Bicircular Laser Field Research Articles

Comparative study on atomic ionization in bicircular laser fields by length and velocity gauges S-matrix theory**Project supported by the Key Laboratory Project of Computational Physics of National Defense Science and Technology of China (Grant No. 6142A05180401), the National Key Program for S&T Research and Development of China (Grant Nos. 2019YFA0307700 and 2016YFA0401100), and the National Natural Science Foundation of China

Ionization of atoms in counter-rotating and co-rotating bicircular laser fields is studied using the S-matrix theory in both length and velocity gauges. We show that for both the bicircular fields, ionization rates are enhanced when the two circularly polarized lights have comparable intensities. In addition, the curves of ionization rate versus the field amplitude ratio of the two colors for counter-rotating and co-rotating fields coincide with each other in the length gauge case at the total laser intensity 5 × 1014 W/cm2, which agrees with the experimental observation. Moreover, the degree of the coincidence between the ionization rate curves of the two bicircular fields decreases with the increasing field amplitude ratio and decreasing total laser intensity. With the help of the ADK theory, the above characteristics of the ionization rate curves can be well interpreted, which is related to the transition from the tunneling to multiphoton ionization mechanism.

Role of the relative phase and intensity ratio in electron-ion recombination assisted by a bicircular laser field

S-matrix theory is used in order to analyze the energy spectra of electron-ion radiative recombination assisted by a bicircular (two-component circularly polarized) laser field. The analysis includes the cases of corotating and counterrotating laser-field components. The scattering effects are also included by applying the second Born approximation in the expansion of the S-matrix element. We have investigated the sensitivity of the emitted photon energy spectra to the relative phase and intensity ratio of the laser-field components. The obtained energy spectra show the plateau-like oscillatory structures with abrupt cutoffs. Positions of these cutoffs in the energy spectra are confirmed by a classical analysis. Scattering effects may be observed in the calculated energy spectra not only in the case of counterrotating, but also in the case of corotating laser-field components. This is different from the process of above-threshold ionization/detachment by a bicircular laser field, where the scattering effects in the photoelectron energy spectra may be observed only in the case of counterrotating laser-field components.

Selection of right-circular-polarized harmonics from p orbital of neon atom by two-color bicircular laser fields**Project supported by the National Natural Science Foundation of China (Grant Nos. 11974229, 11504221, and 11404204), the Scientific and Technological Innovation Program of Higher Education Institutions in Shanxi Province, China (Grant No. 2019L0452), the Natural Science Foundation for Young Scientists of Shanxi Province, China (Grant No.

The polarization properties of high-order harmonic generation (HHG) in the two-color circularly polarized laser fields are investigated by numerically solving the two-dimensional time-dependent Schrödinger equation. By adding a wavelength of 1600-nm right-circular-polarized field to an 800-nm left-circular-polarized field, HHG is simulated from a real model of neon atom with p orbital, but not from a hydrogen-like atom model with s orbital. The orders of 3n+1 can be selected while the orders of 3n+2 are suppressed by adjusting the intensities of the two pulses. The physical mechanism is analyzed by time–frequency analysis and semiclassical model.

Controlling polarization of high-order harmonic generation by molecular alignment in a bicircular laser field

The polarization of high-order harmonic generation in ${\mathrm{H}}_{2}{}^{+}$ driven by a counterrotating bichromatic $\ensuremath{\omega}/3\ensuremath{\omega}$ circularly polarized laser field is investigated theoretically. The elliptically polarized harmonics are characterized by ellipticity and rotation angle, where the latter is expanded to $[\ensuremath{-}\ensuremath{\pi}/2,\ensuremath{\pi}/2]$ by an optimized definition. We find a critical alignment angle where the ellipticity of high-energy harmonics is almost the same, so an attosecond pulse train with a high degree of polarization can be synthesized. In addition, we find that the rotation angle is always different from the alignment angle by ${90}^{\ensuremath{\circ}}$ with a small deflection. Thus, we propose a scheme to control the harmonic polarization state by the molecular alignment angle.

