Coherent Electron Wave Packet, CEWP, Interference in Attosecond Photoionization with Ultrashort Circularly Polarized XUV Laser Pulses

Abstract

Effects of coherent electron interference in attosecond photoionization with intense ultrashort circularly polarized extreme ultraviolet (XUV) laser pulses are studied. Simulations are performed on oriented one electron molecular ions H$_2^+$ and H$_3^{2+}$ by numerically solving appropriate time-dependent Schrödinger equations (TDSEs). It is found that due to interference of coherent continuum scattering electron wave packets, momentum stripes with intervals $\varDelta p^s =2\pi /R$ are observed in angular distributions. The momentum stripes are independent of the laser polarization and wavelength, and these are always perpendicular to the molecular internuclear axis. Ionization with two color circularly polarized XUV laser pulses produces an asymmetry in angular distributions resulting from interference of coherent electron wave packets (CEWPs) from multiple pathway ionization. The multiple ionization pathway interference is shown to be sensitive to the pulse relative carrier envelope phase (CEP) and photoelectron kinetic energies. We describe these phenomena by a perturbation multi-photon ionization model, thus suggesting imaging and controlling methods for electrons in molecules on attosecond time scale.

Appendix 1

To understand the sensitivity of the asymmetry in the photoionization momentum and angular distributions to the relative phase $\phi$, we adopt a perturbative theoretical model of a multi-photon ionization. Figure 7.10 illustrates the possible photoionization paths in the two color laser pulses. For the ionization at photoelectron kinetic energies $E_{e1}=2\omega _1-I_p$, the photoionization proceeds by two simultaneous processes, a direct one-photon transition at $\omega _2$ and two one-photon transitions at $\omega _1+\omega _1$. Usually for direct one-photon ionization processes by laser pulses, the ionization differential probability can be expressed simply in the dipole form [67]:

$$\begin{aligned} \frac{dP_\mathrm{ion }}{d\Omega }&\propto | \langle \psi _c^{E_e} (\mathbf r ) | T_H|\psi _0(\mathbf r )\rangle | ^2 =|\langle \psi _c^{E_e} (\mathbf r ) | \mathbf r \cdot \mathbf E |\psi _0(\mathbf r )\rangle | ^2, \end{aligned}$$

(7.16)

where $\psi _c^{E_e} (\mathbf r )$ is the continuum electron wave function with energy $E_e$ and $\psi _0(\mathbf r )$ is the initial state. $T_H$ is a transition operator corresponding to the transition from the initial state $|\psi _0\rangle$ to the continuum state $|\psi _c^{E_e}\rangle$. Under conditions $\lambda _e>R$ no diffraction of electron occurs. The circular polarization MPADs are simply superpositions of the linear parallel and perpendicular (to the molecular axis) polarization distributions [22, 23].

Fig. 7.10

Schematic illustrations of multi-photon ionization pathways from the initial H$_2^+$ $1s\sigma _g$ ground state at $R_e=2$ a.u. in the ultrashort two color laser fields for the MATI peaks at energies a $E_{e1}=2\omega _1-I_p$, b $E_{e2}=3\omega _1-I_p$, and c $E_{e3}=4\omega _1-I_p$. ${\fancyscript{A}}$ is the corresponding transition matrix element. Short and long arrows denote $\omega _1$ and $\omega _2$ photons ($\omega _2=2\omega _1$)

7.1.1 One-Color Pulse Ionization Processes

In a one-color laser pulse, the angular distributions of electrons emitted in one- photon ionization processes have the simple forms respectively ${\fancyscript{P}}_\parallel {(\omega _1)}\propto \alpha _0+\alpha _1\cos ^2\theta$ for the parallel $x$ polarization, and ${\fancyscript{P}}_ \perp {(\omega _1)}\propto \alpha _0+\alpha _1\sin ^2\theta$ for the perpendicular $y$ polarization, where coefficients $\alpha _{i}$, $i=0,1,2,\ldots$, depend on laser pulses and the initial state. Each transition matrix amplitude shows that the CEP $\phi$ has no influence on the multi-photon ionization angular distributions of the photoelectron spectra.

