Paul Kinsler

Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides

We describe generalized nonlinear envelope equation modeling of sub-cycle dynamics on the underlying electric field carrier during one-dimensional propagation in fused silica. Generalized envelope equation simulations are shown to be in excellent quantitative agreement with the numerical integration of Maxwells equations, even in the presence of shock dynamics and carrier steepening on a sub-50 attosecond timescale. In addition, by separating the effects of self-phase modulation and third harmonic generation, we examine the relative contribution of these effects in supercontinuum generation in fused silica nanowire waveguides.

Causality-Based Criteria for a Negative Refractive Index Must Be Used With Care

Using the principle of causality as expressed in the Kramers-Kronig relations, we derive a generalized criterion for a negative refractive index that admits imperfect transparency at an observation frequency omega. It also allows us to relate the global properties of the loss (i.e., its frequency response) to its local behavior at omega. However, causality-based criteria rely on the group velocity, not the Poynting vector. Since the two are not equivalent, we provide some simple examples to compare the two criteria.

Few-cycle pulse propagation

We present a comprehensive framework for treating the nonlinear interaction of few-cycle pulses using an envelope description that goes beyond the traditional slowly varying envelope approximation method. This is applied to a range of simulations that demonstrate how the effect of a {chi}{sup (2)} nonlinearity differs between the many-cycle and few-cycle cases. Our approach, which includes diffraction, dispersion, multiple fields, and a wide range of nonlinearities, builds upon the work of Brabec and Krausz [T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997)] and Porras [M. A. Porras, Phys. Rev. A 60, 5069 (1999)]. No approximations are made until the final stage when a particular problem is considered.

Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion

We model the propagation and interaction of few-cycle pulses using a variant of the finite-difference time-domain (FDTD) method that incorporates a novel pseudospectral technique. Our pseudospectral spatial-domain (PSSD) method offers significant advantages over the standard FDTD approach. Run time is comparable to the analogous FDTD and pseudospectral time-domain (PSTD) methods, but our technique is far more flexible. Representative results are presented for self-phase modulation, second and third harmonic generation and parametric generation, in both real and designer media. We also include results that illustrate the interplay of third harmonic generation and self-phase modulation in the formation of carrier shocks.

Intersubband terahertz lasers using four-level asymmetric quantum wells

We demonstrate the potential for laser operation at far-infrared wavelengths (30–300 μm, 1–10 THz) by using intersubband emission in four-level GaAs/AlGaAs asymmetric (stepped) quantum wells. Achieving population inversion in these devices depends critically on the lifetimes of the nonradiative intersubband transitions, and so we have performed detailed calculations of electron–electron and electron–phonon scattering rates. Our four-subband structures show potential for the realization of room temperature lasing, unlike previously considered three-subband structures which did not give population inversions except at impractically low electron densities and temperatures. Auger-type electron–electron interactions involving the highly populated ground subband effectively destroyed the population inversion in three level systems, but in these four subband structures the inversion is maintained by strong phonon-mediated depopulation of the lower laser level. The largest population inversions are calculated at ...

A uni-directional optical pulse propagation equation for materials with both electric and magnetic responses

I derive unidirectional wave equations for fields propagating in materials with both electric and magnetic dispersion and nonlinearity. The derivation imposes no conditions on the pulse profile except that the material modulates the propagation slowly, that is, that loss, dispersion, and nonlinearity have only a small effect over the scale of a wavelength. It also allows a direct term-to-term comparison of the exact bidirectional theory with its approximate unidirectional counterpart.

