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

Optical implementation ofpropagation-invariant pulsed free-spacewave fieldsKaido ReiveltTartu, April 4, 2003

The study was carried out at the Institute of Experimental Physics and Technology andthe Institute of Physics, University of Tartu.The Dissertation was admitted on January 16, 2003, in partial fulfilment of the requirementsfor the degree of Doctor of Philosophy in physics (optics and spectroscopy), andallowed for defence by the Council of the department of Physics, University of Tartu.Supervisor:Opponents:Prof. Peeter Saari, University of Tartu, Tartu, EstoniaProf. Amr Mohmaed Ali Shaarawi, The American University in Cairo,Cairo, EgyptProf. Aleksei P. Kiselev, Steklov Mathematical Institute, St.Petersburg,RussiaDefence:March 24, 2003 at the University of Tartu, Tartu, Estoniac°Kaido Reivelt, 2003Tartu Ülikooli KirjastusTiigi 78, Tartu 50410Tellimus nr. 572

CONTENTLIST OF PUBLICATIONS INCLUDED IN THE THESIS 5NOTATION 71. INTRODUCTION 92. INTEGRAL REPRESENTATIONS OF FREE-SPACE ELECTROMAG-NETIC WAVE FIELDS 152.1 Solutions of the Maxwell equations in free space 152.2 Plane wave expansions of scalar wave fields 162.3 Bidirectional plane wave decomposition 183. A PRACTICAL APPROACH TO SCALARFWM’S 203.1 Propagation invariance of scalar wave fields 203.2 Few remarks on properties of FWM’s 313.3 Alternate derivations of scalar FWM’s 344. AN OUTLINE OF SCALAR LOCALIZED WAVES STUDIED INLITERATURE SO FAR 454.1 Introduction 454.2 The original FWM’s 464.3 Bessel-Gauss pulses 484.4 X-type wave fields 514.5 Two limiting cases of the propagation-invariance 574.6 Physically realizable approximations to FWM’s 614.7 Several more LW’s 674.8 On the transition to the vector theory 674.9 Conclusions. 705. LOCALIZED WAVES IN THE THEORY OF PARTIALLY COHERENTWAVE FIELDS 715.1 Propagation-invariance in domain of partially coherent fields insecond order coherence theory 715.2 Special cases of partially coherent FWM’s 765.3 Conclusions 816. OPTICAL GENERATION OF LW’S 826.1 Introduction 826.2 Feasible approach to optical generation FWM’s 846.3 Finite energy approximations to FWM’s 916.4 Optical generation of partially coherent LW’s 966.5 Conclusions. Optical generation of general LW’s 986.6 On the physical nature of propagation-invariance of pulsed wave fields 993

7. THE EXPERIMENTS 1037.1 FWM’s in interferometric experiments 1037.2 Experiment on optical Bessel-X pulses 1057.3 Experiment on optical FWM’s 1098. SELF-IMAGING OF PULSED WAVEFIELDS 1198.1 Monochromatic self-imaging 1198.2 Self-imaging of pulsed wave fields 121SUMMARY 127REFERENCES 129SUMMARYINESTONIAN 135ACKNOWLEDGEMENTS 1374

NOTATIONρ, z, ϕ – cylindrical coordinates systemk x ,k y ,k z– the Cartesian components of the wave vector kk 0– carrier wave numberk, θ, φ – spherical coordinates in k-space;χ = k ρ = k sin θγ, γ ρ – see Eq. (3.4), (3.25)β,ξ – see Eqs. (3.5) and (3.8)µ = z + ctζ = z − ctv g , v p– group velocity (3.4) and phase velocitya 1 ,a 2 ,κ,b,p,q – parameters, see Eqs. (4.3), (4.6), (4.43)θ F (k) ≡ θ F (k, β), θ (ρ)F(k) – see Eqs. (3.6), (3.27)θ G (k) – see Eq. (6.10)k F (θ)¡– see Eq. (3.7)˜α = 1 ω2 ¡ c + k z¢˜β = 1 ω2 c − k z¢τ = t 2 − t 1 , ∆z– the time- and z coordinate differenceτ s , σ k , σ z , σ ρ– see Eqs. (3.37), (3.38a), (3.38b)∆t, z ∆ , z m – see Eqs. (7.2), (7.12), (7.11)ψ (k,ω)=F [Ψ (r,t)] – see Eq. (2.9)A (k x ,k y, k z ) – see Eq. (2.13)A n (k, θ) – see Eq. (2.17)A z (k x ,k y ), A xy (k z ) – see Eq. (3.31), (3.35)A wen (k, χ) – see Eqs. (2.21)B n (k), B 0 (k) ≡ B (k) – see Eq. (3.12)à (k x ,k y, k z ), à n (k, θ), ˜B n (k) – see note after Eq. (3.57)C n³˜α, ˜β,χ´– see Eq. (2.22)Ξ (χ, β) – see Eq. (3.83)a(k, θ, φ)– stochastic angular spectrum of plane wavesΨ 0 (x, y, z, t), Ψ (ρ, z, ϕ, t) – see Eqs. (2.10), (2.19)E (r,t), H (r,t),A (r,t), Π (e) , Π (m) – see Sec. 2.1Ψ F (ρ, z, ϕ, t), Ψ f (ρ, z, ϕ, t) – see Eqs. (3.14), (4.1)T (x, y, z, t; φ), F (x, z, t) – see Eqs. (3.60), (3.62)Γ (r 1 , r 2 ,t 1 ,t 2 ) – mutual coherence function, Eq. (5.1)W (r 1 , r 2 ,k)– cross-spectral densityA (k 1 ,k 2 ,θ 1 ,θ 2 ,φ 1 ,φ 2 , )– see Eq. (5.14), angular correlation functionC (k 1 ,k 2 ,φ 1 ,φ 2 ) – see Eq. (5.23)V (k, φ) – see Eq. (5.23)s (k)– see Eq. (3.36), frequency spectrum of light sourceS (k) =|s (k)| 2– spectral density (power spectrum) of light source7

