Optical implementation of propagation-invariant pulsed free ... - Tartu

distribution of the FWM’s can be expressed as [see Eqs. (2.15) and (3.12)]Ψ F (ρ, z, t) ==Z π0Z π0dφ [T (x, y, z, t; φ)+T (x, y, z, t; φ + π)] (3.59)dφF 0 (x, y, z, t; φ) ,where T (x, y, z, t; φ) is the spatial amplitude distribution of the tilted plane wave pulses,that in the spectral representation are given byT (x, y, z, t; φ) =Z ∞0dk à (k, θ F (k) ,φ) (3.60)× exp [ik (x cos φ sin θ F (k)+y sin φ sin θ F (k)+z cos θ F (k) − ct)] ,where à (k, θ F (k) ,φ) is the angular spectrum of plane waves of the wave field and theangular function θ F (k) is defined by Eq. (3.6). From Eqs. (3.59) and (3.60) we getF 0 (x, y, z, t; φ) =2Z ∞0dk à (k, θ F (k) ,φ) (3.61)× cos [k sin θ F (k)(x cos φ + y sin φ)] exp [ik (z cos θ F (k) − ct)] .An example of the spatial amplitude distribution of a tilted pulse with Gaussian frequencyspectrum corresponding to approximately τ s ∼ 4fs in Eq. (3.60) is depicted in Fig. 3.7a,the spatial amplitude distributions of the corresponding superposition of two tilted pulsesin Eq. (3.61) and FWM in Eq. (3.14) are depicted in Fig. 3.7b).In this representation the properties of FWM’s can be given the following interpretation:1. The localized central peak of FWM’s is simply the well-known consequence of takingthe axially symmetric superposition of a harmonic function. Indeed, the interferenceof the two transform-limited tilted pulses in Eq. (3.61) gives rise to the harmonic interferencepattern, the transversal width of which is proportional to the temporal lengthof the tilted pulses (3.60). The central peak arises due to the constructive interferenceof the tilted pulses along the optical axis, formally, the cos () function in Eq. (3.61) isreplaced by J 0 () in Eq. (3.14) [see Fig. 3.7b];2. The nondispersing propagation of the optical FWM’s wave fields can be given analternate wave-optical interpretation. Namely, it can be seen from Fig. 3.8, that inlarge scale the longitudinal length of the tilted pulses depends on the distance fromthe optical axis so that the tilted pulses have a ”waist” (this claim is identical to thatgiven in section 3.1.4 that the radial wave propagating toward the z axis and backis not propagation-invariant). The relation (3.6) essentially guarantees, that the waistpropagates along the optical axis and do not spread – in this case the central peak of thecorresponding cylindrically symmetric superpositions, FWM’s (3.14), also remainstransform-limited;3. The local variations of the central peak of the wave field, noted for example in Ref. [6],can be explained as the result of the difference between the phase and group velocitiesalong the optical axis – as can be seen from Fig. 3.7c the pulse and phase fronts of the35