General Definitions

For Attosecond Streaking the definitions can be found here.

Nonlinear Methods

  1. SHG

    \(G(E) = E\)


  1. THG

    \(G(E) = E^2\)


  1. PG

    \(G(E) = |E|^2\)


  1. SD (This definition causes a flip in delay and frequency-sign)

    \(G(E) = (E^*)^2\)


  1. nth-HG

    \(G(E) = E^{n-1}\)

Indices and Symbols

\(\quad k,j\quad\rightarrow\quad\) Time Domain
\(\quad n,p\quad\rightarrow\quad\) Frequency Domain
\(\quad m\quad\rightarrow\quad\) Transformation (e.g. shift in time, applied chirp, … )

\(\quad E_k\quad\rightarrow\quad\) pulse in time domain
\(\quad A_{mk}\quad\rightarrow\quad\) transformed pulse in time domain

\(\quad D_{kn} \quad\rightarrow\quad\) fourier matrix (inverse)
\(\quad D_{nk} \quad\rightarrow\quad\) fourier matrix (forward)

Transformations

  1. (Inverse) Fourier Transform

    \(E_n = \sum_k D_{nk} E_n = \sum_k e^{-i \omega_n t_k} E_k\)
    \(E_k = \sum_n D_{kn} E_n = \sum_n e^{i \omega_n t_k} E_n\)


  1. Method dependent transformation

    \(A_{mk} = \sum_n D_{kn} E_n \cdot e^{i\phi_{mn}}\)

    \(\phi_{mn} = -\tau_m \cdot \omega_n\) (linear phase)

    \(\phi_{mn} = f(\theta_m, \omega_n)\) (arbitrary phase, through material or pulse-shaper)

Definition of nonlinear signal in time domain

  1. Frog

    \(S_{mk} = E_k \cdot G(A_{mk})\)

  2. Chirp-Scan

    \(S_{mk} = A_{mk}\cdot G(A_{mk})\)

  3. Interferometric Frog

    \(S_{mk} = (E_k + A_{mk})\cdot G(E_k + A_{mk})\)

  4. Time-Domain-Ptychography

    Same as FROG or I-FROG only \(A_{mk}\) is different.
    \(A_{mk}' = \sum_n D_{kn}B_n A_{mn}\quad\) (\(B_n\) represents a known spectral filter, can be complex valued)

  5. 2D-SI

    \(S_{mk} = G(E_k'\cdot e^{-i\tau_0\omega_n} + A_{mk}'') \cdot E_k \quad\) ( \(\tau_0\) is a fixed delay of \(E_k\) and \(E_k'\) )
    \(E_k' = \sum_n D_{kn}B_n E_n\quad\) ( \(B_n\) represents a known spectral filter, can be complex valued)

  6. Vampire

    \(S_{mk} = G(E_k\cdot e^{-i\tau_0\omega_n} + E_k') \cdot A_{mk}\quad\) (Original definition, pushing \(m\) onto gate makes it easier with PIE.)
    \(S_{mk} = G_m(E_k\cdot e^{-i\tau_0\omega_n} + E_k') \cdot E_k = G(A_{mk}\cdot e^{-i\tau_0\omega_n} + A_{mk}')\cdot E_k\)
    \(E_k' = \sum_n D_{kn}E_ne^{i\phi_n}\quad\) (\(\phi_n\) represents a material dispersion)
    \(\tau_0\) is a fixed delay in the interferometer.

Error

  1. G-Error (Trace Error)

    \(L = \frac{1}{N_m\cdot N_n}\sum\limits_{nm} (T_{mn}^{\mathrm{exp}} - \mu \cdot T_{mn})^2\)

  2. Z-Error

    \(Z = \sum\limits_{km} | S_{mk}' - S_{mk}|^2\)

  3. PIE-Error

    \(L = \frac{1}{N_m\cdot N_n}\sum\limits_{nm} (\sqrt{T_{mn}^{\mathrm{exp}}} - \mu\cdot|S_{mn}|)^2\)

Calculation of \(\mu\)

  1. Frequency independent

    \(\mu = \frac{\sum_{nm} T_{mn} \cdot T_{mn}^{\mathrm{exp}}}{\sum_{nm} T_{mn}^2}\)

  2. Frequency dependent

    \(\mu_n = \frac{\sum_{m} T_{mn} \cdot T_{mn}^{\mathrm{exp}}}{\sum_{m} T_{mn}^2}\)


  1. For PIE-Error

    \(T_{mn}\quad\rightarrow\quad|S_{mn}|\)

    \(T_{mn}^{\mathrm{exp}}\quad\rightarrow\quad\sqrt{T_{mn}^{\mathrm{exp}}}\)

Z-Gradient

\(\nabla_n Z_m = -2\cdot\sum_k \Delta S_{mk}^*\cdot\frac{dS_{mk}}{dE_n^*} + \Delta S_{mk}\cdot\left(\frac{dS_{mk}}{dE_n}\right)^*\)

\(\nabla_n Z = \sum_m \nabla_n Z_m\)

Z-Pseudo-Hessian

\(H_{zz} = U_{zz} - V_{zz}\)