The two-photon ($\omega _1+\omega _1$) transition matrix element ${\fancyscript{A}}_{2\omega _1}$ reads in the perturbative dipole approximation as:

$$\begin{aligned} {\fancyscript{A}}_{2\omega _1}&= \langle \psi _c^{E_e}|T_H|\psi _0 \rangle \nonumber \\&= \int _{\lim \varepsilon \rightarrow 0} dE_n^e \frac{\langle \psi _c^{E_e}|\mathbf r \cdot \mathbf E |\psi _n\rangle \langle \psi _n|\mathbf r \cdot \mathbf E |\psi _0\rangle }{E_0^{e}-E_n^{e}+\omega _1+i\varepsilon } \nonumber \\&= \mathrm{PP } \int dE_n^e \frac{\langle \psi _c^{E_e}|\mathbf r \cdot \mathbf E |\psi _n\rangle \langle \psi _n|\mathbf r \cdot \mathbf E |\psi _0\rangle }{E_0^{e}-E_n^{e}+\omega _1}\nonumber \\&-i\pi \delta (E_0^{e}-E_n^{e}+\omega _1){\langle \psi _c^{E_e}|\mathbf r \cdot \mathbf E |\psi _n\rangle }\nonumber {\langle \psi _n|\mathbf r \cdot \mathbf E |\psi _0\rangle } \nonumber \\&= Re ({\fancyscript{A}}_{2\omega _1})+iIm ({\fancyscript{A}}_{2\omega _1}), \end{aligned}$$

(7.17)

where the integral sums over all intermediate (virtual) states $|\psi _n\rangle$. Equation (7.17) applies to continuum intermediate states $|\psi _n\rangle$ and/or high density Rydberg states [81]. The transition amplitude in (7.17) can thus be written as

$$\begin{aligned} {\fancyscript{A}}_{2\omega _1}&= {\fancyscript{R}}_{2\omega _1} e^{i\eta _{2\omega _1}}, \\ \nonumber {\fancyscript{R}}_{2\omega _1}&= \sqrt{[Re ({\fancyscript{A}}_{2\omega _1})]^2+ [Im ({\fancyscript{A}}_{2\omega _1})]^2}, \\ \nonumber \eta _{2\omega _1}&= \tan ^{-1}\left[ \displaystyle {Im ({\fancyscript{A}}_{2\omega _1})}/ { Re ({\fancyscript{A}}_{2\omega _1})} \right] , \end{aligned}$$

(7.18)

where $\eta _{2\omega _1}$ is the phase of the transition amplitude of the two $\omega _1$ photon ionizations. The total transition matrix element is separated into a nonresonant (virtual) principle part $\mathrm{PP }$ integral and a resonant transition $\omega _1=E_n^{e}-E_0^{e}$, where $E_0^{e}$ and $E_n^{e}$ is the energies of the initial and intermediate electronic states. For continuum energies or dense states such as Rydberg states, the PP integral becomes negligible due to cancellation from fluctuations of the denominators $E_0^{e}-E_n^{e}+\omega _1>0$ and $E_0^{e}-E_n^{e}+\omega _1<0$ in (7.17), i.e.

$$\begin{aligned} \mathrm{PP } \int \limits _{-\infty }^\infty dE_n^e \frac{\langle \psi _c^{E_e}|\mathbf r \cdot \mathbf E |\psi _n\rangle \langle \psi _n|\mathbf r \cdot \mathbf E |\psi _0\rangle }{E_0^{e}-E_n^{e}+\omega _1}=0. \end{aligned}$$

(7.19)

The nonresonant (virtual) processes in (7.19) occur on a time scale $\tau _{nr}= {1}/{|E_0^{e}-E_n^{e}+\omega _1|}$. In this limit can one assume that the total transition element depends on the resonant process, i.e. the imaginary fact of the transition amplitude in (7.17) [28, 81],

$$\begin{aligned} {\fancyscript{A}}_{2\omega _1}&\propto {\langle \psi _c^{E_e}|\mathbf r \cdot \mathbf E |\psi _n\rangle \langle \psi _n|\mathbf r \cdot \mathbf E |\psi _0\rangle } \end{aligned}$$

(7.20)

In polar coordinates the parallel/perpendicular dipole interaction simply takes the form $\mathbf r \cdot \mathbf E =rE (\mathbf r _0\cdot \mathbf e _{x/y})$, where $\mathbf r _0$ is the unit position vector, and then the radial and angular variables in the transition matrix element are easily separated. The electric field is defined as $E(\omega _1)=E_0(\omega _1)\exp (i\phi _{\omega _1})$, where $E_0(\omega _{1})$ is the electric strength and $\phi _{\omega _{1}}$ is the pulse phase. The transition matrix element ${\fancyscript{A}}_{2\omega _1}$ can then be written as [82]

$$\begin{aligned} {\fancyscript{A}}_{2\omega _1}&\propto T_{2\omega _1}f_{2\omega _1}(\theta )e^{i\xi _{2\omega _1}}E(\omega _1)^2 \nonumber \\&= T_{2\omega _1}f_{2\omega _1}(\theta )E_0(\omega _1)^2e^{i\xi _{2\omega _1}+2\phi _{\omega _1}}, \end{aligned}$$