Theory of directional pulse propagation

We study analytically the photoionization of a coherent superposition of atomic and molecular electronic states by an ultrashort, attosecond x-ray pulse. We show that the broad photoelectron spectrum contains detailed information about the time-dependent electron wave packet. The asymmetry of the photoelectron momentum distribution measures the momentum asymmetry of the initial bound-state wave packet. We show that molecular interference modulates the time-dependent photoelectron spectrum and asymmetry. The modulation also depends on the internuclear separation. The photoelectron spectrum and its asymmetry monitors coherent electron motion and in principle electron transfer on the attosecond time scale.We present an exact study of the finite-temperature properties of hard-core bosons (HCBs) confined on one-dimensional optical lattices. Our solution of the HCB problem is based on the Jordan-Wigner transformation and properties of Slater determinants. We analyze the effects of the temperature on the behavior of the one-particle correlations, the momentum distribution function, and the lowest natural orbitals. In addition, we compare results obtained using the grand-canonical and canonical descriptions for systems like the ones recently achieved experimentally. We show that even for such small systems, as small as 10 HCBs in 50 lattice sites, there are only minor differences between the energies and momentum distributions obtained within both ensembles.Fault-tolerant logical operations for qubits encoded by Calderbank-Shor-Steane codes are discussed, with emphasis on methods that apply to codes of high rate, encoding k qubits per block with k>1. It is shown that the logical qubits within a given block can be prepared by a single recovery operation in any state whose stabilizer generator separates into X and Z parts. Optimized methods to move logical qubits around and to achieve controlled-NOT and Toffoli gates are discussed. It is found that the number of time steps required to complete a fault-tolerant quantum computation is the same when k>1 as when k=1.Using numerical solutions to relativistic quantum field theory with space-time resolution, we illustrate how an incoming electron wave packet with a definite spin scatters off a supercritical potential step. We show that the production rate is reduced of only those electrons that have the same spin as the incoming electron is reduced. This spin-resolved result further clarifies the importance of the Pauli-exclusion principle for the Klein paradox.Theoretical investigations on single charge-transfer processes in collisions of F{sup 2+}+H{yields}F{sup +}+H{sup +} and its reverse process have been carried out at collision energies from 20 eV/u to 10 keV/u. The molecular orbital expansion method within the semiclassical impact parameter formalism has been employed for the scattering dynamics, while the ab initio multireference single- and double-excitation configuration interaction (MRD-CI) method was adopted for determination of molecular electronic states. The initial channels correspond to the quintet and triplet states for the corresponding collision processes, respectively. Four molecular states in the quintet manifold and eight molecular states in the triplet manifold were coupled. In the quintet manifold, the charge-transfer cross sections for F{sup 2+}+H{yields}F{sup +}+H{sup +} range from 1.3x10{sup -22} cm{sup 2} at 20 eV/u to 2.5x10{sup -15}cm{sup 2} at 10 keV/u. The cross sections of the reverse process, F{sup +}+H{sup +}{yields}F{sup 2+}+H, range from 3.0x10{sup -22} cm{sup 2}to 2.3x10{sup -15} cm{sup 2} in the same energy range. In the triplet states, the charge-transfer cross sections for F{sup 2+}+H{yields}F{sup +}+H{sup +} range from 1.1x10{sup -18} cm{sup 2} to 2.5x10{sup -16} cm{sup 2}, and its reverse process gives charge-transfer cross sections ranging from 1.7x10{sup -24} cm{sup 2} to 1.5x10{sup -17} cm{sup 2}.We optimize the turning on of a one-dimensional optical potential, V{sub L}(x,t)=S(t)V{sub 0} cos{sup 2}(kx) to obtain the optimal turn-on function S(t) so as to load a Bose-Einstein condensate into the ground state of the optical lattice of depth V{sub 0}. Specifically, we minimize interband excitations at the end of the turn-on of the optical potential at the final ramp time t{sub r}, where S(t{sub r})=1, given that S(0)=0. Detailed numerical calculations confirm that a simple unit cell model is an excellent approximation when the turn-on time t{sub r} is long compared with the inverse of the band excitation frequency and short in comparison with nonlinear time ({Dirac_h}/2{pi})/{mu} where {mu} is the chemical potential of the condensate. We demonstrate using the Gross-Pitaevskii equation with an optimal turn-on function S(t) that the ground state of the optical lattice can be loaded with no significant excitation even for times t{sub r} on the order of the inverse band excitation frequency.We have calculated the electronic stopping power and the energy-loss straggling parameter of swift He, Li, B, and N ions moving through several oxides, namely SiO{sub 2}, Al{sub 2}O{sub 3}, and ZrO{sub 2}. The evaluation of these stopping magnitudes was done in the framework of the dielectric formalism. The target properties are described by means of a combination of Mermin-type energy-loss functions that characterize the response