High-resolution Interferometric Diagnostics for Ultrashort Pulses

6. HIGH-HARMONIC GENERATION• Saddle points of the birth time integral yield∂ S ω (t r ,t b ,p)= v 2 (t b )+ I p = 0 (6.17)∂ t b 2which under a classical interpretation suggests that electrons are born with negative kineticenergy. Clearly, (6.17) has no real solutions. However, by analytically continuing the domainof all the functions to the entire complex plane, complex-valued solutions can be found. Theclassically nonsensical state of possessing negative kinetic energy is interpreted as tunnelingof the evanescent quantum wavefunction under a classically disallowed barrier. The imaginarypart of the birth time is interpreted as a tunneling time [368, 370]. The assumption ofzero kinetic energy at birth in the classical picture is “as close as possible” to this intrinsicallynonclassical process.• Saddle points of the recombination time integral give∂ S ω (t r ,t b ,p)= v 2 (t r )+ I p − ω = 0 (6.18)∂ t r 2which shows that the photon energy equals the total energy relinquished by the electronupon recombination.The equivalence of the physical interpretations of the stationary-point conditions and the assumptionsof the classical model lends a comforting mutual consistency between these two approximatepictures of high-harmonic generation.In general, for a given ω, the stationary-point equations will have multiple solutions, which Ilabel with superscripts so that the solution j is {t (j )b(j )(ω),t r (ω),p (j ) (ω)}. For notational simplicityI define S (j )ω as the corresponding action, and (j ) (j )(ω), rec(ω) the corresponding ionization andionrecombination amplitudes. The stationary points may be systematically classified according totheir birth and return times. I shall use the superscript label j = αβm, similar to references [373–375]. In this labelling system, α = S or L denotes whether the orbit is a short or a long trajectory.Next, β = 1,2... indicates the number of returns the electron has made to the core, and hence140