{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e6eceb79",
   "metadata": {},
   "source": [
    "# Attosecond Streaking (Definitions, Z-Gradients and Z-Pseudo-Hessians)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ff00c22",
   "metadata": {},
   "source": [
    "## Definitions  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37d3fb52",
   "metadata": {},
   "source": [
    "### Symbols and Indices\n",
    "\n",
    "The SFA introduces a momentum axis `k`. In the numerical implementation a conjugate spatial axis `r` is also relevant. \n",
    "These two axis require new index convetions.  \n",
    "\n",
    "$\\quad q\\quad\\rightarrow\\quad$ Spatial Domain  \n",
    "$\\quad b\\quad\\rightarrow\\quad$ Momentum Domain  \n",
    "\n",
    "Its a bit annoying that im already using $k$ and $p$ as indices. If its not an index $k$ refers to the momentum.\n",
    "\n",
    "In addition the channel-dependent transition-dipole-matrix-element (DTME or $\\rho^C$) and the channel dependent ionization potential $IP^C$ are introduced.  \n",
    "In Streaking `A` always refers to the vectorpotential of the femtosecond pulse, while `E` refers to the field of the EUV-pulse."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "611c3d1e",
   "metadata": {},
   "source": [
    "### Streaking Amplitude (within Strong-Field-Approximation)  \n",
    "\n",
    "For a single-channel the SFA is implemented as:\n",
    "\n",
    "$S_{mb}^C = -i \\cdot \\sum\\limits_k dt\\cdot \\rho_{bk}^C \\cdot E_{mk} \\cdot e^{-i t_k\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)}\\cdot e^{-i\\phi_{bk}}$  \n",
    "\n",
    "Where $S_{mb}$ is the streaking amplitude in the momenum domain, it can be fourier transform to the spatial domain to yield $S_{mq}$. The multi-channel SFA is simply a coherent sum over all channels, which translates to all gradients:\n",
    "\n",
    "$S_{mb} = \\sum\\limits_C S_{mb}^C$  \n",
    "\n",
    "$\\frac{d S_{mb}}{dX} = \\sum\\limits_C \\frac{d S_{mb}^C}{dX}$  \n",
    "\n",
    "\n",
    "The vectorpotential-dependent part of the Volkov-Phase is:\n",
    "\n",
    "$\\phi_{bk} = \\sum\\limits_{j'=k}^{k_{\\mathrm{max}}} dt\\cdot \\left(k_b A_{j'} + \\frac{A_{j'}^2}{2}\\right)$  \n",
    "\n",
    "The DTME is momentum dependent and thus modulated by the vectorpotential. This can be numerically implemented using the Fourier-shift theorem.\n",
    "\n",
    "$\\rho_{bk}^C = \\sum\\limits_q D_{bq} \\rho_q^C \\cdot e^{i\\cdot 2\\pi r_q\\cdot A_k}$  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a6bf04f",
   "metadata": {},
   "source": [
    "### Z-Error Definition  \n",
    "\n",
    "$Z = \\sum\\limits_{mb} |S_{mb}' - S_{mb}|^2$  \n",
    "\n",
    "$\\nabla_{n}Z_m = -2\\sum\\limits_b \\Delta S_{mb} \\left(\\frac{d S_{mb}}{d E_n}\\right)^* + \\Delta S_{mb}^*\\frac{d S_{mb}}{d E_n^*}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b549f82",
   "metadata": {},
   "source": [
    "## Z-Gradients"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "79e184c9",