of valence-band electrons, together with generalized oscillator strengths to take into account the ionization of inner-shell electrons. We have considered the different charge states that the projectile can have, as a result of electron capture and loss processes, during its motion through the target. The electron density for each charge state was described using the Brandt-Kitagawa statistical model and, for He and Li ions, also hydrogenic orbitals. This procedure provides a realistic representation of both the excitation properties of the target electrons and the projectile charge density, yielding stopping powers that compare reasonably well with available experimental data above a few tens of keV/amu.We analyze how a maximally entangled state of two qubits (e.g., the singlet {psi}{sub s}) is affected by the action of local channels described by completely positive maps E. We analyze the concurrence and the purity of states {rho}{sub E}=ExI[{psi}{sub s}]. Using the concurrence-versus-purity phase diagram we characterize local channels E by their action on the singlet state {psi}{sub s}. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.Resonant formation of the muonic molecule dt{mu} in t{mu} atom collision with condensed H-D-T targets is considered. A specific resonance correlation function, which is a generalization of the Van Hove single-particle correlation function, is introduced to calculate the resonant-formation rate in such targets. This function is derived in the case of a polycrystalline harmonic solid. Also, a general asymptotic form of the resonance correlation function for high momentum transfers is found, which is valid for any solid or dense-fluid hydrogen-isotope target. Numerical calculations of the rates are performed for solid hydrogen isotopes at zero pressure, using the isotropic Debye model of a solid. It is shown that condensed-matter effects in resonant formation are strong, which explains some unexpected experimental results. In particular, the resonance profiles are affected by large zero-point vibrations of the hydrogen-isotope molecules bound in the considered crystals, even for high ({approx}1 eV) collision energies. This is important for explaining the time-of-flight measurements of the dt{mu}-formation rate, carried out at TRIUMF. The calculated mean values of the formation rate in solid D-T targets, for fixed target temperatures and steady-state conditions, are in good agreement with the PSI and RIKEN-RAL experiments.To find the criterion of a set of positive-definite matrices which can be written as reduced density matrices of a multipartite matrix is a hard and important problem. When the problem is concerned with multiparty density matrices, it is much more significant for computational many-body physics and many-body quantum entanglement which is one of the focuses of current quantum information theory. We give several results on the necessary compatibility relations between a set of two-party reduced density matrices and a global state in Hilbert space N{sub A}xN{sub B}xN{sub C} where N{sub A}, N{sub B}, and N{sub C} are arbitrary.Radiative lifetimes are measured for metastable levels in the iron charge states Fe{sup 9+},Fe{sup 10+}, and Fe{sup 13+}. The ions are generated in a 14 GHz electron cyclotron resonance ion source and trapped in a Kingdon ion trap. The Fe levels and their measured lifetimes are (a) 73.0{+-}0.8 ms for the 3s{sup 2}3p{sup 4}({sup 3}P)3d {sup 4}F{sub 7/2} level in Fe{sup 9+} (b) 9.91{+-}0.5 ms for the 3s{sup 2}3p{sup 4} {sup 1}D{sub 2} level in Fe{sup 10+}, and (c) 17.0{+-}0.2 ms for the 3s{sup 2}3p {sup 2}P{sup o}{sub 3/2} level in Fe{sup 13+}. Comparisons are made to other measured results using a Kingdon trap, an ion storage, and an electron-beam ion trap (EBIT)Quantum mechanics provides us with probability densities-wave functions modulus squared. Such a probability density is experimentally recovered as an average over many repeated measurements performed on a system in a given wave function. Sometimes it is important to be able to theoretically predict not just the average but also a possible outcome of a single measurement. It is very difficult to make exact predictions of this kind in the case of many-body systems due to correlations in the corresponding many-body wave functions. Here I propose an approximate way of simulating the outcomes of a single-experiment density measurement that is performed on variety of states of N bosons. The approximation is accurate if occupation of single-particle modes is macroscopic.Recently Barrett and Kok proposed an elegant method for entangling separated matter qubits. They outlined a strategy for using their entangling operation (EO) to build graph states, the resource for one-way quantum computing. Here I argue that their strategy considerably underrates the power and utility of their EO. By viewing their EO as a graph fusion event, one perceives that each successful event introduces an ideal redundant graph edge, which growth strategies should exploit. For example, if each EO succeeds with probability p > or approx. 