modify Moharam's derivation part

zhaonat · May 23, 2022 · 88a205f · 88a205f
1 parent b4a755b
commit 88a205f
Showing 1 changed file with 13 additions and 13 deletions.
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"
    ]
@@ -19,7 +19,7 @@
    "source": [
     "We start with a system of four equations:\n",
     "\n",
-    "$\\begin{align}\n",
+    "$\\begin{align*}\n",
     "\\begin{bmatrix}\n",
     "\\delta_{i0}  \\\\ jn_Icos(\\theta)\\delta_{i0}\n",
     "\\end{bmatrix} +\n",
@@ -46,17 +46,17 @@
     "c^{+} \\\\\n",
     "c^{-} \\\\\n",
     "\\end{bmatrix}\n",
-    "\\end{align}$\n",
+    "\\end{align*}$\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",
     "$\n",
-    "\\begin{align}\n",
+    "\\begin{align*}\n",
     "\\delta_{i0} + R &= Wc^{+}+WXc^{-} &(1)\\\\\n",
     "jn_{I}cos(\\theta) -jY_IR &= Vc^{+}-VXc^{-} &(2)\\\\\n",
     "T &= WXc^{+} + Wc^{-} &(3)\\\\\n",
     "jY_{II} &= VXc^{+} - Vc^{-} &(4)\n",
-    "\\end{align}\n",
+    "\\end{align*}\n",
     "$"
    ]
   },
@@ -78,15 +78,15 @@
     "Now we can solve $c^{+}$ using the expression (3) and (4)\n",
     "\n",
     "$\n",
-    "\\begin{align}\n",
+    "\\begin{align*}\n",
     "jY_{II}T &= VXC^+ -V(W^{-1}(T-WXC^+)) \\\\\n",
     "jY_{II}T &= VXC^+ -VW^{-1}T+VW^{-1}WXC^+ \\\\\n",
     "&=(VX+VX)c^+ -VW^{-1}T \\\\\n",
     "2VXc^+ &= jY_{II}T + VW^{-1}T \\\\\n",
     "c^+ &= 0.5X^{-1}V^{-1}(jY_{II}T + VW^{-1}T) \\\\\n",
     "&= 0.5X^{-1}V^{-1}(jY_{II} + VW^{-1})T \\\\\n",
     "&= 0.5X^{-1}(W^{-1} + jV^{-1}Y_{II})T \\\\\n",
-    "\\end{align}\n",
+    "\\end{align*}\n",
     "$"
    ]
   },
@@ -95,14 +95,14 @@
    "source": [
     "We can substitute this back into the expression for $c^{-}$ <br>\n",
     "$\n",
-    "\\begin{align}\n",
+    "\\begin{align*}\n",
     "c^{-} &= W^{-1}\\bigg[T - WX\\big( 0.5X^{-1}V^{-1}(jY_{II}+VW^{-1})T\\big)\\bigg]\\\\\n",
     "&= W^{-1}\\bigg[T - 0.5WV^{-1}(jY_{II}+VW^{-1})T\\bigg] \\\\\n",
     "&= W^{-1}T - 0.5V^{-1}(jY_{II}+VW^{-1})T \\\\\n",
     "&= W^{-1}T - 0.5(V^{-1}jY_{II}+W^{-1})T \\\\\n",
     "&= 0.5W^{-1}T - 0.5V^{-1}jY_{II}T \\\\\n",
     "&= 0.5(W^{-1} -jV^{-1}Y_{II})T\n",
-    "\\end{align}\n",
+    "\\end{align*}\n",
     "$"
    ],
    "metadata": {
@@ -115,21 +115,21 @@
     "#### Now we mark the steps that substitutes our expressions above into the reflection equations\n",
     "First we rewrite the two reflection equations: <br>\n",
     "$\n",
-    "\\begin{align}\n",
+    "\\begin{align*}\n",
     "\\delta_{i0} + R &= Wc^{+}+WXc^{-} \\\\\n",
     "jn_{I}cos(\\theta) -jY_IR &= Vc^{+}-VXc^{-}\n",
-    "\\end{align}\n",
+    "\\end{align*}\n",
     "$\n",
     "\n",
     "Now we begin substitution:<br>\n",
     "$\n",
-    "\\begin{align}\n",
+    "\\begin{align*}\n",
     "\\begin{matrix}\n",
     "\\delta_{i0} + R = W\\bigg(0.5X^{-1}(W^{-1} +jV^{-1}Y_{II})T \\bigg)+WX\\bigg(0.5(W^{-1} -jV^{-1}Y_{II})T \\bigg)  \\\\\n",
     "jn_{I}cos(\\theta) -jY_IR = V\\bigg(0.5X^{-1}(W^{-1} +jV^{-1}Y_{II})T \\bigg)-VX\\bigg(0.5(W^{-1} -jV^{-1}Y_{II})T \\bigg) \\\\\n",
     "\\vdots\n",
     "\\end{matrix}\n",
-    "\\end{align}\n",
+    "\\end{align*}\n",
     "$"
    ],
    "metadata": {