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

3.3.2 FWM’s as the moving, modulated Gaussian beamsIn literature the closed-form expression (3.1) for the original FWM’s have been derivedwith the use of the anzatz [3, 5, 6]Ψ (x, y, z, t) =exp[iβµ] F 0 (x, y, ζ) , (3.65)where µ = z + ct and ζ = z − ct. With (3.65) the wave equation (2.8) reduces to theSchrödinger equation for F 0 (∆ ⊥ +4iβ∂ ζ ) F 0 (x, y, ζ) =0 (3.66)which, assuming axial symmetry, has a solution of the form [6]· ¸F 0 1(ρ, ζ) =4πi (a 1 + iζ) exp −βρ2 , (3.67)a 1 + iζso that one can write the solution similar to the FWM’s in Eq. (3.1)· ¸a 1Ψ f (ρ, µ, ζ) =exp[iβµ]4πi (a 1 + iζ) exp −βρ2 . (3.68)a 1 + iζTo give the FWM a more convenient form one can use the transform1a 1 + iζ = 1βa 2 1 (ζ) − i 1R (ζ)with which the Eq. (3.68) can be shown to yield(3.69)whereΨ f (ρ, z, ζ) = W 0exp [−iβζ] (3.70)4πa 1 (ζ)µ µ ¸× exp·− ρ2 βρ2ζa 2 + i1 (ζ) R (ζ) − i arctan − 2βz ,a 1µ # 12 2ζa 1 (ζ) = W 0"1+a 1"R (ζ) = ζ 1+µ # 2 a1ζ(3.71a)(3.71b)andra1W 0 =β . (3.72)If one compares the Eqs. (3.70) - (3.72) to those of the monochromatic Gaussian beam(see Ref. [161] for example) one can see that, the FWM’s can be interpreted as moving,modulated Gaussian beams for which a 1 (ζ) and R (ζ) are the beam width and radius ofcurvature respectively and W 0 is the beam waist at ζ =0(see Refs. [3, 5, 6] for relevantdescriptions).Now, several interesting consequences can be drawn at this point. Most importantly,this formal analogy between the FWM’s and Gaussian beams is very conditional and even39