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

FWM’s have often been termed as carrier-free wave fields (see Refs. [47, 49] for example).In lights of (4.12) we can conclude that the carrier-free behaviour of the FWM’sis indeed caused by the integral transform table, not by physical arguments.In the numerical example in Fig. 4.14 we have optimized the parameters of the wavefield as to match the spectral band specified in the introduction of this section. Again, theparameter β is determined by the bandwidth and the cone angle of the central frequencyas described above. Thus we got: σ = 40000 radm , a 1 =5× 10 −6 m, β =40 radm , γ =1.The evaluation of the Eq. (4.12) yields k 0z =9.80004 × 10 6 radmand this result is in goodcorrespondence with the numerical simulations.In conclusion, the Bessel-Gauss pulses are obviously much more appropriate for modelingrealistic experimental situations.4.4 X-type wave fieldsThe X-type localized wave fields are characterized by that for their angular spectrumof plane waves β =0in Eq. (3.6) [or ξ =0in Eq. (3.9)]. This choice implies, thattheir support of angular spectrum of plane waves is a cone in k-space (see Fig. 3.2).Consequently, the phase and group velocity of X-type pulses are equal (both necessarilysuperluminal) and the field propagates without any local changes along the optical axis.4.4.1 Bessel beamsThe Bessel beams [52–92] are the simplest special case of the propagation-invariant wavefields. Being the exact solutions to the Helmholtz equation in cylindrical coordinates theirwave amplitude reads asΨ B (ρ, z, t) = X nc n J n (kρ sin θ 0 )exp[inφ]exp[ik (z cos θ 0 − ct)] (4.13)so that for the zeroth order Bessel beam we haveΨ B (ρ, z, t) =J 0 (kρ sin θ 0 )exp[ik (z cos θ 0 − ct)] . (4.14)In the Fourier picture, the zeroth–order Bessel beam is the cylindrically symmetric superpositionof the monochromatic plane waves propagating at angles θ 0 relative to z axis,correspondingly, their angular spectrum of plane waves in Eq. (2.18) readsA (B)0 (k, θ) ∼ δ (k − k 0 ) δ (θ − θ 0 ) . (4.15)The bidirectional representation of the Bessel beam can be found in Ref. [14].The properties of Bessel beams have been discussed in many publications both in termsof angular spectrum of plane waves [52–64] and diffraction theory [65–92] and their propertiesare very well understood today. The interest has been triggered in Refs. [52, 53]where Durnin et al presented them as "nondiffracting" solutions of the homogeneousscalar wave equation – they demonstrated experimentally, that the central maximum of51