Improved strong-field approximation and quantum-orbit theory: Application to ionization by a bicircular laser field

A theory of above-threshold ionization of atoms by a strong laser field is formulated. Two versions of the strong-field approximation (SFA) are considered, the direct SFA and the improved SFA, which do not and do, respectively, take into account rescattering of the freed electron off the parent ion. The atomic bound state is included in two different ways: as an expansion in terms of Slater-type orbitals or as an asymptotic wave function. Even though we are using the single-active-electron approximation, multielectron effects are taken into account in two ways: by a proper choice of the ground state and by an adequate definition of the ionization rate. For the case of the asymptotic bound-state wave functions, using the saddle-point method, a simple expression for the $T$-matrix element is derived for both the direct and the improved SFA. The theory is applied to ionization by a bicircular field, which consists of two coplanar counterrotating circularly polarized components with frequencies that are integer multiples of a fundamental frequency $\omega$. Special emphasis is on the $\omega \text{−}2\omega$ case. In this case, the threefold rotational symmetry of the field carries over to the velocity map of the liberated electrons, for both the direct and the improved SFA. The results obtained are analyzed in detail using the quantum-orbit formalism, which gives good physical insight into the above-threshold ionization process. For this purpose, a specific classification of the saddle-point solutions is introduced for both the backward-scattered and the forward-scattered electrons. The high-energy backward-scattering quantum orbits are similar to those discovered for high-order harmonic generation. The short forward-scattering quantum orbits for a bicircular field are similar to those of a linearly polarized field. The conclusion is that these orbits are universal, i.e., they do not depend much on the shape of the laser field.

Figure 1

Electric-field vector $\mathbf{E}\left(t\right),0\le t\le T$ (solid lines), and the corresponding vector potential $\mathbf{A}\left(t\right)$ [dashed (red) lines] and the vector $\mathbf{\alpha }\left(t\right)$ [dot-dashed (cyan) line] of the $r\omega \text{−}s\omega$ bicircular laser field (31). The intensities of the field components are equal and the results are presented in arbitrary units. The electric-field vector starts from the point $\mathbf{E}\left(0\right)=(0,0)$ and develops in the clockwise direction for $t>0$, while the vector potential develops counterclockwise as indicated by the arrows in the top left panel. The various panels depict the field for different combinations of the values of $r$ and $s$ as indicated.

Figure 2

Logarithm of the differential ionization rate (in a.u.) of Ne atoms presented in false colors in the electron momentum plane for ionization by a bicircular $r\omega \text{−}s\omega$ laser field with values of $r$ and $s$ as in Fig. 1, equal intensity of both components $I_1=I_2=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, and the fundamental wavelength of 800 nm. The results are obtained by numerical integration of the direct SFA matrix element and using the exact wave functions. The false color scale covers four orders of magnitude.

Figure 3

Shown in the right-hand panel is a comparison of the differential ionization rate obtained by numerical integration (black dashed curve) with the rate obtained using the SPM (red solid curve), presented as a function of the electron energy $E_{\mathbf{p}}$ (in units of the ponderomotive energy of the $\omega$ field component). The top middle panel shows three complex saddle-point solutions for the ionization time $t_0$ (in units of the optical period $T$; the electron energy on each curve changes continuously from $0.2{U_p}_1$ to $8{U_p}_1$). The left panel shows electron trajectories for the saddle-point solutions from the top middle panel and fixed electron energy $E_{\mathbf{p}}=2{U_p}_1$. The laser and atomic parameters are as in Fig. 2 and the electron emission angle is $\theta ={50}^{\circ }$.

Figure 4

Differential ionization rate of Ar atoms presented in false colors in the electron momentum plane for ionization by a bicircular $\omega \text{−}2\omega$ laser field with equal intensity of both components $I_1=I_2=5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, the fundamental wavelength of 790 nm, $h_1=-1,h_2=1,{\varphi }_1=0$, and ${\varphi }_2=\pi$. The results are obtained using the SPM with the asymptotic wave function. The false color scale is linear and normalized to 1. The corresponding bicircular field and vector potential, in arbitrary units and normalized to one, are shown in the top panel.

Figure 5

Logarithm of the differential ionization rate for the same parameters as in Fig. 2 and presented similarly. The ISFA result (23) is obtained by two-dimensional numerical integration and using four $2p$ Slater-type orbitals (16) to represent the ground state of the Ne atom.

Figure 6

Electric-field vector $\mathbf{E}\left(t\right),0\le t\le T$ (solid lines), and the corresponding vector potential $\mathbf{A}\left(t\right)$ [dashed (red) lines] of the $r\omega \text{−}s\omega$ bicircular laser field (31), presented similarly to Fig. 1. For the 1-1 case a small ellipticity is introduced $\left({\varepsilon }_1=0.99\right)$ in order to visually distinguish the field in the direction of the $x$ axis and in the opposite direction (notice the different scale of the $y$ axis for the bottom right panel).

