{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cherenkov Radiation\n",
    "Charged particles radiate when travelling faster than light in a medium. This is because the region of non-zero field grows with time, so it occurs even when the particle is in uniform motion.\n",
    "In contrast, if the particle moves slower than the speed of light in the medium, it can't outrun its own electromagnetic field."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 1,
     "metadata": {
      "image/png": {
       "width": 700
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(filename=\"cherenkov.png\", width = 700)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To change the properties of the plot, adjust the function `R(T)` and the evaluation time, `time`, near the bottom of the program. This code won't handle superluminal velocities on curved paths since the potential is computed from only the most recent point on the particle's worldline which satisfies the lightcone condition."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import linalg as la\n",
    "\n",
    "#speed of light in medium\n",
    "cn = 1.0\n",
    "#charge of particle\n",
    "q = 1.0\n",
    "\n",
    "#time discretisation\n",
    "dt = 0.005\n",
    "Nsteps = 200\n",
    "t = [dt*i for i in range(Nsteps)]\n",
    "\n",
    "#space discretisation\n",
    "dx = 0.01\n",
    "N = 100\n",
    "x = np.linspace(-0.5*N*dx, 0.5*N*dx, N)\n",
    "y = np.linspace(-0.5*N*dx, 0.5*N*dx, N)\n",
    "\n",
    "#discretisation error\n",
    "square_error = dt**2 + 2*(dx/cn)**2\n",
    "\n",
    "#define particle position in xy plane\n",
    "def R(T):\n",
    "    #particle at rest at (0,0) for T<0\n",
    "    X = 0.0\n",
    "    Y = 0.0\n",
    "    if T >= 0:\n",
    "        X = min(x) + 1.0*T**2\n",
    "        Y = max(y) - 1.0*T**2\n",
    "    return np.array([X,Y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To find the electric scalar potential at a given position and time, we search the worldline of the particle to find emission times satisfying the lightcone condition. \n",
    "For general time-varying charge distributions it can be calculated using the retarded potential\n",
    "\n",
    "$$\\phi(\\vec{r}, t) = \\int \\frac{\\rho(\\vec{r}, t - \\frac{|\\vec{r}-\\vec{R}|}{c})}{|\\vec{r}-\\vec{R}|} d^3\\vec{R}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#find the difference between time and space\n",
    "#interval for obversation and emission.\n",
    "#off_worldline=0 is the lightcone condition.\n",
    "def off_worldline(emission_time, position, time):\n",
    "    return (time - emission_time - la.norm(position - R(emission_time))/cn)**2\n",
    "\n",
    "#for a given position and time, find the time\n",
    "#at which the received radiation was emitted\n",
    "def get_emission_time(position,time):\n",
    "    emission_time = 0.0\n",
    "    #check all emission times < time in reverse order\n",
    "    for T in (t[0:t.index(time)])[::-1]:\n",
    "        ow = off_worldline(T, position, time)\n",
    "        if (ow < square_error):\n",
    "            emission_time = T\n",
    "            break\n",
    "    return emission_time\n",
    "\n",
    "#calculate the scalar potential\n",
    "def get_phi(position, time):\n",
    "    T = get_emission_time(position, time)\n",
    "    return q/(4.0*np.pi*la.norm(position - R(T)))\n",
    "    \n",
    "#scalar potential\n",
    "phi = np.zeros((N,N))\n",
    "time = t[-1]\n",
    "it = np.nditer(phi, ['multi_index'])\n",
    "for element in it:\n",
    "    i,j = it.multi_index\n",
    "    position = np.array([x[j],y[i]])\n",
    "    phi[i,j] = get_phi(position, time)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Particles satisfying the Cherenkov condition (i.e. travelling faster than light does) emit cones of radiation around their paths. The angle at which the radiation is emitted is\n",
    "$$cos\\theta = \\frac{1}{n \\beta},$$\n",
    "where $\\beta = \\nu/c$ and $\\nu$ is the particle speed. Since refractive index is a function of wavelength, different wavelengths are emitted in different directions. Also, for a given particle speed, the Cherenkov condition may be satisfied for only a finite range of wavelengths. This often gives it a characteristic colour."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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pGxDVViapkJTAp9Rt5KzfxP5TAtDqrp+rraDL3Dqk7D+lDYeUuSvluFLez3ZP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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1ccac5b9ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "#plot the electric scalar potential\n",
    "fig = plt.figure(figsize = (12,9))\n",
    "plt.pcolor(x,y,np.log10(phi), cmap = 'jet')\n",
    "plt.xlim(min(x), max(x))\n",
    "plt.ylim(min(y), max(y))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title(\"Electric Scalar Potential of a Moving Point Charge\")\n",
    "plt.colorbar()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The particle in the plot started at rest in the top left and accelerated under a constant force to the bottom right. It was initially subliminal, so there is no Cherenkov cone there. Instead, the field is diffuse as for a charge at rest. Once the charge had a speed greater than $c/n$, any emitted radiation was confined to the Cherenkov cone, as in the bottom right.\n",
    "\n",
    "The transition between these two regimes created the gradual gathering of radiation into a cone."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [Root]",
   "language": "python",
   "name": "Python [Root]"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}