Ellipticity Dependence Research Articles

Geometrical and physical optics analysis for mm-wavelength refractor telescopes designed to map the cosmic microwave background.

We present a compact two-lens high-density polyethylene (HDPE) $ f/1.6 $f/1.6 refractor design that is capable of supporting a 28-deg diffraction-limited field of view at 1-mm wavelengths and contrast it to a similar two-lens refractor using silicon lenses. We compare the optical properties of these two systems as predicted by both geometrical and physical optics. The presented analysis suggests that by relaxing telecentricity requirements, a plastic two-lens refractor system can be made to have a surprisingly large field of view. Furthermore, this HDPE system is found to perform comparably to a similar silicon system across a wide field of view and wavelengths up to 1 mm. We show that for both telescope designs, cold stop spillover changes significantly across the field of view in a way that is somewhat inconsistent with Gaussian beam formalism and simple $ f $f-number scaling. We present results that highlight beam ellipticity dependence on both pixel location and pixel aperture size-an effect that is challenging to reproduce in standard geometrical optics. We show that a silicon refractor design suffers from larger cross-polarization compared to the HDPE design. Our results address the limitations of relying solely on geometrical optics to assess relative performance of two optical systems. We discuss implications for future refractor designs.

High harmonics diffraction caused by an ellipticity grating

We study the generation of high-harmonic beams by the non-collinear wave mixing of two circularly or elliptically polarized fundamental beams. Changing the ellipticity of the two fundamental beams changes the electric field structure in the focal plane, generating multiple high harmonics beamlets. We show that the ellipticity dependence of the harmonic dipole modulates the near-field harmonic intensity, resulting in the diffraction of the generated harmonics in the far-field.

Ellipticity dependence of nonperturbative harmonic generation in few-layer MoS2

High-order harmonic generation in solid state materials is viewed as a means of studying strong field nonperturbative physics. Researching ellipticity dependence can help understand the dynamics of electron–hole pairs in the presence of strong laser field. In this work, ellipticity dependence of high harmonic generation in few layer MoS2 is experimentally investigated with a midinfrared laser centered at 3.9 um. We irradiate monolayer, bilayer and trilayer MoS2 with intense mid-infrared light to generate non-perturbative harmonics up to 11th order. We find that the normalized yield of each harmonic descends in a form of Gaussian-like profile as increasing the ellipticity of driving pulse. It is astonishing to see that ellipticity dependence of monolayer, bilayer and trilayer MoS2 follows the same rules. We ascribe this similarity to the same deviation distance of electron–hole pairs in three sets of samples. Meanwhile, the higher the harmonic order is, the more sensitive it is to ellipticity, which shows agreement with gas-phase harmonics. To explain the phenomena, we explore the role of intraband and interband dynamics with multiband semiconductor Bloch equations, which helps to delve deeper into ultrafast and strong-field features of MoS2.

Optimization of N2+ lasing through population depletion in the X2Σg+ state using elliptically modulated ultrashort intense laser fields

The authors generate air lasing by elliptically-modulated intense femtosecond laser pulses and show that the lasing intensities of N${}_{2}$${}^{+}$ at 391 and 428 nm can be manipulated by adjusting the temporal separation between the two polarization components in the elliptically-modulated laser pulse and their relative amplitudes. The authors interpret this characteristic ellipticity dependence in terms of the post-ionization coupling occurring in the later part of the laser pulse, which depletes efficiently the population in the X${}_{2}$ ${\mathrm{\ensuremath{\Sigma}}}_{g}^{+}$ (v'' = 0) state.

Phase invariance of the semiconductor Bloch equations

We investigate the gauge invariance of the semiconductor Bloch equations (SBEs) in solid high-order harmonic generation (HHG). It is found that the gauge dependence of the SBEs can be attributed to the absence of Berry connection terms in the SBEs in previous studies. When the Berry connection terms are considered, the gauge invariance of the SBEs can be naturally preserved under an arbitrary global phase transformation. To satisfy the demand of the continuity of transition dipole matrix elements in one-dimensional calculations, we propose a gauge that is easy to implement numerically. Finally, the saddle point analysis shows that Berry connections can influence many properties of HHG, such as ellipticity dependence and even harmonic generation. This work uncovers the importance of the Berry connection terms that have been neglected for a long time in simulations of solid HHG. © 2019 American Physical Society.