0.4 then a highly connected graph can be formed with an overhead of only about ten EO attempts per graph edge. The Barrett and Kok (BK) scheme then becomes competitive with the more elaborate entanglement procedures designed to permit p to approach unity [Phys. Rev. Lett. 95, 030505 (2005)].Recent experimental data for fully differential cross sections have been compared to various continuum-distorted-wave eikonal-initial-state models without much success, despite good agreement with double-differential cross sections. A four-body model is formulated here and results are presented both when the internuclear potential is omitted and when it is included. They are compared with recent experimental data for fully differential cross sections for 3.6 MeV/u Au{sup Z{sub P}}{sup +}+He collisions, Z{sub P}=24,53.Relative cross sections for the 4 MeV H{sup +}+D{sub 2} ({sup 1}{sigma}{sub g}{sup +}){yields}H{sup +}+D{sub 2}{sup +}(1s{sigma})+e{sup -} ionization process were measured as a function of the molecular alignment during the interaction. The angle between the molecular axis and the projectile was obtained by using a momentum imagining technique and isolating the events in which the D{sub 2}{sup +}(1s{sigma}) ions are excited to the vibrational continuum and subsequently dissociate. While anisotropic cross sections have been observed in the past for a number of collision processes involving both target electrons, the one electron process investigated here is isotropic within our experimental uncertainties.We propose a method to probe Landau and Beliaev processes in dilute trapped atomic condensates with a multiple-state structure using electromagnetically induced transparency configurations. Under certain conditions, damping rates from these collisional processes are directly proportional to the dephasing rates, making it possible to determine damping rates through measurement of the dephasing. In the systems we consider, Landau decay rates are enhanced at low momenta, which allows one to distinguish between Landau-dominated and Beliaev-dominated regimes at the same temperature. Furthermore, the enhancement of Landau rates potentially provides a way to measure low temperatures (T<<T{sub c}) in dilute condensates more accurately than current methods permit.We report here the measurements of the complete valence shell binding energy spectra and the valenceorbital momentum profiles of butanone using the binary e,2e electron momentum spectroscopy. The impact energy was 1200 eV plus the binding energy and the symmetric noncoplanar kinematics was used. The experimental momentum profiles of the valence orbitals are compared with the theoretical momentum distributions calculated using Hartree-Fock and density functional theory methods with various basis sets. The experimental measurements are generally described by theoretical calculations except for summed 4a ,1 5a, 3a, and 14a orbital and summed 8a ,7 a, and 6a orbitals.We have performed calculations of two successive charge transfers from Rydberg states in a strong magnetic field. In the first charge transfer, a positron interacts with a highly excited atom to form positronium. In the second stage, the positronium interacts with an antiproton to give antihydrogen. For many parameters, our results are in qualitative agreement with previous calculations with no magnetic field. However, we do find that there are important changes which may affect the usefulness of the method for efficient formation of antihydrogen that can be trapped.We have investigated the dependence of two electron processes leading to dissociation on the orientation of the H{sub 2} molecule, by measuring differential cross sections for proton fragment emission in coincidence with the outgoing projectile charge state. Proton energy spectra (4-15 eV) emerging at angles 10 deg. and 90 deg. were obtained for He{sup +} and He{sup 0} charge states from He{sup 2+}+H{sub 2} collisions at E{sub P}=25 and 100 keV/amu (v{sub P}=1 and 2 a.u.). By means of the Franck-Condon approximation we found the contribution to the proton emission from the 2p{pi}{sub u}, 2s{sigma}{sub g}, 2p{sigma}{sub u}, and Coulomb explosion dissociation channels, allowing us to obtain cross sections for double capture, transfer ionization, and transfer excitation processes. Cross sections for double ionization and ionization plus excitation were also obtained from the measured data. The results were discussed on the basis of a two-step model within the independent electron approximation, using a perturbative approach for the single capture process.A general multistep linear state symmetrization device for photonic qubits is presented together with the experimental realizations of the 1→3 and 2→3 universal optimal quantum cloning machines and of a 3-qubit purification procedure. Since the present method exploits the bosonic nature of the photons, it can be applied to any particle obeying to the Bose statistics. On a technological perspective, the present protocol is expected to find relevant applications as a multiqubit symmetrization device to be used in modern quantum-information networks.Photodissociation of D{sub 2}{sup +} molecular ions in a beam from an ion source has been studied with 785 nm intense femtosecond laser pulses. Using a high-resolution velocity imaging technique, the neutral fragments from single vibrational levels of the D{sub 2}{sup +} molecules have been resolved. The effects of one- and net two-photon bond softening: level shifting, vibrational trapping, and molecular alignment are observed in the kinetic energy and angular distributions and are discussed in detail. In comparison with our previous study on H{sub 2}{sup +}, we observe