{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "83da74b6",
   "metadata": {},
   "source": [
    "# Time-Domain-Ptychography"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb8d9557",
   "metadata": {},
   "source": [
    "## Auto-Correlation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5d62d6e",
   "metadata": {},
   "source": [
    "### Z-Gradient  \n",
    "\n",
    "1. SHG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = D_{kn}\\cdot(A_{mk}' + B_n\\cdot e^{-i\\tau_m\\omega_n} E_k)$\n",
    "\n",
    "<br>\n",
    "\n",
    "2. THG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = D_{kn}\\cdot(A_{mk}'^2 + 2B_n\\cdot e^{-i\\tau_m\\omega_n} E_k A_{mk})$\n",
    "\n",
    "<br>\n",
    "\n",
    "3. PG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = D_{nk} \\cdot B_n^*\\cdot e^{i\\tau_m\\omega_n} E_k A_{mk}$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = D_{kn} A_{mk}'^* \\cdot (A_{mk}' + B_n\\cdot e^{-i\\tau_m\\omega_n} E_k)$\n",
    "\n",
    "<br>\n",
    "\n",
    "4. SD  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 2 D_{nk} B_n^*\\cdot E_k A_{mk}'^* e^{i\\tau_m\\omega_n}$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = D_{kn} (A_{mk}'^*)^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "5. nth-HG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = D_{kn}\\cdot(A_{mk}'^{n-1} + (n-1)B_n\\cdot e^{-i\\tau_m\\omega_n} E_k A_{mk}^{n-2})$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35171ff9",
   "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  \n",
    "    $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = D_{pk} D_{kn}\\cdot \\left( A_{mk}'^* B_p e^{-i\\tau_m\\omega_p} + E_k^* B_n^*B_p e^{i\\tau_m\\omega_n} e^{-i\\tau_m\\omega_p} \\right)$\n",
    "\n",
    "    $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = D_{nk} D_{kp} \\cdot \\left( A_{mk}'^* B_n e^{-i\\tau_m\\omega_n} + E_k^* B_nB_p^* e^{i\\tau_m\\omega_p} e^{-i\\tau_m\\omega_n} \\right)$\n",
    "    \n",
    "\n",
    "<br>\n",
    "\n",
    "4. SD  \n",
    "    $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 2\\cdot D_{pk}D_{kn} \\cdot A_{mk}' B_p e^{-i\\tau_m\\omega_p}$\n",
    "\n",
    "    $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 2\\cdot D_{nk}D_{kp} \\cdot A_{mk}' B_n e^{-i\\tau_m\\omega_n}$\n",
    "    \n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "5. nth-HG  \n",
    "    \n",
    "    $V_{zz} = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cfe92742",
   "metadata": {},
   "source": [
    "## Cross-Correlation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3cfe173",
   "metadata": {},
   "source": [
    "### Z-Gradient (with respect to pulse)  \n",
    "\n",
    "Same for all nonlinear methods  \n",
    "\n",
    "$\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "$\\frac{dS_{mk}}{dE_n} = D_{kn} G(A_{mk}')$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd6e85de",
   "metadata": {},
   "source": [
    "### Z-Gradient (with respect to gate-pulse)  \n",
    "\n",
    "$\\frac{dS_{mk}}{dE_n} = \\frac{dS_{mk}}{dA_{mk}'}\\frac{dA_{mk}'}{dE_n}$  \n",
    "\n",
    "$\\frac{dS_{mk}}{dE_n^*} = \\frac{dS_{mk}}{dA_{mk}'^*}\\left(\\frac{dA_{mk}'}{dE_n}\\right)^*$\n",
    "\n",
    "<br>\n",
    "\n",
    "1. SHG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = D_{kn} B_n\\cdot E_k e^{-i\\tau_m\\omega_n}$\n",
    "\n",
    "<br>\n",
    "\n",
    "2. THG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2 \\cdot D_{kn} B_n\\cdot E_k A_{mk}' e^{-i\\tau_m\\omega_n}$\n",
    "\n",
    "<br>\n",
    "\n",
    "3. PG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = D_{nk} \\cdot B_n^*\\cdot E_k A_{mk}' e^{i\\tau_m\\omega_n}$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = D_{kn} \\cdot B_n\\cdot E_k A_{mk}'^* e^{-i\\tau_m\\omega_n}$\n",
    "\n",
    "<br>\n",
    "\n",
    "4. SD  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 2\\cdot D_{nk} B_n^*\\cdot E_k A_{mk}'^* e^{i\\tau_m\\omega_n}$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 0$\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "5. nth-HG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = (n-1) \\cdot D_{kn} B_n\\cdot E_k \\left(A_{mk}'\\right)^{n-2} e^{-i\\tau_m\\omega_n}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b798ebd0",
   "metadata": {},
   "source": [
    "### Z-Pseudo-Hessian (with respect to pulse)  \n",
    "\n",
    "Same for all nonlinear methods.  \n",
    "\n",
    "$V_{zz} = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78685feb",
   "metadata": {},
   "source": [
    "### Z-Pseudo-Hessian (with respect to gate-pulse)  \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  \n",
    "    $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = D_{nk}D_{kp} E_k^* B_n^*B_p e^{i\\tau_m\\omega_n} e^{-i\\tau_m\\omega_p}$\n",
    "\n",
    "    $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = D_{kn}D_{pk} E_k^* B_nB_p^* e^{-i\\tau_m\\omega_n} e^{i\\tau_m\\omega_p}$\n",
    "    \n",
    "\n",
    "<br>\n",
    "\n",
    "4. SD  \n",
    "    $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 0$\n",