(7.21)

where $\xi _{2\omega _1}=\eta _{2\omega _1}$ is the continuum electron wave function phase, $\phi _{2\omega _1}$ and $\phi _{\omega _1}$ are the pulse CEPs. $T_{2\omega _1}$ and $f_{2\omega _1}(\theta )$ are respectively the radial and angular parts of the reduced transition moments. In the case of bound state resonances only, $Im ({\fancyscript{A}}_{2\omega _1})=0$ in (7.17), so the corresponding amplitude of $Re ({\fancyscript{A}}_{2\omega _1})$ is the same as the continuum transition amplitude in (7.21) since the final state $|\psi _c\rangle$ is the same.

7.1.2 Two-Color Photoionization Processes

For a combination of bichromatic laser pulses of frequencies $\omega _1$ and $\omega _2=2\omega _1$, the electron can be ionized via multiple pathways to reach the same final energies in the continuum simultaneously. We below consider respectively the photoionization at different kinetic energies by bichromatic laser pulses.

7.1.2.1 Interference of CEWPs at Energy $E_{e1}=2\omega _1-I_p$

For the MPADs with energy $E_{e1}=2\omega _1-I_p$, i.e., simultaneous two-photon ($\omega _1\,+\,\omega _1$) in (7.20) and one-photon ($\omega _2=2\omega _1$) ionizations in (7.16), the total transition probability is the square of the two amplitudes with an interference term of the cross products of the two one- and two-photon ionization amplitudes, i.e., $\displaystyle {d\sigma ({E_{e1}})}/{d\Omega }={\fancyscript{P}}{(2\omega _1)}+{\fancyscript{P}}{(\omega _2)}+ {\fancyscript{P}}{(2\omega _1,\omega _2)}$, where ${\fancyscript{P}}{(2\omega _1,\omega _2)}$ is the interference term which can be simply written as [78, 79, 82]

$$\begin{aligned} {\fancyscript{P}}{(2\omega _1,\omega _2)}&\propto {\fancyscript{A}}^*_{2\omega _1} \cdot {\fancyscript{A}}_{\omega _2}+ {\fancyscript{A}}_{2\omega _1}\cdot {\fancyscript{A}}^{*}_ {\omega _2} \nonumber \\&= T_{2\omega _1}f_{2\omega _1}(\theta )E_0(\omega _1)^2E_0(\omega _2)\cos (\varDelta \eta ), \end{aligned}$$

(7.22)

where ${\fancyscript{A}}_{2\omega _1}$ and ${\fancyscript{A}}_{\omega _2}$ are respectively the matrix element of the two- and one-photon absorptions, and the angular factors are obtained in the perturbative limit $f_{2\omega _1}(\theta )=\langle (\mathbf r _0\cdot \mathbf e )^3\rangle$. The two field amplitudes are defined as $E(\omega _2)=E_0(\omega _2)\exp (i\phi _{\omega _2})$ and $E(\omega _1)=E_0(\omega _1)\exp (i\phi _{\omega _1})$, where $E_0(\omega _{1,2})$ is the electric strength and $\phi _{\omega _{1,2}}$ is the pulse phase. The total phase difference $\varDelta \eta$ between the transition amplitudes for the two-pathway ionizations is the sum of difference phases of the laser pulses $\varDelta \phi$ and of the continuum electron wave functions $\varDelta \xi$, i.e., $\varDelta \eta =\varDelta \phi +\varDelta \xi$ with $\varDelta \phi =\phi _{\omega _2}-2\phi _{\omega _1}$ and $\varDelta \xi = \xi _{\omega _2}-\xi _{2\omega _1}$, where $\xi _{\omega _2}$ and $\xi _{2\omega _1}$ are respectively the phases of continuum electron wave functions for direct one $\omega _2$ and two $\omega _1$ photon ionizations. Then the total parallel MPADs can be finally written as sums of direct and interfering photoionization distributions,

$$\begin{aligned} \frac{d\sigma _\parallel ({E_{e1}})}{d\Omega }&\propto \alpha _0+\alpha _1\cos ^2\theta +\alpha _2\cos ^4\theta +\beta \cos ^3\theta \cos (\varDelta \eta ). \end{aligned}$$