1 INTRODUCTIONThe birth of the diffraction theory of light dates back to the works of Francis MariaGrimaldi (1618 – 1663), Robert Hook (1635 – 1703), Christiaan Huygens (1629 – 1695)and Thomas Young (1773 – 1829) and was mathematically formulated by Augustin JeanFresnel (1788 – 1827). Over the two centuries it has been considered as a very successfultheory – it indeed very precisely describes the propagation of light in linear media. Thefoundation of the diffraction theory is the principle of Huygens which states that (i) allthe points of a wavefront act as the sources of secondary wavelets and (ii) the field at allthe subsequent points is determined by the superposition of those wavelets.The topic of free-space propagation of wave fields has attracted a renewed interestin 1983 when James Neill Brittingham claimed [1] that he discovered a family of threedimensional,nondispersive, source-free, free-space, classical electromagnetic pulses whichpropagate in a straight line in free space at light velocity (in this work he also introducedthe term focus wave mode (FWM) for those wave fields). Now, the very idea of secondaryspherical sources in the classical diffraction theory implies that the spatial amplitudedistribution of any optical wave field suffers from lateral and longitudinal spread inthe course of propagation in free space, whereby the diffraction angle of the spread is thelarger the narrower is the field radius. In the view of this general principle the Brittingham’sstatement is an astounding one and he quite rightly used the formula "to convincethe scientific community" in its arguments. However, the original focus wave mode wasindeed the solutions of the Maxwell’s equations and the scientific community had to resolvethis apparent contradiction. As to give the reader an idea of the initial problemthe theoreticians had to tackle with, we reproduce here the original definition of Brittinghamwhich he deduced by "a very extensive heuristical fit of various differential equationsolutions": given the Maxwell equations (SI)∇ × E = − ∂B∂t∇ × H = ∂D∂t∇ · D = 0∇ · B = 0,where E, D, H, B and t are electric field, electric flux density, magnetic field, magneticinduction and time variable, respectively and using cylindrical coordinate system(ρ, ϕ, z − ct) the mathematical formulation of the original FWM reads asD ρ (ρ, ϕ, z, t) = Ψ 1 + Ψ ∗ 1D ϕ (ρ, ϕ, z, t) = Ψ 2 + Ψ ∗ 2H ρ (ρ, ϕ, z, t) = Ψ 3 + Ψ ∗ 3H ϕ (ρ, ϕ, z, t) = Ψ 4 + Ψ ∗ 4H z (ρ, ϕ, z, t) = Ψ 5 + Ψ ∗ 5,9

where the functions Ψ q (q =1, 2...5) are written asfor q =1and 4,andΨ q = A q (ρ, z − ct) G 1 (ρ, z − ct) G 2 (z − ct) G 3 (z − c 1 t) Φ 0 (φ)Ψ q = A q (ρ, z − ct) G 1 (ρ, z − ct) G 2 (z − ct) G 3 (z − c 1 t) Φ (φ)for q =2, 3, 5. In those equationsG 1 =¸exp·− ρ24FG 2 = exp[−ik 1 (z − ct)]G 3 = exp[ik 2 (z − c 1 t)]andF = ig (z − ct)+ξ .The remaining definitions read as·(nA 1 = DTE +1)cgρn−1c 2 F n+2 − cgρn+14F n+3 + (k 1c − k 2 c) ρ n−1 ¸F n+2A 2 = − DTEc 2·n (n +1)cgρn−1−F n+2+ n (k 1c − k 2 c 1 ) ρ n−1F n+1+ 2cgρn+316F n+4 − 2(k 1c − k 2 c 1 ) ρ n+14F n+2(3n +4)cgρn+14F n+3·A 3 = − DTE n (n +1)gρn−1 (3n +4)gρn+1c 2 −F n+2 +4F n+3+ n (−k 1 + k 2 ) ρ n−1F n+1− 2gρn+316F n+4 − 2(−k 1 + k 2 ) ρ n+14F n+2·A 4 = −D TE (n +1)gρn−1−F n+2+ (−k 1 + k 2 ) ρ n−1F n+1¸¸¸+ gρn+14F n+3· ¸A 5 = −iD TE ρn+2 (n +1)ρn−4Fn+3F n+2 ,10

where D TE , g, ξ, k 1 and k 2 are constants. The Φ functions are defined as½ ¾ sin (nϕ)Φ (ϕ) =cos (nϕ)½¾Φ 0 n cos (nϕ)(ϕ) =−n sin (nϕ)and the supplemental conditions read2gk 2 d 1 = 1k2d 2 2 = k 1g ,where³d 1 = 1 − c ´1µcd 2 = 1 − c2 1c 2 .One has to agree, that the physical idea is very much hidden behind this mathematicalformulation.Brittingham claimed, that this mathematical formulation (i) satisfy the homogeneousMaxwell’s equations, (ii) is continuous and nonsingular, (iii) has a three-dimensionalpulse structure, (iv) is nondispersive for all time, (v) move at light velocity in straightlines, and (vi) carry finite electromagnetic energy. Thus, the formulas above give a mathematicalformulation of a free-space wave field that can be described as a ”light bullet”and, though the proof of the last claim was shown to be faulty by Wu and King [2], thewhole idea was very intricate and rose a considerable scientific interest [3–144].The theoretical work of following years could be divided into the following topics (seealso Ref. [35] for an overview):In the following publications [3, 6] the original vector field was reduced to its scalarcounterpart and the dominant part of the research work that followed has been formulatedin terms of solutions to homogeneous scalar wave equation.The close connection between the FWM’s and the solutions of the paraxial wave equationand Schrödinger’s equation (which both allow localized solutions) has been established[3,5,6] – it has been shown that in terms of the variables z +ct and z −ct,ifthesolutionof the scalar wave equation is given by the anzatz exp [β (z + ct)] F (x, y, z − ct),the problem can be reduced to one of those equations.The infinite energy content of the original FWM’s has been addressed in several publications(see Refs. [3, 4, 6, 14, 16, 31, 42, 44–47] and references therein). First of all,Sezginer [3] and Wu and Lehmann [4] proved that any finite energy solution of the waveequation irreversibly leads to dispersion and to spread of the energy. Then Ziolkowski[6] pointed out, that the superpositions of the infinite energy FWM’s could result in finiteenergy solutions and in following publications a number of finite energy solutions to thescalar wave equation and Maxwell equations were deduced – "electromagnetic directedenergypulse trains" (EDEPT) [31, 42], "acoustic directed-energy pulse trains" (ADEPT)11

