High-resolution Interferometric Diagnostics for Ultrashort Pulses

B.4 Propagation modelamount to prevent the algorithm becoming “stuck”. If the solutions are valid, then the step is takenand the step-size is increased slightly. The maximum permitted distance for a valid step is half thedistance between the solutions, or some user-determined distance — whichever is the smaller.When scanning over a range of laser intensities, I use a similar adaptive method to control theintensity step-size ∆I .B.4 Propagation modelI wrote a propagation code to simulate the spatial profiles observed in chapter 7. The model was anapproximate one in which the only variations considered in the laser pulse are those of intensityand arrival time. Since the single-atom response E S (I ,q) is only affected trivially by the arrivaltime, it only needs to be calculated over the two-dimensional parameter range (I ,q), where q is theharmonic order. The laser beam is defined at its centre frequency, and can be assigned any spatialintensity profile I L (r,z ) and phase profile φ L (r,z ). Cylindrical symmetry is assumed throughout.The phase of the laser profile is treated as a time delay φ L (r,z )/ω L on the pulse, resulting in aphase qφ L (r,z ) in the q-th harmonic. This approximation is equivalent to ignoring variations inthe carrier-envelope phase of the pulse, and is valid for many-cycle pulses.The model includes the refractive index contributions of the gas at both the laser and harmonicwavelengths, and the absorption of the harmonics. The gas density is low enough that itsabsorption and refractive index contribution are proportional to the local pressure g (z ). The harmonicstherefore experience complex-valued wavenumber β q g (z ), where β q =(n q −1+i α q /2)qk Lwhich includes both the refractive index n q and absorption α q . It is convenient to define these valuesat one atmosphere, in which case the units of g (z ) are atmospheres. Likewise, the laser haswavenumber β L g (z ) where β L =(n L − 1)k L . Putting this all together, the Hankel transform of themacroscopic response E M (k T , L,ω) is written (neglecting constant factors) asE M (k T , L,ω)= ∞−∞e −i k 2 T2qk L(L−z )+i[ḡ (∞)−ḡ (z )]β qHT E SĨL (r,z ),q e iq[β Lḡ (z )+φ L (r,z )] g (z ) dz (B.1)where ḡ is the “effective cumulative pressure-thickness” of the gas ḡ (z )= z−∞ g (z ) dz .227