{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Using the RCW module"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The RCW module of pySCATMECH implements the `SCATMECH::RCW_Model` class, which is a rigorous coupled wave (RCW) solver for one-dimensionally periodic gratings. This notebook will demonstrate a number of its features.\n",
"\n",
"First, we import the module. Since we are also using `numpy` and `matplotlib`, we will import those, too."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"from pySCATMECH.rcw import *"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Single line grating\n",
"\n",
"We will start our demonstration with a single line grating. We can create a `RCW_Model` with parameters:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"{None : 'RCW_Model',\n",
"'order' : '25',\n",
"'type' : '0',\n",
"'lambda' : '0.532',\n",
"'thetai' : '70',\n",
"'rotation' : '0',\n",
"'grating' : {None : 'Single_Line_Grating',\n",
" 'period' : '0.2',\n",
" 'medium_i' : '(1,0)',\n",
" 'medium_t' : 'silicon',\n",
" 'material' : 'silicon',\n",
" 'space' : '(1,0)',\n",
" 'height' : '0.2',\n",
" 'topwidth' : '0.1',\n",
" 'bottomwidth' : '0.15',\n",
" 'offset' : '0',\n",
" 'nlevels' : '10'}}\n"
]
}
],
"source": [
"gratingParameters = {\n",
" None : 'Single_Line_Grating', # The Grating model can be assigned with the `None` key\n",
" 'period' : 0.2, # The period \n",
" 'medium_i' : 1, # The incident medium0\n",
" 'medium_t' : 'silicon', # The transmitting (substrate) material\n",
" 'material' : 'silicon', # The grating material\n",
" 'space' : 1, # The space material\n",
" 'height' : 0.2, # The height\n",
" 'topwidth' : 0.1, # Top width\n",
" 'bottomwidth' : 0.15, # Bottom width\n",
" 'offset' : 0, # Offset of top, relative to bottom\n",
" 'nlevels' : 10} # Number of layers in the staircase approximation\n",
"\n",
"rcwParameters = {\n",
" 'order' : 25, # number of Floquet orders considered\n",
" 'type' : 0, # 0 indicates reflection\n",
" 'lambda' : 0.532, # wavelength\n",
" 'thetai' : 70, # incident angle in degrees \n",
" 'rotation' : 0, # rotation angle in degrees\n",
" 'grating' : gratingParameters\n",
" }\n",
"\n",
"model = RCW_Model(rcwParameters)\n",
"\n",
"print(model)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's show the NCS parameters for the 0th order diffraction (specular) as a function of wavelength:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\r\n",
"\r\n",
"\r\n",
"\r\n"
],
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"# 250 wavelengths from 0.2 to 1.5...\n",
"wavelengths = np.linspace(0.2, 1.5, 250)\n",
"\n",
"def rcw(wavelength):\n",
" rcwParameters['lambda'] = wavelength\n",
" model.setParameters(rcwParameters)\n",
" return model.DiffractionEfficiency(0)\n",
"\n",
"muellerMatrices = [rcw(wavelength) for wavelength in wavelengths]\n",
"N = [m[0,1] for m in muellerMatrices]\n",
"C = [m[2,2] for m in muellerMatrices]\n",
"S = [m[2,3] for m in muellerMatrices]\n",
"\n",
"plt.plot(wavelengths*1000, N, label=\"N\")\n",
"plt.plot(wavelengths*1000, C, label=\"C\")\n",
"plt.plot(wavelengths*1000, S, label=\"S\")\n",
"plt.xlabel('$\\lambda$ / nm')\n",
"plt.legend()\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's do the same thing, but rotate the sample 45 degrees. The Mueller matrices are no longer block diagonal."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\r\n",
"\r\n",
"\r\n",
"\r\n"
],
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"wavelengths = np.linspace(0.2, 1.5, 250)\n",