[41] , splash pulses [6], modified power spectrum (MPS) [31] pulses, electromagneticmissiles [26, 27], various super- and subluminal pulses [15] etc. In correspondence with[3,4] this broader class of localized waves (LW) have generally extended but finite rangesof localizations. Also, several alternate infinite energy LW’s (Bessel-Gauss pulses [33]for example) were deduced.In Ref. [14] Besieris et al introduced a novel integral representation for synthesizingthose LW’s. This bidirectional plane wave decomposition is based on a decomposition ofthe solutions of the scalar wave equations into the forward and backward traveling planewave solutions and it has been shown to be a very natural basis for description of LW’s(see Ref. [31] for example).The FWM’s have been interpreted as being related in a special way to the field of asource, moving on a complex trajectory parallel to the real axis of propagation [6, 10, 12,31]. This observation linked the FWM’s with the works by Deschamps [145] and Felsen[146] where the Gaussian beams have been described as being paraxially equivalent tospherical waves with centers at stationary complex locations.There has been a considerable effort in finding the LW solutions in other branches ofphysics, spanning various differential equations like spinor wave equation [22], fist-orderhyperbolic systems like cold-plasma equation [23], Klein-Gordon equation [14, 15].In 1988 Durnin [52] published his paper on so called Bessel beams (see for exampleRef. [93] for an earlier publication on the topic). The idea attracted much interest andthe Bessel beams and their pulsed counterparts – X-waves [99–107] and Bessel-X waves[116–118] – became the research field of its own rights. In this context the issue of thesuperluminal propagation of a class of LW’s has been considered in Refs. [104,109–114].It has been shown that the FWM’s can be described as monochromatic Gaussian beamsobserved in a moving relativistic inertial reference frame [7, 35].Propagation of optical pulses or beams without any appreciable drop in the intensityand spread over long distances would be highly desirable in many applications. Theobvious uses could be in fields like optical communication, monitoring, imaging, andfemtosecond laser spectroscopy, also in laser acceleration of charged particles. Due tothis general interest the experimental generation FWM’s and LW’s has been discussed innumerous publications (see Refs. [39–51] and references therein). However, the progressin this topic has been moderate – prior to this thesis there has been very few experimentalresults on LW’s and those have been obtained by means of acoustical sources.A good understanding of the experimental difficulties can be given if we account formethods that have been proposed in literature:The most widely discussed approach has been to use directly the principle of Huygensand launch the LW’s from planar sources [43]. However, it appears that each point sourcein such array must (i) have ultra-wide bandwidth and (ii) be independently drivable asthe temporal evolution of the LW’s generally is of the non-separable nature. Due to thepresent state of the experiment this approach has not been realized even in radio-frequencydomain (it has been realized in acoustics [40, 41]).12

In an another approach it has been shown that the LW’s can be launched by the so-calledGaussian dynamic apertures, that are characterized by an effective radius that shrinks froman infinite extensions at t →−∞to a finite value at t → 0, then expands once more toan infinite dimension as t →∞[45] or by the spectrally depleted (finite excitation time)Gaussian apertures [46–49].It has been shown, that the field from an infinite line source contains a FWM component[17] and the LW’s can be generated by a disk source moving "more slowly than the speedof light" [50, 51].It can be easily understood that none of those approaches is feasible in optical domain.The main aim of this thesis is to interpret the published theoretical work in termsof optical feasibility, provide a realizable means of launching FWM’s and LW’s ingeneral and provide the experimental proof of the feasibility of the optical generationof the wave fields.The thesis is organized as follows:In the preliminary Chapter 2 we introduce the necessary integral representations for thesolutions of the Maxwell’s equations and scalar homogeneous wave equation. Predominantlywe will use the Fourier representation of the free-space wave fields.In Chapter 3 we deduce what in our opinion is the physically most comprehensiverepresentation of the FWM’s and LW’s – we will show, that the necessary and sufficientcondition for a free-space wave field to be propagation-invariant is that its support ofangular spectrum of plane waves is of a specific form. Several additional conclusions onthe properties of the LW’s will be drawn.In Chapter 4 we give an outline of the properties of the known (published) LW’s. Thematerial in this section is important, because, to our best knowledge, this is the first timewhere the optical feasibility of certain well-known closed-form LW’s is estimated – wewill see that majority of the known LW’s, including the original FWM’s, are not realizablein optical domain.In Chapter 5 we generalize the theory of the propagation-invariant propagation into thedomain of partially coherent wave fields – we define the conditions for the propagationinvarianceof the mutual coherence function of the wideband, stochastic, stationary fields.The theory also gives a means of estimating the effect of spatial and temporal coherenceof the source light on the properties of generated fields and is used in the analysis of theresults of our experiments.In Chapter 6 we present the general idea of the optical generation of LW’s. First ofall, the setup for the generation of simplest special case – optical Bessel-X pulses – isintroduced. Then we show that in Fourier picture the optical generation of FWM’s canbe resolved to applying specific angular dispersion to the Bessel-X pulses and discuss onthe finite energy approximations of the FWM’s. Also, the optical generation of partiallycoherent propagation-invariant wave fields is discussed.In Chapter 7 we present the results of experiments carried out as the part of this thesis.In particular, we report on experimental measurements of the whole three-dimensional13

