Cut-off Law Research Articles

Tuning Overbias Plasmon Energy and Intensity in Molecular Plasmonic Tunneling Junctions by Atomic Polarizability.

Plasmon excitation in molecular tunnel junctions is interesting because the plasmonic properties of the device can be, in principle, controlled by varying the chemical structure of the molecules. The plasmon energy of the excited plasmons usually follows the quantum cutoff law, but frequently overbias plasmon energy has been observed, which can be explained by quantum shot noise, multielectron processes, or hot carrier models. So far, clear correlations between molecular structure and the plasmon energy have not been reported. Here, we introduce halogenated molecules (HS(CH2)12X, with X = H, F, Cl, Br, or I) with polarizable terminal atoms as the tunnel barriers and demonstrate molecular control over both the excited plasmon intensity and energy for a given applied voltage. As the polarizability of the terminal atom increases, the tunnel barrier height decreases, resulting in an increase in the tunneling current and the plasmon intensity without changing the tunneling barrier width. We also show that the plasmon energy is controlled by the electrostatic potential drop at the molecule-electrode interface, which depends on the polarizability of the terminal atom and the metal electrode material (Ag, Au, or Pt). Our results give new insights in the relation between molecular structure, electronic structure of the molecular junction, and the plasmonic properties which are important for the development of molecular scale plasmonic-electronic devices.

Extension of high-order harmonic generation cutoff from laser-ablated tin plasma plumes.

The high-order harmonic spectra from laser-ablated tin plasma plumes are investigated experimentally and theoretically at different laser wavelengths. It is found that the harmonic cutoff is extended to ∼84 eV and the harmonic yield is greatly improved by decreasing the driving laser wavelength from 800 nm to 400 nm. Appling the Perelomov-Popov-Terent'ev theory with the semiclassical cutoff law and one-dimensional time-dependent Schrödinger equation, the contribution of the Sn3+ ion to harmonic generation accounts for the cutoff extension at 400 nm. With the qualitative analysis of the phase mismatching effect, we reveal the phase matching caused by the dispersion of free electrons is greatly optimized in the 400 nm driving field relative to the 800 nm driving field. The high-order harmonic generated from laser-ablated tin plasma plumes driven by the short laser wavelength provides a promising way to extend cutoff energy and generate intensely coherent extreme ultraviolet radiation.

Theory for all-optical responses in topological materials: The velocity gauge picture

High Harmonic Generation (HHG), which has been widely used in atomic gas, has recently expanded to solids as a means to study highly nonlinear electronic response in condensed matter and produce coherent high frequency radiation with new properties. Most recently, attention has turned to Topological Materials (TMs) and the use of HHG to characterize topological bands and invariants. Theoretical interpretation of nonlinear electronic response in TMs, however, presents many challenges. In particular, the Bloch wavefunction phase of TMs has undefined points in the Brillouin Zone. This leads to singularities in calculating the inter-band and intra-band transition dipole matrix elements of Semiconductor Bloch Equations (SBEs). Here, we use the laser-electromagnetic velocity gauge ${\boldsymbol p}\cdot {\bf A}(t)$ to numerically integrate the SBEs and treat the singularity in the production of the electrical currents and HHG spectra. We use a prototype of Chern Insulators (CIs), the Haldane model, to demonstrate our approach. We find good qualitative agreement of the velocity gauge compared to the length gauge and the Time-Dependent Density Functional theory in the case of topologically trivial materials such as MoS$_2$. For velocity gauge and length gauge, our two-band Haldane model reproduces key HHG spectra features: ($\textit i$) The selection rules for linear and circular light drivers, ($\textit ii$) The linear cut-off law scaling and ($\textit iii$) The anomalous circular dichroism. We conclude that the velocity-gauge approach captures experimental observations and provides theoretical tools to investigate topological materials.

