Bicircular Field Research Articles

Single and double ionization of helium in intense bicircular laser fields

The single and double ionization of helium in intense 2ω:ω bicircular laser fields is analyzed with a classical ensemble method. A high-performance computational cluster is utilized to generate high-resolution data and to probe low-rate events. It is found that both counter- and co-rotating fields support rescattering and non-sequential double ionization (NSDI). The effects of scattering off of the ionic core are evident in the single-ionization energy spectra, and the role of the Coulomb potential is clearly visible in the electron momentum distributions. The dynamics of the double ionization processes are investigated by analyzing ionization phase, electron energy and momenta, and rescattering timing. For corotating fields, rescattering is only possible for ionization near the field minimum, placing limits on the efficiency of NSDI. For counterrotating fields, the electron trajectories that contribute to rescattering are found to fall into only three categories (looping, triangular, or a combination of the two), and the contributions from these trajectories is distinguishable in the resulting double-ion electron momentum distributions. This study explores the full range of field amplitude ratios that contribute to the dynamics of rescattering, and therefore provides an overall view of the limitations of using bicircular fields for applications in strong-field physics.

Evidence of ac-Stark-shifted resonances in intense two-color circularly polarized laser fields

We report on the appearance of a structure at low energies in the photo-electron momentum distribution of the hydrogen atom exposed to two-color counter-rotating bi-circular laser fields. These structures, which arise due to AC-Stark shifted resonances, break the three-fold symmetry, typical for the $\omega-2\omega$ bi-circular fields. We discuss the physical origin of this structure in terms of partial-wave interference between direct ionization channels and a resonant pathway, that passes through the AC-Stark shifted state and show how the underlying Rydberg state population depend on the field strength and pulse duration.

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.

Attosecond pulse generation from the interaction of Bromine molecule with bicircularly-polarized laser pulses

In this paper, high-order harmonic generation by a bicircular field, which consists of two coplanar counterrotating circularly polarized fields of frequency ω and 3ω, is investigated for bromine (Br2) molecule. This field possesses dynamical symmetry, which can be adapted to the symmetry of the highest occupied molecular orbital (HOMO) and used to investigate the molecular symmetry. An accurate three dimensional(3D) calculation of molecular dynamics through time dependent density- functional theory (TDDFT)is considered to explore broad harmonic plateau with high efficiency. We show that an attosecond pulse with a pulse duration of 349as from the superposition of several optimum harmonics is generated. Furthermore, in order to study of the quantum trajectory of electrons, time-frequency analysis is utilized. This work is proved to be a potential way for applications in coherent imaging and photoelectron spectroscopy.

Controlling circularly polarized high-order harmonic generation in molecules by intense tricircular laser pulses

We present a scheme to control circularly polarized high-order harmonic generation (HHG) in molecules, the source of extreme ultraviolet and x-ray pulses, by multiple frequency co- and counter-rotating circularly polarized laser pulses. Simulations are performed on the oriented single electron molecular ions with cyclic equilateral geometry, ${\mathrm{H}}_{3}^{2+}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{\mathrm{H}}_{4}^{3+}$, as benchmark models, by numerically solving time-dependent Schr\"odinger equations. Results show that circularly polarized HHG can be produced with extended harmonic order cutoffs, which depend on the intensity ratio of fields and the field-molecule symmetry. We describe the laser induced electron dynamics in a rotating frame to examine recollision dynamics and Coriolis effects. The resulting modulation of HHG spectra arises from a combination of two bicircular fields with counter-rotating and co-rotating components. It is also found that harmonic polarization is a function of the pulse frequencies and helicities, illustrating the importance of matching the field-molecule symmetry. The results in principle illustrate a method for controlling circular polarization properties of harmonics in molecules.

Analytic description of high-order harmonic generation in the adiabatic limit with application to an initial s state in an intense bicircular laser pulse

An analytic description of high-order harmonic generation (HHG) is proposed in the adiabatic (low-frequency) limit for an initial $s$ state and a laser field having an arbitrary wave form. The approach is based on the two-state time-dependent effective range theory and is extended to the case of neutral atoms and positively charged ions by introducing ad hoc the Coulomb corrections for HHG. The resulting closed analytical form for the HHG amplitude is discussed in terms of real classical trajectories. The accuracy of the results of our analytic model is demonstrated by comparison with numerical solutions of the time-dependent Schr\odinger equation for a strong bicircular field composed of two equally intense components with carrier frequencies $\ensuremath{\omega}$ and $2\ensuremath{\omega}$ and opposite helicities. In particular, we demonstrate the effect of ionization gating on HHG in a bicircular field, both for the case that the two field components are quasimonochromatic and for the case that the field components are time-delayed short pulses. We show how ionization in a strong laser field not only smooths the usual peak structures in HHG spectra but also changes the positions and polarization properties of the generated harmonics, seemingly violating the standard dipole selection rules. These effects appear for both short and long incident laser pulses. In the case of time-delayed short laser pulses, ionization gating provides an effective tool for control of both the HHG yield and the harmonic polarizations [Frolov et al., Phys. Rev. Lett. 120, 263203 (2018)]. For the case of short laser pulses, we introduce a simple two-dipole model that captures the physics underlying the harmonic emission process, describing both the oscillation patterns in HHG spectra and also the dependence of the harmonic polarizations on the harmonic energy.