distribution of the field of optical X waves – Bessel-X pulses – and provide the experimentalverification of the optical feasibility of FWM’s.In Chapter 8 we give an outline of our work on self-imaging pulsed wave fields – itappears, that certain discrete superpositions of the FWM’s can be used to compose spatiotemporallyself-imaging wave fields that carry non-trivial three-dimensional images.14

2 INTEGRAL REPRESENTATIONS OFFREE-SPACE ELECTROMAGNETIC WAVEFIELDSIn this preliminary chapter we introduce the necessary integral representations for the solutionsof the homogeneous Maxwell’s equations and scalar homogeneous wave equation.Only the free-space wave fields are considered, i.e., the wave fields under investigation donot have any sources (except perhaps at infinity) and they do not interact with any materialobjects. As we will see, such an approach is suitable for our purposes.2.1 Solutions of the Maxwell equations in free spaceIn SI units the source-free Maxwell equations can be written as∂H∇ × E = −µ 0 (2.1a)∂t∂E∇ × H = ε 0 (2.1b)∂t∇ · E = 0 (2.1c)∇ · H = 0, (2.1d)E and H being the electric and magnetic field vectors respectively, µ 0 is the magneticpermittivity of free space, ε 0 is the electric permittivity of free space. As it is well know,in this special case the components of the electric and magnetic field vectors satisfy thehomogeneous wave equationµ∇ 2 − 1 ∂ 2 c 2 ∂t 2 E (r,t) = 0 (2.2a)µ∇ 2 − 1 ∂ 2 c 2 ∂t 2 H (r,t) = 0. (2.2b)In Eqs. (2.2a) and (2.2b) only two of the six field variables are independent and theMaxwell equations have to used to solve for the other, dependent field components.The general solution of the scalar wave equations (2.2b) can be expressed as the Fourierdecomposition asZ1 ∞ Z Z Z ∞E (r,t) =(2π) 4 dω dk E (k,ω)exp[ikr − iωt] (2.3a)−∞−∞Z1 ∞ Z Z Z ∞H (r,t) =(2π) 4 dω dk H (k,ω)exp[ikr − iωt] , (2.3b)−∞−∞where E =(E x , E y , E z ) and H =(H x , H y , H z ) are plane wave spectrums of the electricand magnetic field. Specifying, for example E x E y as two solutions of the scalar wave15

equation we get from ∇ · E =0thatE z (k,ω)=− 1 k z[k x E x (k,ω)+k y E y (k,ω)] (2.4)∂Hand from ∇ × E = −µ 0∂tH x (k,ω) = − 1 £kx k y E x (k,ω)+ ¡ k 2 − k 2 ¢x Ey (k,ω) ¤ ωk z µ 0(2.5a)H y (k,ω) =1 £¡k 2 − k 2 ¢y Ex (k,ω)+k x k y E y (k,ω) ¤ ωk z µ 0(2.5b)H z (k,ω) =1k y E x (k,ω) − k x E y (k,ω) .ωµ 0(2.5c)If we substitute the Eqs. (2.4) – (2.5c) in (2.3a) and (2.3b) we have a general solution offree-space Maxwell equations as a superposition of monochromatic plane waves.The other approach is to determine the vector potential A as the solution of the homogeneouswave equation – if we use the Coulomb gauge and no sources are present thescalar potential is zero and the fields are given by [159]E (r,t) = − ∂ A (r,t)∂t(2.6a)B (r,t) = ∇ × A (r,t) (2.6b)Alternatively, we can determine the Hertz vectors Π from the homogeneous wave equation,then the fields are given by [160]E (r,t) = ∇³∇ (e)´ · Π − µ 0 ∇ × ∂ ∂t Π(m) − 1 ∂ 2c 2 ∂t 2 Π(e) (2.7a)H (r,t) = ∇³∇ (m)´ · Π − ε 0 ∇ × ∂ ∂t Π(e) − 1 ∂ 2c 2 ∂t 2 Π(m) . (2.7b)The choice of the vector components of the Hertz vectors and vector potential generallydetermine the polarization properties of the resulting vector field.2.2 Plane wave expansions of scalar wave fieldsIf we assume, that the general solution Ψ (r,t) of the scalar homogeneous wave equationµ∇ 2 − 1 ∂ 2 c 2 ∂t 2 Ψ 0 (r,t)=0 (2.8)can be decomposed into the Fourier superposition of plane waves asZ ∞ Z Z Z ∞ψ (k,ω)= dt dr Ψ 0 (r,t)exp[−ikr + iωt] , (2.9)the inverse transform yields−∞Ψ 0 (r,t)= 1(2π) 4 Z ∞−∞−∞Z Z Z ∞dω dk ψ (k,ω)exp[ikr − iωt] . (2.10)−∞16