Extended high-harmonic spectra through a cascade resonance in confined quantum systems

The study of high-harmonic generation in confined quantum systems is vital to establishing a complete physical picture of harmonic generation from atoms and molecules to bulk solids. Based on a multilevel approach, we demonstrate how intraband resonances significantly influence the harmonic spectra via charge pumping to the higher subbands and, thus, redefine the cutoff laws. As a proof of principle, we consider the interaction of graphene nanoribbons, with zigzag as well as armchair terminations, and resonant fields polarized along the cross-ribbon direction. Here, this effect is particularly prominent due to many nearly equi-separated energy levels. In such a scenario, a cascade resonance effect can take place in high-harmonic generation when the field strength is above a critical threshold, which is completely different from the harmonic generation mechanism of atoms, molecules and bulk solids. We further discuss the implications not only for other systems in a nanoribbon geometry, but also systems where only a few subbands (energy levels) meet this frequency-matching condition by considering a generalized multilevel Hamiltonian. Our study highlights that cascade resonance bears fundamentally distinct influence on the laws of harmonic generation, specifically the cutoff laws based on laser duration, field strength, and wavelength, thus unraveling new insights in solid-state high-harmonic generation.

Investigation of high-harmonic cutoff of metal ions driven by near-infrared laser.

The cutoff-energies of high-harmonic generation in the laser-ablated plumes of various metal targets (Ca, Mg, Fe, Zn, Ta, Mo, Al, W, In, Cu, Au, Ti, Ag) driven by near-infrared (0.8-µm) femtosecond laser are investigated and compared. Due to the low ionization potentials of metal atoms, it is believed that the observed high-harmonic cutoffs are contributed by the singly charged or even doubly charged ions. Ionization calculations using Perelomov-Popov-Terent'ev theory are performed to estimate the laser intensities at which saturation of ionization occur for different ions. Treating the calculated values as the effective driving laser intensities, the observed cutoffs from most of the targets are in reasonable agreement with the predictions of the semi-classical cutoff law.

Cutoff for continuum-continuum high-order harmonic generation

We establish and analytically derive the cutoff law for high harmonics generated with two interfering electron wave packets in the continuum. The phenomenon occurs for laser intensities high enough to spark substantial ionization, yet low enough that depletion of the ground state remains moderate to ensure a large interference yield. The continuum-continuum (CC) cutoff emerges from the maximal action difference of the dominant classical trajectories in the phase of the time-dependent dipole response. This action difference translates into an interference of a maximally time-delayed, so called stranded continuum trajectory with the trajectory of a high energy continuum electron. The time delay renders the CC harmonics more sensitive to electron wave-packet dispersion than standard high harmonics. Hence, CC harmonics should be most prominent in dimensionally reduced systems.

Huygens-Fresnel Picture for High Harmonic Generation in Solids.

High harmonic generation (HHG) is usually described by the laser-induced recollision of particlelike electrons, which lies at the heart of attosecond physics and also inspires numerous attosecond spectroscopic methods. Here, we demonstrate that the wavelike behavior of electrons plays an important role in solid HHG. By taking an analogy to the Huygens-Fresnel principle, an electron wave perspective on solid HHG is proposed by using the wavelet stationary-phase method. From this perspective, we have explained the deviation between the cutoff law predicted by the particlelike recollision model and the numerical simulation of semiconductor Bloch equations. Moreover, the emission times of HHG can be well predicted with our method involving the wave property of electrons. However, in contrast, the prediction with the particlelike recollision model shows obvious deviations compared to the semiconductor Bloch equations simulation. The wavelike properties of the electron motion can also be revealed by the HHG in a two-color field.

Does starting school before labor day affect high school retention and graduation: evidence from Virginia’s kings dominion law

ABSTRACT Several states have required K-12 public schools to start after Labour Day in an effort to aid the tourism and hospitality industry. However, little is known about how these policies impact educational outcomes. We examine the impact of Virginia’s post-Labour Day school start law on high school retention and graduation rates. We use a difference-in-differences model to exploit exogenous variation in school division start dates. Our results show small differences of up to three weeks have little effect on high school dropout and graduation rates. Our findings inform the debate on post-Labour Day school start laws and compulsory attendance age cut-off laws.

