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

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frequency spectrum of the wave fields extends from ∼ 0.5 × 10 7 rad/m (∼ 1100nm ofwavelength) to ∼ 2 × 10 7 rad/m (∼ 300nm of wavelength), so that σ k ∼ 3.8 × 10 −7 m(3.37) for which the length of the corresponding plane wave pulse is ∼ 3fs– the shortestpossible pulse length available, (ii) the plane wave with central wavelength propagateapproximately at the angle ≈ 0.2 ◦ relative to the propagation axis, giving σ z /σ ρ ≈ 0.003for the approximate ratio of pulse widths in xy and z direction. Note, that specifyingthe frequency range and cone angle of the Bessel beam of central wavelength completelydetermines the parameter β –forγ =1we have β ≈ 40rad/m (clarify section 3.1.1).4.2 The original FWM’sWith Eqs. (2.24a) and (2.24b) the angular spectrum of plane waves of the original scalarFWM’s in Eq. (3.1)· ¸a 1Ψ f (ρ, z, t) =exp[iβµ]4πi (a 1 + iζ) exp −βρ2(4.1)a 1 + iζcan be derived from its bidirectional plane wave representation [14]C 0³˜α, ˜β,χ´= π ³˜β2 δ − 2β´exp [−˜αa 1 ] , (4.2)givingA 0 (k, θ) = π ·2 exp − a ¸1k (1 + cos θ)δ (k − k cos θ − 2β) (4.3)2(see Fig. 4.13a). Inserting the angular spectrum (4.3) into integral representation of thetype (2.17) yieldsZ ∞·Ψ f (ρ, z, t) = dk kexp − a ¸1k (1 + cos θ F (k))02×J 0 [kρ sin θ F (k)] exp [ik (z cos θ F (k) − ct)] (4.4)(as compared to (2.17) here we have taken into account the 1/k term that appears in(2.12) as to be consistent with [18] for example) so that the frequency spectrum of thesuperposition can be written as·˜B (k) =k exp − a ¸1k (1 + cos θ F (k))(4.5)2[the significance of the factor k sin θ F (k) will be discussed in following sections].The frequency spectrum ˜B (k) in Eq. (4.5) has two free parameters, a 1 and β, the latterhaving the same definition as in Eq. (3.2) of section 3.1.1. As we already noted in theintroduction of this chapter, the choice γ =1together with the frequency range and coneangle of the Bessel beam of central wavelength determines β =40rad/m. The single freeparameter is a 1 and a single parameter does not allow to approximate for any realistic lightsources – Fig. 4.13b shows a typical spectrum that can be modeled in terms of Eq. (4.5)as compared to the optically feasible frequency spectrum specified in the introduction ofthis overview and it can be seen, that the bandwidth of the wave field is far beyond the46
4k x(10 rad/m)12 3 4 57k z(10 rad/m)8arb. units~B(k)(a)θF(k)deg1arb. units0.512 3 4 5(b)7k (10 rad/m)Re(Ψ f )5×10 -3 m(c)xz4×10 -5 mFig. 4.13 A numerical example of a FWM with the parameters γ = 1, β = 40 radm ,a 1 =1.4 × 10 −7 m: (a) The angular spectrum of plane waves in two perspectives; (b) The frequencyspectrum of the FWM (black line), the frequency spectrum of an optically feasible wavefield (green line), the angle θ F (k) as the function of the wave number (dashed blue line); (c) Thespatial amplitude of the FWM.47
Page 1 and 2: Optical implementation ofpropagatio
Page 3 and 4: CONTENTLIST OF PUBLICATIONS INCLUDE
Page 7: NOTATIONρ, z, ϕ - cylindrical coo
Page 10 and 11: where the functions Ψ q (q =1, 2..
Page 12 and 13: [41] , splash pulses [6], modified
Page 14 and 15: distribution of the field of optica
Page 16 and 17: equation we get from ∇ · E =0tha
Page 18 and 19: adially symmetric case only the ter
Page 20 and 21: 3 A PRACTICAL APPROACH TO SCALARFWM
Page 22 and 23: orξ (γ ± 1)k F (θ) = (3.10)γ
Page 24 and 25: (a)0χk z χk z k zk xkk 0kk zgγ<
Page 26 and 27: chromatic Bessel beam can be repres
Page 28 and 29: 3.1.5 The spatial localization of F
Page 30 and 31: of angular spectrum of plane waves
Page 32 and 33: θ ΙD + D -π/2θ RFig. 3.6 The in
Page 34 and 35: In the Fourier picture, the Parseva
Page 36 and 37: xv g tθ F(k 0)ϑphasefrontszλ 0v
Page 38 and 39: xv p t =v g tλ 0zθ 0v p t 1ctk 0x
Page 40 and 41: Re(Ψ F)arb. unitsz5×10 -3 mx4×10
Page 42 and 43: monochromatic Gaussian beams the te
Page 44 and 45: 6k x(10 rad/m)V = 0.999c3V = 0.9c V
Page 48 and 49: each of any realistic light source.
Page 50 and 51: Eq. (3.9)]. The change of variables
Page 52 and 53: 4k x(10 rad/m)12 3 4 57k z(10 rad/m
Page 54 and 55: 84k x(10 rad/m)12 3 4 57k z(10 rad/
Page 56 and 57: 84k x(10 rad/m)12 3 4 57k z(10 rad/
Page 58 and 59: time84k x(10 rad/m)12 3 4 57k z(10
Page 60 and 61: 4k x(10 rad/m)12 3 4 57k z(10 rad/m
Page 62 and 63: k xk zΛ(β)k xk z(a)k xk zk xΛ(β
Page 64 and 65: 8arb. units4k x(10 rad/m)12 3 4 57k
Page 66 and 67: 4k x(10rad/m)12 3 4 57k z(10 rad/m)
Page 68 and 69: Ziolkowski [31] where he used the H
Page 70: tions of different order. (see Refs

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

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