Wave interference and diffraction.ipynb

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   "source": [
    "# Wave Interference\n",
    "\n",
    "The plotted function is the superimposition of $n$ 2d-sine  functions centered in $(0,Y_i)$:\n",
    "$$f(x,y)=\\sin\\big(p\\cdot\\sqrt{x^2+(y-Y_i)^2}\\big)$$\n",
    "\n",
    "The parameters that can be changed are:\n",
    "\n",
    "-- $n$ the number of sources (start with 2)\n",
    "\n",
    "-- $d$ the distance between sources\n",
    "\n",
    "-- $p$ the periodicity. Higher periodicity means shorter wavelength. $p =2\\pi / \\lambda$. Note that in electron microscopy the usual convention is to define the wave vector as $k=1/\\lambda$. \n",
    "\n",
    "-- $f$ the field of view.\n",
    "\n",
    "One sees the directions of space where intereferences are constructive / destructive, with the angle between the outgoing beams increasing with decreasing $p$, and decreasing source spacing.\n",
    "\n",
    "## Hints for the use\n",
    "\n",
    "Start with $n=2$. \n",
    "\n",
    "Press on \"run interact\" to initiate the plotting after changing the parameters. \n",
    "\n",
    "With many sources, the intereference patterns close to the source is quite intricated. One needs to go to the far field (increase $f$ the field of view) to see the formation of the outgoing beams.\n",
    "\n",
    "With more sources (try up to 10 sources), the outgoing beams are getting thinner."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import matplotlib\n",
    "\n",
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "import ipywidgets as widgets\n",
    "matplotlib.rcParams['figure.figsize'] = (16,16)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from bokeh.plotting import figure, show, gridplot\n",
    "from bokeh.io import output_notebook, push_notebook\n",
    "from bokeh.palettes import Viridis256\n",
    "output_notebook()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
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   "source": [
    "# n is the number of sources\n",
    "# d distance between sources\n",
    "# p changes periodicity\n",
    "# f field of view\n",
    "\n",
    "def waveplot2d(n,d,p,f):\n",
    "    ###################\n",
    "    #Calculation part #\n",
    "    ###################\n",
    "    \n",
    "    #2D grid definition\n",
    "    x=np.linspace(0,f,2001)\n",
    "    y=np.linspace(-f/2,f/2,2001)\n",
    "    X, Y = np.meshgrid(x, y)\n",
    "    \n",
    "    #Layout of the sources\n",
    "    c=np.linspace(-(n-1)*d/2,(n-1)*d/2,n)\n",
    "    c0=np.linspace(0,0,n)\n",
    "    print(c)\n",
    "    \n",
    "    #Far field distance\n",
    "    far=4*(d*n)**2/p\n",
    "    \n",
    "    #Calculation of the wave equations\n",
    "    Z=0*X+0*Y\n",
    "    Z_far=0*y\n",
    "    for i in c:\n",
    "        #2D representation\n",
    "        Zi=(np.sin(p*np.sqrt(X**2+(Y-i)**2)))**1\n",
    "        Z=Z+Zi\n",
    "        #Far field\n",
    "        Zi_far=(np.sin(p*np.sqrt(far**2+(y-i)**2)))**1\n",
    "        Z_far+=Zi_far\n",
    "    int_Z_far=np.power(Z_far,2)\n",
    "    \n",
    "    #############\n",
    "    #Bokeh part #\n",
    "    #############\n",
    "    \n",
    "    #2D map\n",
    "    tooltips=[(\"value\",\"@image\"),(\"measured intensity\",\"@intensity\"),(\"x\", \"$x\"),(\"y\", \"$y\")]\n",
    "    dict_Z={\"image\":[Z],\"intensity\":[np.power(Z,2)],\"x\":[0],\"y\":[-f/2]}\n",
    "    p=figure(tools=\"hover,pan,wheel_zoom,box_zoom,reset\",\n",
    "             tooltips=tooltips,\n",
    "             title=\"2D map of the wave interference between {} sources\".format(n))\n",
    "    p.image(source=dict_Z,image=\"image\",x=\"x\",y=\"y\",dw=f,dh=f,palette=Viridis256)\n",
    "    \n",
    "    #Far field line\n",
    "    pv = figure(toolbar_location=None,\n",
    "                width=200, height=p.height,\n",
    "                x_range=(min(int_Z_far)-0.05*abs(min(int_Z_far)),max(int_Z_far)+0.05*abs(max(int_Z_far))),\n",
    "                y_range=p.y_range,\n",
    "                min_border=10,\n",
    "                y_axis_location=\"right\",\n",
