Viewing_ABCD_matrices.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The goal of this notebook is to examine the ABCD matrices for a specific example and confirm several relationships between them as generated by the Potapov code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import Roots\n",
    "import Potapov\n",
    "import Time_Delay_Network\n",
    "import Time_Sims\n",
    "import functions\n",
    "import tests\n",
    "\n",
    "import numpy as np\n",
    "import numpy.linalg as la\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def contour_plot(Mat,func = abs):\n",
    "    '''\n",
    "    Make a simple plot to view a matrix\n",
    "\n",
    "    Args:\n",
    "        Mat (compelx-valued matrix): a matrix to view\n",
    "        func (optional[function]): a function to apply to each component\n",
    "\n",
    "    Generates a plot of the matrix.\n",
    "    '''\n",
    "    fig = plt.figure()\n",
    "    ax = fig.add_subplot(111)\n",
    "    cax = ax.matshow(func(Mat), interpolation='nearest')\n",
    "    fig.colorbar(cax)\n",
    "    plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Ex = Time_Delay_Network.Example3(r1 = 0.9, r3 = 0.9, max_linewidth=35.,max_freq=25.)\n",
    "Ex.run_Potapov()\n",
    "E = Ex.E\n",
    "roots = Ex.roots\n",
    "M1 = Ex.M1\n",
    "delays = Ex.delays\n",
    "modes = functions.spatial_modes(roots,M1,E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-0.36514939263783186-10.901513851604911j),\n",
       " (-0.33972600185569823+20.178467995423109j),\n",
       " (-0.36514939263783186+10.901513851604911j),\n",
       " (-0.33972600185569823-20.17846799542311j),\n",
       " (-0.3376937738186606-4.1474624048492497e-23j)]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "roots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A,B,C,D = Potapov.get_Potapov_ABCD(Ex.roots,Ex.vecs,Ex.T,z=0.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Omega = (A-A.H)/(2j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To obtain the SLH from the ABCD, we can use that for a passive system $\\Omega = (A - A^H)/2i$.\n",
    "\n",
    "We generate $B$ and $C$ in such a way that $B = C^H$.\n",
    "\n",
    "\n",
    "Also notice that $(A+A^H)/2 = BC/2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e633250>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "contour_plot(Omega)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e3a6b90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## Let's look at the off-diagonal elements\n",
    "Omega2 = - (A.imag+A.imag.H)/2\n",
    "for i in range(Omega2.shape[0]):\n",
    "    Omega2[i,i]=0\n",
    "    \n",
    "contour_plot(Omega2)\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Yl5klx2NcZpYcT4cws+S4q2hmyXFX0cyS47uKZpYcJy4zS864TVxxylXtPF3nPZ5WvOKG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\nxGVmyXHiMrPk/D99oFH6XT9PIQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x103da6190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "contour_plot((A+A.H)/2.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Yl5klx2NcZpYcT4cws+S4q2hmyXFX0cyS47uKZpYcJy4zS864TVxxylXtPF3nPZ5WvOKG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\nxGVmyXHiMrPk/D99oFH6XT9PIQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b0f4190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "contour_plot(B*C/2.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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bbU/OarpjnhB9D/AQ8IJe12sy6+C6Xm0iIoZt+lhfI59rgIdtPwogaRtwDTA7aOf8m7yk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\n6S0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b0083d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "contour_plot(B.H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b0f3c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "contour_plot(C)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}