(7.23)

The coefficient $\beta$ is determined by the intensities of the two laser pulses. Similarly, the total perpendicular MPADs can be written as

$$\begin{aligned} \frac{d\sigma _\perp ({E_{e1}})}{d\Omega }&\propto \alpha _0+\alpha _1\sin ^2\theta +\alpha _2\sin ^4\theta +\beta \sin ^3\theta \cos (\varDelta \eta ). \end{aligned}$$

(7.24)

We note that (7.23) and (7.24) contain odd order terms in both $\cos \theta$/$\sin \theta$ and $\cos (\varDelta \eta )=\cos (\varDelta \phi +\varDelta \xi )$. The simultaneous interference effect will therefore break the symmetry of the electron flux in forward/upward and backward/downward directions and the MPADs will vary periodically with the phase difference $\varDelta \phi$ of the laser pulses. At $\varDelta \eta =\pi /2$ there is no interference and at $\varDelta \eta =0$ and $\pi$ one gets maximum asymmetry.

7.1.2.2 Interference of CEWPs at Energy $E_{e2}=3\omega _1-I_p$

Photoionization processes can also occur at higher kinetic energy $E_{e2}=3\omega _1-I_p$, as schematically illustrated in Fig. 7.10b, where electron ionizes through channels after absorption of three $\omega _1$ photons ($\omega _1\,+\,\omega _1\,+\,\omega _1$), giving a direct transition ${\fancyscript{P}}{(3\omega _1)}$ a one $\omega _1$ and one $\omega _2$ photon ${\fancyscript{P}}{(\omega _1\,+\,\omega _2)}$ transition. Therefore the simultaneous interference term between these two multi-photon ionization pathways can be written as

$$\begin{aligned} {\fancyscript{P}}{(3\omega _1,\omega _1+\omega _2)}&\propto {\fancyscript{A}}^*_{3\omega _1}\cdot {\fancyscript{A}}_{\omega _1+\omega _2}+{\fancyscript{A}}_{3\omega _1}\cdot {\fancyscript{A}}^*_{\omega _1+\omega _2} \nonumber \\&= T_{3\omega _1}f_{3\omega _1}(\theta )E_0(\omega _1)^4E_0(\omega _2)\cos (\varDelta \phi ). \end{aligned}$$

(7.25)

${\fancyscript{A}}_{3\omega _1}$ and ${\fancyscript{A}}_{\omega _1+\omega _2}$ are transition amplitudes from $\omega _1+\omega _1+\omega _1$ and $\omega _1+\omega _2$ pathway ionizations. The corresponding angular factor is given by $f_{3\omega _1}(\theta )=\langle (\mathbf r _0\cdot \mathbf e )^5\rangle$. Then one can express the overall parallel polarization angular distributions

$$\begin{aligned} \frac{d\sigma _\parallel ({E_{e2}})}{d\Omega }&\propto \alpha _0+\alpha _1\cos ^2\theta +\alpha _2\cos ^4\theta +\alpha _3\cos ^6\theta \nonumber \\&+\beta \cos ^5\theta \cos (\varDelta \eta ), \end{aligned}$$

(7.26)

For the perpendicular $y$ polarization angular distributions, the corresponding expression is

$$\begin{aligned} \frac{d\sigma _\perp ({E_{e2}})}{d\Omega }&\propto \alpha _0+\alpha _1\sin ^2\theta +\alpha _2\sin ^4\theta +\alpha _3\sin ^6\theta \nonumber \\&+\beta \sin ^5\theta \cos (\varDelta \eta ), \end{aligned}$$

(7.27)