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.

Strong-Field Ionization of Linear Molecules by a Bichromatic Elliptically Polarized Laser Field with Coplanar Counterrotating or Corotating Components of Different Frequencies

We investigate strong-field ionization of linear molecules by a two-color laser field of frequencies rω and sω having coplanar counterrotating or corotating elliptically polarized components (ω is the fundamental laser field frequency and r and s are integers). Using the improved molecular strong-field approximation we analyze direct above-threshold ionization (ATI) and high-order ATI (HATI) spectra. More precisely, reflection and rotational symmetries of these spectra for linear molecules aligned in the laser-field polarization plane are considered. The reflection symmetries for particular molecular orientations, known to be valid for a bicircular field (this is the field with circularly polarized counterrotating components), are valid also for arbitrary component ellipticities. However, specific rotational symmetries that are satisfied for HATI by a bicircular field, are violated for an arbitrary elliptically polarized field with counterrotating components. For the corotating case and the N2 molecule we analyze molecular-orientation-dependent interferences and plateau structures for various ellipticities.

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.

Method of XUV attosecond pulse characterization using two-photon resonant Raman-type transition

We suggest a new method of attosecond pulse train measurement based on two-photon Raman-type resonant interaction of the XUV with an atom (Attosecond pulse reconstruction using Raman-type Transitions, ART). We show analytically and numerically that the attosecond pulse auto-correlation function and chirp can be measured using this process. Its resonant nature allows obtaining reasonable signal even for relatively low XUV intensity. The interpretation of the results is remarkably simple for the XUV generated by bi-circular field. The experimental realization of the method can be based either on measuring the excitation of the gas or on four-wave mixing fully optical scheme.

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.

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.

Spectral control of high harmonics from relativistic plasmas using bicircular fields.

We introduce two-color counterrotating circularly polarized laser fields as a way to spectrally control high harmonic generation (HHG) from relativistic plasma mirrors. Through particle-in-cell simulations, we show that only a selected group of harmonic orders can appear owing to the symmetry of the laser fields and the related conservation laws. By adjusting the intensity ratio of the two driving field components, we demonstrate the overall HHG efficiency, the relative intensity of allowed neighboring harmonic orders, and that the polarization state of the harmonic source can be tuned. The HHG efficiency of this scheme can be as high as that driven by a linearly polarized laser field.

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.

Control of attosecond light polarization in two-color bicircular fields

We develop theoretically and confirm both numerically and experimentally a comprehensive analytical model which describes the propensity rules in the emission of circularly polarized high harmonics by systems driven by two-color counter-rotating fields, a fundamental and its second harmonic. We identify and confirm the three propensity rules responsible for the contrast between the 3N+1 and 3N+2 harmonic lines in the HHG spectra of noble gas atoms. We demonstrate how these rules depend on the laser parameters and how they can be used in the experiment to shape the polarization properties of the emitted attosecond pulses.

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).

Strong-field approximation in a rotating frame: High-order harmonic emission from p states in bicircular fields

High-order harmonic generation with bicircular fields - the combination of counter-rotating circularly polarized pulses at different frequencies - results in a series of short-wavelength XUV harmonics with alternating circular polarizations, and experiments show that there is an asymmetry in the emission between the two helicities: a slight one in helium, and a larger one in neon and argon, where the emission is carried out by p-shell electrons. Here we analyze this asymmetry by switching to a rotating frame in which the field is linearly polarized; this induces an effective magnetic field which lowers the ionization potential of the $p_+$ orbital that co-rotates with the lower-frequency driver, enhancing its harmonic emission and the overall helicity of the generated harmonics, while also introducing nontrivial effects from the transformation to a non-inertial frame in complex time. In addition, this analysis directly relates the small asymmetry produced by s-shell emission to the imaginary part of the recollision velocity in the standard strong-field-approximation formalism.

Spin-dependent effects in high-order above-threshold ionization: spin–orbit interaction and exchange effects

Spin-dependent effects in atomic processes can be caused by the spin–orbit interaction or/and by the requirement that the wave function of identical electrons is antisymmetric. Such effects are usually neglected in strong-field physics. We show two examples, supported by theoretical results and numerical calculations, in which these effects are important. The first one is based on strong-field ionization of Xe atoms by a bicircular field. The corresponding momentum distribution of spin-polarized electrons emitted in the above-threshold ionization process exhibits wavelength-dependent fast oscillations. For longer wavelengths and small electron emission angle there is a wide photoelectron kinetic energy region in which the spin asymmetry parameter changes continuously from large positive to large negative values. In addition, the emission time of such electrons in high-order above-threshold ionization is determined on the attosecond time scale. The second process considered is high-order above-threshold ionization of excited Li+ ions. We have found that in this case, even in the absence of spin–orbit coupling, the photoelectron momentum distribution strongly depends on the initial spin state. In the singlet state we have characteristic minima, which are caused by the destructive interference of the direct and exchange rescattering amplitudes. Such minima are absent in the triplet state.