   "metadata": {},
   "source": [
    "### With respect to the real-part of vectorpotential in frequency domain ($A_n$)  \n",
    "\n",
    "Vectorpotential is required to be real for streaking. Writing this out is omitted here for the sake of readability.  \n",
    "\n",
    "$\\nabla_{n}Z_m = \\mathrm{FT}\\{\\nabla_{j}Z_m\\}$\n",
    "\n",
    "$\\nabla_{j}Z_m = -1\\cdot\\sum\\limits_b \\Delta S_{mb} \\left(\\frac{d S_{mb}}{d A_j}\\right)^* + \\Delta S_{mb}^*\\frac{d S_{mb}}{d A_j^*}= -2\\cdot\\Re\\left(\\sum\\limits_{b, C}\\Delta S_{mb}\\left(\\frac{d S_{mb}^C}{d A_j}\\right)^*\\right)$\n",
    "\n",
    "\n",
    "$\\frac{d S_{mb}^C}{d A_j} = -i\\cdot\\sum\\limits_k dt \\cdot E_{mk} \\cdot e^{-i t_k\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bk}}\\cdot\\left(-i\\cdot \\rho_{bk}^C\\cdot \\frac{d \\phi_{bk}}{d A_j} + \\frac{d\\rho_{bk}^C}{d A_j} \\right)$  \n",
    "\n",
    "For the derivatives of the inner funcition one gets:\n",
    "\n",
    "$\\frac{d \\rho_{bk}^C}{d A_j} = i\\cdot 2\\pi \\cdot \\delta_{kj} \\sum\\limits_q D_{bq}\\rho_q^C\\cdot e^{i\\cdot 2\\pi r_q\\cdot A_k}r_q$  \n",
    "\n",
    "The $\\delta_{kj}$ does not appear in the other derivative, a different one with different indices appears and gets rid of a different sum.\n",
    "\n",
    "$\\frac{d \\phi_{bk}}{d A_j} = \\begin{cases}\n",
    "dt \\cdot (k_b + A_j) & \\text{if } k \\le j \\le k_{max} \\\\\n",
    "0 & \\text{else}\n",
    "\\end{cases}$ \n",
    "\n",
    "Thus the sum over k survives in this term. The entire gradient reads:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c8bbdba2",
   "metadata": {},
   "source": [
    "$\\frac{d S_{mb}^C}{d A_j} = g_1 + g_2$\n",
    "\n",
    "$g_1 = -1\\cdot dt^2 \\cdot \\sum\\limits_k \\begin{cases}\n",
    "E_{mk} \\cdot e^{-i t_k\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bk}}\\cdot\\rho_{bk}^C\\cdot (k_b + A_j) & \\text{if } k \\le j \\le k_{max} \\\\\n",
    "0 & \\text{else}\n",
    "\\end{cases}$ \n",
    "\n",
    "$g_2 = 2\\pi dt \\cdot E_{mj} \\cdot e^{-i t_j\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bj}}\\cdot \\left(\\sum\\limits_q D_{bq}\\rho_q^C \\cdot r_q\\cdot e^{i\\cdot 2\\pi r_q\\cdot A_j}\\right)$  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6816eab2",
   "metadata": {},
   "source": [
    "### With respect to the attosecond pulse in frequency domain $\\left(E_n\\right)$  \n",
    "\n",
    "$\\nabla_{n}Z_m = -2\\sum\\limits_b \\Delta S_{mb} \\left(\\frac{d S_{mb}}{d E_n}\\right)^* = -2\\sum\\limits_{b,C}\\Delta S_{mb} \\left(\\frac{d S_{mb}^C}{d E_n}\\right)^*$\n",
    "\n",
    "$\\frac{d S_{mb}^C}{d E_n} = -i \\cdot dt\\cdot e^{-i\\omega_n\\tau_m}\\cdot \\sum\\limits_k D_{kn}\\cdot \\rho_{bk}^C \\cdot e^{-i t_k\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)}\\cdot e^{-i\\phi_{bk}}$  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8245d066",
   "metadata": {},
   "source": [
    "### With respect to the dipole transition matrix element in momentum domain ($\\rho_b^C$)  \n",
