{
"cells": [
{
"cell_type": "markdown",
"id": "9c7ae500",
"metadata": {},
"source": [
"# FROG"
]
},
{
"cell_type": "markdown",
"id": "38407f2d",
"metadata": {},
"source": [
"## Auto-Correlation"
]
},
{
"cell_type": "markdown",
"id": "a852b310",
"metadata": {},
"source": [
"### Z-Gradient \n",
"\n",
"\n",
"1. SHG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = D_{kn}\\cdot(A_{mk} + e^{-i\\tau_m\\omega_n} E_k)$\n",
"\n",
"
\n",
"\n",
"2. THG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = D_{kn}\\cdot(A_{mk}^2 + 2e^{-i\\tau_m\\omega_n} E_k A_{mk})$\n",
"\n",
"
\n",
"\n",
"3. PG \n",
" $\\frac{dS_{mk}}{dE_n^*} = D_{nk} \\cdot e^{i\\tau_m\\omega_n} E_k A_{mk}$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = D_{kn} A_{mk}^* \\cdot (A_{mk} + e^{-i\\tau_m\\omega_n} E_k)$\n",
"\n",
"
\n",
"\n",
"4. SD \n",
" $\\frac{dS_{mk}}{dE_n^*} = 2 D_{nk} E_k A_{mk}^* e^{i\\tau_m\\omega_n}$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = D_{kn} (A_{mk}^*)^2$\n",
"\n",
"
\n",
"\n",
"5. nth-HG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = D_{kn}\\cdot(A_{mk}^{n-1} + (n-1)e^{-i\\tau_m\\omega_n} E_k A_{mk}^{n-2})$"
]
},
{
"cell_type": "markdown",
"id": "03fd35a4",
"metadata": {},
"source": [
"### Z-Pseudo-Hessian \n",
"\n",
"1. SHG \n",
"\n",
" $V_{zz} = 0$\n",
" \n",
"
\n",
"\n",
"2. THG \n",
"\n",
" $V_{zz} = 0$\n",
"\n",
"
\n",
"\n",
"3. PG \n",
" $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = D_{pk} D_{kn}\\cdot \\left( A_{mk}^* e^{-i\\tau_m\\omega_p} + E_k^* e^{i\\tau_m\\omega_n} e^{-i\\tau_m\\omega_p} \\right)$\n",
"\n",
" $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = D_{nk} D_{kp} \\cdot \\left( A_{mk}^* e^{-i\\tau_m\\omega_n} + E_k^* e^{i\\tau_m\\omega_p} e^{-i\\tau_m\\omega_n} \\right)$\n",
" \n",
"\n",
"
\n",
"\n",
"4. SD \n",
" $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 2\\cdot D_{pk}D_{kn} \\cdot A_{mk} e^{-i\\tau_m\\omega_p}$\n",
"\n",
" $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 2\\cdot D_{nk}D_{kp} \\cdot A_{mk} e^{-i\\tau_m\\omega_n}$\n",
" \n",
"\n",
"\n",
"
\n",
"\n",
"5. nth-HG \n",
" \n",
" $V_{zz} = 0$\n"
]
},
{
"cell_type": "markdown",
"id": "37f0d8b9",
"metadata": {},
"source": [
"## Cross-Correlation"
]
},
{
"cell_type": "markdown",
"id": "e86f6286",
"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",
"$\\frac{dS_{mk}}{dE_n} = D_{kn} G(A_{mk})$"
]
},
{
"cell_type": "markdown",
"id": "193313b9",
"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",
"
\n",
"\n",
"1. SHG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = D_{kn} E_k e^{-i\\tau_m\\omega_n}$\n",
"\n",
"
\n",
"\n",
"2. THG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = 2 \\cdot D_{kn} E_k A_{mk} e^{-i\\tau_m\\omega_n}$\n",
"\n",
"
\n",
"\n",
"3. PG \n",
" $\\frac{dS_{mk}}{dE_n^*} = D_{nk} \\cdot E_k A_{mk} e^{i\\tau_m\\omega_n}$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = D_{kn} \\cdot E_k A_{mk}^* e^{-i\\tau_m\\omega_n}$\n",
"\n",
"
\n",
"\n",
"4. SD \n",
" $\\frac{dS_{mk}}{dE_n^*} = 2\\cdot D_{nk} E_k A_{mk}^* e^{i\\tau_m\\omega_n}$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = 0$\n",
"\n",
"
\n",
"\n",
"5. nth-HG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = (n-1) \\cdot D_{kn} E_k A_{mk}^{n-2} e^{-i\\tau_m\\omega_n}$"
]
},
{
"cell_type": "markdown",
"id": "884e085c",
"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": "8ccf84a2",
"metadata": {},
"source": [
"### Z-Pseudo-Hessian (with respect to gate-pulse) \n",
"\n",
"1. SHG \n",
"\n",
" $V_{zz} = 0$\n",
" \n",
"
\n",
"\n",
"2. THG \n",
"\n",
" $V_{zz} = 0$\n",
"\n",
"
\n",
"\n",
"3. PG \n",
" $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = D_{nk}D_{kp} E_k^* e^{i\\tau_m\\omega_n} e^{-i\\tau_m\\omega_p}$\n",
"\n",
" $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = D_{kn}D_{pk} E_k^* e^{-i\\tau_m\\omega_n} e^{i\\tau_m\\omega_p}$\n",
" \n",
"\n",
