{ "cells": [ { "cell_type": "markdown", "id": "edbd7f21", "metadata": {}, "source": [ "# Two-Dimensional Spectral-Shearing Interferometry (2D-SI)" ] }, { "cell_type": "markdown", "id": "a0b12ae8", "metadata": {}, "source": [ "## Auto-Correlation" ] }, { "cell_type": "markdown", "id": "00c85be7", "metadata": {}, "source": [ "### Z-Gradient \n", "\n", "1. SHG \n", "\n", " $\\frac{dS_{mk}}{dE_n^*} = 0$ \n", "\n", " $\\frac{dS_{mk}}{dE_n} = D_{kn}\\left((E_k'+A_{mk}'')+E_k\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')\\right)$\n", "\n", "
\n", "\n", "2. THG \n", "\n", " $\\frac{dS_{mk}}{dE_n^*} = 0$ \n", "\n", " $\\frac{dS_{mk}}{dE_n} = D_{kn}\\left((E_k'+A_{mk}'')^2+2E_k\\cdot(E_k'+A_{mk}'')(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')\\cdot E_k\\right)$\n", "\n", "
\n", "\n", "3. PG \n", "\n", " $\\frac{dS_{mk}}{dE_n^*} = D_{nk}E_k(E_k'+A_{mk}'')\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')^*$\n", "\n", " $\\frac{dS_{mk}}{dE_n} = D_{kn}\\left(|E_k'+A_{mk}''|^2 + E_k\\cdot(E_k'+A_{mk}'')^*\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')\\right)$ \n", "\n", "
\n", "\n", "4. SD \n", "\n", " $\\frac{dS_{mk}}{dE_n^*} = 2D_{nk}E_k\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')^*\\cdot(E_k' + A_{mk}'')^*$\n", "\n", " $\\frac{dS_{mk}}{dE_n} = D_{kn}\\left((E_k'+A_{mk}'')^*\\right)^2$\n", "\n", "
\n", "\n", "5. nth-HG \n", "\n", " $\\frac{dS_{mk}}{dE_n^*} = 0$ \n", "\n", " $\\frac{dS_{mk}}{dE_n} = D_{kn}\\left((E_k'+A_{mk}'')^{n-1}+(n-1)E_k\\cdot(E_k'+A_{mk}'')^{n-2}(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')\\cdot E_k\\right)$" ] }, { "cell_type": "markdown", "id": "d79f7707", "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", "\n", " $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = D_{nk}D_{kp}\\cdot\\left((E_k'+A_{mk}'')^*(B_p'e^{-i\\tau_0\\omega_p}+B_p''e^{-i\\tau_m\\omega_p}) + E_k^*(B_n'e^{-i\\tau_0\\omega_n}+B_n''e^{-i\\tau_m\\omega_n})^*(B_p'e^{-i\\tau_0\\omega_p}+B_p''e^{-i\\tau_m\\omega_p})\\right)$\n", "\n", " $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = D_{pk}D_{kn}\\cdot\\left((E_k'+A_{mk}'')^*(B_n'e^{-i\\tau_0\\omega_n}+B_n''e^{-i\\tau_m\\omega_n}) + E_k^*(B_n'e^{-i\\tau_0\\omega_n}+B_n''e^{-i\\tau_m\\omega_n})(B_p'e^{-i\\tau_0\\omega_p}+B_p''e^{-i\\tau_m\\omega_p})^*\\right)$\n", "\n", "
\n", "\n", "4. SD\n", "\n", " $\\frac{d}{dE_p}\\left(\\frac{dS_{mk}}{dE_n}\\right)^* = D_{nk}D_{kp}\\cdot 2(E_k'+A_{mk}'')(B_p'e^{-i\\tau_0\\omega_p}+B_p''e^{-i\\tau_m\\omega_p})$\n", "\n", " $\\frac{d}{dE_p^*}\\left(\\frac{dS_{mk}}{dE_n^*}\\right)^* = D_{pk}D_{kn}\\cdot 2(E_k'+A_{mk}'')(B_n'e^{-i\\tau_0\\omega_n}+B_n''e^{-i\\tau_m\\omega_n})$\n", "\n", "\n", "\n", "
\n", "\n", "5. nth-HG \n", " \n", " $V_{zz} = 0$" ] }, { "cell_type": "markdown", "id": "aa8422c2", "metadata": {}, "source": [ "## Cross-Correlation" ] }, { "cell_type": "markdown", "id": "eccbbf4e", "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", "$\\frac{dS_{mk}}{dE_n} = D_{kn}G_{mk}$" ] }, { "cell_type": "markdown", "id": "1e1909a2", "metadata": {}, "source": [ "### Z-Gradient (with respect to gate-pulse) \n", "\n", "1. SHG \n", "\n", " $\\frac{dS_{mk}}{dA_n^*} = 0$ \n", "\n", " $\\frac{dS_{mk}}{dA_n} = D_{kn}E_k\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')$\n", "\n", "
\n", "\n", "2. THG \n", "\n", " $\\frac{dS_{mk}}{dA_n^*} = 0$ \n", "\n", " $\\frac{dS_{mk}}{dA_n} = 2D_{kn}E_k\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')\\cdot(A_k + A_{mk})$\n", "\n", "
\n", "\n", "3. PG \n", "\n", " $\\frac{dS_{mk}}{dA_n^*} = D_{nk}E_k\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')^*\\cdot(A_k + A_{mk})$ \n", "\n", " $\\frac{dS_{mk}}{dA_n} = D_{kn}E_k\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')\\cdot(A_k + A_{mk})^*$\n", "\n", "
\n", "\n", "4. SD \n", "\n", " $\\frac{dS_{mk}}{dA_n^*} = 2D_{nk}E_k\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')^*\\cdot(A_k + A_{mk})^*$\n", "\n", " $\\frac{dS_{mk}}{dA_n} = 0$\n", "\n", "
\n", "\n", "5. nth-HG \n", "\n", " $\\frac{dS_{mk}}{dE_n^*} = 0$ \n", "\n", " $\\frac{dS_{mk}}{dE_n} = (n-1)D_{kn}E_k\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')\\cdot(A_k + A_{mk})^{n-2}$\n" ] }, { "cell_type": "markdown", "id": "292bb278", "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": "360191f6", "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", "\n", " $\\frac{d}{dA_p}\\left(\\frac{dS_{mk}}{dA_n}\\right)^* = D_{nk}D_{kp}\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')^*(B_p'e^{-i\\tau_0\\omega_p}+e^{-i\\tau_m\\omega_p}B_p'')\\cdot E_k^*$\n", "\n", " $\\frac{d}{dA_p^*}\\left(\\frac{dS_{mk}}{dA_n^*}\\right)^* = D_{kn}D_{pk}\\cdot(B_n'e^{-i\\tau_0\\omega_n}+e^{-i\\tau_m\\omega_n}B_n'')(B_p'e^{-i\\tau_0\\omega_p}+e^{-i\\tau_m\\omega_p}B_p'')^*\\cdot E_k^*$\n", "\n", "
\n", "\n", "4. SD\n", "\n", " $V_{zz}=0$\n", "\n", "
\n", "\n", "5. nth-HG \n", " \n", " $V_{zz} = 0$" ] } ], "metadata": { "language_info": { "name": "python" } }, "nbformat": 4, "nbformat_minor": 5 }