zhaonat · yonghakim · May 23, 2022 · May 23, 2022 · May 23, 2022 · May 31, 2022
diff --git a/RCWA_1D_examples/1D_Grating_Gaylord_TE.py b/RCWA_1D_examples/1D_Grating_Gaylord_TE.py
@@ -237,6 +237,11 @@ def grating_fft(eps_r):
     #print(np.linalg.cond(X@b)); #not well conditioned.
 
     term = X@a@fbiX; # THIS IS SHITTILY CONDITIONED
+    # This is different from the paper which is X @ b @ a_i @ f @ X.
+    # k_I and k_II have positive sign in imaginary part and it's the opposite of the paper.
+    # And because of this, a and b are switched here so it works.
+    # To correct this, use conjugate() method to inverse the sign of the imaginary part.
+
     # print((np.linalg.cond(X), np.linalg.cond(term)))
     # print(np.linalg.cond(I+term)); #but this is EXTREMELY WELL CONDITIONED.
     f = np.matmul(W, I+term);

diff --git a/RCWA_1D_examples/1D_Grating_Gaylord_TM.py b/RCWA_1D_examples/1D_Grating_Gaylord_TM.py
@@ -229,6 +229,11 @@ def grating_fft(eps_r):
     b = ab[PQ:,:];
 
     term = X @ a @ np.linalg.inv(b) @ X;
+    # This is different from the paper which is X @ b @ a_i @ f @ X.
+    # k_I and k_II have positive sign in imaginary part and it's the opposite of the paper.
+    # And because of this, a and b are switched here so it works.
+    # To correct this, use conjugate() method to inverse the sign of the imaginary part.
+
     f = W @ (I+term);
     g = V@(-I+term);
     T = np.linalg.inv(np.matmul(j*Z_I, f) + g);

diff --git a/notebooks/RCWA/RCWA_derivation.ipynb b/notebooks/RCWA/RCWA_derivation.ipynb
@@ -8,7 +8,7 @@
    "source": [
     "# Note on the derivation of the reflection and transmission coefficents in\n",
     "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings\n",
-    "M. G. Moharam, Eric B. Grann, Drew A. Pommet, and T. K. Gaylord </br>\n",
+    "M. G. Moharam, Eric B. Grann, Drew A. Pommet, and T. K. Gaylord <br>\n",
     "\n",
     "The final step in this paper seems straightforward but is actually non-trivial to work out to get the final working RCWA code"
    ]
@@ -17,47 +17,46 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We start with a system of four equations: \n",
-    "\n",
-    "$\\begin{bmatrix} \n",
-    "\\delta_{i0}  \\\\ \n",
-    "jn_Icos(\\theta)\\delta_{i0}  \\\\   \n",
-    "\\end{bmatrix}$ + \n",
-    "$\\begin{bmatrix} \n",
-    "I  \\\\ \n",
-    "-jY_I \\\\   \n",
-    "\\end{bmatrix}$[R] = \n",
-    "$\\begin{bmatrix} \n",
-    "W  & WX\\\\ \n",
-    "V & -VX  \\\\   \n",
-    "\\end{bmatrix}$$\\begin{bmatrix} \n",
-    "c^{+}  \\\\ \n",
-    "c^{-}  \\\\   \n",
-    "\\end{bmatrix}$\n",
-    "\n",
-    "\n",
-    "$\\begin{bmatrix} \n",
-    "I  \\\\ \n",
-    "jY_{II} \\\\   \n",
-    "\\end{bmatrix}$[T] = \n",
-    "$\\begin{bmatrix} \n",
-    "WX  & W\\\\ \n",
-    "VX & -V  \\\\   \n",
-    "\\end{bmatrix}$$\\begin{bmatrix} \n",
-    "c^{+}  \\\\ \n",
-    "c^{-}  \\\\   \n",
-    "\\end{bmatrix}$\n",
+    "We start with a system of four equations:\n",
+    "\n",
+    "$\\begin{align*}\n",
+    "\\begin{bmatrix}\n",
+    "\\delta_{i0}  \\\\ jn_Icos(\\theta)\\delta_{i0}\n",
+    "\\end{bmatrix} +\n",
+    "\\begin{bmatrix}\n",
+    "I  \\\\\n",
+    "-jY_I\n",
+    "\\end{bmatrix}R &= \\begin{bmatrix}\n",
+    "W & WX \\\\\n",
+    "V & -VX\n",
+    "\\end{bmatrix}\n",
+    "\\begin{bmatrix}\n",
+    "c^{+} \\\\\n",