Polarization ellipticity dependence of ${\rm N}_{2}^{+}$ air lasing: the role of coupling between the ground state and a photo-excited intermediate state

We experimentally investigate the dependence of N2+ lasing radiation on the ellipticity of an 800 nm pump laser. The results show that 391 and 428 nm lasing signals exhibit different behaviors with the change of ellipticity. The 391 nm lasing signals show abnormal ellipticity dependence regardless of the pump intensity, whereas for the 428 nm lasing signals, the abnormal characteristic only occurs at a high laser intensity. The qualitative analysis reveals that the physical origin of the phenomenon is the postionization coupling between the ground X2Σg+ state and the intermediate A2Π u state of N2+ ions.

Resolving multi-orbital effects on high harmonic generation from aligned N2 molecules in linearly and elliptically polarized intense laser fields**Project supported by the National Natural Science Foundation of China (Grant Nos. 91750104, 11127403, and 11474130) and the Natural Science Foundation of Jilin Province, China (Grant No. 20160101332JC).

We perform an experimental study of the multi-orbital effect on the high-order harmonic generation (HHG) from aligned N2 molecules in both linearly and elliptically polarized intense laser fields. Measured by a home-built extreme ultraviolet (XUV) flat grating spectrometer with the pump–probe method, the angular distributions of different orders of HHG are obtained, which show distinctive behaviors for harmonics in the plateau and the cut-off regions. The ellipticity dependence of HHG is investigated by aligning the molecular axis parallel or perpendicular to the laser polarization. Our results indicate that both the highest occupied molecular orbital (HOMO) as well as the lower one (HOMO-1) contribute to the HHG of N2 molecules, in either linearly or elliptically polarized intense laser field. The study paves the way for understanding the ultrafast electron dynamics of molecules exposed to an intense laser field.

Frustrated tunneling ionization in the elliptically polarized strong laser fields.

We theoretically investigated frustrated tunneling ionization (FTI) in the interaction of atoms with elliptically polarized laser pulses by a semiclassical ensemble model. Our results show that the yield of frustrated tunneling ionization events exhibits an anomalous behavior which maximizes at the nonzero ellipticity. By tracing back the initial tunneling coordinates, we show that this anomalous behavior is due to the fact that the initial transverse velocity at tunneling of the FTI events is nonzero in the linear laser pulses and it moves across zero as the ellipticity increases. The FTI yield maximizes at the ellipticity when the initial transverse momentum for being trapped is zero. Moreover, the angular momentum distribution of the FTI events and its ellipticity dependence are also explored. The anomalous behavior revealed in our work is very similar to the previously observed ellipticity dependence of the near- and below-threshold harmonics, and thus our work may uncover the mechanism of the below-threshold harmonics which is still a controversial issue.

Reciprocal-Space-Trajectory Perspective on High-Harmonic Generation in Solids.

We revisit the mechanism of high-harmonic generation (HHG) from solids by comparing HHG in laser fields with different ellipticities but a constant maximum amplitude. It is shown that the cutoff of HHG is strongly extended in a circularly polarized field. Moreover, the harmonic yield with large ellipticity is comparable to or even higher than that in the linearly polarized field. To understand the underlying physics, we develop a reciprocal-space-trajectory method, which explains HHG in solids by a trajectory ensemble from different ionization times and different initial states in the reciprocal space. We show that the cutoff extension is related to an additional preacceleration step prior to ionization, which has been overlooked in solids. By analyzing the trajectories and the time-frequency spectrogram, we show that the HHG in solids cannot be interpreted in terms of the classical recollision picture alone. Instead, the radiation should be described by the electron-hole interband polarization, which leads to the unusual ellipticity dependence. We propose a new four-step model to understand the mechanism of HHG in solids.

Ellipticity dependence transition induced by dynamical Bloch oscillations

The dependence of high-harmonic generation (HHG) on laser ellipticity is investigated using a modified ZnO model. In the driving of relatively weak field, we reproduce qualitatively the ellipticity dependence as observed in the HHG experiment of wurtzite ZnO. When increasing the field strength, the HHG shows an anomalous ellipticity dependence, similar to that observed experimentally in the single-crystal MgO. With the help of a semiclassical analysis, it is found that the key mechanism inducing the change of ellipticity dependence is the interplay between the dynamical Bloch oscillation and the anisotropic band structure. The dynamical Bloch oscillation contributes additional quantum paths, which are less sensitive to ellipticity. The anisotropic band-structure make the driving pulse with finite ellipticity be able to drive the pairs to the band positions with larger gap, which extends the harmonic cutoff. The combination of these two effects leads to the anomalous ellipticity dependence. The result reveals the importance of dynamical Bloch oscillations for the ellipticity dependence of HHG from bulk ZnO.

