Attosecond Streaking (Definitions, Z-Gradients and Z-Pseudo-Hessians)

Definitions

Symbols and Indices

The SFA introduces a momentum axis k. In the numerical implementation a conjugate spatial axis r is also relevant. These two axis require new index convetions.

\(\quad q\quad\rightarrow\quad\) Spatial Domain
\(\quad b\quad\rightarrow\quad\) Momentum Domain

Its a bit annoying that im already using \(k\) and \(p\) as indices. If its not an index \(k\) refers to the momentum.

In addition the channel-dependent transition-dipole-matrix-element (DTME or \(\rho^C\)) and the channel dependent ionization potential \(IP^C\) are introduced.
In Streaking A always refers to the vectorpotential of the femtosecond pulse, while E refers to the field of the EUV-pulse.

Streaking Amplitude (within Strong-Field-Approximation)

For a single-channel the SFA is implemented as:

\(S_{mb}^C = -i \cdot \sum\limits_k dt\cdot \rho_{bk}^C \cdot E_{mk} \cdot e^{-i t_k\cdot \left(IP^C + \frac{k_b^2}{2}\right)}\cdot e^{-i\phi_{bk}}\)

Where \(S_{mb}\) is the streaking amplitude in the momenum domain, it can be fourier transform to the spatial domain to yield \(S_{mq}\). The multi-channel SFA is simply a coherent sum over all channels, which translates to all gradients:

\(S_{mb} = \sum\limits_C S_{mb}^C\)

\(\frac{d S_{mb}}{dX} = \sum\limits_C \frac{d S_{mb}^C}{dX}\)

The vectorpotential-dependent part of the Volkov-Phase is:

\(\phi_{bk} = \sum\limits_{j'=k}^{k_{\mathrm{max}}} dt\cdot \left(k_b A_{j'} + \frac{A_{j'}^2}{2}\right)\)

The DTME is momentum dependent and thus modulated by the vectorpotential. This can be numerically implemented using the Fourier-shift theorem.

\(\rho_{bk}^C = \sum\limits_q D_{bq} \rho_q^C \cdot e^{i\cdot 2\pi r_q\cdot A_k}\)

Z-Error Definition

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

\(\nabla_{n}Z_m = -2\sum\limits_b \Delta S_{mb} \left(\frac{d S_{mb}}{d E_n}\right)^* + \Delta S_{mb}^*\frac{d S_{mb}}{d E_n^*}\)

Z-Gradients

With respect to the real-part of vectorpotential in frequency domain (\(A_n\))

Vectorpotential is required to be real for streaking. Writing this out is omitted here for the sake of readability.

\(\nabla_{n}Z_m = \mathrm{FT}\{\nabla_{j}Z_m\}\)

\(\nabla_{j}Z_m = -1\cdot\sum\limits_b \Delta S_{mb} \left(\frac{d S_{mb}}{d A_j}\right)^* + \Delta S_{mb}^*\frac{d S_{mb}}{d A_j^*}= -2\cdot\Re\left(\sum\limits_{b, C}\Delta S_{mb}\left(\frac{d S_{mb}^C}{d A_j}\right)^*\right)\)

\(\frac{d S_{mb}^C}{d A_j} = -i\cdot\sum\limits_k dt \cdot E_{mk} \cdot e^{-i t_k\cdot \left(IP^C + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bk}}\cdot\left(-i\cdot \rho_{bk}^C\cdot \frac{d \phi_{bk}}{d A_j} + \frac{d\rho_{bk}^C}{d A_j} \right)\)

For the derivatives of the inner funcition one gets:

\(\frac{d \rho_{bk}^C}{d A_j} = i\cdot 2\pi \cdot \delta_{kj} \sum\limits_q D_{bq}\rho_q^C\cdot e^{i\cdot 2\pi r_q\cdot A_k}r_q\)

The \(\delta_{kj}\) does not appear in the other derivative, a different one with different indices appears and gets rid of a different sum.