Generating attosecond pulses with high ellipticity by use of He atoms

ABSTRACTTo control the polarization of attosecond pulse trains (APTs), the most widely used way is to adjust the intensity ratio between the two components in a bicircular laser field. Here we demonstrate another effective means to obtain highly elliptically polarized APTs. For this purpose, we propose a circular-elliptical laser field, which consists of a circularly polarized field with its coplanar counter-rotating elliptical harmonic pulse, in order to generate high-order harmonics by He atom. We show that the polarization of APTs depends evidently on the ellipticity ξ of harmonic laser field. Most importantly, nearly circular APTs can be obtained with a proper ξ for driving laser field. Our results contribute to the generation of high elliptically polarized APTs with s-state atom.

The influence of the driving-bicircular-field component intensities on the helicities of emitted high-order harmonics

High-order harmonics generated by a linearly polarized laser field are also linearly polarized. Having in mind that for various application, such as the exploration of magnetic materials, chiral molecules etc., we need circularly polarized high harmonics which serve as coherent soft x-rays, we explore high-order harmonic generation by the so-called bicircular laser field. This field consists of two coplanar counter-rotating circularly polarized fields of different frequencies equal to integer multiples of a fundamental frequency ω. High harmonics generated by such field are circularly polarized with helicity alternating between +1 and −1. Combining a group of such harmonics, instead of obtaining a circularly polarized attosecond pulse train, one obtains a pulse with unusual polarization properties. But, if the harmonics of particular helicity are stronger, i.e., if we have helicity asymmetry in a high-harmonic energy interval, then it is possible to generate an elliptical or even circular pulse train. We theoretically investigated a wide range of bicircular field-component intensities (I1 and I2) and found regions where both the harmonic intensity is high and the helicity asymmetry is large. Particular attention is devoted to the ω−2ω and ω−3ω bicircular fields and atoms having the s and p ground states. In our calculations we use strong-field approximation and quantum-orbit theory. We show that, even in the extreme case of I2 = 8I1, for an ω−3ω bicircular field, high-order harmonic generation is more efficient than in the I2 = I1 case. The obtained results are explained analyzing the relevant electron trajectories and velocities, which follow from the quantum-orbit theory. For the atoms having p ground state the helicity asymmetry parameter is large for a wide range of high-harmonic photon energies, while for the atoms having s ground state the helicity asymmetry parameter can be large only for low harmonics. We confirm this by averaging the obtained results over the intensity distribution in the laser focus.

Attoclock with counter-rotating bicircular laser fields

The attoclock technique which maps the emission time of a photoelectron to its detection angle is an important tool in strong-field physics. Previously, it was implemented only with circularly or elliptically polarized laser fields. Here, we show how counter-rotating bicircular laser fields can be used as an attoclock to investigate the ionization dynamics in quasilinear polarization. This is achieved by choosing the ratio of the two field strengths in a way such that the vector potential has aspects of the attoclock and time is mapped directly to the photoelectron momentum, but the shape of the electric field corresponds to approximately linear polarization during three intervals per optical cycle. We report momentum distributions calculated by solving the time-dependent Schr\odinger equation for a model helium atom and obtain the mapping from photoelectron momentum to ionization time using a trajectory-free method. Unlike circular polarization where the time of maximal ionization rate typically deviates less than 5 attoseconds from the maximium of the electric field, we find positive ionization times of more than 10 attoseconds in the quasilinear case.

Circular dichroism in angular distribution of electron-hydrogen scattering in a two-color bicircular laser field

We study the origin of dichroic effects in elastic scattering of high-energy electrons by hydrogen atoms in the presence of a two-color bicircular laser field of commensurate frequencies, in the domain of moderate intensities below $10\phantom{\rule{0.28em}{0ex}}\mathrm{TW}/{\mathrm{cm}}^{2}$. We use a semiperturbative approach in which the interaction of the hydrogen atom with the laser field is treated in second-order perturbation theory, while the interaction of the projectile electron with the laser field is described by Gordon-Volkov wave functions. An analytical formula of circular dichroism in the angular distribution of scattered electrons is derived in the weak-field domain for a two-color laser field that is a combination of the fundamental and its third harmonic. A comparison between the two-photon differential cross sections for two-color co- and counter-rotating circularly polarized laser fields is made, and the effect of the intensity ratio of the monochromatic field components on the circular dichroism is investigated. The dichroic effect in the angular distribution of scattered electrons for two-photon absorption is analyzed as a function of the scattering and azimuthal angles. We show that the two-color bicircular laser field can induce a strong circular dichroism in the angular distribution of scattered electrons at small scattering angles where the atomic dressing effects are important, as well at larger scattering angles. At small scattering angles we demonstrate that the dichroic effect for two-photon transitions can be predicted under the following conditions: the scattering process is treated in fist-order Born approximation, and the dressing of the atomic states by the laser field is carried out at least in first-order time-dependent perturbation theory.

