Two-Dimensional Spectral-Shearing Interferometry (2D-SI)

Auto-Correlation

Z-Gradient

1. SHG

   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}\left((E_k'+A_{mk}'')+E_k\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')\right)\)

2. THG

   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}\left((E_k'+A_{mk}'')^2+2E_k\cdot(E_k'+A_{mk}'')(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')\cdot E_k\right)\)

3. PG

   \(\frac{dS_{mk}}{dE_n^*} = D_{nk}E_k(E_k'+A_{mk}'')\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')^*\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}\left(|E_k'+A_{mk}''|^2 + E_k\cdot(E_k'+A_{mk}'')^*\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')\right)\)

4. SD

   \(\frac{dS_{mk}}{dE_n^*} = 2D_{nk}E_k\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')^*\cdot(E_k' + A_{mk}'')^*\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}\left((E_k'+A_{mk}'')^*\right)^2\)

5. nth-HG

   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn}\left((E_k'+A_{mk}'')^{n-1}+(n-1)E_k\cdot(E_k'+A_{mk}'')^{n-2}(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')\cdot E_k\right)\)

Z-Pseudo-Hessian

1. SHG

   \(V_{zz}=0\)

2. THG

   \(V_{zz}=0\)

3. PG

   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = D_{nk}D_{kp}\cdot\left((E_k'+A_{mk}'')^*(B_p'e^{-i\tau_0\omega_p}+B_p''e^{-i\tau_m\omega_p}) + E_k^*(B_n'e^{-i\tau_0\omega_n}+B_n''e^{-i\tau_m\omega_n})^*(B_p'e^{-i\tau_0\omega_p}+B_p''e^{-i\tau_m\omega_p})\right)\)

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = D_{pk}D_{kn}\cdot\left((E_k'+A_{mk}'')^*(B_n'e^{-i\tau_0\omega_n}+B_n''e^{-i\tau_m\omega_n}) + E_k^*(B_n'e^{-i\tau_0\omega_n}+B_n''e^{-i\tau_m\omega_n})(B_p'e^{-i\tau_0\omega_p}+B_p''e^{-i\tau_m\omega_p})^*\right)\)

4. SD

   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = D_{nk}D_{kp}\cdot 2(E_k'+A_{mk}'')(B_p'e^{-i\tau_0\omega_p}+B_p''e^{-i\tau_m\omega_p})\)

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = D_{pk}D_{kn}\cdot 2(E_k'+A_{mk}'')(B_n'e^{-i\tau_0\omega_n}+B_n''e^{-i\tau_m\omega_n})\)

5. nth-HG

   \(V_{zz} = 0\)

Cross-Correlation

Z-Gradient (with respect to pulse)

Same for all nonlinear methods

\(\frac{dS_{mk}}{dE_n^*} = 0\)

\(\frac{dS_{mk}}{dE_n} = D_{kn}G_{mk}\)

Z-Gradient (with respect to gate-pulse)

1. SHG

   \(\frac{dS_{mk}}{dA_n^*} = 0\)

   \(\frac{dS_{mk}}{dA_n} = D_{kn}E_k\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')\)

2. THG

   \(\frac{dS_{mk}}{dA_n^*} = 0\)

   \(\frac{dS_{mk}}{dA_n} = 2D_{kn}E_k\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')\cdot(A_k + A_{mk})\)

3. PG

   \(\frac{dS_{mk}}{dA_n^*} = D_{nk}E_k\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')^*\cdot(A_k + A_{mk})\)

   \(\frac{dS_{mk}}{dA_n} = D_{kn}E_k\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')\cdot(A_k + A_{mk})^*\)

4. SD

   \(\frac{dS_{mk}}{dA_n^*} = 2D_{nk}E_k\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')^*\cdot(A_k + A_{mk})^*\)

   \(\frac{dS_{mk}}{dA_n} = 0\)

5. nth-HG

   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = (n-1)D_{kn}E_k\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')\cdot(A_k + A_{mk})^{n-2}\)

Z-Pseudo-Hessian (with respect to pulse)

Same for all nonlinear methods

\(V_{zz} = 0\)

Z-Pseudo-Hessian (with respect to gate-pulse)

1. SHG

   \(V_{zz}=0\)

2. THG

   \(V_{zz}=0\)

3. PG

   \(\frac{d}{dA_p}\left(\frac{dS_{mk}}{dA_n}\right)^* = D_{nk}D_{kp}\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')^*(B_p'e^{-i\tau_0\omega_p}+e^{-i\tau_m\omega_p}B_p'')\cdot E_k^*\)

   \(\frac{d}{dA_p^*}\left(\frac{dS_{mk}}{dA_n^*}\right)^* = D_{kn}D_{pk}\cdot(B_n'e^{-i\tau_0\omega_n}+e^{-i\tau_m\omega_n}B_n'')(B_p'e^{-i\tau_0\omega_p}+e^{-i\tau_m\omega_p}B_p'')^*\cdot E_k^*\)

4. SD

   \(V_{zz}=0\)

5. nth-HG

   \(V_{zz} = 0\)