Three alternative forms of harmonic spectra, based on the dipole moment, dipole velocity, and dipole acceleration, are compared by a numerical solution of the Schrödinger equation for a hydrogen atom interacting with a linearly polarized laser pulse, whose electric field is given by E͑t͒ = E 0 f͑t͒cos͑ 0 t + ͒ with Gaussian carrier envelope f͑t͒ = exp͑−t 2 / ␦ 2 ͒. The carrier frequency 0 is fixed to correspond to a wavelength of 800 nm. Spectra for a selection of pulses, for which the intensity I 0 = c 0 E 0 2 , duration T ϰ ␦, and carrier-envelope phase are systematically varied, show that, depending on , all three forms are in good agreement for "weak" pulses with I 0 Ͻ I b , the over-barrier ionization threshold, but that marked differences among the three appear as the pulse becomes shorter and stronger ͑I 0 Ͼ I b ͒. Except for scalings by powers of the harmonic frequency, the three forms differ from one another only by "limit contributions" proportional to the expectation values of the dipole moment ͗z͑t f ͒͘ or dipole velocity ͗ż͑t f ͒͘ at the end ͑t f ͒ of the pulse. For long, weak pulses the limit contributions are negligible, whereas for short, strong ones they are not. In the short, strong limit, where ͗ż͑t f ͒͘ 0 and therefore ͗z͑t͒͘ may increase without bound ͑i.e., the atom may ionize͒, depending on , an "infinite-time" spectrum based on the acceleration form provides a convenient computational pathway to the corresponding infinite-time dipole-velocity spectrum, which is related directly to the experimentally measured "harmonic photon number spectrum" ͑HPNS͒. For short, intense pulses the HPNS is quite sensitive to and exhibits not only the usual odd harmonics but also even ones. The analysis also reveals that most of the harmonic photons are emitted during the passage of the pulse. Because of the divergence of ͗z͑t͒͘ the dipolemoment form does not provide a numerically reliable route to the harmonic spectrum for very short ͑few-cycle͒, very intense laser pulses.

We present a method for producing a single circularly polarized attosecond pulse by an intense few cycle elliptically polarized laser pulse combined with a terahertz field from numerical solutions of the time-dependent Schrödinger equation for the molecular ion H2(+). It is found that in the presence of a 62.5 THz (λ=4800  nm) field at an intensity of ∼10(14)  W/cm2, a single circularly polarized 114 as pulse can be generated by an elliptical polarized laser pulse at a wavelength of 400 nm with an ellipticity of ϵ=0.59. The efficiency of circular polarization attosecond pulse generation is interpreted based on a classical model of single electron recollision with the parent ion.

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Numerical solutions of the time-dependent Schrödinger equation, using a new unitary numerical method, are used to obtain gauge-independent photoionization angular distributions for two-dimensional (planar) models of aligned H 2 and H 2 + molecules in different electronic states. Simulations are performed with few-cycle attosecond (as) extreme-ultraviolet (xuv) laser pulses at wavelengths λ = 10 nm [ω = 4.56 a.u. (atomic units), τ = 2π/ω = 1.38 a.u. = 33.4 as], and λ = 5 nm (ω = 9.11 a.u., τ = 16.7 as) for linear and circular polarization ionizations at R e = 1.675 a.u., where molecular orbital configurations dominate for which the electron wavelengths λ e R e and at large internuclear distance R = 10 a.u. (λ e < R) where Heitler-London atomic configurations including ionic states H + H − are more appropriate. The simulations allow one to investigate the effects of electron correlation and entanglement in H 2 at different R due to electron spin exchange as compared to the delocalized one-electron H 2 + system. An ultrashort (delta function) pulse model is used to interpret interferences in momentum distributions.