where $\alpha _i$ and $\beta$ are pulse dependent coefficients with transition amplitude phase differences $\varDelta \eta =\varDelta \phi +\varDelta \xi$, $\varDelta \phi =\phi _{\omega _2}-2\phi _{\omega _1}$ and $\varDelta \xi = \xi _{\omega _1+\omega _2}-\xi _{3\omega _1}$. $\xi _{3\omega _1}$ and $\xi _{\omega _1+\omega _2}$ are the phases of the continuum electron wave functions for the $3\omega _1$ and $\omega _1+\omega _2$ pathways ionizations. As illustrated in Fig. 7.10a, the ionization is in fact a three-pathway ionization process. The $\omega _1+\omega _1+\omega _1$ transition corresponds to three successive one photon transitions giving rise to a single transition amplitude ${\fancyscript{A}}_{3\omega _1}={\fancyscript{R}}_{3\omega _1}e^{i\eta _{3\omega _1}}$ whereas the $\omega _1+\omega _2$ transition corresponds to two amplitudes, ${\fancyscript{A}}'_{\omega _1+\omega _2}$ and ${\fancyscript{A}}'_{\omega _2+\omega _1}$ whose sum is defined as ${\fancyscript{A}}_{\omega _1+\omega _2}={\fancyscript{R}}_{\omega _1+\omega _2}e^{i\eta _{\omega _1+\omega _2}}$. Again, a phase difference dependent asymmetry in the MPADs is obtained due to the odd order terms in both $\cos \theta$/$\sin \theta$ and $\cos (\varDelta \eta )$ in (7.26) and (7.27).

7.1.2.3 Interference of CEWPs at Energy $E_{e3}=4\omega _1-I_p$

Similar results for total MPADs can be obtained for the ionization processes with next higher kinetic energy $E_{e3}=4\omega _1-I_p$ via five-pathway transitions after direct absorptions of four $\omega _1$ photons ($\omega _1+\omega _1+\omega _1+\omega _1$), and two $2\omega$ photons ($\omega _2+\omega _2$), and one $2\omega$ and two $\omega$ photons ($\omega _1+\omega _1+\omega _2$), as shown in Fig. 7.10c. For this multi-pathway ionization, the interference is simply the sum of the contributions from all pathways. From Fig. 7.10c we note that in the ionization processes three-pathway interference occurs, i.e., the four $\omega _1$ photon ionization with transition ${\fancyscript{A}}_{4\omega _1}={\fancyscript{R}}_{4\omega _1} e^{i\eta _{4\omega _1}}$, two $\omega _2$ photon ionization with transition ${\fancyscript{A}}_{2\omega _2}={\fancyscript{R}}_{2\omega _2} e^{i\eta _{2\omega _2}}$, and the three photon ionization processes ${\fancyscript{A}}_{2\omega _1+\omega _2}={\fancyscript{R}}_{2\omega _1+\omega _2} e^{i\eta _{2\omega _1+\omega _2}}$ corresponding to three amplitudes ${\fancyscript{A}}'_{2\omega _1+\omega _2}$, ${\fancyscript{A}}'_{\omega _1+\omega _2+\omega _1}$, and ${\fancyscript{A}}'_{\omega _2+2\omega _1}$ interfere with each other. Then we get respectively ${\fancyscript{A}}_{4\omega _1}$ and ${\fancyscript{A}}_{2\omega _2}$ interference,

$$\begin{aligned} {\fancyscript{P}}(4\omega _1,2\omega _2)&\propto {\fancyscript{A}}^*_{4\omega _1}\cdot {\fancyscript{A}}_{2\omega _2}+{\fancyscript{A}}_{4\omega _1}\cdot {\fancyscript{A}}^*_{2\omega _2} \nonumber \\&= T_{4\omega _1}f_{4\omega _1}(\theta ) E(\omega _1)^4E(\omega _2)^2 \cos (2\varDelta \eta _0), \end{aligned}$$

(7.28)

${\fancyscript{A}}_{4\omega _1}$ and ${\fancyscript{A}}_{2\omega _1+\omega _2}$ interference,

$$\begin{aligned} {\fancyscript{P}}(4\omega _1,2\omega _1+\omega _2)&\propto {\fancyscript{A}}^*_{4\omega _1}\cdot {\fancyscript{A}}_{2\omega _1+\omega _2}+{\fancyscript{A}}_{4\omega _1}\cdot {\fancyscript{A}}^*_{2\omega _1+\omega _2} \nonumber \\&= T'_{4\omega _1}f'_{4\omega _1}(\theta ) E(\omega _1)^6E(\omega _2) \cos (\varDelta \eta _1), \end{aligned}$$

(7.29)

and ${\fancyscript{A}}_{2\omega _2}$ and ${\fancyscript{A}}_{2\omega _1+\omega _2}$ interference,

$$\begin{aligned} {\fancyscript{P}}(2\omega _2,2\omega _1+\omega _2)&\propto {\fancyscript{A}}^*_{2\omega _2}\cdot {\fancyscript{A}}_{2\omega _1+\omega _2}+{\fancyscript{A}}_{2\omega _2}\cdot {\fancyscript{A}}^*_{2\omega _1+\omega _2} \nonumber \\&= T''_{4\omega _1}f''_{4\omega _1}(\theta ) E(\omega _1)^2E(\omega _2)^2 \cos (\varDelta \eta _2), \end{aligned}$$