"\n",
"rcwParameters['rotation'] = 45\n",
"model.setParameters(rcwParameters)\n",
"\n",
"def rcw(wavelength):\n",
" rcwParameters['lambda'] = wavelength\n",
" model.setParameters(rcwParameters)\n",
" return model.DiffractionEfficiency(0)\n",
"\n",
"muellerMatrices = [rcw(wavelength) for wavelength in wavelengths]\n",
"\n",
"plt.figure(figsize = [13,13])\n",
"for element in range(16):\n",
" plt.subplot(4,4,element+1)\n",
" if element==0:\n",
" plt.ylim(0,1)\n",
" else:\n",
" plt.ylim(-1,1)\n",
" plt.plot(wavelengths,[m[element//4,element%4] for m in muellerMatrices],'b-')\n",
"\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What does this grating look like?"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\r\n",
"\r\n",
"\r\n",
"\r\n"
],
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"def showGrating(model,x,z):\n",
" xx, zz = np.meshgrid(x, z, sparse=True)\n",
"\n",
" def eps(x,z):\n",
" return model.getEpsilon(x,z).real\n",
"\n",
" epsilon = np.array([[eps(x0,z0) for x0 in x] for z0 in z])\n",
"\n",
" fig = plt.figure(figsize=(6,5))\n",
" ax = fig.add_subplot(111)\n",
" h = ax.contourf(x,z,epsilon,256,cmap=\"brg\")\n",
" plt.colorbar(h,ax=ax)\n",
"\n",
" plt.show()\n",
" \n",
"showGrating(model,np.linspace(-0.2, 0.2, 200),np.linspace(-0.3,.1,200))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One can view the diffraction pattern:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\r\n",
"\r\n",
"\r\n",
"\r\n"
],
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"gratingParameters = {\n",
" None : 'Single_Line_Grating',\n",
" 'period' : 10,\n",
" 'medium_i' : 1,\n",
" 'medium_t' : 'silicon', \n",
" 'material' : 'silicon',\n",
" 'space' : 1,\n",
" 'height' : '0.5',\n",
" 'topwidth' : '0.4',\n",
" 'bottomwidth' : '0.15',\n",
" 'offset' : '0',\n",
" 'nlevels' : '10'}\n",
"\n",
"rcwParameters = {\n",
" None : 'RCW_Model',\n",
" 'order' : '25',\n",
" 'type' : '0',\n",
" 'lambda' : '0.532',\n",
" 'thetai' : 45,\n",
" 'rotation' : 45,\n",
" 'grating' : gratingParameters\n",
" }\n",
"\n",
"model.setParameters(rcwParameters)\n",
"\n",
"def plotDiffractionPattern(model):\n",
" order = int(model.getParameter(\"order\"))\n",
" x, y, I = [], [], []\n",
" for i in range(-order,order):\n",
" direction = model.Direction(i)\n",
" if direction[2] != 0 :\n",
" x.append(direction[0])\n",
" y.append(direction[1])\n",
" I.append(model.DiffractionEfficiency(i)[0,0])\n",
" \n",
" thetas = np.linspace(0, 2*pi, 100)\n",
" circlex = np.cos(thetas)\n",
" circley = np.sin(thetas)\n",
" \n",
" plt.figure(figsize = (10,10))\n",
" ax = plt.subplot(111)\n",
" ax.scatter(x, y, s=np.array(I)*100) \n",
" ax.plot(circlex, circley, c = 'r')\n",
" plt.show()\n",
"\n",
"plotDiffractionPattern(model)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Blazed reflection grating"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\r\n",
"\r\n",
"\r\n",
"\r\n"
],
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/svg+xml": [
"\r\n",
"\r\n",
"\r\n",
"\r\n"
],
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"# Set the lines per millimeter:\n",
"lpm = 600\n",
"# Set the nominal wavelength in µm:\n",
"blazeLambda = 1.00\n",
"\n",
"##########################\n",
"period = 1000/lpm\n",
"sinBlaze = blazeLambda/2/period\n",
"height = period*sinBlaze\n",
"##########################\n",
"\n",
"# Description of the grating...\n",
"gratingParameters = {None: 'Triangular_Grating', \n",
" 'period': 1000/lpm, \n",