Does Starting School Before Labor Day Affect High School Retention and Graduation: Evidence from Virginia's Kings Dominion Law

Several states have required K-12 public schools to start after Labor Day in an effort to aid the tourism and hospitality industry. However, little is known about how these policies impact educational outcomes. We examine the impact of Virginia's post-Labor Day school start law on high school retention and graduation rates. We use a difference-in-differences model to exploit exogenous variation in school division start dates. Our results show small differences of up to three weeks have little effect on high school dropout and graduation rates. Our findings inform the debate on post-Labor Day school start laws and compulsory attendance age cutoff laws.

Diversity time-period and diversity-time-area relationships exemplified by the human microbiome

We extend the ecological laws of species-time relationship (STR) and species-time-area relationship (STAR) to general diversity time-period relationship (DTR) and diversity-time-area relationship (DTAR), and test the extensions with the human vaginal microbiome datasets by building 1460 DTR/DTAR models. Our extensions were inspired by the observation that Hill numbers, well regarded as the most appropriate measure of alpha-diversity and also particularly suitable for multiplicative beta-diversity partitioning, are actually in the units of effective species, and therefore, should be able to substitute for species in the STR and STAR. We found that the traditional power law (PL) model is only applicable for DTR at diversity order zero (i.e., species richness); at higher diversity orders (q = 1–4), the power law with exponent cutoff (PLEC) and power law with inverse exponent cutoff (PLIEC) are more appropriate. In particular, PLEC has an advantage over PLIEC in predicting maximal accumulation diversity (MAD) over time. In fact, with the DTR extensions, we can construct DTR and MAD profiles. To the best of our knowledge, this is the first comprehensive investigation of the DTR/DTAR in human microbiome. Methodologically, our DTR/DTAR profiles can characterize general diversity scaling beyond species richness, covering both alpha- and beta-diversity regimes across different diversity orders.

High-order harmonic generation driven by inhomogeneous plasmonics fields spatially bounded: influence on the cut-off law

We study high-order harmonic generation (HHG) in model atoms driven by plasmonic-enhanced fields. These fields result from the illumination of plasmonic nanostructures by few-cycle laser pulses. We demonstrate that the spatial inhomogeneous character of the laser electric field, in a form of Gaussian-shaped functions, leads to an unexpected relationship between the HHG cutoff and the laser wavelength. Precise description of the spatial form of the plasmonic-enhanced field allows us to predict this relationship. We combine the numerical solutions of the time-dependent Schrödinger equation (TDSE) with the plasmonic-enhanced electric fields obtained from 3D finite element simulations. We additionally employ classical simulations to supplement the TDSE outcomes and characterize the extended HHG spectra by means of their associated electron trajectories. A proper definition of the spatially inhomogeneous laser electric field is instrumental to accurately describe the underlying physics of HHG driven by plasmonic-enhanced fields. This characterization opens up new perspectives for HHG control with various experimental nano-setups.

High harmonic generation from bulk diamond driven by intense femtosecond laser pulse

We present theoretical results on the high-harmonic generation (HHG) in bulk diamond induced by intense laser pulse of wavelength 800nm and duration 15fs. For laser intensity in the range 1 ≤ I ≤ 50 TW/cm2 above bandgap harmonics are generated after the pulse peak. We find that the intensity of individual harmonics increases non-linearly with the peak laser intensity, following a non-perturbative trend. For moderate intensity the HHG spectrum exhibits a primary plateau with noisy odd order harmonic structure and a cutoff regime. For increased laser intensity a secondary plateau emerges with quasi-continuous spectrum of harmonics extending beyond the 50th order. Consistently with experimental observations, it is found that the cutoff energy for HHG scales linearly with the peak field strength and derivation of this cutoff law is provided.

Discrete power law with exponential cutoff and Lotka's law

One of the first bibliometric laws appeared in Alfred J. Lotka's 1926 examination of author productivity in chemistry and physics. The result was a productivity distribution described by a power law. In this paper, Lotka's original data on author productivity in chemistry are reconsidered. We define a discrete power law with exponential cutoff, test Lotka's data, and compare the fit to the discrete power law.

