Theory

This page provides an overview of the theoretical foundations behind StrongFieldDynamics.jl.

Strong Field Approximation (SFA)

The SFA is a widely used approach for modeling ionization of atoms in intense laser fields. It treats the interaction of the electron with the laser field non-perturbatively, while the atomic potential is included approximately.

• Direct Ionization: The electron is ionized directly by the laser field.

• Rescattering: After ionization, the electron can be driven back to the parent ion by the laser field, leading to rescattering phenomena.

Partial Wave Expansion

The electron continuum in the laser field is given by Volkov-type states which, similarly, can be expanded into (distorted) partial waves

$$|{\chi }_{\mathbf{p}}(t)⟩=\sqrt{\frac{2}{\pi }}{e}^{-i{S}_V(t)}\sum _{{\ell }_p=0}^{\mathrm{\infty }}\sum _{m_p=-{\ell }_p}^{{\ell }_p}{Y}_{{\ell }_p{m}_p}^{\ast }({\vartheta }_p,{\phi }_p)|{\epsilon }_p{\ell }_p{m}_p⟩\otimes |{\chi }_{f,m_s^{\prime }}⟩,$$

The plane wave expassion of the continuum electron state is given by

$$4\pi \sum _{l_p=0}^{\mathrm{\infty }}\sum _{m_p=-l_p}^{l_p}{i}^{l_p}{j}_{l_p}(pr)Y_{l_p{m}_p}^{\ast }({\theta }_p,{\varphi }_p)Y_{l_p{m}_p}(\theta ,\varphi )$$

For the case of distorted waves, the effect of the Atomic potential is considered hence the distortion of continuum electron. The distorted wave expansion is given by

$$\frac{4\pi }{r}\sum _{{\ell }_p=0}^{\mathrm{\infty }}\sum _{m_p=-{\ell }_p}^{{\ell }_p}{i}^{{\ell }_p}{e}^{\pm i{\mathrm{\Delta }}_{{\ell }_p}}{P}_{{\epsilon }_p{\ell }_p}(r)Y_{{\ell }_p{m}_p}^{\ast }({\vartheta }_p,{\phi }_p)Y_{{\ell }_p{m}_p}({\vartheta }_p,{\phi }_p)$$

where ${\mathrm{\Delta }}_{{\ell }_p}$ is the Coulomb phase shift, which accounts for the influence of the atomic potential on the ionized electron and $P_{{\epsilon }_p{l}_p}(r)$ is the radial wave function of the distorted wave.

In the velocity gauge, the interaction between the continuum electron and laser field is thereby accounted for by the so-called Volkov phase

$$S_V(t)=\frac{1}{2}{\int }^{t}d\tau {[\mathbf{p}+\mathbf{A}(\tau )]}^2$$

This expansion is crucial for capturing the angular structure of the ionization process.

Atomic Potential Effects

The package provides several methods to include the influence of the atomic potential on the ionized electron, improving upon the plain SFA.

Direct Ionization Amplitude

The direct ionization amplitude is computed using the partial wave expansion of the Volkov states. The amplitude for direct ionization can be expressed as:

$$\begin{array}{rl}{T}_0(\mathbf{p},m_j,m_s^{\prime })=& -\frac{i}{\sqrt{2\pi }}{\mathcal{F}}_1[\omega ;f;\mathbf{p}](\sum _{{\ell }_p=0}^{\mathrm{\infty }}\sum _{j_p\ge 1/2}\sum _{q=0,\pm 1}(-1{)}^q{u}_q{Y}_{{\ell }_p,m_j-m_s^{\prime }-q}({\vartheta }_p,{\phi }_p)\\ & \times ⟨{\ell }_p(m_j-m_s^{\prime }-q),\frac{1}{2}{m}_s^{\prime }|j_p(m_j-q)⟩⟨j{m}_j,1(-q)|j_p(m_j-q)⟩⟨{\epsilon }_p{\ell }_p{j}_p\Vert \mathbf{p}\Vert n\ell j⟩)\\ & -\frac{i}{\sqrt{2\pi }}{\mathcal{F}}_1[-\omega ;f;\mathbf{p}](\sum _{{\ell }_p=0}^{\mathrm{\infty }}\sum _{j_p\ge 1/2}\sum _{q=0,\pm 1}{u}_q^{\ast }{Y}_{{\ell }_p,m_j-m_s^{\prime }+q}({\vartheta }_p,{\phi }_p)\\ & \times ⟨{\ell }_p(m_j-m_s^{\prime }+q),\frac{1}{2}{m}_s^{\prime }|j_p(m_j+q)⟩⟨j{m}_j,1q|j_p(m_j+q)⟩⟨{\epsilon }_p{\ell }_p{j}_p\Vert \mathbf{p}\Vert n\ell j⟩)\\ & -\frac{i}{\sqrt{2\pi }}{\mathcal{F}}_2[f;\mathbf{p}]Y_{\ell ,m_j-m_s^{\prime }}({\vartheta }_p,{\phi }_p)⟨\ell (m_j-m_s^{\prime }),\frac{1}{2}{m}_s^{\prime }|j{m}_j⟩⟨{\epsilon }_p\ell j{m}_j|n\ell j{m}_j⟩,\end{array}$$

And the Ionization Probability is given by

$$\mathrm{P}(\mathbf{p})==\frac{p}{2j+1}\sum _{m_j=-j}^j\sum _{m_s^{\prime }=\pm 1/2}{\left|T(\mathbf{p},m_j,m_s^{\prime })\right|}^2$$

References

• Birger Böning, Stephan Fritzsche, "Partial-wave representation of the strong-field approximation", Phys. Rev. A 102, 053108 (2020).

• Birger Böning, Stephan Fritzsche, "Partial-wave representation of the strong-field approximation. II. Coulomb asymmetry in the photoelectron angular distribution of many-electron atoms", Phys. Rev. A 107, 023108 (2023).

For more details, see the source code and comments in the package.