\(U_{zz}^m = \frac{1}{2}\cdot\sum_k\left(\left(\frac{dS_{mk}}{dE_n}\right)^*\frac{dS_{mk}}{dE_p} + \left(\left(\frac{dS_{mk}}{dE_n^*}\right)^*\frac{dS_{mk}}{dE_p^*}\right)^*\right)\)

\(V_{zz}^m = \frac{1}{2}\cdot\sum_k\left( \frac{d}{dE_p}\left(\frac{dS_{mk}}{dE_n}\right)^*\cdot \Delta S_{mk} + \left(\frac{d}{dE_p^*}\left(\frac{dS_{mk}}{dE_n^*}\right)^*\cdot \Delta S_{mk}\right)^* \right)\)

Z-Gradient and Pseudo-Hessian (with respect to spectral phase factor)

\(E_n = |E_n| \cdot \Psi_n\)

\(\frac{d E_n}{d \Psi_n} = \frac{d E_n^*}{d \Psi_n^*} = |E_n|\)

Z-Gradient

\(\frac{d Z_m}{d\Psi_n^*} = \frac{d E_n^*}{d\Psi_n^*}\frac{d Z_m}{d E_n^*} = |E_n|\cdot\frac{d Z_m}{d E_n^*}\)

Z-Pseudo Hessian

\(\frac{d}{d\Psi_p}\left(\frac{d S_{mk}}{d \Psi_n}\right)^* = \frac{d}{d\Psi_p}\left(\frac{d E_n}{d\Psi_n}\frac{d S_{mk}}{d E_n}\right)^* = \frac{d E_p}{d\Psi_p}\frac{d E_n^*}{d\Psi_n^*}\frac{d}{dE_p}\left(\frac{d S_{mk}}{d E_n}\right)^* = |E_p||E_n|\cdot \frac{d}{dE_p}\left(\frac{d S_{mk}}{d E_n}\right)^*\)

\(\qquad \Rightarrow \quad H_{\Psi} = H_{E} \cdot |E_p| |E_n|\)

I am assuming that all other terms from the product-rule are zero. Maybe thats wrong?

PIE Gradient

\(\nabla_k L_m = \frac{d |S_{mn}|}{d O_k^*}\frac{d L_m}{d |S_{mn}|}\)

\(\nabla_k L = \sum_m \nabla_k L_m\)

PIE Pseudo-Hessian

\(H_{zz} = U_{zz} - V_{zz}\)

\(U_{zz}^m = \frac{1}{2}\cdot\sum_n\left(\left(\frac{d|S_{mn}|}{dE_j}\right)^*\frac{d|S_{mn}|}{dE_k} + \left(\left(\frac{d|S_{mn}|}{dE_j^*}\right)^*\frac{d|S_{mn}|}{dE_k^*}\right)^*\right)\)

\(V_{zz}^m = \frac{1}{2}\cdot\sum_n\left( \frac{d}{dE_k}\left(\frac{d|S_{mn}|}{dE_j}\right)^*\cdot (\sqrt{T_{mn}^{\mathrm{exp}}} - |S_{mn}|) + \left(\frac{d}{dE_k^*}\left(\frac{d|S_{mn}|}{dE_j^*}\right)^*\cdot (\sqrt{T_{mn}^{\mathrm{exp}}} - |S_{mn}|)\right)^* \right)\)

PIE Gradient and Pseudo-Hessian (with respect to spectral phase factor)

\(O_n = |O_n| \cdot \Psi_n\)

\(\frac{d O_n}{d \Psi_n} = \frac{d O_n^*}{d \Psi_n^*} = |O_n|\)

PIE Gradient

\(\frac{d L_m}{d\Psi_n^*} = \frac{d O_n^*}{d\Psi_n^*}\frac{d L_m}{d O_n^*} = |O_n|\cdot\frac{d L_m}{d O_n^*}\)

\(\frac{d L_m}{d O_n^*} = \frac{d O_k^*}{d O_n^*}\frac{d L_m}{d O_k^*} = \mathrm{FFT}\{{\frac{d L_m}{d O_k^*}}\}\)

PIE Pseudo-Hessian

\(\frac{d|S_{mn}|}{d\Psi_n} = \frac{d E_n}{d \Psi_n}\frac{d E_k}{d}\frac{d |S_{mn}|}{d E_k}\)

\(\frac{d}{d \Psi_n}\left(\frac{d |S_{mn}|}{d \Psi_p}\right)^* = \frac{d E_p}{d \Psi_p}\frac{d E_j}{d E_p}\left(\frac{d E_n}{d \Psi_n}\frac{d E_k}{d E_n}\right)^*\frac{d}{d E_j}\left(\frac{d |S_{mn}|}{d E_k}\right)^*\)

\(\qquad \Rightarrow \quad H_{\Psi} = H_{E} \cdot \frac{d E_p}{d \Psi_p}\frac{d E_j}{d E_p}\left(\frac{d E_n}{d \Psi_n}\frac{d E_k}{d E_n}\right)^*\)

I am assuming that all other terms from the product-rule are zero. Maybe thats wrong?