\(\frac{d \phi_{bk}}{d A_j} = \begin{cases} dt \cdot (k_b + A_j) & \text{if } k \le j \le k_{max} \\ 0 & \text{else} \end{cases}\)

Thus the sum over k survives in this term. The entire gradient reads:

\(\frac{d S_{mb}^C}{d A_j} = g_1 + g_2\)

\(g_1 = -1\cdot dt^2 \cdot \sum\limits_k \begin{cases} E_{mk} \cdot e^{-i t_k\cdot \left(IP^C + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bk}}\cdot\rho_{bk}^C\cdot (k_b + A_j) & \text{if } k \le j \le k_{max} \\ 0 & \text{else} \end{cases}\)

\(g_2 = 2\pi dt \cdot E_{mj} \cdot e^{-i t_j\cdot \left(IP^C + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bj}}\cdot \left(\sum\limits_q D_{bq}\rho_q^C \cdot r_q\cdot e^{i\cdot 2\pi r_q\cdot A_j}\right)\)

With respect to the attosecond pulse in frequency domain \(\left(E_n\right)\)

\(\nabla_{n}Z_m = -2\sum\limits_b \Delta S_{mb} \left(\frac{d S_{mb}}{d E_n}\right)^* = -2\sum\limits_{b,C}\Delta S_{mb} \left(\frac{d S_{mb}^C}{d E_n}\right)^*\)

\(\frac{d S_{mb}^C}{d E_n} = -i \cdot dt\cdot e^{-i\omega_n\tau_m}\cdot \sum\limits_k D_{kn}\cdot \rho_{bk}^C \cdot e^{-i t_k\cdot \left(IP^C + \frac{k_b^2}{2}\right)}\cdot e^{-i\phi_{bk}}\)

With respect to the dipole transition matrix element in momentum domain (\(\rho_b^C\))

Here one wants the gradient with respect to the DTME of each channel.

\(\nabla_{b}Z_m = FFT_{bq} \left(\nabla_{q}Z_m\right)\)

\(\nabla_{q'}^{C'}Z_m = -2\sum\limits_{b,C} \Delta S_{mb} \left(\frac{d S_{mb}^C}{d \rho_{q'}^{C'}}\right)^*\)

\(\frac{d S_{mb}^C}{d \rho_{q'}^{C'}} = -i \delta_{CC'}\delta_{qq'}\cdot dt \cdot D_{bq}\cdot \sum\limits_k E_{mk} \cdot e^{-i t_k\cdot \left(IP^C + \frac{k_b^2}{2}\right)}\cdot e^{-i\phi_{bk}} \cdot e^{i\cdot 2\pi r_q A_k}\)

Z-Pseudo-Hessian

With respect to the real-part of vectorpotential in frequency domain (\(A_n\))

\(U_{zz} = \sum\limits_b \Re\left\{\left(\frac{d S_{mb}}{d A_{n'}}\right)^*\cdot\frac{d S_{mb}}{d A_n}\right\}\)

\(\frac{d S_{mb}}{d A_n} = FT_{nj}\left\{\frac{d S_{mb}}{d A_j}\right\}\)

\(V_{zz} = \sum\limits_b \Re\left\{\frac{d}{d A_n}\left(\frac{dS_{mb}}{d A_{n'}}\right)^*\cdot \Delta S_{mb}\right\} = \sum\limits_b \Re\left\{\left(\frac{d}{d A_n}\frac{dS_{mb}}{d A_{n'}}\right)^*\cdot \Delta S_{mb}\right\} = \sum\limits_b \Re\left\{\left( D_{jn} D_{j'n'} \frac{d}{d A_j}\frac{dS_{mb}}{d A_{j'}}\right)^*\cdot \Delta S_{mb}\right\}\)

Writing the full hessian out isnt fun. So i wont.

\(\frac{d S_{mb}}{d A_j} = -i\cdot dt \sum\limits_{Ck} E_{mk}\cdot e^{-i t_k\cdot\left(IP^C+\frac{k_b^2}{2}\right)}\cdot \frac{d}{dA_j}\left(\rho_{bk}^C\cdot e^{-i\phi_{bk}}\right)\)

\(\frac{d}{dA_j}\left(\rho_{bk}^C\cdot e^{-i\phi_{bk}}\right) = e^{-i\phi_{bk}}\cdot\left(-i\cdot\rho_{bk}^C\cdot\frac{d\phi_{bk}}{dA_j} + \frac{d\rho_{bk}^C}{dA_j}\right)\)

\(\frac{d}{dA_{j'}}\frac{d S_{mb}}{d A_j} = -i\cdot dt \sum\limits_{Ck} E_{mk}\cdot e^{-i t_k\cdot\left(IP^C+\frac{k_b^2}{2}\right)}\cdot \frac{d}{dA_{j'}}\frac{d}{dA_j}\left(\rho_{bk}^C\cdot e^{-i\phi_{bk}}\right)\)

Below are the critial derivatives.

