqalJTU1NFW/H3L3yjZh1kDRR0mR3HxPx/O8kzXL3v6XuL5HUz91XRZT1OOoEAADQ0sxM7m6lvi6u\nIcg/SXo2KnylTJD0dUkysz6S3osKXwAAAG1BxT1gZtZX0hxJT0ny1M8VknpKcncfmyp3o6TBktZI\nOt/dH8uxPXrAAABAXSi3ByyWIcg4EcAAAEC9qPYQJAAAAIpEAAMAAEgYAQwAACBhBDAAAICEEcAA\nAAASRgCrYY2NjRo4cLgGDhyuxsbGalcHAADEhGUoalRjY6POPHOE1q27TpLUqdMojR8/ToMGDapy\nzQAAQBrrgLUyAwcO17RpQyWNSD0yTgMGTNDUqfdUs1oAACAD64ABAADUiQ7VrgCijRx5gebNG6F1\n68L9Tp1GaeTIcdWtFAAAiAVDkDWssbFR118/VlIIZMz/AgCgtjAHDAAAIGHMAQMAAKgTBDAAAICE\nEcAAAAASRgADAABIGAEMAAAgYQQwAACAhBHAAAAAEkYAAwAASBgBDAAAIGEEMAAAgIQRwAAAABJG\nAAMAAEgYAQwAACBhBDAAAICEEcAAAAASRgADAABIGAEMAAAgYQQwAACAhBHAAAAAEkYAAwAASBgB\nDAAAIGEEMAAAgIQRwAAAABJGAAMAAEgYAQwAACBhBDAAAICEEcAAAAASRgADAABIGAEMAAAgYQQw\nAACAhMUSwMzsZjNbZWZP5ni+n5m9Z2aPpX6uimO/bUVTU1O1q1CTaJdotEs02qU52iQa7RKNdolX\nXD1gt0gaVKDMHHc/KvXz85j22yZw0EejXaLRLtFol+Zok2i0SzTaJV6xBDB3nyfp3QLFLI59AQAA\n1Lsk54AdZ2aLzGySmR2c4H4BAABqirl7PBsy6ynpfnc/LOK5zpI2uftaMxsiaYy7H5BjO/FUCAAA\nIAHuXvIoX4eWqEg2d1+dcXuymf3WzHZ093ciyjJUCQAAWrU4hyBNOeZ5mdmuGbd7K/S8NQtfAAAA\nbUEsPWBmdrukBkndzOxlSVdL2lqSu/tYSf9iZhdJWi9pnaRz4tgvAABAPYptDhgAAACKU9WV8M1s\nBzObambPmVmjmXXNUe6fZvaEmT1uZg8nXc+kmNlgM1tiZs+b2agcZW4ws6WpM0qPSLqO1VCoXdri\nQr+FFj9OlWmLxwqLQmcxsz3MbKaZPWNmT5nZJTnKtanjpZh2aaPHS0czW5D6vH3GzH6Ro1ybOV6K\naZOyjhV3r9qPpOsk/Th1e5Ska3OUe0nSDtWsawJt0U7SC5J6StpK0iJJB2aVGSJpUur2sZIeqna9\na6Rd+kmaUO26JtwuJ0g6QtKTOZ5vc8dKke3SFo+V3SQdkbrdWdJz/N9SdLu0ueMl9b63Sf3bXtJD\nkvpyvBRsk5KPlWpfC3KYpHGp2+MknZGjnKn1X7eyt6Sl7r7c3ddLukOhfTINk3SrJLn7AkldM09w\naKWKaRepjS3064UXP26Lx0ox7SK1vWPldXdflLq9WtJiST2yirW546XIdpHa2PEiSe6+NnWzo8Jn\nb8nt4boAAAKeSURBVPbfVFs8Xgq1iVTisVLtULOLu6+Swh+DpF1ylHNJ08zsETP7TmK1S1YPSa9k\n3H9Vzf8zyC6zIqJMa1NMu0gs9JutLR4rxWqzx4qZ7a3QQ7gg66k2fbzkaRepDR4vZtbOzB6X9Lqk\nJnd/NqtImzteimgTqcRjpcXXATOzaZIyk7EpBKqo8dFcZwT0dfeVZrazQhBbnPqmC0jSo5L28s0L\n/d4rKXKhX7R5bfZYSS2IfbekH3jG2oxtXYF2aZPHi7tvknSkmW0naaqZ9XP32dWuVzUV0SYlHyst\n3gPm7gPc/bCMn0NT/06QtCrdbWlmu0l6I8c2Vqb+fVPSeIVhqdZmhaS9Mu7vkXosu8yeBcq0NgXb\nxd1Xp7uH3X2ypK3MbMfkqliT2uKxUlBbPVbMrINCyLjN3e+LKNImj5dC7dJWj5c0d/9A0iRJR2c9\n1SaPFyl3m5RzrFR7CHKCpG+kbo+Q1OwPwMy2SX1DkZltK2mgpKeTqmCCHpG0n5n1NLOtJZ2r0D6Z\nJkj6uiSZWR9J76WHcFuxgu1ibXeh35yLH6ttHitpLArd3J8kPevuY3I831aPl7zt0haPFzPbyVIr\nEphZJ0kDFE5+ytSmjpdi2qScYyWRSxHlcZ2kO83sm5KWS/qSJJlZd0l/cPfTFYYvx1u4RmQHSX9x\n96nVqnBLcfeNZnaxpKkKwfhmd19sZhcqtaCtuz9gZqea2QuS1kg6v5p1TkIx7aI2uNCvFVj8uC0e\nKxKLQkcxs76SviLpqdQcFpd0hcKZxW32eCmmXdQGjxdJ3SWNM7P0yW+3ufuMNv5ZVLBNVMaxwkKs\nAAAACav2ECQAAECbQwADAABIGAEMAAAgYQQwAACAhBHAAAAAEkYAAwAASBgBDAAAIGH/H18MSoRm\nR25rAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108cd5208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.special import fresnel, fresnel_zeros\n",
    "from scipy.optimize import minimize\n",
    "from mpmath import fresnels, fresnelc\n",
    "\n",
    "x = np.linspace(0, 10, 1000)\n",
    "def A(x):\n",
    "    f_sin, f_cos = fresnel(x)\n",
    "    return np.sqrt(f_sin**2 + f_cos**2)\n",
    "\n",
    "@np.vectorize\n",
    "def A2(x):\n",
    "    return np.sqrt(fresnels(x)**2 + fresnelc(x)**2)\n",
    "\n",
    "\n",
    "f, (ax1, ax2) = plt.subplots(2)\n",
    "ax1.plot(x, A(x))\n",
    "minima_guesses = (1.9,2.75, 3.4, 4.12)\n",
    "minima = np.array([minimize(A2, x0 = guess, tol = 0.000001).x for guess in minima_guesses])\n",
    "ax1.scatter(minima_guesses, [0.50] * len(minima_guesses))\n",
    "ax1.scatter(minima, [0.55] * len(minima_guesses))\n",
    "\n",
    "n = range(len(minima))\n",
    "ax2.scatter(n,minima)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'f_sin' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-5-6ae6771a66af>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf_sin\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mf_cos\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m\"b\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      2\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mf_sin\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0mf_cos\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m\"b\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNameError\u001b[0m: name 'f_sin' is not defined"
     ]
    }
   ],
   "source": [
    "plt.plot(f_sin, f_cos, \"b\")\n",
    "plt.plot(-f_sin, -f_cos, \"b\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}