+    "c^{-} \\\\\n",
+    "\\end{bmatrix} \\\\\n",
+    "\\begin{bmatrix}\n",
+    "I  \\\\\n",
+    "jY_{II} \\\\\n",
+    "\\end{bmatrix}T &=\n",
+    "\\begin{bmatrix}\n",
+    "WX & W \\\\\n",
+    "VX & -V \\\\\n",
+    "\\end{bmatrix}\n",
+    "\\begin{bmatrix}\n",
+    "c^{+} \\\\\n",
+    "c^{-} \\\\\n",
+    "\\end{bmatrix}\n",
+    "\\end{align*}$\n",
     "\n",
     "This is the original form written in the paper, but it is more transparent to write them out so you see all four equations\n",
     "\n",
     "$\n",
-    "\\delta_{i0} + R = Wc^{+}+WXc^{-} \n",
-    "$ <br\\> $\n",
-    "jn_{I}cos(\\theta) -jY_IR = Vc^{+}-VXc^{-} \n",
-    "$<br\\> $\n",
-    "T = WXc^{+} + Wc^{-}\n",
-    "$<br\\> $\n",
-    "jY_{II} = VXc^{+} - Vc^{-}\n",
+    "\\begin{align*}\n",
+    "\\delta_{i0} + R &= Wc^{+}+WXc^{-} &(1)\\\\\n",
+    "jn_{I}cos(\\theta) -jY_IR &= Vc^{+}-VXc^{-} &(2)\\\\\n",
+    "T &= WXc^{+} + Wc^{-} &(3)\\\\\n",
+    "jY_{II} &= VXc^{+} - Vc^{-} &(4)\n",
+    "\\end{align*}\n",
     "$"
    ]
   },
@@ -72,84 +71,93 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "First we solve for $c^{-}$<br\\>\n",
-    "$\n",
-    "c^{-} = W^{-1}(T-Xc^{+})\n",
-    "$\n",
-    "<br\\> Now we can solve $c^{+}$ using the expression above <br\\>\n",
-    "$\n",
-    "c^{+} = F^{-1}(jY_{II} + VW^{-1})T\n",
-    "$\n",
-    "<br\\> where the term $F$ is given by: <br\\>\n",
-    "$\n",
-    "F = V(I+W^{-1})X\n",
-    "$\n",
-    "<br\\> We can substitute this back into the expression for $c^{-}$ <br\\>\n",
-    "$\n",
-    "c^{-} = W^{-1}\\bigg(I-XF^{-1}(iY_{II}+VW^{-1})\\bigg)T\n",
+    "First we rewrite (3) for $c^{-}$\n",
+    "\n",
+    "$c^{-} = W^{-1}(T-Xc^{+})$\n",
+    "\n",
+    "Now we can solve $c^{+}$ using the expression (3) and (4)\n",
+    "\n",
     "$\n",
-    "<br\\>\n",
-    "We will call the expression inside the big parentheses $G$ to keep notation simple"
+    "\\begin{align*}\n",
+    "jY_{II}T &= VXC^+ -V(W^{-1}(T-WXC^+)) \\\\\n",
+    "jY_{II}T &= VXC^+ -VW^{-1}T+VW^{-1}WXC^+ \\\\\n",
+    "&=(VX+VX)c^+ -VW^{-1}T \\\\\n",
+    "2VXc^+ &= jY_{II}T + VW^{-1}T \\\\\n",
+    "c^+ &= 0.5X^{-1}V^{-1}(jY_{II}T + VW^{-1}T) \\\\\n",
+    "&= 0.5X^{-1}V^{-1}(jY_{II} + VW^{-1})T \\\\\n",
+    "&= 0.5X^{-1}(W^{-1} + jV^{-1}Y_{II})T \\\\\n",
+    "\\end{align*}\n",
+    "$"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
-    "#### Now we mark the steps that substitutes our expressions above into the reflection equations\n",
-    "First we rewrite the two reflection equations: <br\\>\n",
-    "$\n",
-    "\\delta_{i0} + R = Wc^{+}+WXc^{-} \n",
-    "$ <br\\> $\n",
-    "jn_{I}cos(\\theta) -jY_IR = Vc^{+}-VXc^{-} \n",
-    "$<br\\> \n",
-    "\n",
-    "Now we begin substitution:<br\\>\n",
-    "$\n",
-    "\\delta_{i0} + R = W\\bigg[F^{-1}(iY_{II}+VW^{-1}) +XG \\bigg]T\n",
-    "$ <br\\>\n",
-    "$\n",
-    "jn_Icos(\\theta)\\delta_{i0} -jY_IR = V\\bigg[ F^{-1}(iY_{II}+VW^{-1}) - XG\\bigg]T\n",
-    "$\n",
-    "<br\\> Now we do two notation simplifications: <br\\>\n",
-    "$\n",
-    "A = F^{-1}(iY_{II}+VW^{-1}) +XG \n",
-    "$ <br\\>\n",
-    "$\n",
-    "B =  F^{-1}(iY_{II}+VW^{-1}) - XG\n",
-    "$\n"
-   ]
+    "We can substitute this back into the expression for $c^{-}$ <br>\n",
+    "$\n",
+    "\\begin{align*}\n",
+    "c^{-} &= W^{-1}\\bigg[T - WX\\big( 0.5X^{-1}V^{-1}(jY_{II}+VW^{-1})T\\big)\\bigg]\\\\\n",