\(\frac{d}{dA_{j'}}\frac{d}{dA_j}\left(\rho_{bk}^C\cdot e^{-i\phi_{bk}}\right) =-i e^{-i\phi_{bk}}\cdot\left(\rho_{bk}^C\cdot\left(\frac{d}{dA_{j'}}\frac{d\phi_{bk}}{dA_j} -i\frac{d\phi_{bk}}{dA_{j'}}\frac{d\phi_{bk}}{dA_j}\right) + \frac{d\phi_{bk}}{dA_{j'}}\frac{d\rho_{bk}^C}{dA_j}+\frac{d\rho_{bk}^C}{dA_{j'}}\frac{d\phi_{bk}}{dA_j}+i\frac{d}{dA_{j'}}\frac{d\rho_{bk}^C}{dA_j}\right)\)

\(\frac{d \phi_{bk}}{d A_j} = \begin{cases} dt \cdot (k_b + A_j) & \text{if } k \le j \le k_{max} \\ 0 & \text{else} \end{cases}\)

\(\frac{d \rho_{bk}^C}{d A_j} = i\cdot 2\pi \cdot \delta_{kj} \sum\limits_q D_{bq}\rho_q^C\cdot e^{i\cdot 2\pi r_q\cdot A_k}r_q\)

\(\frac{d}{dA_{j'}}\frac{d \phi_{bk}}{d A_j} = \begin{cases} dt^2\delta_{jj'} & \text{if } k \le j \le k_{max} \\ 0 & \text{else} \end{cases}\)

\(\frac{d}{dA_{j'}}\frac{d \rho_{bk}^C}{d A_j} = -1\cdot 4\pi^2 \cdot \delta_{kj}\delta_{kj'} \sum\limits_q D_{bq}\rho_q^C\cdot e^{i\cdot 2\pi r_q\cdot A_k}\cdot r_q^2\)

With respect to EUV-Pulse in frequency domain (\(E_n\))

\(\frac{d S_{mb}}{d E_n} = -i\cdot e^{-i\tau_m\omega_n} \cdot \sum\limits_{Ck} D_{kn}\cdot dt\cdot \rho_{bk}^C \cdot e^{-i t_k\cdot \left(IP^C + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bk}}\)

\(\frac{d S_{mb}}{d E_n^*} = 0\)

\(U_{zz} = \frac{dt^2}{2} \cdot e^{i\tau_m\omega_n}\cdot e^{-i\tau_m\omega_p}\sum\limits_b \left(\left(\sum\limits_{Ck}D_{kn}\cdot \rho_{bk}^C \cdot e^{-i t_k\cdot \left(IP^C + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bk}}\right)^*\cdot\left(\sum\limits_{C'j}D_{jp}\cdot \rho_{bj}^{C'} \cdot e^{-i t_j\cdot \left(IP^{C'} + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bj}}\right)\right)\)

\(V_{zz} = 0\)

With respect to the dipole transition matrix element in momentum domain (\(\rho_b^C\))

\(a\) is used as an additional index for momentum, because \(b\) and \(b'\) are also already used here.

\(\frac{d S_{mb}^C}{d \rho_{b'}^{C'}} = -i\cdot dt\cdot\sum\limits_{k} D_{kn}\cdot E_{mk} \cdot e^{-i t_k\cdot \left(IP^{C} + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bk}} \cdot \frac{d\rho_{bk}^{C}}{d\rho_{b'}^{C'}}\)

\(\frac{d\rho_{bk}^{C}}{d\rho_{b'}^{C'}} = \delta_{CC'}\sum\limits_q D_{b'q}D_{qb}e^{i\cdot 2\pi r_q A_k}\)

\(\frac{d S_{mb}}{d E_n^*} = 0\)

\(U_{zz} = \frac{dt^2}{2}\sum\limits_b \left(\left(\sum\limits_{k}D_{kn}\cdot E_{mk} \cdot e^{-i t_k\cdot \left(IP^C + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bk}} \cdot\frac{d\rho_{bk}^{C}}{d\rho_{b'}^{C}}\right)^*\cdot\left(\sum\limits_{j}D_{jp}\cdot E_{mj} \cdot e^{-i t_j\cdot \left(IP^{C} + \frac{k_b^2}{2}\right)} \cdot e^{-i\phi_{bj}}\cdot\frac{d\rho_{bj}^{C}}{d\rho_{a}^{C}}\right)\right)\)

\(V_{zz} = 0\)

PIE Gradient

TODO

One could use a FROG-CRAB approach and decompose the DTME/Vectorpotential part into its components each iteration and update via usage of the SFA.