Coulomb potential effects in strong-field atomic ionization under elliptical polarization

We experimentally and theoretically investigate strong-field ionization of noble-gas atoms in elliptically polarized laser fields at 2000 nm. The ellipticity dependence of the ion yields has been measured at a series of intensities and compared to various theoretical model predictions [i.e., the Ammosov-Delone-Krainov (ADK), strong-field approximation (SFA), and Coulomb-Volkov distorted-wave approximation (CVA) models]. The SFA and ADK calculations match each other, which alludes to the negligible contribution of nonadiabatic effect. In the meantime, they both deviate significantly from the measurements. On the other hand, the good agreement between CVA calculations and measurements reveals the salient role played by the Coulomb potential in atomic ionization. The Coulomb potential effect is further confirmed by comparison of CVA calculations and measurements at 800 nm. Our work demonstrates the significant Coulomb potential effect on the ionization yields, the experimental evidence of which has rarely been documented.

Electron-wave-packet dynamics extracted from the ellipticity dependence of high-order harmonic generation in benzene molecules

We measure the ellipticity dependence of high-order harmonic intensities generated from benzene $({\mathrm{C}}_{6}{\mathrm{H}}_{6})$ molecules with and without a yttrium aluminum garnet (YAG) laser field. We successfully extract the expansion dynamics of the electron wave packet by analyzing the ellipticity dependence based on the semiclassical electron trajectory. Without the YAG laser pulse, we find that the ellipticity dependence reflects the expansion dynamics of the electron wave packet and the difference in highest occupied molecular orbital compared to nitrogen $({\mathrm{N}}_{2})$ molecules. We also measure the ellipticity dependence under the YAG laser field and show that the difference in the ellipticity dependence with and without the YAG laser field is qualitatively explained by the harmonic-order-dependent efficiency of the sum and difference frequency generation, whose efficiency is higher as the harmonic order becomes higher.

Driving-laser ellipticity dependence of high-order harmonic generation in graphene

High-order harmonic generation in graphene driven by the elliptically polarized midinfrared driving pulse is theoretically investigated by solving the two-band semiconductor Bloch equation in the Houston basis under the tight-binding approximation. It is found that the maximal harmonic yield occurs at a specific driver ellipticity whose value is determined by relaxation processes, so that the proper choice of the dephasing time is key to quantitatively reproducing the recent experiment. We show that the ellipticity dependence mainly originates from the interband harmonic emission, and exhibits an almost stable behavior against the crystal orientation. Furthermore, we also demonstrate that both the large magnitude and the circular vortex structure of the transition dipole near the graphene Dirac point are responsible for forming the unique characteristics. These results demonstrate that high-harmonic generation by the elliptically polarized driving field carries significant information on the dephasing time and the electronic structure signature in the graphene.

Effects of nodal planes on strong-field ionization and high-order-harmonic generation in ring-type molecules

We measure the ellipticity dependence of high-harmonic generation (HHG) from unaligned gas-phase ensembles of the five-membered ring molecules 2,3-dihydrofuran (${\mathrm{C}}_{4}{\mathrm{H}}_{6}\mathrm{O}$), furan (${\mathrm{C}}_{4}{\mathrm{H}}_{4}\mathrm{O}$), and thiophene (${\mathrm{C}}_{4}{\mathrm{H}}_{4}\mathrm{S}$). As is normally the case, the HHG emission is suppressed for increased ellipticity since the recollision wave packet leading to HHG is steered away from the parent ion for large ellipticity. However, through comparison with computations of the first step in HHG, namely, strong-field ionization (SFI), we show that the observed differences in the ellipticity dependence for the three molecular species reflect the lateral momentum distributions of the continuum electron responsible for HHG, which in turn provides information about the particular orientation between the molecular axis and the laser field that maximizes the SFI probability. Strikingly, and contrary to the conventional wisdom in the strong-field community, we find that for furan and thiophene the most probable orientation for SFI occurs when the electric field of the laser is aligned near a nodal plane of the corresponding ionizing orbital. 2,3-dihydrofuran, on the other hand, follows the standard expectation that the most probable orientation for SFI occurs when the electric field is aligned away from any nodal plane.

Ellipticity dependence of high-harmonic generation in solids originating from coupled intraband and interband dynamics

The strong ellipticity dependence of high-harmonic generation (HHG) in gases enables numerous experimental techniques that are nowadays routinely used, for instance, to create isolated attosecond pulses. Extending such techniques to solids requires a fundamental understanding of the microscopic mechanism of HHG. Here we use first-principles simulations within a time-dependent density-functional framework and show how intraband and interband mechanisms are strongly and differently affected by the ellipticity of the driving laser field. The complex interplay between intraband and interband effects can be used to tune and improve harmonic emission in solids. In particular, we show that the high-harmonic plateau can be extended by as much as 30% using a finite ellipticity of the driving field. We furthermore demonstrate the possibility to generate, from single circularly polarized drivers, circularly polarized harmonics. Our work shows that ellipticity provides an additional knob to experimentally optimize HHG in solids.