    "\n",
    "Here one wants the gradient with respect to the DTME of each channel.  \n",
    "\n",
    "$\\nabla_{b}Z_m = FFT_{bq} \\left(\\nabla_{q}Z_m\\right)$\n",
    "\n",
    "$\\nabla_{q'}^{C'}Z_m = -2\\sum\\limits_{b,C} \\Delta S_{mb} \\left(\\frac{d S_{mb}^C}{d \\rho_{q'}^{C'}}\\right)^*$  \n",
    "\n",
    "$\\frac{d S_{mb}^C}{d \\rho_{q'}^{C'}} = -i \\delta_{CC'}\\delta_{qq'}\\cdot dt \\cdot D_{bq}\\cdot \\sum\\limits_k E_{mk} \\cdot e^{-i t_k\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)}\\cdot e^{-i\\phi_{bk}} \\cdot e^{i\\cdot 2\\pi r_q A_k}$  \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2dd91862",
   "metadata": {},
   "source": [
    "## Z-Pseudo-Hessian  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5eec26c",
   "metadata": {},
   "source": [
    "### With respect to the real-part of vectorpotential in frequency domain ($A_n$)  \n",
    "\n",
    "$U_{zz} = \\sum\\limits_b \\Re\\left\\{\\left(\\frac{d S_{mb}}{d A_{n'}}\\right)^*\\cdot\\frac{d S_{mb}}{d A_n}\\right\\}$  \n",
    "\n",
    "$\\frac{d S_{mb}}{d A_n} = FT_{nj}\\left\\{\\frac{d S_{mb}}{d A_j}\\right\\}$\n",
    "\n",
    "$V_{zz} = \\sum\\limits_b \\Re\\left\\{\\frac{d}{d A_n}\\left(\\frac{dS_{mb}}{d A_{n'}}\\right)^*\\cdot \\Delta S_{mb}\\right\\} = \\sum\\limits_b \\Re\\left\\{\\left(\\frac{d}{d A_n}\\frac{dS_{mb}}{d A_{n'}}\\right)^*\\cdot \\Delta S_{mb}\\right\\} = \\sum\\limits_b \\Re\\left\\{\\left( D_{jn} D_{j'n'} \\frac{d}{d A_j}\\frac{dS_{mb}}{d A_{j'}}\\right)^*\\cdot \\Delta S_{mb}\\right\\}$  \n",
    "\n",
    "Writing the full hessian out isnt fun. So i wont. \n",
    "\n",
    "$\\frac{d S_{mb}}{d A_j} = -i\\cdot dt \\sum\\limits_{Ck} E_{mk}\\cdot e^{-i t_k\\cdot\\left(IP^C+\\frac{k_b^2}{2}\\right)}\\cdot \\frac{d}{dA_j}\\left(\\rho_{bk}^C\\cdot e^{-i\\phi_{bk}}\\right)$  \n",
    "\n",
    "$\\frac{d}{dA_j}\\left(\\rho_{bk}^C\\cdot e^{-i\\phi_{bk}}\\right) =  e^{-i\\phi_{bk}}\\cdot\\left(-i\\cdot\\rho_{bk}^C\\cdot\\frac{d\\phi_{bk}}{dA_j} + \\frac{d\\rho_{bk}^C}{dA_j}\\right)$  \n",
    "\n",
    "$\\frac{d}{dA_{j'}}\\frac{d S_{mb}}{d A_j} = -i\\cdot dt \\sum\\limits_{Ck} E_{mk}\\cdot e^{-i t_k\\cdot\\left(IP^C+\\frac{k_b^2}{2}\\right)}\\cdot \\frac{d}{dA_{j'}}\\frac{d}{dA_j}\\left(\\rho_{bk}^C\\cdot e^{-i\\phi_{bk}}\\right)$  \n",
    "\n",
    "Below are the critial derivatives.\n",
    "\n",
    "$\\frac{d}{dA_{j'}}\\frac{d}{dA_j}\\left(\\rho_{bk}^C\\cdot e^{-i\\phi_{bk}}\\right) =-i e^{-i\\phi_{bk}}\\cdot\\left(\\rho_{bk}^C\\cdot\\left(\\frac{d}{dA_{j'}}\\frac{d\\phi_{bk}}{dA_j} -i\\frac{d\\phi_{bk}}{dA_{j'}}\\frac{d\\phi_{bk}}{dA_j}\\right) + \\frac{d\\phi_{bk}}{dA_{j'}}\\frac{d\\rho_{bk}^C}{dA_j}+\\frac{d\\rho_{bk}^C}{dA_{j'}}\\frac{d\\phi_{bk}}{dA_j}+i\\frac{d}{dA_{j'}}\\frac{d\\rho_{bk}^C}{dA_j}\\right)$  \n",
    "\n",
    "$\\frac{d \\phi_{bk}}{d A_j} = \\begin{cases}\n",
    "dt \\cdot (k_b + A_j) & \\text{if } k \\le j \\le k_{max} \\\\\n",
    "0 & \\text{else}\n",
    "\\end{cases}$  \n",
    "\n",