    "                title=\"intensity at {:2f} distance\".format(far))\n",
    "    \n",
    "    #Comment out this out to plot the wave in the far field\n",
    "    #pv.line(Z_far,y,legend_label='value', color='navy')\n",
    "    \n",
    "    pv.line(int_Z_far,y,legend_label='intensity', color='red')\n",
    "    \n",
    "    #Comment out this line to display the legend\n",
    "    #pv.legend.location = \"top_left\"\n",
    "    \n",
    "    #Layout and plotting\n",
    "    layout = gridplot([[p, pv], [None, None]], merge_tools=False)\n",
    "    show(layout,notebook_handle=True)\n",
    "    push_notebook()\n",
    "\n",
    "    \n",
    "    \n",
    "#Interactive \n",
    "n=2\n",
    "#interact(waveplot2d,n=(1,10,1),d=(2,20,1),p=(1,5,1),f=(50,900,50))\n",
    "interact(waveplot2d,n=(2,2,1),d=(5,5,1),p=(0.8*np.pi/5,0.8*np.pi/5,1),f=(50,900,50))"
   ]
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   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "# Version chauves souris\n",
    "# 2 sources (narines)\n",
    "# d distance entre les sources (mm)entre 2 et 3\n",
    "# longueur d'onde: 5mm\n",
    "\n",
    "lam = 5 #longueur d'onde\n",
    "p = 2*np.pi/lam #période de l'onde\n",
    "n = 2 #nombre de narines\n",
    "f = 450 #champ de vue\n",
    "\n",
    "\n",
    "def waveplot2d(n,d,p,f):\n",
    "    ###################\n",
    "    #Calculation part #\n",
    "    ###################\n",
    "    \n",
    "    #2D grid definition\n",
    "    x=np.linspace(0,f,2001)\n",
    "    y=np.linspace(-f/2,f/2,2001)\n",
    "    X, Y = np.meshgrid(x, y)\n",
    "    \n",
    "    #Layout of the sources\n",
    "    c=np.linspace(-(n-1)*d/2,(n-1)*d/2,n)\n",
    "    c0=np.linspace(0,0,n)\n",
    "    print(c)\n",
    "    \n",
    "    #Far field distance\n",
    "    far=4*(d*n)**2/p\n",
    "    \n",
    "    #Calculation of the wave equations\n",
    "    Z=0*X+0*Y\n",
    "    Z_far=0*y\n",
    "    for i in c:\n",
    "        #2D representation\n",
    "        Zi=(np.sin(p*np.sqrt(X**2+(Y-i)**2)))**1\n",
    "        Z=Z+Zi\n",
    "        #Far field\n",
    "        Zi_far=(np.sin(p*np.sqrt(far**2+(y-i)**2)))**1\n",
    "        Z_far+=Zi_far\n",
    "    int_Z_far=np.power(Z_far,2)\n",
    "    \n",
    "    #############\n",
    "    #Bokeh part #\n",
    "    #############\n",
    "    \n",
    "    #2D map\n",
    "    tooltips=[(\"value\",\"@image\"),(\"measured intensity\",\"@intensity\"),(\"x\", \"$x\"),(\"y\", \"$y\")]\n",
    "    dict_Z={\"image\":[Z],\"intensity\":[np.power(Z,2)],\"x\":[0],\"y\":[-f/2]}\n",
    "    p=figure(tools=\"hover,pan,wheel_zoom,box_zoom,reset\",\n",
    "             tooltips=tooltips,\n",
    "             title=\"2D map of the wave interference between {} sources\".format(n))\n",
    "    p.image(source=dict_Z,image=\"image\",x=\"x\",y=\"y\",dw=f,dh=f,palette=Viridis256)\n",
    "    \n",
    "    #Far field line\n",
    "    pv = figure(toolbar_location=None,\n",
    "                plot_width=200, plot_height=p.plot_height,\n",
    "                x_range=(min(int_Z_far)-0.05*abs(min(int_Z_far)),max(int_Z_far)+0.05*abs(max(int_Z_far))),\n",
    "                y_range=p.y_range,\n",
    "                min_border=10,\n",
    "                y_axis_location=\"right\",\n",
    "                title=\"intensity at {:2f} distance\".format(far))\n",
    "    \n",
    "    #Comment out this out to plot the wave in the far field\n",
    "    #pv.line(Z_far,y,legend_label='value', color='navy')\n",
    "    \n",
    "    pv.line(int_Z_far,y,legend_label='intensity', color='red')\n",
    "    \n",
    "    #Comment out this line to display the legend\n",
    "    #pv.legend.location = \"top_left\"\n",
    "    \n",
    "    #Layout and plotting\n",
    "    layout = gridplot([[p, pv], [None, None]], merge_tools=False)\n",
    "    show(layout,notebook_handle=True)\n",
    "    push_notebook()\n",
    "\n",
    "    \n",
    "    \n",
    "\n",
    "#interact(waveplot2d,n=2,p=2*np.pi/lam,f=100,d=(1,50,.5))\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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