High-resolution Interferometric Diagnostics for Ultrashort Pulses

B. SIMULATION CODES FOR HIGH-HARMONIC GENERATIONB.2 Classical recollision solverThe classical recollision solver finds all solutions (t r ,t b ) to the recollision equation i.e. the rootsof (6.1). Its inputs are the vector potential A(t ) and a set of initial guesses of (t r ,t b ) pairs, onefor each half-cycle in which an electron “birth” is required. As guesses, I use the cutoff birth andreturn times for a sinusoidal field — this is close enough even for few-cycle pulses. The algorithmmoves through a range of return times in regular steps of size ∆t r . At each step, the birth time iscomputed using the MATLABfunction with the previously calculated birth time as a guess.Starting from the initial guess, close to the cutoff, the algorithm steps backwards in return time,moving through the short trajectories. It stops when the computed birth time equals the returntime. It then returns to the initial guess but steps forward in return time, passing through thelong trajectories and then on to the multiple-return trajectories. It stops when a given maximumexcursion time is reached.B.3 Quantum-orbits modelThe quantum orbits model solves for stationary points (t r ,t b ) of the action i.e. (6.17) and (6.18).(The stationary momentum is easily calculated using (6.16) and not therefore not treated as anunknown.) Its inputs are the vector potential A(t ), a set of initial guesses at a given emissionfrequency ω and the ionization potential. Each guess corresponds to one trajectory αβm. Thealgorithm steps through the desired range of emission frequencies, with step size ∆ω. At eachfrequency (t r ,t b ) are found using Newton’s method. A linear extrapolation of the previous twosolutions is used for the initial guess at each frequency.Near the cutoff, the solutions of the long and short trajectories become very close, and theNewton’s method solver tends to jump between them. I overcame this with an adaptive stepsizemethod which takes smaller steps when the solutions become close. When moving from frequencyω to ω + ∆ω, the solutions obtained using Newton’s method at ω + ∆ω are compared tothe solutions at ω. If the solutions have moved by too great a distance, they are considered invalid,∆ω is reduced and the procedure is repeated. The guess is also modified by a small random226