" 'medium_i': 1, \n",
" 'medium_t': 'aluminum', \n",
" 'material': 'aluminum',\n",
" 'amplitude' : height,\n",
" 'aspect' : 1,\n",
" 'nlevels' : 20}\n",
"\n",
"# The RCW parameters\n",
"rcwParameters = {\n",
" None : 'RCW_Model',\n",
" 'order' : 25,\n",
" 'type' : 0,\n",
" 'lambda' : 0.500,\n",
" 'thetai' : 0,\n",
" 'rotation' : 0,\n",
" 'grating' : gratingParameters\n",
" }\n",
"model = RCW_Model(rcwParameters)\n",
"\n",
"def efficiency(wavelength):\n",
" \"\"\"Set the wavelength and calculate the efficiency\"\"\"\n",
" thetai=math.asin(wavelength/2/period)\n",
" model.setParameters(wavelength=wavelength,thetai=thetai)\n",
" return model.DiffractionEfficiency(1)\n",
"\n",
"wavelengths = np.arange(0.2, 1.5, 0.02)\n",
"\n",
"eff = [efficiency(L) for L in wavelengths]\n",
"effs = [Sensitivity('u') @ e @ Polarization('s') for e in eff]\n",
"effp = [Sensitivity('u') @ e @ Polarization('p') for e in eff]\n",
"\n",
"plt.figure()\n",
"plt.plot(wavelengths, effs, label = \"s-pol\")\n",
"plt.plot(wavelengths, effp, label = \"p-pol\")\n",
"plt.xlabel(\"$\\lambda$/µm\")\n",
"plt.ylabel(\"Diffraction efficiency\")\n",
"plt.legend()\n",
"plt.show()\n",
"\n",
"showGrating(model,np.linspace(-6, 6, 200),np.linspace(-6,6,200))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Sinusoidal transmission grating "
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"image/svg+xml": [
"\r\n",
"\r\n",
"\r\n",
"\r\n"
],
"text/plain": [
""
]
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"needs_background": "light"
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"\r\n",
"\r\n",
"\r\n",
"\r\n"
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""
]
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"output_type": "display_data"
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],
"source": [
"# Set the lines per millimeter:\n",
"lpm = 1000\n",
"# Sinusoid amplitude:\n",
"amplitude = .5\n",
"# Incident angle:\n",
"thetai = 45*deg\n",
"\n",
"\n",
"period = 1000/lpm\n",
"from pySCATMECH.fresnel import OpticalFunction\n",
"\n",
"FusedSilica = OpticalFunction(lambda L: 1.4580 + 0.00354/L**2, np.arange(0.2,1.5,0.05))\n",
"\n",
"# Description of the grating...\n",
"gratingParameters = {None: 'Sinusoidal_Relief_Grating', \n",
" 'period': period, \n",
" 'medium_i': 1, \n",
" 'medium_t': FusedSilica, \n",
" 'material': FusedSilica,\n",
" 'amplitude' : amplitude,\n",
" 'base' : 0,\n",
" 'option' : 0,\n",
" 'nlevels' : 20}\n",
"\n",
"# The RCW parameters\n",
"rcwParameters = {\n",
" None : 'RCW_Model',\n",
" 'order' : 10,\n",
" 'type' : 1,\n",
" 'lambda' : 0.500,\n",
" 'thetai' : 0,\n",
" 'rotation' : 0,\n",
" 'grating' : gratingParameters\n",
" }\n",
"model = RCW_Model(rcwParameters)\n",
"\n",
"def efficiency(wavelength):\n",
" \"\"\"Set the wavelength and calculate the efficiency\"\"\"\n",
" rcwParameters['lambda'] = wavelength\n",
" model.setParameters(rcwParameters)\n",
" return model.DiffractionEfficiency(1)\n",
"\n",
"wavelengths = np.arange(0.2, 1.5, 0.02)\n",
"\n",
"eff = [efficiency(L) for L in wavelengths]\n",
"\n",
"effs = [Sensitivity('u') @ e @ Polarization('s') for e in eff]\n",
"effp = [Sensitivity('u') @ e @ Polarization('p') for e in eff]\n",
"\n",
"plt.figure()\n",
"plt.plot(wavelengths, effs, label = \"s-pol\")\n",
"plt.plot(wavelengths, effp, label = \"p-pol\")\n",
"plt.legend()\n",
"plt.show()\n",
"\n",
"showGrating(model,np.linspace(-6, 6, 200),np.linspace(-6,6,200))"
]
}
],
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