{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b410d49a",
   "metadata": {},
   "source": [
    "# General Definitions  \n",
    "\n",
    "For Attosecond Streaking the definitions can be found [here](https://pulse-retrieval-with-jax.readthedocs.io/en/latest/eqs_streaking.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b369d93",
   "metadata": {},
   "source": [
    "## Nonlinear Methods\n",
    "\n",
    "1. SHG\n",
    "\n",
    "    $G(E) = E$\n",
    "\n",
    "<br>\n",
    "\n",
    "2. THG\n",
    "\n",
    "    $G(E) = E^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "3. PG\n",
    "\n",
    "    $G(E) = |E|^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "4. SD (This definition causes a flip in delay and frequency-sign)\n",
    "\n",
    "    $G(E) = (E^*)^2$\n",
    "\n",
    "<br>\n",
    "\n",
    "5. nth-HG\n",
    "\n",
    "    $G(E) = E^{n-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b534791b",
   "metadata": {},
   "source": [
    "## Indices and Symbols\n",
    "\n",
    "$\\quad k,j\\quad\\rightarrow\\quad$ Time Domain  \n",
    "$\\quad n,p\\quad\\rightarrow\\quad$ Frequency Domain  \n",
    "$\\quad m\\quad\\rightarrow\\quad$ Transformation (e.g. shift in time, applied chirp, ... )\n",
    "\n",
    "$\\quad E_k\\quad\\rightarrow\\quad$ pulse in time domain  \n",
    "$\\quad A_{mk}\\quad\\rightarrow\\quad$ transformed pulse in time domain  \n",
    "\n",
    "\n",
    "$\\quad D_{kn} \\quad\\rightarrow\\quad$ fourier matrix (inverse)  \n",
    "$\\quad D_{nk} \\quad\\rightarrow\\quad$ fourier matrix (forward)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d2c28db6",
   "metadata": {},
   "source": [
    "## Transformations \n",
    "\n",
    "1. (Inverse) Fourier Transform  \n",
    "\n",
    "    $E_n = \\sum_k D_{nk} E_n = \\sum_k e^{-i \\omega_n t_k} E_k$  \n",
    "    $E_k = \\sum_n D_{kn} E_n = \\sum_n e^{i \\omega_n t_k} E_n$\n",
    "\n",
    "<br>\n",
    "\n",
    "2. Method dependent transformation  \n",
    "\n",
    "    $A_{mk} = \\sum_n D_{kn} E_n \\cdot e^{i\\phi_{mn}}$  \n",
    "\n",
    "    $\\phi_{mn} = -\\tau_m \\cdot \\omega_n$ (linear phase)  \n",
    "\n",
    "    $\\phi_{mn} = f(\\theta_m, \\omega_n)$ (arbitrary phase, through material or pulse-shaper)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4eb501c8",
   "metadata": {},
   "source": [
    "## Definition of nonlinear signal in time domain\n",
    "\n",
    "1. Frog  \n",
    "\n",
    "    $S_{mk} = E_k \\cdot G(A_{mk})$\n",
    "\n",
    "\n",
    "2. Chirp-Scan  \n",
    "\n",
    "    $S_{mk} = A_{mk}\\cdot G(A_{mk})$\n",
    "\n",
    "\n",
    "3. Interferometric Frog  \n",
    "\n",
    "    $S_{mk} = (E_k + A_{mk})\\cdot G(E_k + A_{mk})$\n",
    "\n",
    "\n",
    "4. Time-Domain-Ptychography  \n",
    "\n",
    "    Same as FROG or I-FROG only $A_{mk}$ is different.  \n",
    "    $A_{mk}' = \\sum_n D_{kn}B_n A_{mn}\\quad$ ($B_n$ represents a known spectral filter, can be complex valued)\n",
    "\n",
    "\n",
    "5. 2D-SI  \n",
    "\n",
    "    $S_{mk} = G(E_k'\\cdot e^{-i\\tau_0\\omega_n} + A_{mk}'') \\cdot E_k \\quad$ ( $\\tau_0$ is a fixed delay of $E_k$ and $E_k'$ )  \n",