Figure 7

Examples of the notation $(\alpha ,\beta ,m)$ used to label the solutions of the system of the saddle-point equations (28) and (29) for linear polarization (which is equivalent with the $\omega \text{−}\omega$ bicircular field). The solid, dotted, long-dashed, and dot-dashed curves in the right-hand part $\left(t>0\right)$ of the figure $\left(0\le \mathrm{Re}{t}_r\le T\right)$ specify the real part of the rescattering times for the four pairs of orbits with the shortest travel times. In the left-hand part of the figure $\left(t<0\right)$, the counterpart of each curve identifies the corresponding real part of the ionization times $t_0$. The emitted electron energy in multiples of $U_p$ is plotted on the ordinate. Horizontal lines at constant energy (not shown) relate the real parts of the ionization and rescattering times for the respective orbits. There are infinitely many further solutions that have their real parts of the ionization time beyond the left-hand margin of the figure. The curves have been calculated for Ne, for emission in the direction $\theta ={50}^{\circ }$ (rather than $0^{\circ }$, which is why the maximal energy is lower than $10U_p$), and for a laser field having the intensity $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and the fundamental wavelength of 800 nm.

Figure 8

Solutions of the system of the saddle-point equations (28) and (29) for the parameters of Fig. 7 and for the fixed electron emission angle $\theta ={50}^{\circ }$. For each of the presented solutions the electron energy $E_{\mathbf{p}}$ (in units of the ponderomotive energy $U_p$) is presented as a function of the real part of the electron ionization time $\mathrm{Re}{t}_0$ (top) and rescattering time $\mathrm{Re}{t}_r$ (bottom), in units of the optical period $T$. The solutions whose contribution should be discarded after the cutoff are represented by the dashed lines. Each solution is denoted by the corresponding index $(\beta ,m)$. In the bottom panel all solutions on the left (right) have the index $\alpha =+1 \left(\alpha =-1\right)$. There are infinitely many further solutions that have $\mathrm{Re}{t}_0/T$ beyond the left margin of the top panel.

Figure 9

Comparison of the differential ionization rates as a function of the electron energy (in units of the ponderomotive energy) for the same parameters as in Fig. 8. The results obtained by numerical integration are represented by black solid lines with circles, while the results obtained as a coherent sum of the contributions of 14 saddle-point solutions are shown by the solid orange line with squares. The contributions of the divergent solutions are neglected after the corresponding cutoff values. The partial rates for each of the 14 solutions $(\alpha ,\beta ,m)$ are presented and identified by their colors, as explained in the right margin.

Figure 10

Logarithm of the differential ionization rate (in a.u.) of Ne atoms represented in false colors in the electron momentum plane for ionization by a bicircular $\omega \text{−}2\omega$ field. The results are obtained using the SPM with the solutions $(\alpha ,\beta ,m)=(\pm 1,1,0)$, as explained in the text.

Figure 11

Electron trajectories for some of the quantum-orbit solutions presented in Fig. 8. The electron energies for the presented pairs $\alpha =\pm 1$ of solutions are $8U_p$ for the solutions $(\beta ,m)=(1,0)$ and $(2,0),5.6U_p$ for the solutions $(3,0)$, and $7.2U_p$ for the solutions $(1,1)$.

Figure 12

Electric-field vector $\mathbf{E}\left(t\right)$ and the corresponding vectors $\mathbf{A}\left(t\right)$ and $\mathbf{\alpha }\left(t\right)$ of the $\omega \text{−}2\omega$ bicircular laser field (31), presented as in Fig. 1. The field vectors at the ionization times $\left(I\right)$ and rescattering times $\left(R\right)$, for the electron trajectories that are represented by the solid lines in Fig. 11, are indicated by the symbols in the legend.

Figure 13

The $\omega \text{−}2\omega$ bicircular electric-field vector (top left), vector potential (bottom left), electron trajectory (top right), and velocity (bottom right) between the ionization times $\left(I\right)$ and rescattering times $\left(R\right)$ that correspond to the orbit $(\alpha ,\beta ,m)=(1,1,0)$ and the energy $8U_p$.

Figure 14

Comparison of the differential ionization rates as a function of the electron energy, presented similarly to Fig. 9, but for different saddle-point solutions, as denoted in the legend. See the text for further explanation.