    "\n",
    "    $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 0$\n",
    "    \n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "5. nth-HG  \n",
    "    \n",
    "    $V_{zz} = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "007c8241",
   "metadata": {},
   "source": [
    "# Interferometric Time-Domain-Ptychography"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08d6ad51",
   "metadata": {},
   "source": [
    "## Auto-Correlation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "43713f0d",
   "metadata": {},
   "source": [
    "### Z-Gradient  \n",
    "\n",
    "1. SHG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2D_{kn}(1+B_n\\cdot e^{-i\\tau_m\\omega_n})\\cdot (E_k + A_{mk}')$\n",
    "    \n",
    "\n",
    "<br>\n",
    "\n",
    "2. THG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 3D_{kn}(1+B_n\\cdot e^{-i\\tau_m\\omega_n})\\cdot (E_k + A_{mk}')^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "3. PG/SD  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = D_{nk}(1+B_n^*\\cdot e^{i\\tau_m\\omega_n})\\cdot (E_k + A_{mk}')^2$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2D_{kn}(1+B_n\\cdot e^{-i\\tau_m\\omega_n})\\cdot |E_k + A_{mk}'|^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "4. nth-HG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = n\\cdot D_{kn}(1+B_n\\cdot e^{-i\\tau_m\\omega_n})\\cdot (E_k + A_{mk}')^{n-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "260499fc",
   "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",
    "    $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 2D_{nk}D_{kp} (E_k+A_{mk}')^* (1+B_n^*e^{i\\tau_m\\omega_n}) (1+B_p e^{-i\\tau_m\\omega_p})$\n",
    "\n",
    "    $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 2D_{kn}D_{pk} (E_k+A_{mk}')^* (1+B_n e^{-i\\tau_m\\omega_n}) (1+B_p^*e^{i\\tau_m\\omega_p})$\n",
    "    \n",
    "\n",
    "<br>\n",
    "\n",
    "5. nth-HG  \n",
    "    \n",
    "    $V_{zz} = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd80a721",
   "metadata": {},
   "source": [
    "## Cross-Correlation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8264a7b9",
   "metadata": {},
   "source": [
    "### Z-Gradient (with respect to pulse)   \n",
    "\n",
    "1. SHG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2D_{kn}(E_k + A_{mk}')$\n",
    "    \n",
    "\n",
    "<br>\n",
    "\n",
    "2. THG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 3D_{kn} (E_k + A_{mk}')^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "3. PG/SD  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = D_{nk}(E_k + A_{mk}')^2$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2D_{kn}|E_k + A_{mk}'|^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "4. nth-HG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = n\\cdot D_{kn} (E_k + A_{mk}')^{n-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1406a51f",
   "metadata": {},
   "source": [
    "### Z-Gradient (with respect to gate-pulse)  \n",
    "\n",
    "1. SHG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2D_{kn}B_n\\cdot e^{-i\\tau_m\\omega_n}(E_k + A_{mk}')$\n",
    "    \n",
    "\n",
    "<br>\n",
    "\n",
    "2. THG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 3D_{kn}B_n\\cdot e^{-i\\tau_m\\omega_n}(E_k + A_{mk}')^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "3. PG/SD  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = D_{nk}B_n^*\\cdot e^{i\\tau_m\\omega_n}(E_k + A_{mk}')^2$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = 2D_{kn}B_n\\cdot e^{-i\\tau_m\\omega_n}|E_k + A_{mk}'|^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "4. nth-HG  \n",
    "    $\\frac{dS_{mk}}{dE_n^*} = 0$  \n",
    "\n",
    "    $\\frac{dS_{mk}}{dE_n} = n\\cdot D_{kn}B_n\\cdot e^{-i\\tau_m\\omega_n}(E_k + A_{mk}')^{n-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c822119",
   "metadata": {},
   "source": [
    "### Z-Pseudo-Hessian (with respect to pulse)  \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",
    "    $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 2D_{nk}D_{kp} (E_k+A_{mk}')^*$\n",
    "\n",
    "    $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 2D_{kn}D_{pk} (E_k+A_{mk}')^*$\n",
    "    \n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "5. nth-HG  \n",
    "    \n",
    "    $V_{zz} = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b069981e",
   "metadata": {},
   "source": [
    "### Z-Pseudo-Hessian (with respect to gate-pulse)  \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",
    "    $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 2D_{nk}D_{kp} (E_k+A_{mk}')^* B_n^*B_p e^{i\\tau_m\\omega_n}e^{-i\\tau_m\\omega_p}$\n",
    "\n",
    "    $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 2D_{kn}D_{pk} (E_k+A_{mk}')^* B_nB_p^* e^{-i\\tau_m\\omega_n}e^{i\\tau_m\\omega_p}$\n",
    "    \n",
    "\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "4. nth-HG  \n",
    "    \n",
    "    $V_{zz} = 0$"
   ]
  }
 ],
 "metadata": {
  "language_info": {
   "name": "python"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}