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

x(mm)80z40µmFig. 3.8The large-scale behaviour of the spatial shape of the modulus of the tilted pulses.tilted pulses are not parallel;4. The group velocity of the wave field can be set by changing the parameter γ inEq. (3.6). The Fig. 3.7c gives this effect a wave optical interpretation – it can beseen, that the on-axis group velocity of the wave field directly depends on the anglebetween the phase front and pulse front and on the direction of the wave vector of themean frequency.It is easy to see, that all the presented arguments are equally valid for the superpositionsof tilted pulses in Eq. (3.61) and for its cylindrically symmetric counterparts – FWM’s.Thus, we can state that the defined interfering pair of tilted pulses possess all the characteristicproperties of FWM’s. In fact, the physics behind the two wave fields is similar tothe degree, that we will call the wave field (3.61)F (x, z, t) =Z ∞0dk ˜B 0 (k)cos[kx sin θ F (k)] exp [ik (z cos θ F (k) − ct)] (3.62)as two-dimensional FWM (2D FWM) in what follows.We end this section by noting that the special case of this approach can be used todiscuss the properties of X-type pulses (Ref. I). In this case θ F (k) =const = θ 0 and wehave the interference of two plane wave pulses:so thatT (x, y, z, t; φ) =(see Fig. 3.9)F (x, z, t) =Z ∞0Z ∞0dk à (k, θ F (k) ,φ) (3.63)× exp [ik (x cos φ sin θ 0 y sin φ sin θ 0 + z cos θ 0 − ct)] ,dk ˜B 0 (k)cos[kx sin θ 0 ]exp[ik (z cos θ 0 − ct)] . (3.64)37