The Eq. (2.10) together with the condition³ ωkx 2 + ky 2 + kz 2 = k 2 ´2=c(2.11)which assures, that the Fourier representation satisfies the wave equation (2.8), is thegeneral source-free solution of the scalar homogeneous wave equation that will be usedin this thesis. The representation (2.10) leads to Whittaker and Weyl type plane waveexpansions (for the discussions on this topic see for example Refs. [178–181] and [158]).2.2.1 Whittaker type plane wave expansionThe dispersion relation (2.11) can be inserted into (2.10) as a delta function δ(k 2 − k 2 x +k 2 y + k 2 z) so that the integration over ω yieldsΨ 0 (r,t)= 1(2π) 4 Z Z Z ∞−∞dk x dk y dk zc2k×A 0 (k x ,k y ,k z )exp[i (k x x + k y y + k z z − kct)] .(2.12)orΨ 0 (r,t)= 1(2π) 4 Z Z Z ∞−∞dk x dk y dk z×A (k x ,k y ,k z )exp[i (k x x + k y y + k z z − kct)] . (2.13)whereA (k x ,k y ,k z )= c2k A0 (k x ,k y ,k z ) (2.14)If we also introduce the cylindrical coordinate system (ρ, z, ϕ) in real space and sphericalcoordinate system (k, θ, φ) in k-space the Eq. (2.13) yieldsZ ∞ Z π Z 2πΨ 0 (r,t)= dkk 2 dθ sin θ dφ A (k sin θ cos φ, k sin θ cos φ, k cos θ)000× exp [ik (x sin θ cos φ + y sin θ sin φ + z cos θ − ct)] (2.15)(here and hereafter we omit the normalizing constants in front of the integrals of thistype). We can also expand the radial dependence of the angular spectra as the Fourierseries∞XA (k sin θ cos φ, k sin θ cos φ, k cos θ) = A n (k, θ)exp[inφ] (2.16)and get another form of (2.15)Ψ (ρ, z, ϕ, t) =∞Xexp [±inϕ]n=0n=−∞Z ∞0Z πdkk 2 dθ sin θ×A n (k, θ) J n (kρ sin θ)exp[ik (z cos θ − ct)] , (2.17)where J n () is the n-th order Bessel function of the first kind and we introduced the polarcoordinates in real space (ρ, z, ϕ), so that Ψ (ρ, z, ϕ, t) =Ψ 0 (ρ cos ϕ, ρ sin ϕ, z, t). Inthe017

adially symmetric case only the term n =0is taken into account in Eq. (2.17) and wehaveZ ∞ Z πΨ (ρ, z, ϕ, t) = dkk 2 dθ sin θA 0 (k, θ)00×J 0 (kρ sin θ)exp[ik (z cos θ − ct)] . (2.18)If we define χ = k sin θ and again use the Fourier series expansion of the radial dependenceof the angular spectrum, the representation (2.13) yields∞XZ ∞ Z ∞µ pkΨ (ρ, z, ϕ, t) = dk z dχχA2 n z + χ 2 χ, arcsin √k 2 z +χ 2n=0−∞×J n (χρ)exp[±inϕ]exp0h ³i k z z − ct p ´iχ 2 + kz2 . (2.19)Again, in the radially symmetric case only the term n =0is taken into account and wehaveZ ∞ Z ∞µ pkΨ (ρ, z, ϕ, t) = dk z dχχA2 0 z + χ 2 , arcsin √χk 2−∞ 0z +χ 2h ³×J 0 (χρ)exp i k z z − ct p ´iχ 2 + kz2 . (2.20)2.2.2 Weyl type plane wave expansionIf we use the dispersion relation (2.11) to eliminate the variable k z instead, then Eq. (2.10)can be given the following form∞XZ ∞ Z ∞Ψ (ρ, z, ϕ, t) = exp [±inϕ] dk dχχn=0×A wen (k, χ) J n (χρ)exphi00³z p k 2 − χ 2 − kct´i, (2.21)which is the Weyl type superposition over the plane waves (see for example Ref. [158]for a thorough treatment).The Weyl type spectrum of plane waves is often derived as the Fourier transform of thewave field in plane z =0. In contrary, the Whittaker type superposition is calculated asits three-dimensional Fourier transform over the space. Note however, that the distinctionbetween the two is not clear for wideband wave fields, as the calculation of Weyl representationrequires the knowledge of the evolution of the amplitude of the wave field onthe z =0plane for all times [see Eq. (2.9)].2.3 Bidirectional plane wave decompositionThe bidirectional plane wave decomposition was introduced by Besieris et al in Ref. [14]and it has been proved to be useful for description of LW’s. It is based on a decompositionof the solutions of the scalar wave equations into the forward and backward traveling plane18

wave solutions, in this representation the general solution to the scalar wave equation canbe written in the form (Eq. 2.22 of Ref. [14])Ψ (ρ, ζ, η, ϕ) = 1(2π) 2X ∞ Z ∞n=00d˜αZ ∞×J n (χρ)exp[±inϕ]exp[−i˜αζ]exp0d˜βZ ∞0h ii˜βηdχχC n³˜α, ˜β,χ´δµ˜α˜β − χ2, (2.22)4where η = z + ct and ζ = z − ct. Even though the Eq. (2.22) differs noticeably from theFourier decomposition, there is one to one correspondence between these two through thechange of variablesor inverselyk z = ˜α − ˜β (2.23a)ωc= ˜α + ˜β, (2.23b)˜α = 1 ³ ω´2 c + k z(2.24a)˜β = 1 ³ ωz´2 c − k . (2.24b)Consequently we can writeψ (k,ω)∝∞Xexp [±inϕ]n=0×C n·12Note that the delta function constraint³ ωz´c + k , 1 ³ ω´ q ¸2 c − k z , kx 2 + ky2 . (2.25)4˜α˜β = χ 2 (2.26)in Eq. (2.22) in the Fourier picture reduces to³ ω´2− k2c z − χ 2 =0. (2.27)For circularly symmetric wave fields the bidirectional expansions yieldsZ1 ∞ Z ∞ Z ∞Ψ (ρ, ζ, η, ϕ) =(2π) 3 d˜α d˜β dχχC 0³˜α, ˜β,χ´J 0 (χρ)0 0 0h i× exp [−i˜αζ]exp i˜βη δµ˜α˜β − χ2. (2.28)419