smaller widths of the peaks in the kinetic energy distributions of the D{sub 2}{sup +} fragments from single vibrational levels. We attribute this to the longer lifetimes of D{sub 2}{sup +} vibrational states in the light-induced potentials. The width of the angular distributions increases for lowest fragment energies, which suggests vibrational trapping of the levels close to the three-photon crossing.Charge-transfer processes in collisions of H{sup +} ions with C{sub 2}H{sub 6} molecules are investigated theoretically below 10-keV collision energies within a molecular representation. Converged total as well as differential cross sections are obtained in this energy range within a discrete basis of electronic states computed by ab inito methods. The present collision system suggests that the combination of the Demkov-type and Landau-Zener-type mechanisms primarily governs the scattering dynamics for the flux exit from the initial channel. The present charge-transfer cross sections determined are found to agree very well with all available experimental data below a few keV, but begin to deviate above 3 keV, in which the present results slowly decrease, while measurements stay nearly constant. From the study of the electronic state calculation, we provide some information on fragmented species, which should help shed some light on the fragmentation mechanism and process of C{sub 2}H{sub 6}{sup +} ions produced after charge transfer. In addition, the vibrational effect of the initial target to charge transfer is examined.The s-wave elastic phase shifts and cross sections for the H-Li system are predicted using an ab initio and nonadiabatic quantal method at very low energies. The smooth and monotonic change of the phase shifts with added eigenstates indicates that the leptonic potential for the present system has no barrier as noticed in the case of the H-He system, or even a hump which is present in H-H interaction. The very high value of the s-wave elastic cross section at zero or near zero energy implies that Li will be a more efficient buffer gas in cooling the antiatom to ultralow temperatures. This study may stimulate new experiments in thermalizing H atom.We propose an easy implementable prepare-and-measure protocol for robust quantum key distribution with photon polarization. The protocol is fault tolerant against collective random unitary channel noise. The protocol does not need any collective quantum measurement or quantum memory. A security proof and a specific linear optical realization using spontaneous parametric down conversion are given.Here, I discuss the propagation of an ultrashort pulse through a collection of harmonic multilevel systems. In the limit of weak excitation and a large number of excited states, I show that the amplitude of the input driving pulse decays exponentially with propagation distance. The absorption coefficient associated with this decay is determined by the characteristic time of the manifold of excited states, instead of the polarization decay time as in the conventional absorption coefficient of a two-level atom. The input ultrashort pulse creates in the excited states a wave packet, which oscillates emitting secondary pulses in the process. Analytic solutions are obtained that describe the propagation of individual wave-packet re-emission pulses, and it is shown that their phase depends on the detuning of the input pulse.High-order harmonic generation (HHG) from a single hydrogen atom is studied analytically and numerically in the regime of small Keldysh parameter. The HHG spectra from different Coulomb-like model potentials, such as soft-core and/or one-dimensional (1D) potentials are compared to the three-dimensional (3D) Coulomb potential. It is shown, using analytic arguments, that the famous plateau in the HHG spectrum owes its existence to the Coulomb singularity, whereas soft-core potentials give spectra that fall off exponentially with increasing frequency. The idea is demonstrated numerically on a 3D soft-core potential that has the same long-range asymptotic behavior and ground-state energy as hydrogen. In addition, a number of widely used 1D Coulomb-like potentials are discussed. It is shown that in order that a 1D potential be a reasonable substitute for the 3D Coulomb potential, it must have a cusp singularity. A specific potential satisfying this criterion is proposed.Electron capture and loss cross sections have been measured for fast light projectile ions of 0.5 MeV H{sup 0,1+} and 0.5-2.0 MeV {sup 4}He{sup 0,1+,2+} in collisions with C{sub 60}. The gaseous target of C{sub 60} was prepared by heating C{sub 60} powder in a target cell to temperatures of 300-500 deg. C, and outgoing charge fractions were measured as a function of the cell temperature. Absolute cross sections are deduced by using two different vapor-pressure data available in literature. Experimental cross sections are examined in comparison with theoretical values obtained from various conventional formulas proposed for atomic targets. In addition, single- and double-electron capture cross sections are also calculated on the basis of a classical model by taking account of the local electron density of C{sub 60}. From a complete set of our experimental cross sections, equilibrium charge fractions are also deduced and found to be essentially the same as carbon-foil data, indicating no gas-solid difference.

