Electron interference in atomic ionization by two crossing polarized ultrashort pulses

Formation of geometrically regular interference patterns in the photoelectron momentum distributions (PMDs) corresponding to the photoionization of atoms by two single-color, crossing ultrashort pulses is investigated both analytically and numerically. It is shown that, in contrast to the photoionization by monochromatic pulses, PMDs for the ionization by crossing and co-propagating broadband pulses are essentially different (unless both pulses are linearly polarized), namely, when one pulse is linearly polarized along the propagation direction, $\widehat{\mathbf{k}}$, of the circularly polarized (CP) pulse, then interference maxima (minima) of the ionization probability have the form of three-dimensional single-arm regular spirals which are wound along $\widehat{\mathbf{k}}$. Next, the interference maxima (minima) of the ionization probability by a pair of crossing elliptically polarized pulses have the form of either Newton's rings or two-arm Fermat's spirals, depending on the position of a detection plane. Remarkably, these regular patterns occur only for certain values of the pulse ellipticities, and they become distorted for CP pulses. For both above-mentioned pulse configurations, the features of interference patterns depend on the time delay between pulses, their relative electric field amplitude, and relative carrier-envelope phase. Our predictions, illustrated by the numerical results for the ionization of H and He atoms by two orthogonal pulses, are quite general and we expect them to be valid for the ionization of any randomly oriented atomic or molecular target.

Figure 1

Detection scheme for the observation of single-arm spiral patterns in PMD of atomic photoionization by perpendicular LP and CP pulses. The wave vector ${\mathbf{k}}_1$ defines the propagation direction of the LP pulse with the polarization vector ${\mathit{\epsilon }}_1\parallel \widehat{\mathbf{z}}$, while ${\mathbf{k}}_2$ defines the direction of the right-hand CP pulse whose polarization ellipse is defined by the unit vectors ${\mathit{\epsilon }}_x$, ${\mathit{\epsilon }}_y$. The solid circle $A$ denotes an atomic target. Detected are the photoelectrons with momentum $\mathbf{p}$ emitted along the cone surface with the opening angle $\theta$. The PMD on the surface of the lower cone is a reflection of the PMD on the upper cone with respect to point $A$.

Figure 2

(a) Geometry for the observation of two-arm spiral fringes for atomic photoionization by two short elliptically polarized pulses propagating in perpendicular directions. The wave vectors ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ define the directions of the laser beams. The unit vectors ${\widehat{\mathit{\epsilon }}}_y$, ${\widehat{\mathit{\epsilon }}}_z$ are directed along the major and minor axes of the polarization ellipse of the first pulse propagating along the $x$ axis. Similarly, the unit vectors ${\widehat{\mathit{\epsilon }}}_x$, ${\widehat{\mathit{\epsilon }}}_z$ define the polarization ellipse of the second pulse. Arrows on the polarization ellipses correspond to positive ellipticities, ${\eta }_1,{\eta }_2>0$. (b) Top view of the geometry on panel (a), including the detection plane. The $z$ axis is directed toward the reader, the detection $\left(z{x}^{\prime }\right)$ plane is tilted at the angle $\psi$ with respect to the $\left(zx\right)$ plane, $-\pi /2\le \psi \le \pi /2$. $\mathbf{p}$'s show possible directions of the photoelectron momentum vector.

Figure 3

TDP of the ionization of the H atom by a pair of orthogonal linear/left-circularly polarized (LP-LCP) pulses corresponding to the detection geometry defined in Fig. 1 for two different time delays in (a) and (b). The intensity of each pulse is ${10}^{14}$ W/${\mathrm{cm}}^2$. Pulses have identical envelopes, zero relative CEP, ${\varphi }_{12}=0$, and their duration is 1.551 fs, which is six optical cycles of the carrier frequency equal to 16 eV. The magnitudes of the TDP (in units of ${10}^{-2}$ a.u.) are indicated by the color scales in each panel.