"
\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",
"
\n",
"\n",
"5. nth-HG \n",
" \n",
" $V_{zz} = 0$\n",
" "
]
},
{
"cell_type": "markdown",
"id": "e2425966",
"metadata": {},
"source": [
"# Interferometric FROG"
]
},
{
"cell_type": "markdown",
"id": "6fb5a307",
"metadata": {},
"source": [
"## Auto-Correlation"
]
},
{
"cell_type": "markdown",
"id": "bfd762b3",
"metadata": {},
"source": [
"### Z-Gradient \n",
"\n",
"\n",
"1. SHG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = 2D_{kn}(1+e^{-i\\tau_m\\omega_n})\\cdot (E_k + A_{mk})$\n",
" \n",
"\n",
"
\n",
"\n",
"2. THG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = 3D_{kn}(1+e^{-i\\tau_m\\omega_n})\\cdot (E_k + A_{mk})^2$\n",
"\n",
"
\n",
"\n",
"3. PG/SD \n",
" $\\frac{dS_{mk}}{dE_n^*} = D_{nk}(1+e^{i\\tau_m\\omega_n})\\cdot (E_k + A_{mk})^2$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = 2D_{kn}(1+e^{-i\\tau_m\\omega_n})\\cdot |E_k + A_{mk}|^2$\n",
"\n",
"
\n",
"\n",
"4. nth-HG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = n\\cdot D_{kn}(1+e^{-i\\tau_m\\omega_n})\\cdot (E_k + A_{mk})^{n-1}$"
]
},
{
"cell_type": "markdown",
"id": "122010af",
"metadata": {},
"source": [
"### Z-Pseudo-Hessian \n",
"\n",
"1. SHG \n",
"\n",
" $V_{zz} = 0$\n",
" \n",
"
\n",
"\n",
"2. THG \n",
"\n",
" $V_{zz} = 0$\n",
"\n",
"
\n",
"\n",
"3. PG/SD \n",
" $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 2D_{nk}D_{kp} (E_k+A_{mk})^* (1+e^{i\\tau_m\\omega_n}) (1+e^{-i\\tau_m\\omega_p})$\n",
"\n",
" $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 2D_{kn}D_{pk} (E_k+A_{mk})^* (1+e^{-i\\tau_m\\omega_n}) (1+e^{i\\tau_m\\omega_p})$\n",
" \n",
"\n",
"
\n",
"\n",
"4. nth-HG \n",
" \n",
" $V_{zz} = 0$"
]
},
{
"cell_type": "markdown",
"id": "c4f568a9",
"metadata": {},
"source": [
"## Cross-Correlation"
]
},
{
"cell_type": "markdown",
"id": "6510250c",
"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",
"
\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",
"
\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",
"
\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": "f0662bd6",
"metadata": {},
"source": [
"### Z-Gradient (with respect to gate-pulse) \n",
"\n",
"1. SHG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = 2D_{kn}e^{-i\\tau_m\\omega_n}(E_k + A_{mk})$\n",
" \n",
"\n",
"
\n",
"\n",
"2. THG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = 3D_{kn}e^{-i\\tau_m\\omega_n}(E_k + A_{mk})^2$\n",
"\n",
"
\n",
"\n",
"3. PG/SD \n",
" $\\frac{dS_{mk}}{dE_n^*} = D_{nk}e^{i\\tau_m\\omega_n}(E_k + A_{mk})^2$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = 2D_{kn}e^{-i\\tau_m\\omega_n}|E_k + A_{mk}|^2$\n",
"\n",
"
\n",
"\n",
"4. nth-HG \n",
" $\\frac{dS_{mk}}{dE_n^*} = 0$ \n",
"\n",
" $\\frac{dS_{mk}}{dE_n} = n\\cdot D_{kn}e^{-i\\tau_m\\omega_n} (E_k + A_{mk})^{n-1}$"
]
},
{
"cell_type": "markdown",
"id": "e97cb0de",
"metadata": {},
"source": [
"### Z-Pseudo-Hessian (with respect to pulse) \n",
"\n",
"1. SHG \n",
"\n",
" $V_{zz} = 0$\n",
" \n",
"
\n",
"\n",
"2. THG \n",
"\n",
" $V_{zz} = 0$\n",
"\n",
"
\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",
"
\n",
"\n",
"4. nth-HG \n",
" \n",
" $V_{zz} = 0$"
]
},
{
"cell_type": "markdown",
"id": "a967cdfc",
"metadata": {},
"source": [
"### Z-Pseudo-Hessian (with respect to gate-pulse) \n",
"\n",
"1. SHG \n",
"\n",
" $V_{zz} = 0$\n",
" \n",
"
\n",
"\n",
"2. THG \n",
"\n",
" $V_{zz} = 0$\n",
"\n",
"
\n",
"\n",
"3. PG/SD \n",
" $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = 2D_{nk}D_{kp} (E_k+A_{mk})^* e^{i\\tau_m\\omega_n}e^{-i\\tau_m\\omega_p}$\n",
"\n",
" $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = 2D_{kn}D_{pk} (E_k+A_{mk})^*e^{-i\\tau_m\\omega_n}e^{i\\tau_m\\omega_p}$\n",
" \n",
"\n",
"\n",
"
\n",
"\n",
"4. nth-HG \n",
" \n",
" $V_{zz} = 0$"
]
}
],
"metadata": {
"language_info": {
"name": "python"
}
},
"nbformat": 4,
"nbformat_minor": 5
}