Chirp-Scan

Z-Gradient

1. SHG

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

   \(\frac{dS_{mk}}{dE_n} = 2 D_{kn} A_{mk} e^{i\phi_{mn}}\)

2. THG

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

   \(\frac{dS_{mk}}{dE_n} = 3 D_{kn} A_{mk}^2 e^{i\phi_{mn}}\)

3. PG/SD

   \(\frac{dS_{mk}}{dE_n^*} = D_{nk} A_{mk}^2 e^{-i\phi_{mn}}\)

   \(\frac{dS_{mk}}{dE_n} = 2 D_{kn} |A_{mk}|^2 e^{i\phi_{mn}}\)

5. nth-HG

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

   \(\frac{dS_{mk}}{dE_n} = n\cdot D_{kn} A_{mk}^{n-1} e^{i\phi_{mn}}\)

Z-Pseudo-Hessian

1. SHG

   \(V_{zz}=0\)

2. THG

   \(V_{zz}=0\)

3. PG/SD

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

   \(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^* = 2D_{kn}D_{pk} A_{mk}^* e^{i \phi_{mn}} e^{-i \phi_{mp}}\)

4. nth-HG

   \(V_{zz} = 0\)