+    "&= W^{-1}\\bigg[T - 0.5WV^{-1}(jY_{II}+VW^{-1})T\\bigg] \\\\\n",
+    "&= W^{-1}T - 0.5V^{-1}(jY_{II}+VW^{-1})T \\\\\n",
+    "&= W^{-1}T - 0.5(V^{-1}jY_{II}+W^{-1})T \\\\\n",
+    "&= 0.5W^{-1}T - 0.5V^{-1}jY_{II}T \\\\\n",
+    "&= 0.5(W^{-1} -jV^{-1}Y_{II})T\n",
+    "\\end{align*}\n",
+    "$"
+   ],
+   "metadata": {
+    "collapsed": false
+   }
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
-    "Now we can finally solve for R: <br\\>\n",
-    "$\n",
-    "T = A^{-1}(\\delta_{i0} + R)\n",
+    "#### Now we mark the steps that substitutes our expressions above into the reflection equations\n",
+    "First we rewrite the two reflection equations: <br>\n",
     "$\n",
-    "<br\\>\n",
+    "\\begin{align*}\n",
+    "\\delta_{i0} + R &= Wc^{+}+WXc^{-} \\\\\n",
+    "jn_{I}cos(\\theta) -jY_IR &= Vc^{+}-VXc^{-}\n",
+    "\\end{align*}\n",
     "$\n",
-    "(jn_Icos(\\theta) - VBA^{-1})\\delta_{i0} = (jY_I + VBA^{-1})R\n",
-    "$ <br\\>\n",
+    "\n",
+    "Now we begin substitution:<br>\n",
     "$\n",
-    "R = \\frac{(jn_Icos(\\theta) - VBA^{-1})}{(jY_I + VBA^{-1})}\\delta_{i0} \n",
-    "$ "
-   ]
+    "\\begin{align*}\n",
+    "\\begin{matrix}\n",
+    "\\delta_{i0} + R = W\\bigg(0.5X^{-1}(W^{-1} +jV^{-1}Y_{II})T \\bigg)+WX\\bigg(0.5(W^{-1} -jV^{-1}Y_{II})T \\bigg)  \\\\\n",
+    "jn_{I}cos(\\theta) -jY_IR = V\\bigg(0.5X^{-1}(W^{-1} +jV^{-1}Y_{II})T \\bigg)-VX\\bigg(0.5(W^{-1} -jV^{-1}Y_{II})T \\bigg) \\\\\n",
+    "\\vdots\n",
+    "\\end{matrix}\n",
+    "\\end{align*}\n",
+    "$"
+   ],
+   "metadata": {
+    "collapsed": false
+   }
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
    "source": [
-    "At this point we are done, we can now backsubstitute our solution for R into T and then finally into $c^{+}$ and $c^{-}$\n",
+    "## This may be able to solve, but it's not the point of this paper.\n",
+    "The point is to remove $X^{-1}$ which may have numerical instability problem.\n",
     "\n",
-    "THIS METHOD FAILS: TOO MUCH ILL CONDITIONING"
-   ]
+    "BTW, the code in this repo is implemented as the paper suggested(with minor difference)."
+   ],
+   "metadata": {
+    "collapsed": false,
+    "pycharm": {
+     "name": "#%% md\n"
+    }
+   }
   },
   {
    "cell_type": "markdown",
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "pycharm": {
+     "name": "#%% md\n"
+    }
    },
    "source": [
     "# Potential Failure Case in RCWA with Scattering Matrices\n",
@@ -180,7 +188,7 @@
     "K_z = \\bigg( \\sqrt{k_0^2n - K_x^2 - K_y^2} \\bigg)^*\n",
     "$\n",
     "\n",
-    "THE ISSUE: WE CAN ENGINEER SITUATIONS WHERE KZ IS SINGULAR <br\\>\n",
+    "THE ISSUE: WE CAN ENGINEER SITUATIONS WHERE KZ IS SINGULAR <br>\n",
     "typically, for our gap media, we pick $n_g=1$\n",
     "Additionally, we normalize by $k_0 = \\frac{2\\pi}{\\lambda_0}$\n",
     "\n",
@@ -196,20 +204,11 @@
     "\n",
     "This problem is not just for gap media, but also if any of the layers or reflection/transmission regions have such properties...As a result, it isn't obvious to me that scattering matrix formalism is 'unconditionally stable'"
    ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -223,9 +222,9 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.6.3"
+   "version": "3.9.6"
   }
  },
  "nbformat": 4,
  "nbformat_minor": 1
-}
+}