Figure 15

Quantum-orbit solutions and trajectories for the same parameters as in Fig. 8. The left-hand panels shows the $(\alpha ,\beta ,m)=(\pm 1,0,0)$ saddle-point solutions for the complex ionization time $t_0$ and rescattering time $t_r$. The $\alpha =-1 \left(\alpha =+1\right)$ solutions are denoted by the red dashed (black solid) lines. The electron energy changes from a minimum value to the cutoff value as a continuous parameter along each curve. The complex time $t_r$ of the solution $(-1,0,0)$ (red dashed line) is presented in the top left panel, while the corresponding time $t_0$ is shown in the bottom left panel. The complex time that corresponds to the electron energy $E_{\mathbf{p}}=1.2U_p$ is denoted by diamonds (circle) in the top (bottom) left panel. The right-hand panels show electron trajectories obtained using the saddle-point solutions of the left-hand panels. The electron energy is $E_{\mathbf{p}}=1.2U_p$. The exit of the tunnel and the rescattering point are denoted by the corresponding symbols. The electron leaves in the direction $\theta ={50}^{\circ }$.

Figure 16

Quantum-orbit solutions $(\nu ,\rho ,\mu )=(\pm 1,0,0)$ (left) and the corresponding trajectories for the electron energy $E_{\mathbf{p}}=1.2U_p$ (right) for the same parameters as in Fig. 8 and presented similarly to Fig. 15. The rescattering times that correspond to the energy $E_{\mathbf{p}}=1.2U_p$ are denoted by diamonds in the left-hand panel.

Figure 17

Same as in Fig. 8 but for the forward-scattering-like saddle-point solutions characterized by the multiple index $(\nu ,\rho ,\mu )$.

Figure 18

Electron trajectories for five short forward-scattering-like quantum orbits and the energies as denoted in the legend. Laser and atomic parameters are as in the previous figures.

Figure 19

Vectors $\mathbf{E}\left(t\right),\mathbf{A}\left(t\right)$, and $\mathbf{\alpha }\left(t\right)$, presented as in Fig. 12. The ionization times $\left(I\right)$ and rescattering times $\left(R\right)$ for those electron trajectories that correspond to the solid lines in Fig. 18 are denoted by the corresponding symbols having the same color, as explained in the legend.

Figure 20

Same as in Fig. 10 but obtained using the SPM with the solutions $(\nu ,\rho ,\mu )=(1,0,0)$ (top), $(\nu ,\rho ,\mu )=(\pm 1,2,0)$ (middle), and $(\nu ,\rho ,\mu )=(\pm 1,3,0)$ (bottom).

Figure 21

Same as in Fig. 10 but obtained using the SPM with the coherent sum of the solutions $(\alpha ,\beta ,m)=(\pm 1,1,0),(\nu ,\rho ,\mu )=(1,0,0)$, and $(\nu ,\rho ,\mu )=(\pm 1,2,0)$.

Figure 22

Classification of the backward-scattering-like $(\alpha ,\beta ,m)$ (left panels) and forward-scattering-like $(\nu ,\rho ,\mu )$ (right panels) saddle-point solutions for the ionization time (top panels) and the rescattering time (middle panels) for the $\omega \text{−}3\omega$ bicircular field presented similarly and for the same parameters as in Fig. 8. The corresponding spectra are shown in the bottom panels, which are the analogs of Fig. 9. The spectrum that includes both the direct and the rescattered electron, obtained by numerical integration, is presented by small diamonds and denoted by $D+R$. In the bottom right panel the spectrum obtained as a coherent sum of the relevant contributions of the $(\nu ,\rho ,\mu )$ solutions is denoted by ${\mathrm{\Sigma }}^{\prime }$, while the sum that corresponds to both solutions $(\nu ,\rho ,\mu )$ and $(\alpha ,\beta ,m)$ is denoted by $\mathrm{\Sigma }$.

Figure 23

Electron trajectories for some of the quantum orbits at the given energies. Laser and atomic parameters are as in previous figures.

Figure 24

Same as in Fig. 10 but for the bicircular $\omega \text{−}3\omega$ field. Shown on the left are the backward-scattering-like solutions $(\alpha ,\beta ,m)=(1,0,0)$ (top), $(1,1,0)$ (middle), and $(-1,2,0)$ (bottom). Shown on the right are the forward-scattering-like solutions $(\nu ,\rho ,\mu )=(1,0,0)$ (top), $(-1,1,0)$ (middle), and $(-1,2,0)$ (bottom). Notice the different color codes of each panel.

Figure 25

Comparison of the velocity maps obtained by numerical integration (top) and using the coherent sum of only four SPM solutions: $(\alpha ,\beta ,m)=(1,0,0)$ and $(1,1,0)$ and $(\nu ,\rho ,\mu )=(1,0,0)$ and $(-1,1,0)$ (bottom). The laser and atomic parameters are as in Fig. 24.