3 A PRACTICAL APPROACH TO SCALARFWM’S3.1 Propagation invariance of scalar wave fields3.1.1 The angular spectrum of plane waves of the FWM’sFirst of all, in literature the term FWM has been used mostly with the following closedformsolution of the scalar homogeneous wave equation:· ¸a 1Ψ f (ρ, z, ϕ, t) =exp[iβ (z + ct)]4πi (a 1 + iζ) exp −βρ2 (3.1)a 1 + iζ(Eq. (2.1) of Ref. [18]). The Weyl and Whittaker type plane wave spectrums of this wavefield have been derived in Refs. [14,18] and, omitting the normalizing constants, the latterreadsA (f)0 (k, θ) = 1 ·k exp − a ¸1k (cos θ +1)δ (k − k cos θ − 2β) . (3.2)2In this respect one can say that the following derivation of the angular spectrum of planewaves of the FWM’s is nothing but the different interpretation of the results already published.However, the alternate emphasis in the theory, described in this section (and publishedin Refs. II and VII), have proved to make the difference if the optical generation ofthe FWM’s is under discussion. Also, the term FWM will be redefined in what follows.Consider the general solution of the free-space wave equation represented as the Whittakertype plane wave decomposition Eq. (2.17)∞XZ ∞ Z πΨ (ρ, z, ϕ, t) = exp [±inϕ] dkk 2 dθ sin θn=00×A n (k, θ) J n (kρ sin θ)exp[ikz cos θ − iωt] . (3.3)The integral representation of fundamental FWM’s can be derived from the conditionthat the superposition of Bessel beams in Eq.(3.3) should form a nondispersing pulsepropagating along the z axis. In terms of group velocity dispersion of wave packets thiscondition means that the on-axis group velocity v g =dω/dk z should be constant overthe whole spectral range. This restriction allows non-trivial solutions only if we assumethat the cone angle in relation k z = k cos θ is a function of the wave number, i.e., onecan write θ (k). The corresponding support of the angular spectrum of the plane waveconstituents of the pulse, i.e., the volume of the k-space where the angular spectrum ofplane waves of the wave field is not zero, is a cylindrically symmetric surface in the k-space and the angular spectrum can be expressed by means of Dirac delta function asA n (k, θ) =B n (k) δ [θ − θ (k)].020

k x2βξθ F(k 0)k zFig. 3.1 On the geometrical interpretation of the parameters β and ξ of the supports of angularspectrum of plane waves of FWM’s (gray line).The conditionv g = c dk¸−1 ·1 d=dk z c dk (k cos θ F (k)) = c γ , (3.4)where constant γ determines the group velocity, yieldsk z = γk − 2βγ, (3.5)where the integration constant 2β is defined as the wave number of the plane wave componentpropagating perpendicularly to the z axis, i.e., θ F (2β) =90 ◦ (see Fig. 3.1 forthe geometrical interpretation of the parameter β in k-space, the choice is consistent with[14, 18] for example). Thus, we can writeγ (k − 2β)cos θ F (k) = (3.6)kork F (θ) =2βγγ − cos θ . (3.7)It appears in section 3.1.3 that for a subclass of special cases the above definitions arenot appropriate as the corresponding supports of the angular spectrum of plane waves donot intersect with the k x axis. Then one should determine an alternate integration constantξ from the condition θ F (ξ) = 180 ◦ , this choice yields½ γk − ξ (γ +1), if ξ ≥ 0k z (k) =(3.8)γk − ξ (γ − 1) , if ξ

orξ (γ ± 1)k F (θ) = (3.10)γ − cos θso that2β = ξ γ +1(3.11)γ(aswealwayshaveβ ≥ 0).The definitions (3.6) or (3.7) give the angular spectrum of plane waves in Eq. (3.3) theformA (F n ) (k, θ) =B n (k) δ [θ − θ F (k)] (3.12)andA (F n ) (k, θ) =B n [θ F (k)] δ [k − k F (θ)] , (3.13)correspondingly (see Fig. 3.2 of the section 3.1.3 for the set of special cases).As k z = k cos θ, our result (3.12) is consistent with the support of angular spectrum ofplane waves of the original FWM’s in Eq. (3.2), the constant γ just generalizes to includealso FWM’s of different group velocities. Thus, we can conclude that the physicallytransparent condition (3.4) indeed determines the support of the angular spectrum of planewaves of the FWM’s.3.1.2 Integral expressions for the field amplitude of the scalar FWM’sWith the angular spectrum of plane waves (3.12) we can eliminate variable θ in integral(3.3) and get∞XZ ∞Ψ F (ρ, z, ϕ, t) = exp [±inϕ] dkk 2 sin θ F (k)n=00×B n (k) J n [kρ sin θ F (k)] exp [ik (z cos θ F (k) − ct)] , (3.14)using (3.6) we can write∞XZ ∞Ψ F (ρ, z, ϕ, t) =exp[−i2γβz] exp [±inϕ] dkk 2 sin θ F (k)n=0⎡ s⎤µ 2 γ (k − 2β)×B n (k) J n⎣kρ 1 −⎦ exp [ik (γz − ct)] . (3.15)kAlternatively, we can eliminate k by means of Eqs. (3.13) and get∞XZ πΨ F (ρ, z, ϕ, t) = exp [±inϕ] dθ sin θkF 2 (θ)n=00×B n [θ F (k)] J n [k F (θ) ρ sin θ]exp[ik F (θ)(z cos θ − ct)] , (3.16)022