Bifurcation of ion momentum distributions in sequential double ionization by elliptically polarized laser pulses

Using a fully classical model, we have studied sequential double ionization (SDI) of argon driven by elliptically polarized laser pulses at intensities well in the over-barrier ionization region. The results show that the width of the ion momentum along the major elliptical axis increases with the ellipticity while that in the laser propagation direction decreases with the ellipticity. The ion momentum in the minor elliptical axis bifurcates from one peak structure to three-peak, then to four-peak, finally to six-peak structure. Analysis shows that this ellipticity dependence of the ion momentum distribution is a result of the subcycle nature of the ionization dynamics in SDI. By changing the ellipticity and wavelength of the driving pulses, this subcycle ionization dynamics is more observable. This provides a simple and efficient way to study the subcycle ionization dynamics in strong field processes.

Semiclassical modeling of high-order harmonic generation driven by an elliptically polarized laser field: the role of recolliding periodic orbits

We model high-order harmonic generation (HHG) from an argon atom driven by an elliptically polarized laser field, using classical electron trajectories that are initialized from a discrete set of microcanonical distributions. Analysis of the returning trajectories reveals that when the atomic potential is accounted for, there are two possible ionization pathways leading to the same photon energy for both the short and the long trajectories found in the recollision model of HHG. We also demonstrate that the threshold ellipticity, a measure of the sensitivity to ellipticity, decreases as the harmonic order increases for both short and long trajectories, in qualitative agreement with recent experimental results (Larsen et al 2015 arXiv:1506.08660). We use a nonlinear dynamical analysis to show that this behavior is regulated by a set of recolliding periodic orbits (RPOs) that survive at ellipticities greater than 0.2 and can therefore drive the recollision dynamics even at relatively high ellipticity. The connection between the threshold ellipticity dependence we find and RPOs means that the effect should be general.

Sub-cycle ionization dynamics revealed by trajectory resolved, elliptically-driven high-order harmonic generation.

The sub-cycle dynamics of electrons driven by strong laser fields is central to the emerging field of attosecond science. We demonstrate how the dynamics can be probed through high-order harmonic generation, where different trajectories leading to the same harmonic order are initiated at different times, thereby probing different field strengths. We find large differences between the trajectories with respect to both their sensitivity to driving field ellipticity and resonant enhancement. To accurately describe the ellipticity dependence of the long trajectory harmonics we must include a sub-cycle change of the initial velocity distribution of the electron and its excursion time. The resonant enhancement is observed only for the long trajectory contribution of a particular harmonic when a window resonance in argon, which is off-resonant in the field-free case, is shifted into resonance due to a large dynamic Stark shift.

Ellipticity dependence of neutral Rydberg excitation of atoms in strong laser fields

Rydberg state excitation (RSE) of different atoms in elliptically polarized strong 800 nm laser fields is investigated experimentally, and the results are compared with calculations of the strong-field approximation (SFA) model and the semiclassical model. It is observed that the RSE probability declines with increasing laser ellipticity for all of the He, Ar, and Kr atoms. While the measured ellipticity dependence of He RSE is very consistent with the predictions of both the SFA and semiclassical calculations, the width of the ellipticity dependence for Ar and Kr atoms is wider than that of the SFA model but closer to the semiclassical calculations. Analysis indicates that unlike a tunneling-plus-rescattering process, the decline of the RSE yield with increasing ellipticity can be attributed to a decrease of electrons with low kinetic energy that could be captured in the Rydberg states by the Coulomb potential. It indicates that the atomic RSE process could be related to the very low or near-zero energy structure in the photoelectron spectrum in strong laser fields, which would stimulate further experimental and theoretical studies to reveal their underlying mechanisms.

Polarization gating of high harmonic generation in the water window

We implement the polarization gating (PG) technique with a two-cycle, 1.7 μm driving field to generate an attosecond supercontinuum extending to the water window spectral region. The ellipticity dependence of the high harmonic yield over a photon energy range much broader than previous work is measured and compared with a semi-classical model. When PG is applied, the carrier-envelope phase (CEP) is swept to study its influence on the continuum generation. PG with one-cycle (5.7 fs) and two-cycle (11.3 fs) delay are tested, and both give continuous spectra spanning from 50 to 450 eV under certain CEP values, strongly indicating the generation of isolated attosecond pulses in the water window region.