    "    $E_k' = \\sum_n D_{kn}B_n E_n\\quad$ ( $B_n$ represents a known spectral filter, can be complex valued)  \n",
    "\n",
    "\n",
    "6. Vampire  \n",
    "\n",
    "    $S_{mk} = G(E_k\\cdot e^{-i\\tau_0\\omega_n} + E_k') \\cdot A_{mk}\\quad$  (Original definition, pushing $m$ onto gate makes it easier with PIE.)  \n",
    "    $S_{mk} = G_m(E_k\\cdot e^{-i\\tau_0\\omega_n} + E_k') \\cdot E_k = G(A_{mk}\\cdot e^{-i\\tau_0\\omega_n} + A_{mk}')\\cdot E_k$  \n",
    "    $E_k' = \\sum_n D_{kn}E_ne^{i\\phi_n}\\quad$ ($\\phi_n$ represents a material dispersion)   \n",
    "    $\\tau_0$ is a fixed delay in the interferometer."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96378047",
   "metadata": {},
   "source": [
    "## Error \n",
    "\n",
    "1. G-Error (Trace Error)  \n",
    "\n",
    "    $L = \\frac{1}{N_m\\cdot N_n}\\sum\\limits_{nm} (T_{mn}^{\\mathrm{exp}} - \\mu \\cdot T_{mn})^2$\n",
    "\n",
    "\n",
    "2. Z-Error  \n",
    "\n",
    "    $Z = \\sum\\limits_{km} | S_{mk}' - S_{mk}|^2$\n",
    "\n",
    "\n",
    "3. PIE-Error  \n",
    "\n",
    "    $L = \\frac{1}{N_m\\cdot N_n}\\sum\\limits_{nm} (\\sqrt{T_{mn}^{\\mathrm{exp}}} - \\mu\\cdot|S_{mn}|)^2$\n",
    "\n",
    "\n",
    "\n",
    "## Calculation of $\\mu$  \n",
    "\n",
    "1. Frequency independent  \n",
    "\n",
    "    $\\mu = \\frac{\\sum_{nm} T_{mn} \\cdot T_{mn}^{\\mathrm{exp}}}{\\sum_{nm} T_{mn}^2}$  \n",
    "\n",
    "\n",
    "2. Frequency dependent  \n",
    "\n",
    "    $\\mu_n = \\frac{\\sum_{m} T_{mn} \\cdot T_{mn}^{\\mathrm{exp}}}{\\sum_{m} T_{mn}^2}$  \n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "3. For PIE-Error  \n",
    "\n",
    "    $T_{mn}\\quad\\rightarrow\\quad|S_{mn}|$  \n",
    "\n",
    "    $T_{mn}^{\\mathrm{exp}}\\quad\\rightarrow\\quad\\sqrt{T_{mn}^{\\mathrm{exp}}}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f20a65e2",
   "metadata": {},
   "source": [
    "## Z-Gradient  \n",
    "\n",
    "$\\nabla_n Z_m = -2\\cdot\\sum_k \\Delta S_{mk}^*\\cdot\\frac{dS_{mk}}{dE_n^*} + \\Delta S_{mk}\\cdot\\left(\\frac{dS_{mk}}{dE_n}\\right)^*$  \n",
    "\n",
    "$\\nabla_n Z = \\sum_m \\nabla_n Z_m$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "049aca60",
   "metadata": {},
   "source": [
    "## Z-Pseudo-Hessian  \n",
    "\n",
    "$H_{zz} = U_{zz} - V_{zz}$  \n",
    "\n",
    "$U_{zz}^m = \\frac{1}{2}\\cdot\\sum_k\\left(\\left(\\frac{dS_{mk}}{dE_n}\\right)^*\\frac{dS_{mk}}{dE_p} + \\left(\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^*\\frac{dS_{mk}}{dE_p^*}\\right)^*\\right)$  \n",
    "\n",
    "$V_{zz}^m = \\frac{1}{2}\\cdot\\sum_k\\left( \\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^*\\cdot \\Delta S_{mk} + \\left(\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^*\\cdot \\Delta S_{mk}\\right)^* \\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89b64e35",
   "metadata": {},
   "source": [
    "## Z-Gradient and Pseudo-Hessian (with respect to spectral phase factor)\n",
    "\n",
    "$E_n = |E_n| \\cdot \\Psi_n$  \n",
    "\n",
    "\n",
    "$\\frac{d E_n}{d \\Psi_n} = \\frac{d E_n^*}{d \\Psi_n^*} = |E_n|$\n",