Attospin and channel closings in high-order above-threshold ionization by bicircular laser fields

In Phys. Rev. A 93, 051402(R) (2016), we proposed a method to introduce spin into attoscience with spin-polarized electrons produced by a bicircular laser field. In the present paper we further develop this method. We calculate focal-averaged photoelectron spectra and the corresponding spin asymmetry parameter by using the improved strong-field approximation. We show that the spin asymmetry of the rescattered electrons, which is time dependent on the attosecond timescale, can become especially large for particular peak laser intensities. The pertinent jumps in the spin asymmetry occur when the peak intensity changes so that certain above-threshold ionization channel-closing intensities corresponding to the different ionization continua of Xe become covered by the intensity distribution.

Atomic and Molecular Processes in a Strong Bicircular Laser Field

With the development of intense femtosecond laser sources it has become possible to study atomic and molecular processes on their own subfemtosecond time scale. Table-top setups are available that generate intense coherent radiation in the extreme ultraviolet and soft-X-ray regime which have various applications in strong-field physics and attoscience. More recently, the emphasis is moving from the generation of linearly polarized pulses using a linearly polarized driving field to the generation of more complicated elliptically polarized polychromatic ultrashort pulses. The transverse electromagnetic field oscillates in a plane perpendicular to its propagation direction. Therefore, the two dimensions of field polarization plane are available for manipulation and tailoring of these ultrashort pulses. We present a field that allows such a tailoring, the so-called bicircular field. This field is the superposition of two circularly polarized fields with different frequencies that rotate in the same plane in opposite directions. We present results for two processes in a bicircular field: High-order harmonic generation and above-threshold ionization. For a wide range of laser field intensities, we compare high-order harmonic spectra generated by bicircular fields with the spectra generated by a linearly polarized laser field. We also investigate a possibility of introducing spin into attoscience with spin-polarized electrons produced in high-order above-threshold ionization by a bicircular field.

Control of the helicity of high-order harmonics generated by bicircular laser fields

High-order harmonics generated by bicircular laser field have helicities which alternate between $+1$ and $-1$. In order to generate circularly polarized high-harmonic pulses, which are important for applications, it is necessary to achieve asymmetry in emission of harmonics having opposite helicities. We theoretically investigated a wide range of bicircular field component intensities and found areas where both the harmonic intensity is high and the helicity asymmetry is large. We investigated the cases of $\omega$--$2\omega$ and $\omega$--$3\omega$ bicircular fields and atoms having the $s$ and $p$ ground states, exemplified by He and Ne atoms, respectively. We have shown that for He atoms strong high harmonics having positive helicity can be generated using $\omega$--$3\omega$ bicircular field with a much stronger second field component. For Ne atoms the helicity asymmetry can be large in a wider range of the driving field component intensities and for higher harmonic orders. For the stronger second field component the harmonic intensity is higher and the helicity asymmetry parameter is larger for higher harmonic orders. The results for Ne atoms are illustrated with the parametric plots of elliptically polarized attosecond high-harmonic field.

Quantum-orbit analysis of high-order harmonic generation by bicircular field

ABSTRACTQuantum-orbit theory of high-order harmonic generation (HHG) by bicircular laser field is presented. HHG is a strong laser-field-induced process in which the energy absorbed from the laser field is emitted in the form of a high-energy photon. This process can be described using the strong-field approximation and its approximation – the quantum-orbit theory. We develop a classification of quantum orbits for HHG by bicircular field which consists of two coplanar counter-rotating circularly polarized field of frequencies and , where r and s are integers and ω is a fundamental frequency. Analysis of the contributions of particular quantum orbits to high-harmonic intensity enables a better understanding of the HHG process. The cases of the ω– and ω– bicircular fields and of the atoms having the s and the p ground state are analysed in detail. Particular attention is devoted to the influence of the ratio of the intensities and of the bicircular field components. For inert gases having the p ground state the asymmetry in the emission of the left- and right-circularly polarized harmonics can be large. This is explained comparing the partial harmonic intensities for HHG from the ground state having the magnetic quantum number m=+1 and m=−1 and analysing the contributions of particular quantum orbits and the corresponding electron trajectories and velocities. The contribution of the shortest quantum orbit is dominant. It was found that the electron velocity at the ionization time, which still allows the return of the electron to the parent ion, determines the height of the high-energy HHG plateau. For this velocity is large which, in the case of the p ground state, leads to a large helicity asymmetry parameter. On the other hand, for this velocity is small and the intensity of the high-energy plateau harmonics is high.

