High-resolution Interferometric Diagnostics for Ultrashort Pulses

6. HIGH-HARMONIC GENERATIONwhere rec (t ,p)=g ∗ (t )d ∗ (p + A(t )). (6.11)Here, I have ignored dipole transitions between the continuum states because they generally donot contribute significantly to the harmonic emission, although this currently a topic of theoreticalinvestigation [371]. Equation (6.10) may be interpreted in a similar way to (6.7): electrons are ionizedat t b , acquire a phase S(t ,t b ,p) during their excursion in the continuum, and then recombinewith the core at time t with rate rec (t ,p) given by the product of the ground state amplitude andthe dipole transition. All possible combinations of birth time and momenta must be consideredand are therefore integrated over; their contributions add coherently.The phase S(t ,t b ,p) plays a fundamental role in HHG, and it may be interpreted in severalways. One may view it as the accumulated phase difference between the continuum and groundstate wavefunctions during the electron’s sojourn in the continuum. The contribution of the kineticenergy integral is the continuum contribution; it is the integral of the time-dependent eigenvaluecorresponding to the Volkov eigenstate of the field Hamiltonian. Specifically, the solution tothe potential-free TDSE is |p〉e S p(t ) whereS p (t )=− t∞ p + A(tdt ) 2. (6.12)2The contribution to S(t ,t b ,p) proportional to the ionization potential simply arises from the freeoscillation of the ground state during the electron’s motion in the continuum. A second point ofview is to interpret the phase as a semiclassical action [368]. The SFA then folds into the Feynmanpath integral picture of quantum mechanics, in which all possible system evolutions from theunperturbed ground state, through the continuum, and recombining back to the ground state, aresummed together with a phase factor given by the action.The spectrum of the radiation is found by taking the Fourier transform of the dipole expectation:(ω)= −i 2π ∞−∞dt r td 3 p dt b ion (t b ,p)e iS ω(t r ,t b ,p) rec (t r ,p)+c.c. (6.13)−∞138