On the cutoff law of laser induced high harmonic spectra

The currently well accepted cutoff law for laser induced high harmonic spectra predicts the cutoff energy as a linear combination of two interaction energies, the ponderomotive energy U (p) and the atomic biding energy I (p) , with coefficients 3.17 and 1.32, respectively. Even though, this law has been there for twenty years or so, the background information for these two constants, such as how they relate to fundamental physics and mathematics constants, is still unknown. This simple fact, keeps this cutoff law remaining as an empirical one. Based on the cutoff property of Bessel functions and the Einstein photoelectric law in the multiphoton case, we show these two coefficients are algebraic constants, and , respectively. A recent spectra calculation and an experimental measurement support the new cutoff law.

Semiclassical analysis of high harmonic generation in bulk crystals

High harmonic generation (HHG) in solids is investigated. We find that interband emission is dominant for the midinfrared laser driver frequencies, whereas intraband emission dominates the far-infrared range. Interband HHG is similar to atomic HHG and therewith opens the possibility to apply atomic attosecond technology to the condensed matter phase. Interband emission is investigated with a quasiclassical method, by which HHG can be modeled based on the classical trajectory analysis of electron-hole pairs. This analysis yields a simple approximate cutoff law for HHG in solids. Differences between HHG in atoms and solids are identified that are important for adapting atomic attosecond technology to make it applicable to condensed matter.

Guo-Åberg-Crasemann theory for high harmonic generation and its cutoff law

Based on the scattering theory of Guo-Åberg-Crasemann (GAC), which has no artificial assumptions, high harmonic generation (HHG) is studied by using first-principles. The HHG spectra of different rare atoms are also calculated. Using the properties of ordinary Bessel functions and the Einstein photoelectric law in the strong-field case, we reveal a new cutoff law qcħω = (9 -4√2) Up + (2√2-1) Ip ≈ 3.34 Up + 1.83 Ip of HHG based on a mathematical deduction method and a graphical method, which accords well with the Popmintchev’s experimental result published on Science in 2012. This cutoff law also agrees well with our own calculation using the HHG transition rate formula derived from the GAC scattering theory. Thus, we have four pieces of independent evidence for the same cutoff law of HHG. The cutoff orders predicted by this theory are higher due to the absorption of the extra photons. These photons only participate in the photon-mode up-conversion and do nothing in the photoionization process.

A nonperturbative quantum electrodynamic approach to the theory of laser induced high harmonic generation

Using nonperturbative quantum electrodynamics, we develop a scattering theory for high harmonic generation (HHG). A transition rate formula for HHG is obtained. Applying this formula, we calculate the spectra of high harmonics generated from different noble gases shined by strong laser light. We study the cutoff property of the spectra. The data show that the cutoff orders of high harmonics are greater than that predicted by the “3.17” cutoff law. As a numerical experiment, the data obtained from our repeated calculations support the newly derived theoretical expression of the cutoff law. The cutoff energy of high harmonics described by the new cutoff law, in terms of the ponderomotive energy Up and the ionization potential energy Ip, is 3.34Up + 1.83Ip. The higher cutoff orders predicted by this theory are due to the absorption of the extra photons, which participate only the photon-mode up-conversion and do nothing in the photoionization process.