Probing atomic and molecular targets by intense bicircular counter-rotating laser fields

We present an ab initio study of strong-field ionization of H and by two-colour bicircular counter-rotating 1ω–2ω laser pulses containing two and ten optical cycles at the fundamental frequency corresponding to a wavelength of 800 nm. We address effects of laser intensity, pulse duration and in the case of of alignment on photoelectron momentum distributions (PMDs). In the long pulse limit, as expected from a simple classical picture, the PMDs of H and of aligned perpendicular to the polarization plane show a three-fold symmetry imposed by the bicircular laser field. Compared with the expectation for the positions of the peaks in this three-fold structure, we observe off-set angles that depend on the photoelectron energy as well as the laser intensity. For , the symmetry in the PMDs is broken by the anisotropic molecular potential at molecular alignment not perpendicular to the plane of laser polarization. In the short pulse limit, the PMDs are sensitive to the carrier envelope phase of the laser field.

Helicity asymmetry in strong-field ionization of atoms by a bicircular laser field.

Ionization of atoms by an intense bicircular laser field is considered, which consists of two coplanar corotating or counterrotating circularly polarized field components with frequencies that are integer multiples of a fundamental frequency. Emphasis is on the effect of a reversal of the helicities of the two field components on the photoelectron spectra. The velocity maps of the liberated electrons are calculated using the direct strong-field approximation (SFA) and its improved version (ISFA), which takes into account rescattering off the parent ion. Under the SFA all symmetries of the driving field are preserved in the velocity map while the ISFA violates certain reflection symmetries. This allows one to assess the significance of rescattering in actual data obtained from an experiment or a numerical simulation.

Strong-field ionization of linear molecules by a bicircular laser field: Symmetry considerations

Using the improved molecular strong-field approximation, we investigate (high-order) above-threshold ionization [(H)ATI] of various linear polyatomic molecules by a two-color laser field of frequencies $r\ensuremath{\omega}$ and $s\ensuremath{\omega}$ (with integer numbers $r$ and $s$) having coplanar counter-rotating circularly polarized components (a so-called bicircular field). Reflection and rotational symmetries for molecules aligned in the laser-field polarization plane, analyzed for diatomic homonuclear molecules in Phys. Rev. A 95, 033411 (2017), are now considered for diatomic heteronuclear molecules and symmetric and asymmetric linear triatomic molecules. There are additional rotational symmetries for (H)ATI spectra of symmetric linear molecules compared to (H)ATI spectra of the asymmetric ones. It is shown that these symmetries manifest themselves differently for $r+s$ odd and $r+s$ even. For example, HATI spectra for symmetric molecules with $r+s$ even obey inversion symmetry. For ATI spectra of linear molecules, reflection symmetry appears only for certain molecular orientation angles $\ifmmode\pm\else\textpm\fi{}{90}^{\ensuremath{\circ}}\ensuremath{-}jr{180}^{\ensuremath{\circ}}/(r+s)$ ($j$ integer). For symmetric linear molecules, reflection symmetry appears also for the angles $\ensuremath{-}jr{180}^{\ensuremath{\circ}}/(r+s)$. For perpendicular orientation of molecules with respect to the laser-field polarization plane, the HATI spectra are very similar to those of the atomic targets, i.e., both spectra are characterized by the same type of the ($r+s$)-fold symmetry.

Molecules in a bicircular strong laser field

Strong-field ionization of nonlinear planar triatomic molecules by a bicircular laser field is analyzed within the improved molecular strong-field approximation. Our calculations include additional interaction between the liberated electrons and atomic or ionic centers of the parent molecular ion. The used bicircular field consists of two counterrotating circularly polarized fields having angular frequencies $$r \omega$$ and $$s \omega$$ , with integer r and s. In the case when the laser-field-polarization plane is parallel to the plane of the considered molecule (example of ozone molecule is analyzed), the corresponding photoelectron spectra are not rotationally symmetric. On the other hand, when these planes are mutually perpendicular, for the $$(r\omega ,s\omega )=(\omega ,3\omega )$$ bicircular field, the electron spectra satisfy the corresponding rotational symmetries. Analyzing the obtained spectra and the corresponding symmetries, one can extract information about molecular orientation and structure. This technique may also be useful for more complex polyatomic molecules.