again (3.6) transform the equation to∞XZ πµ 2 2βγΨ F (ρ, z, ϕ, t) = exp [±inϕ] dθsin θn=00 γ − cos θ¸ ··2βγρsin θ 2βγ×B n [θ F (k)] J n exp i (z cos θ − ct)¸(3.17)γ − cos θ γ − cos θ[note that analogous expressions can be written using Eqs. (3.8) – (3.10)].The applied condition (3.4) implies that the longitudinal shape of the central peak of thepulsed wave field in Eqs. (3.14) – (3.17) do not spread as it propagates in z axis direction.From the integral expressions it is also obvious that the pulse do not spread in transversaldirection. However, the wave field has what has been called the "local variations" – theterm exp [−i2γβz] in (3.15) implies that only the instantaneous intensity of the wave fieldis independent of the propagation distance, in what follows we refer to such wave fieldsas propagation-invariant.It is important to note that in Eqs. (3.12), (3.13) and (3.14) – (3.17) the frequencyspectrum is arbitrary. Thus, the necessary and sufficient condition for the propagationinvarianceof the general pulsed wave field (3.3) is that its support of angular spectrum ofplane waves should be defined by Eq. (3.6) or (3.9). The statement can also be invertedand one can say that the wave field is strictly propagation-invariant only if its support ofangular spectrum of plane waves is defined by Eq. (3.12) or (3.13) – indeed, in Eq. (3.4)any other choice would lead to the group velocity dispersion and the pulse would inevitablyspread as it propagates. This also implies, that all the possible solutions of scalarhomogeneous wave equation that have extended depth of propagation as compared to ordinaryGaussian pulses (see next chapter) should be considered as certain approximationsto the FWM’s.Now, the closed-form expressions like (3.1) are very convenient in numerical analysis,however, limiting ourselves to the set of available closed-form integrals of (3.14) – (3.17)is not reasonable by any means. In this thesis we use the term "focus wave modes"(FWM)for all the wave fields that can be represented by the integral expressions (3.14) – (3.17),whereas the closed-form expression (3.1) will be called the original FWM.3.1.3 A physical classification of FWM’sThe recognition, that the spatiotemporal behavior of FWM’s is determined only by thesupport of their angular spectrum of plane waves enables one to give a straightforwardgeneral classificationtotheFWM’s(Ref.V).Note, the dispersion relation³ ωχ 2 + kz 2 ´2− =0 (3.18)ccan be interpreted as a definition of a cone in (χ, k z ,k) space [15] (see Fig. 3.2). Inthis context the specific supports of the angular spectrum of plane waves of FWM’s inEqs. (3.4) – (3.10) have a geometrical interpretation as being the cone sections of (3.18)23

(a)0χk z χk z k zk xkk 0kk zgγ< 1, β=0, v >c(b)0χk zk zk xk0kk zgγ=1, β>0, v =c(c)0χk zk0k zkk xk zgγ>1, β>0, v cFig. 3.2 The physical classification of the FWM’s in terms of sections of the cone χ 2 +k 2 z−k 2 =0in (χ, k z,k) space. The first two columns depict the sections of the cone from two viewpoints, thecorresponding supports of angular spectrums of plane waves are depicted in third column.24

along the planesk z = γk − 2βγ (3.19)(3.5) ork z (k) =γk − ξ (γ ± 1) (3.20)(3.8). It can be seen that the possible supports of the angular spectrums of plane wavescan be divided into four explicit special cases (see Fig. 3.2) that can be taken as the naturalclassification of the FWM’s:1. β =0(ξ =0), γ ≤ 1, the support is a cone in k-space, typical examples are Bessel-Xpulse and X-pulse (the case γ =1corresponds to plane wave pulse);2. β 6= 0(ξ 6= 0), γ =1, the support is a paraboloid in k-space, typical example isFWM’s, propagating at velocity of light;3. β 6= 0(ξ 6= 0), γ>1, the support is an ellipsoid in k-space, the group velocity of theFWM’s satisfies v g

chromatic Bessel beam can be represented as a superposition of so-called Hankel waveswhereΨ (1)m (ρ, z, t) = H (1)m (χρ)exp[ik z z − iωt + imϕ] (3.22a)Ψ (2)m (ρ, z, t) = H (2)m (χρ)exp[ik z z − iωt + imϕ] , (3.22b)H m (1) = J m (χρ)+iN m (χρ) (3.23a)H m (2) (χρ) = J m (χρ) − iN m (χρ) (3.23b)are the m-th order Hankel functions and N m denotes the m-th order Neumann function(the Bessel function of the second kind). For monochromatic wave fields the two solutionsdefine the diverging and converging wave in xy plane, in other terms, they form the "sink"and "source" pair. In those terms the m-th order Bessel beam can be written asJ m (χρ)exp[ik z z − iωt + imϕ] = (3.24)hiH m(1) (χρ)+H(2) m (χρ) exp [ik z z − iωt + imϕ]– this is a standing wave that arise in the superposition of the two Hankel waves (notehow the singularity of the Neumann functions at the origin is eliminated).This approach can be easily generalized for the wideband wave fields–inthiscasethe superposition of the monochromatic Hankel beams form a converging or expandingcircular pulse in the xy plane. If we also use condition (3.12) we get the pulse thatcorresponds to the radial evolution of the FWM’s. The results of a numerical simulationof its behavior are depicted in Fig. 3.3a and 3.3b.Note also, that the radial wave that propagates away from the z axis is generally notpropagation invariant. Indeed, if we follow the arguments of the section 3.1.1 for radialpropagation we can write the condition of propagation-invariance asv g = c dk ·1dχ = d³c dkk sin θ (ρ) (k)´¸−1F= cγ ρ, (3.25)where constant γ ρ again determines the group velocity. Specifying the integration constantξ again from the condition θ ρ (ξ) = 180 ◦ we can write for the support of angularspectrum of plane wavesk sin θ (ρ)F (k) =γ ρk − ξ ¡ γ ρ +1 ¢ . (3.26)Thus, we can writesin θ (ρ)F (k) = γ ρk − ξ ¡ γ ρ +1 ¢(3.27)k(note that in this context ξ ≥ 0).A typical support of the angular spectrum of plane waves defined by Eq. (3.26) isdepicted in Fig. 3.3b. So, the FWM is propagation-invariant in both the z axis directionand radial direction only in the special case ξ ≡ 0 where we can write γ ρ = p 1 − γ 2 .This consequence will be given a further interpretation in section 3.3.1.26