(7.30)

where transition amplitude phase differences are $\varDelta \eta _0=\varDelta \xi _0/2+\varDelta \phi$, $\varDelta \eta _1=\varDelta \xi _1-\varDelta \phi$, and $\varDelta \eta _2=\varDelta \xi _2-\varDelta \phi$, and $\varDelta \xi _{0,1,2}$ are phase differences of corresponding continuum electron wave functions. The angular parts of the reduced transition matrix elements in (7.28–7.30) transform as

$$\begin{aligned} \begin{array}{lll} f_{4\omega _1}(\theta ) =\cos ^6\theta ,&f'_{4\omega _1}(\theta ) =\cos ^7\theta ,&f''_{4\omega _1}(\theta ) =\cos ^5\theta . \end{array} \end{aligned}$$

(7.31)

for the parallel $x$ polarization photoionization, and

$$\begin{aligned} \begin{array}{lll} f_{4\omega _1}(\theta ) =\sin ^6\theta ,&f'_{4\omega _1}(\theta ) =\sin ^7\theta ,&f''_{4\omega _1}(\theta ) =\sin ^5\theta . \end{array} \end{aligned}$$

(7.32)

for the perpendicular $y$ polarization photoionization. The total MPADs for the $E_{e3}$ MATI peaks at energy $E_{e3}=4\omega _1-I_p$ are then,

$$\begin{aligned} \frac{d\sigma _\parallel ({E_{e3}})}{d\Omega }&\propto \alpha _0+\alpha _1\cos ^2\theta +\alpha _2\cos ^4\theta +\alpha _3\cos ^6\theta \nonumber \\&+\,\alpha _4\cos ^8\theta +\beta _0 \cos ^6\theta \cos ^2(\varDelta \eta _0)\nonumber \\&+\,\beta _1 \cos ^5\theta \cos (\varDelta \eta _1)+\beta _2 \cos ^7\theta \cos (\varDelta \eta _2), \end{aligned}$$

(7.33)

for the parallel $x$ polarization and

$$\begin{aligned} \frac{d\sigma _\perp ({E_{e3}})}{d\Omega }&\propto \alpha _0+\alpha _1\sin ^2\theta +\alpha _2\sin ^4\theta +\alpha _3\sin ^6\theta \nonumber \\&+\,\alpha _4\sin ^8\theta +\beta _0 \sin ^6\theta \cos ^2(\varDelta \eta _0)\nonumber \\&+\,\beta _1 \sin ^5\theta \cos (\varDelta \eta _1)+\beta _2 \sin ^7\theta \cos (\varDelta \eta _2), \end{aligned}$$

(7.34)

for the perpendicular $y$ polarization. From (7.33) and (7.34) we see that both odd and even powers of $\cos \theta$/$\sin \theta$ and $\cos (\varDelta \eta )$ are obtained in the interference terms, i.e., an even number of transition $\cos ^6\theta$ and $\sin ^6\theta$ terms occur for odd-odd parity interference whereas the terms $\cos ^5(\theta )$/$\sin ^5(\theta )$ and $\cos ^7(\theta )$/$\sin ^7(\theta )$ correspond to odd-even transition interferences. In (7.33) and (7.34), the terms $\cos ^6\theta \cos ^2(\varDelta \eta _0)$ and $\sin ^6\theta \cos ^2(\varDelta \eta _0)$ come from the interference between the four $\omega _1$ and two $\omega _2$ photons ionization processes, which is symmetric in both angle $\theta$ and phase difference $\varDelta \eta _0$. Thus the interference between ${\fancyscript{A}}_{4\omega _1}$ and ${\fancyscript{A}}_{2\omega _2}$ does not contribute to the asymmetry of the angular distribution. The asymmetry from the $\cos (\varDelta \eta )$ [$\cos (\varDelta \phi )$] phase term only appears in the high odd multi-photon terms $\cos ^5\theta$/$\sin ^5\theta$ and $\cos ^7\theta$/$\sin ^7\theta$ via interferences of ${\fancyscript{A}}_{2\omega _1+\omega _2}$ with ${\fancyscript{A}}_{2\omega _2}$ and ${\fancyscript{A}}_{4\omega _1}$ transition amplitudes, respectively.