FROG

Auto-Correlation

Z-Gradient

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

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

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

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

3. PG
   \(\frac{dS_{mk}}{dE_n^*} = D_{nk} \cdot e^{i\tau_m\omega_n} E_k A_{mk}\)

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

4. SD
   \(\frac{dS_{mk}}{dE_n^*} = 2 D_{nk} E_k A_{mk}^* e^{i\tau_m\omega_n}\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn} (A_{mk}^*)^2\)

5. nth-HG
   \(\frac{dS_{mk}}{dE_n^*} = 0\)

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

Z-Pseudo-Hessian

1. SHG

   \(V_{zz} = 0\)

2. THG

   \(V_{zz} = 0\)

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

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

4. SD
   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = 2\cdot D_{pk}D_{kn} \cdot A_{mk} e^{-i\tau_m\omega_p}\)

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = 2\cdot D_{nk}D_{kp} \cdot A_{mk} e^{-i\tau_m\omega_n}\)

5. nth-HG

   \(V_{zz} = 0\)

Cross-Correlation

Z-Gradient (with respect to pulse)

Same for all nonlinear methods

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

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

Z-Gradient (with respect to gate-pulse)

\(\frac{dS_{mk}}{dE_n} = \frac{dS_{mk}}{dA_{mk}}\frac{dA_{mk}}{dE_n}\)

\(\frac{dS_{mk}}{dE_n^*} = \frac{dS_{mk}}{dA_{mk}^*}\left(\frac{dA_{mk}}{dE_n}\right)^*\)

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

   \(\frac{dS_{mk}}{dE_n} = D_{kn} E_k e^{-i\tau_m\omega_n}\)

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

   \(\frac{dS_{mk}}{dE_n} = 2 \cdot D_{kn} E_k A_{mk} e^{-i\tau_m\omega_n}\)

3. PG
   \(\frac{dS_{mk}}{dE_n^*} = D_{nk} \cdot E_k A_{mk} e^{i\tau_m\omega_n}\)

   \(\frac{dS_{mk}}{dE_n} = D_{kn} \cdot E_k A_{mk}^* e^{-i\tau_m\omega_n}\)

4. SD
   \(\frac{dS_{mk}}{dE_n^*} = 2\cdot D_{nk} E_k A_{mk}^* e^{i\tau_m\omega_n}\)

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

5. nth-HG
   \(\frac{dS_{mk}}{dE_n^*} = 0\)

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

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
   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = D_{nk}D_{kp} E_k^* e^{i\tau_m\omega_n} e^{-i\tau_m\omega_p}\)

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = D_{kn}D_{pk} E_k^* e^{-i\tau_m\omega_n} e^{i\tau_m\omega_p}\)

4. SD
   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = 0\)

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = 0\)

5. nth-HG

   \(V_{zz} = 0\)

Interferometric FROG

Auto-Correlation

Z-Gradient

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

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

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

   \(\frac{dS_{mk}}{dE_n} = 3D_{kn}(1+e^{-i\tau_m\omega_n})\cdot (E_k + A_{mk})^2\)

3. PG/SD
   \(\frac{dS_{mk}}{dE_n^*} = D_{nk}(1+e^{i\tau_m\omega_n})\cdot (E_k + A_{mk})^2\)

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

4. nth-HG
   \(\frac{dS_{mk}}{dE_n^*} = 0\)

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

Z-Pseudo-Hessian

1. SHG

   \(V_{zz} = 0\)

2. THG

   \(V_{zz} = 0\)

3. PG/SD
   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = 2D_{nk}D_{kp} (E_k+A_{mk})^* (1+e^{i\tau_m\omega_n}) (1+e^{-i\tau_m\omega_p})\)

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = 2D_{kn}D_{pk} (E_k+A_{mk})^* (1+e^{-i\tau_m\omega_n}) (1+e^{i\tau_m\omega_p})\)

4. nth-HG

   \(V_{zz} = 0\)

Cross-Correlation

Z-Gradient (with respect to pulse)

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

   \(\frac{dS_{mk}}{dE_n} = 2D_{kn}(E_k + A_{mk})\)

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

   \(\frac{dS_{mk}}{dE_n} = 3D_{kn} (E_k + A_{mk})^2\)

3. PG/SD
   \(\frac{dS_{mk}}{dE_n^*} = D_{nk}(E_k + A_{mk})^2\)

   \(\frac{dS_{mk}}{dE_n} = 2D_{kn}|E_k + A_{mk}|^2\)

4. nth-HG
   \(\frac{dS_{mk}}{dE_n^*} = 0\)

   \(\frac{dS_{mk}}{dE_n} = n\cdot D_{kn} (E_k + A_{mk})^{n-1}\)

Z-Gradient (with respect to gate-pulse)

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

   \(\frac{dS_{mk}}{dE_n} = 2D_{kn}e^{-i\tau_m\omega_n}(E_k + A_{mk})\)

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

   \(\frac{dS_{mk}}{dE_n} = 3D_{kn}e^{-i\tau_m\omega_n}(E_k + A_{mk})^2\)

3. PG/SD
   \(\frac{dS_{mk}}{dE_n^*} = D_{nk}e^{i\tau_m\omega_n}(E_k + A_{mk})^2\)

   \(\frac{dS_{mk}}{dE_n} = 2D_{kn}e^{-i\tau_m\omega_n}|E_k + A_{mk}|^2\)

4. nth-HG
   \(\frac{dS_{mk}}{dE_n^*} = 0\)

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

Z-Pseudo-Hessian (with respect to pulse)

1. SHG

   \(V_{zz} = 0\)

2. THG

   \(V_{zz} = 0\)

3. PG/SD
   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = 2D_{nk}D_{kp} (E_k+A_{mk})^*\)

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

4. nth-HG

   \(V_{zz} = 0\)

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

1. SHG

   \(V_{zz} = 0\)

2. THG

   \(V_{zz} = 0\)

3. PG/SD
   \(\frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^* = 2D_{nk}D_{kp} (E_k+A_{mk})^* e^{i\tau_m\omega_n}e^{-i\tau_m\omega_p}\)

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

4. nth-HG

   \(V_{zz} = 0\)