Resonant enhancement of the harmonic-generation spectrum of beryllium

odinger equation in an intense (≈10 13 W/cm 2 ), 30-cycle laser field. As the laser frequency ωL varies from 1. 7t o 1. 8e V (which is in the tunable range of a Ti:sapphire laser), the seventh harmonic becomes resonant sequentially with the transition between the ground state and two doubly excited autoionizing states, 2p4s( 1 P )( atωL = 1.734 eV) and 2p5s( 1 P ) (at ωL = 1.785 eV), while the third harmonic becomes resonant with the 2s2p( 1 P ) singly excited state (at ωL = 1.766 eV). At each of these resonant frequencies, the HHG power spectrum is found to increase by an order of magnitude over a range of harmonics that form a plateau, extending from the resonant harmonic up to a cutoff at the 25th harmonic. In contrast to the well-known rescattering plateau cutoff law appropriate in the tunneling regime (which predicts a cutoff at the fifth or seventh harmonic), the multiphoton regime plateau we find for Be originates from atomic resonance effects. Off resonance, the Be HHG spectrum decreases monotonically with harmonic order. By taking the ratio of the integrated harmonic power of the seventh harmonic to that of the fifth harmonic, one can isolate the resonant effects of the two doubly excited states in the HHG spectrum from those of singly excited resonance states. These ratios exhibit resonance profiles for driving laser-pulse durations much longer than the lifetimes of these autoionizing states. The energy widths of these resonance features are comparable to the widths of the laser pulse and are much smaller than the autoionizing state widths. These results demonstrate an important role for electron correlations in enhancing harmonic-generation rates in the multiphoton regime.

Observational evidence of planetary wave influences on ozone enhancements over upper troposphere North Africa

MOZAIC instrument measured enhanced ozone on two occasions in February, 1996 and 1997 at cruise altitude over North Africa. The cause and source of ozone enhancements over the region are investigated using additional reanalysis data from ERA-Interim. The ERA-Interim reprocessed GOME ozone indicated existence of enhancement as well. Both observational data revealed that the increase in ozone has wider latitudinal coverage extending from North Europe upto North Africa. The geopotential heights and zonal wind from ERA-Interim have indicated existence of planetary-scale flow that allowed meridional airmass exchanges between subtropics and higher latitudes. The presence of troughs-ridge pattern are attributable to large amplitude waves of zonal wavenumber 1–5 propagating eastward in the winter hemisphere westerly current as determined from Hayashi spectra as well as local fractional variance spectra determined from Multitaper Method-Singular Value Decomposition (MTM-SVD) spectral method. MTM-SVD is also used to understand the role of these waves on ozone enhancement and variability during the observation period in a mechanistic approach. A joint analysis of driving field, such as wind and potential vorticity (PV) for which only signals of the dominant zonal wavenumbers of prevailing planetary waves are retained, has revealed strong linkage between wave activity and ozone enhancement over the region at a temporal cycle of 5.8days. One of these features is the displacement of the polar vortex southward during the enhancements, allowing strong airmass, energy and momentum exchanges. Evidence of cutoff laws that are formed within the deep trough, characteristics of Rossby wave breaking, is also seen in the ozone horizontal distribution at different pressure levels during the events. The reconstruction of signals with the cycle of 5.8days has shown that the time and strength of enhancement depend on the circulation patterns dictated by planetary-scale flow relative to the location of observation. The positive PV anomalies upstream or at the observation region bring ozone rich airmass to the region while a negative PV anomaly upstream does the opposite. The position of the anomalies with time changes in accordance with the period of the waves involved. The snap shot of coherent variation of PV and ozone at different time during half cycle of the 5.8-day period has indicated that a region could experience positive (enhancement) or negative (depletion) ozone anomalies of different degree as the wave propagates eastward.

A reduced-geometry independent particle model calculation of high harmonic generation from closed-shell diatomic molecules

We propose a simple model to calculate high harmonic spectra from closed-shell diatomic molecules based on the time-dependent Schrödinger equation. Quasi-Coulomb potentials are used to represent the two-center geometry of a diatomic molecule in two dimensions. A few outer occupied molecular orbitals are evolved independently using a single-electron Hamiltonian, and the harmonic spectra are evaluated from a coherent sum of single-electron dipole accelerations. According to this independent particle model, harmonic spectra from individual orbitals follow the semiclassical cutoff law, but their relative strengths vary depending on molecular orientations. When the contributions from different orbitals are of comparable strength, their net spectrum extends to the inner-orbital cutoffs, and in some cases acquires a local minimum where harmonic spectra from different molecular orbitals interfere destructively because of their phase difference.