    "\n",
    "### Z-Gradient\n",
    "\n",
    "$\\frac{d Z_m}{d\\Psi_n^*} = \\frac{d E_n^*}{d\\Psi_n^*}\\frac{d Z_m}{d E_n^*} = |E_n|\\cdot\\frac{d Z_m}{d E_n^*}$  \n",
    "\n",
    "### Z-Pseudo Hessian\n",
    "\n",
    "$\\frac{d}{d\\Psi_p}\\left(\\frac{d S_{mk}}{d \\Psi_n}\\right)^* = \\frac{d}{d\\Psi_p}\\left(\\frac{d E_n}{d\\Psi_n}\\frac{d S_{mk}}{d E_n}\\right)^* = \\frac{d E_p}{d\\Psi_p}\\frac{d E_n^*}{d\\Psi_n^*}\\frac{d}{dE_p}\\left(\\frac{d S_{mk}}{d E_n}\\right)^* = |E_p||E_n|\\cdot \\frac{d}{dE_p}\\left(\\frac{d S_{mk}}{d E_n}\\right)^*$  \n",
    "\n",
    "$\\qquad \\Rightarrow \\quad H_{\\Psi} = H_{E} \\cdot |E_p| |E_n|$  \n",
    "\n",
    "I am assuming that all other terms from the product-rule are zero. Maybe thats wrong?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8b1b432",
   "metadata": {},
   "source": [
    "## PIE Gradient  \n",
    "\n",
    "$\\nabla_k L_m = \\frac{d |S_{mn}|}{d O_k^*}\\frac{d L_m}{d |S_{mn}|}$  \n",
    "\n",
    "$\\nabla_k L = \\sum_m \\nabla_k L_m$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c043efff",
   "metadata": {},
   "source": [
    "## PIE Pseudo-Hessian  \n",
    "\n",
    "$H_{zz} = U_{zz} - V_{zz}$  \n",
    "\n",
    "$U_{zz}^m = \\frac{1}{2}\\cdot\\sum_n\\left(\\left(\\frac{d|S_{mn}|}{dE_j}\\right)^*\\frac{d|S_{mn}|}{dE_k} + \\left(\\left(\\frac{d|S_{mn}|}{dE_j^*}\\right)^*\\frac{d|S_{mn}|}{dE_k^*}\\right)^*\\right)$  \n",
    "\n",
    "$V_{zz}^m = \\frac{1}{2}\\cdot\\sum_n\\left( \\frac{d}{dE_k}\\left(\\frac{d|S_{mn}|}{dE_j}\\right)^*\\cdot (\\sqrt{T_{mn}^{\\mathrm{exp}}} - |S_{mn}|) + \\left(\\frac{d}{dE_k^*}\\left(\\frac{d|S_{mn}|}{dE_j^*}\\right)^*\\cdot (\\sqrt{T_{mn}^{\\mathrm{exp}}} - |S_{mn}|)\\right)^* \\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0dc13ebd",
   "metadata": {},
   "source": [
    "## PIE Gradient and Pseudo-Hessian (with respect to spectral phase factor)  \n",
    "\n",
    "$O_n = |O_n| \\cdot \\Psi_n$  \n",
    "\n",
    "\n",
    "$\\frac{d O_n}{d \\Psi_n} = \\frac{d O_n^*}{d \\Psi_n^*} = |O_n|$\n",
    "\n",
    "### PIE Gradient  \n",
    "\n",
    "$\\frac{d L_m}{d\\Psi_n^*} = \\frac{d O_n^*}{d\\Psi_n^*}\\frac{d L_m}{d O_n^*} = |O_n|\\cdot\\frac{d L_m}{d O_n^*}$  \n",
    "\n",
    "$\\frac{d L_m}{d O_n^*} = \\frac{d O_k^*}{d O_n^*}\\frac{d L_m}{d O_k^*} = \\mathrm{FFT}\\{{\\frac{d L_m}{d O_k^*}}\\}$  \n",
    "\n",
    "### PIE Pseudo-Hessian  \n",
    "\n",
    "$\\frac{d|S_{mn}|}{d\\Psi_n} = \\frac{d E_n}{d \\Psi_n}\\frac{d E_k}{d}\\frac{d |S_{mn}|}{d E_k}$  \n",
    "\n",
    "$\\frac{d}{d \\Psi_n}\\left(\\frac{d |S_{mn}|}{d \\Psi_p}\\right)^* = \\frac{d E_p}{d \\Psi_p}\\frac{d E_j}{d E_p}\\left(\\frac{d E_n}{d \\Psi_n}\\frac{d E_k}{d E_n}\\right)^*\\frac{d}{d E_j}\\left(\\frac{d |S_{mn}|}{d E_k}\\right)^*$  \n",
    "\n",
    "$\\qquad \\Rightarrow \\quad H_{\\Psi} = H_{E} \\cdot \\frac{d E_p}{d \\Psi_p}\\frac{d E_j}{d E_p}\\left(\\frac{d E_n}{d \\Psi_n}\\frac{d E_k}{d E_n}\\right)^*$  \n",
    "\n",
    "I am assuming that all other terms from the product-rule are zero. Maybe thats wrong?"
   ]
  }
 ],
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