Figure 4

Top row: Variation of the first term ${[g-cot\theta ]}^2$ and second term $4gcot\theta$ in the TDP (27) with the opening cone angle $\theta$ for a fixed value of $g=f/\sqrt{2}=\sqrt{2}$, where $f=F_2/F_1$ is the ratio of the electric field strengths of the circularly polarized and linearly polarized pulses. Panel (b) is a close-up of panel (a), where the arrow therein indicates the value of the opening cone angle ${\theta }_0\approx {35}^{\circ }$ at which the first term of the TDP (27) vanishes, i.e., $cot{\theta }_0=g$. Middle row: TDSE results for the PMD, $\mathcal{W}$ [calculated using Eq. (B3)], recorded along the surface of a cone whose axis is defined by the propagation direction of the circularly polarized beam at the opening angle ${\theta }_0\approx {35}^{\circ }$. The counterclockwise or clockwise one-arm regular Fermat spiral pattern seen in (c) or (d) is produced by two orthogonal linear/right-circularly polarized (LP-RCP) pulses or by two orthogonal linear/left-circularly polarized (LP-LCP) pulses delayed in time by $\tau =T$. Here, each sine-squared pulse has a carrier frequency $\omega =36$ eV, intensity $I={10}^{14}$ W/${\mathrm{cm}}^2$, total duration $T=344$ as corresponding to three cycles. The relative CEP of the two pulses is ${\varphi }_{12}={\varphi }_1-{\varphi }_2=0$. Bottom row: Panel (e) is the same result as in panel (c) but for the opening cone angle $\theta =\pi -{\theta }_0\approx {145}^{\circ }$. Panel (f) is the same result as in panel (c) but for the relative CEP ${\varphi }_{12}=\pi$. In (c)–(f), the magnitudes of the TDP (in units of ${10}^{-3}$ a.u.) are indicated by the color scales.

Figure 5

The TDSE results for PMDs ${\mathcal{W}}_{\psi }^{(\pm )}\left(\rho \right)$ [calculated using Eq. (B3) for the He atom] in the detection plane ($p_{x^{\prime }},p_z$). Shown in (a) is the TDP ${\mathcal{W}}_{\pi /4}^{(+)}\left(1\right)$ and in (b) is the TDP ${\mathcal{W}}_{-\pi /4}^{(+)}\left(1\right)$ of the photoionization produced by two time-delayed, orthogonal EP pulses with the same ellipticity ${\eta }_1={\eta }_2=1/\sqrt{2}$. Shown in (c) is the TDP ${\mathcal{W}}_{\pi /4}^{(-)}(-1)$ of the photoionization produced by two orthogonal EP pulses with opposite ellipticity: ${\eta }_1=-{\eta }_2=1/\sqrt{2}$. In (a)–(c), $F_1=F_2$ (i.e., $f=1$). Panel (d) shows the TDP ${\mathcal{W}}_{\pi /8}^{(-)}(-2)$, corresponding to pulse ellipticities ${\eta }_1=sin\psi \approx 0.38$ and ${\eta }_2=-cos\psi \approx -0.92$, and the relative field amplitude $f\approx 0.5$. The magnitudes of the TDP (in units of ${10}^{-3}$ a.u.) are indicated by the color scales in each panel.

Figure 6

TDP ${\mathcal{W}}_{\pi /4}^{(-)}(-1)$ of the ionization of the H atom by a pair of orthogonal EP pulses corresponding to the detection geometry in Fig. 2. Pulses are elliptically polarized with opposite ellipticities, ${\eta }_1=-{\eta }_2=1/\sqrt{2}$. The intensity of each pulse is ${10}^{14}$ W/${\mathrm{cm}}^2$ ($f=1$). Pulses have identical envelopes, their duration is 1.551 fs, which is six optical cycles of the carrier frequency equal to 16 eV. Panel (a) is for a time delay $\tau =T/2$, while panel (b) is for $\tau =T$. The magnitudes of the TDP (in units of ${10}^{-2}$ a.u.) are indicated by the color scales in each panel.

Figure 7

Time delay for randomly placed atoms subjected to a pair of mutually perpendicular plane electromagnetic waves with wave vectors ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$. Atoms are depicted as solid circles, thick solid lines denote wavefronts. Regular interference patterns occur in PMDs corresponding to detection planes depicted by dashed and dot-dashed lines. For every atom located in the $\left(z{x}^{\prime }\right)$ plane (the $z$ axis is directed towards the reader) defined by the tilt angle $\psi =\pi /4$ (i.e., the dashed line), the time delay is the same as for an atom located in a point $A$. For atoms located out of $\left(z{x}^{\prime }\right)$ plane, the time delay between pulses depends on coordinates $x,y$ of respective atoms.