Strong-field photoelectron holography of atoms by bicircular two-color laser pulses

We study photoelectron holography in strong bicircular two-color laser fields by solving the time-dependent Schr\"odinger equation (TDSE) and a semiclassical rescattering model with implementing interference effect. The holographic patterns observed in the TDSE are well recaptured by the semiclassical rescattering model. Four types of photoelectron holographic interferences between the forward scattered and nonscattered trajectories are predicted by the semiclassical rescattering model in the bicircular two-color laser field. We find that those holographic patterns are spatially separated from each other in the electron momentum distribution. We further show that the dependence of the initial transverse momentum at the tunnel exit on the ionization time for the rescattering electron is recorded by the holographic patterns.

Channel-closing effects in strong-field ionization by a bicircular field

Channel-closing effects, such as threshold anomalies and resonantlike intensity-dependent enhancements in strong-field ionization by a bicircular laser field are analyzed. A bicircular field consists of two coplanar corotating or counter-rotating circularly polarized fields having different frequencies. For the total detachment rate of a negative ion by a bicircular field we observe threshold anomalies and explain them using the Wigner threshold law and energy and angular momentum conservation. For the corotating bicircular case, these effects are negligible, while for the counter-rotating case they are pronounced and their position depends on the magnetic quantum number of the initial state. For high-order above-threshold ionization of rare-gas atoms by a counter-rotating bicircular laser field we observe very pronounced intensity-dependent enhancements. We find all four types of threshold anomalies known from collision theory. Contrary to the case of linear polarization, channel-closing effects for a bicircular field are visible also in the cutoff region of the electron energy spectrum, which is explained using quantum-orbit theory.

Low-frequency approximation for high-order harmonic generation by a bicircular laser field

We present low-frequency approximation (LFA) for high-order harmonic generation (HHG) process. LFA represents the lowest-order term of an expansion of the final-state interaction matrix element in powers of the laser-field frequency $\ensuremath{\omega}$. In this approximation the plane-wave recombination matrix element which appears in the strong-field approximation is replaced by the exact laser-free recombination matrix element calculated for the laser-field dressed electron momenta. First, we have shown that the HHG spectra obtained using the LFA agree with those obtained solving the time-dependent Schr\odinger equation. Next, we have applied this LFA to calculate the HHG rate for inert gases exposed to a bicircular field. The bicircular field, which consists of two coplanar counter-rotating fields having different frequencies (usually $\ensuremath{\omega}$ and $2\ensuremath{\omega}$), is presently an important subject of scientific research since it enables efficient generation of circularly polarized high-order harmonics (coherent soft x rays). Analyzing the photorecombination matrix element we have found that the HHG rate can efficiently be calculated using the angular momentum basis with the states oriented in the direction of the bicircular field components. Our numerical results show that the HHG rate for atoms having $p$ ground state, for higher high-order harmonic energies, is larger for circularly polarized harmonics having the helicity $\ensuremath{-}1$. For lower energies the harmonics having helicity $+1$ prevails. The transition between these two harmonic energy regions can appear near the Cooper minimum, which, in the case of Ar atoms, makes the selection of high-order harmonics having the same helicity much easier. This is important for applications (for example, for generation of attosecond pulse trains of circularly polarized harmonics).

Comparison Study of Strong-Field Ionization of Molecules and Atoms by Bicircular Two-Color Femtosecond Laser Pulses.

We experimentally investigate the single and double ionization of N_{2} and O_{2} molecules in bicircular two-color femtosecond laser pulses, and compare with their companion atoms of Ar and Xe with comparable ionization thresholds. Electron recollision assisted enhanced ionization is observed in N_{2} and Ar by controlling the helicity and field ratio between the two colors, whereas the enhanced ionization via the recollision is almost absent in O_{2} and Xe. Our S-matrix simulations clearly reveal the crucial role of the detailed electronic structures of N_{2} and O_{2} on the two-dimensional recollision of the electrons driven by the bicircular two-color laser fields. As compared to Ar, the resonant multiphoton excitation dominates the double ionization of Xe.