    "$\\frac{d \\rho_{bk}^C}{d A_j} = i\\cdot 2\\pi \\cdot \\delta_{kj} \\sum\\limits_q D_{bq}\\rho_q^C\\cdot e^{i\\cdot 2\\pi r_q\\cdot A_k}r_q$  \n",
    "\n",
    "$\\frac{d}{dA_{j'}}\\frac{d \\phi_{bk}}{d A_j} = \\begin{cases}\n",
    "dt^2\\delta_{jj'} & \\text{if } k \\le j \\le k_{max} \\\\\n",
    "0 & \\text{else}\n",
    "\\end{cases}$  \n",
    "\n",
    "$\\frac{d}{dA_{j'}}\\frac{d \\rho_{bk}^C}{d A_j} = -1\\cdot 4\\pi^2 \\cdot \\delta_{kj}\\delta_{kj'} \\sum\\limits_q D_{bq}\\rho_q^C\\cdot e^{i\\cdot 2\\pi r_q\\cdot A_k}\\cdot r_q^2$  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1271a851",
   "metadata": {},
   "source": [
    "### With respect to EUV-Pulse in frequency domain ($E_n$)  \n",
    "\n",
    "$\\frac{d S_{mb}}{d E_n} = -i\\cdot e^{-i\\tau_m\\omega_n} \\cdot \\sum\\limits_{Ck} D_{kn}\\cdot dt\\cdot \\rho_{bk}^C \\cdot e^{-i t_k\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bk}}$  \n",
    "\n",
    "$\\frac{d S_{mb}}{d E_n^*} = 0$  \n",
    "\n",
    "$U_{zz} = \\frac{dt^2}{2} \\cdot e^{i\\tau_m\\omega_n}\\cdot e^{-i\\tau_m\\omega_p}\\sum\\limits_b \\left(\\left(\\sum\\limits_{Ck}D_{kn}\\cdot \\rho_{bk}^C \\cdot e^{-i t_k\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bk}}\\right)^*\\cdot\\left(\\sum\\limits_{C'j}D_{jp}\\cdot \\rho_{bj}^{C'} \\cdot e^{-i t_j\\cdot \\left(IP^{C'} + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bj}}\\right)\\right)$\n",
    "\n",
    "$V_{zz} = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5338f73c",
   "metadata": {},
   "source": [
    "### With respect to the dipole transition matrix element in momentum domain ($\\rho_b^C$)  \n",
    "\n",
    "$a$ is used as an additional index for momentum, because $b$ and $b'$ are also already used here.\n",
    "\n",
    "$\\frac{d S_{mb}^C}{d \\rho_{b'}^{C'}} = -i\\cdot dt\\cdot\\sum\\limits_{k} D_{kn}\\cdot E_{mk} \\cdot e^{-i t_k\\cdot \\left(IP^{C} + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bk}} \\cdot \\frac{d\\rho_{bk}^{C}}{d\\rho_{b'}^{C'}}$  \n",
    "\n",
    "$\\frac{d\\rho_{bk}^{C}}{d\\rho_{b'}^{C'}} = \\delta_{CC'}\\sum\\limits_q D_{b'q}D_{qb}e^{i\\cdot 2\\pi r_q A_k}$\n",
    "\n",
    "$\\frac{d S_{mb}}{d E_n^*} = 0$  \n",
    "\n",
    "$U_{zz} = \\frac{dt^2}{2}\\sum\\limits_b \\left(\\left(\\sum\\limits_{k}D_{kn}\\cdot E_{mk} \\cdot e^{-i t_k\\cdot \\left(IP^C + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bk}} \\cdot\\frac{d\\rho_{bk}^{C}}{d\\rho_{b'}^{C}}\\right)^*\\cdot\\left(\\sum\\limits_{j}D_{jp}\\cdot E_{mj} \\cdot e^{-i t_j\\cdot \\left(IP^{C} + \\frac{k_b^2}{2}\\right)} \\cdot e^{-i\\phi_{bj}}\\cdot\\frac{d\\rho_{bj}^{C}}{d\\rho_{a}^{C}}\\right)\\right)$  \n",
    "\n",
    "\n",
    "$V_{zz} = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02df70c4",
   "metadata": {},
   "source": [
    "## PIE Gradient  \n",
    "\n",
    "TODO  \n",
    "\n",
    "One could use a FROG-CRAB approach and decompose the DTME/Vectorpotential part into its components each iteration and update via usage of the SFA."
   ]
  }
 ],
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