{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "13af3d73",
   "metadata": {},
   "source": [
    "# Chirp-Scan"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc928827",
   "metadata": {},
   "source": [
    "### Z-Gradient  \n",
    "\n",
    "1. SHG  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2 D_{kn} A_{mk} e^{i\\phi_{mn}}$\n",
    "\n",
    "<br>\n",
    "\n",
    "2. THG  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 3 D_{kn} A_{mk}^2 e^{i\\phi_{mn}}$\n",
    "\n",
    "<br>\n",
    "\n",
    "3. PG/SD  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n^*} = D_{nk} A_{mk}^2 e^{-i\\phi_{mn}}$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2 D_{kn} |A_{mk}|^2 e^{i\\phi_{mn}}$\n",
    "\n",
    "<br>\n",
    "\n",
    "5. nth-HG  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = n\\cdot D_{kn} A_{mk}^{n-1} e^{i\\phi_{mn}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c56c9e0",
   "metadata": {},
   "source": [
    "### Z-Pseudo-Hessian  \n",
    "\n",
    "1. SHG  \n",
    "\n",
    "    $V_{zz}=0$\n",
    "    \n",
    "<br>\n",
    "\n",
    "2. THG  \n",
    "\n",
    "    $V_{zz}=0$\n",
    "\n",
    "<br>\n",
    "\n",
    "3. PG/SD\n",
    "\n",
    "    $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 2D_{nk}D_{kp} A_{mk}^* e^{-i \\phi_{mn}} e^{i \\phi_{mp}}$\n",
    "\n",
    "    $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 2D_{kn}D_{pk} A_{mk}^* e^{i \\phi_{mn}} e^{-i \\phi_{mp}}$\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "4. nth-HG  \n",
    "    \n",
    "    $V_{zz} = 0$"
   ]
  }
 ],
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   "name": "python"
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