Criteria for negative refraction in active and passive media

In 1968, Veselago[2] investigated lossless media with simultaneously negative permittivity and permeability; these are commonly referred to as negative refractive index (NRI) media. In NRI media, the phase velocity of electromagnetic-wave propagation is anti-parallel to the energy flow, and the phrase Negative Phase Velocity (NPV) propagation is sometimes used as an alternative to negative refraction (or NRI) [3]. Note also that some alternative definitions of NRI rely on the opposition of phase and group velocities; hence the use in [4] of the acronyms NPVE (NPV w.r.t. energy velocity) and NPVG (NPV w.r.t. group velocity); here we use just NPV, meaning specifically NPVE. Such materials have even been experimentally realized [5]. As might be expected in a rapidly evolving field of research, a variety of conditions [3, 6, 7] for negative refraction have been proposed in the literature. As summarized in [6], it was also shown that all the different conditions are equivalent in passive media, despite their rather different mathematical forms. In this paper we show that the different conditions are not equivalent when they are extended to active media. Active media that support NPV propagation must (of course) be dispersive, and, on account of the KramersKronig relations, dispersion implies loss. Therefore, by embracing active media, the limitations imposed by loss might be overcome; although Stockman [8] has claimed that there are some important restrictions – albeit restrictions valid only in the case of perfect transparency at the observation frequency. In isotropic media where magneto-electric effects are absent, gain arises from the response to the electric or magnetic fields: i.e. from the permittivity, ǫ = ǫr + ıǫi, or the permeability, μ = μr + ıμi, respectively. Situations where both, or either of these occur must be distinguished. In this paper, if both ǫ and μ are active, we denote that a doubly active medium. Next, if neither ǫ

Four Poynting theorems

The Poynting vector is an invaluable tool for analysing electromagnetic problems. However, even a rigorous stress–energy tensor approach can still leave us with the question: is it best defined as E × H or as D × B? Typical electromagnetic treatments provide yet another perspective: they regard E × B as the appropriate definition, because E and B are taken to be the fundamental electromagnetic fields. The astute reader will even notice the fourth possible combination of fields, i.e. D × H. Faced with this diverse selection, we have decided to treat each possible flux vector on its merits, deriving its associated energy continuity equation but applying minimal restrictions to the allowed host media. We then discuss each form, and how it represents the response of the medium. Finally, we derive a propagation equation for each flux vector using a directional fields approach, a useful result which enables further interpretation of each flux and its interaction with the medium.

Carrier-wave steepened pulses and gradient-gated high-order harmonic generation

We show how to optimize the process of high-order harmonic generation (HHG) by gating the interaction using the field gradient of the driving pulse. Since maximized field gradients are efficiently generated by self-steepening processes, we first present a generalized theory of optical carrier-wave self-steepened (CSS) pulses. This goes beyond existing treatments, which only consider third-order nonlinearity, and has the advantage of describing pulses whose wave forms have a range of symmetry properties. Although a fertile field for theoretical work, CSS pulses are difficult to realize experimentally because of the deleterious effect of dispersion. We therefore consider synthesizing CSS-like profiles using a suitably phased subset of the harmonics present in a true CSS wave form. Using standard theoretical models of HHG, we show that the presence of gradient-maximized regions on the wave forms can raise the spectral cutoff and so yield shorter attosecond pulses. We study how the quality of the attosecond bursts created by spectral filtering depends on the number of harmonics included in the driving pulse.