timeH(1)(χρ)H(2)(χρ)ρρρρρρρ(b)(a)k xξθ ρ(k)k z(c)Fig. 3.3 (a) Typical spatiotemporal amplitude distribution of a FWM; (b) The temporal evolutionof the FWM in radial direction as the superposition of the pulsed Hankel beams (solid lines), theamplitude of the corresponding carrier-frequency monochromatic Hankel beam is added for comparison(dotted line); (c) The support of the angular spectrum of plane waves of a wave field that ispropagation-invariant in radial direction (see text).27

3.1.5 The spatial localization of FWM’sIn this thesis we do not make any efforts to find closed-form integrals to Eq. (3.14). Consequently,we have to deal with integral transforms and the straightforward numericalsimulation of any realistic situation may be a tedious task (this is especially true for generalLW’s where the double integrals have to be computed). However, there is a simplemethod for qualitative estimate of the spatial amplitude of the resulting wave fields, basedon three-dimensional Fourier transforms (the monochromatic case of the approach was introducedby McCutchen in Ref. [183] and has been used for example in Refs. [184,185]).Let us start with Whittaker type plane wave decomposition in Eq. (2.13) and set t =0:Z Z Z ∞Ψ 0 (x, y, z, 0) = dk x dk y dk z−∞×A (k x ,k y, k z )exp[i (k x x + k y y + k z z)] . (3.28)Obviously we can write for the amplitude on z axis the relationZ ∞½Z Z ∞¾Ψ 0 (0, 0,z,0) = dk z exp [ik z z] dk x dk y A (k x ,k y, k z ) , (3.29)−∞−∞so thatZ ∞Ψ 0 (0, 0,z,0) = dk z exp [ik z z] A xy (k z ) , (3.30)where−∞Z Z ∞A xy (k z )= dk x dk y A (k x ,k y, k z ) (3.31)and from the definition of one-dimensional Fourier transform, we can write−∞Ψ 0 (0, 0,z,0) = 2πF −1z [A xy (k z )] . (3.32)Here Fz−1 [...] denotes the inverse Fourier’ transform in k z -direction and the integral(3.31) can be thought of as the projection of the angular spectrum plane waves onto the zaxis (see Fig. 3.4). Similarly we can write for the xy amplitude at z =0Z Z ∞Ψ 0 (x, y, 0, 0) = dk x dk y exp [ik x x + ik y y]−∞½Z ∞¾× dk z A (k x ,k y, k z ) , (3.33)−∞so thatΨ 0 (x, y, 0, 0) = (2π) 2 Fxy −1 [A z (k x ,k y )] , (3.34)whereZ ∞A z (k x ,k y )= dk z A (k x ,k y, k z ) (3.35)−∞and Fxy −1 [...] denotes the two-dimensional inverse Fourier transform.Now, having in mind the table of basic one- and two-dimensional Fourier transformsand the general properties of Fourier transforms, the knowledge of the defined projections28

k xk xk yθ F (k )k z0∆z∆χ∆k zk yFourier transformk zzxxyz∆ρFig. 3.4The Fourier transform estimation of the spatial shape of the FWS’s (see text).29

of angular spectrum of plane waves onto the k z axis and k x k y plane allows one immediatelyestimate the general shape of the amplitude of the wave field on z axis and xy planerespectively. If we also note that in studies of the propagation-invariant wave fields theestimates are valid over the entire z axis (for space-time points γz − ct), the approach canprovetobeveryuseful.Let us specify the frequency spectrum of the light source s (k) as the Gaussian one:s (k) =exp·− 1 2¸2 σ2 k (k − k 0 ) , (3.36)where k 0 denote the mean wave number of the wave field and σ k is determined from thepulse length τ s of the corresponding plane wave pulse asσ k =cτ s2 √ 2ln2 . (3.37)From the known character of the angular spectrum of plane waves of the FWM’s we canapproximate for the Gaussian profiles of the k z and χ projections of the angular spectrumof plane wavesσ z =cos θ F (k 0 )(3.38a)σ ρ =σ ksin θ F (k 0 )(3.38b)respectively.The spectral profile of the k z -projection of the angular spectrum of plane waves thenreadsA xy (k z ) ∝ exp·− 1 2¸2 σ2 z (k z − k z0 ) (3.39)with the FWHM (full width at half-maximum)whereσ k∆k z ≈ ∆k cos θ F (k 0 )= 2√ 2ln2σ z, (3.40)∆k = 2√ 2ln2. (3.41)σ kThe corresponding intensity profile is¸Fz−1 [A xy (k z )] ∝ exp·− z2(3.42)with FWHMcτ s∆z ≈cos θ F (k 0 ) = σ z2 √ 2ln2. (3.43)For the transversal amplitude we can give a good estimate by recognizing that theintensity profile on xy plane has the Bessel profile that is multiplied by an envelope. Theprofile of the latter can be estimated by the 1D Fourier transform of the projection of the2σ 2 z30

k x2βθ F(k 0)k zk z

θ ΙD + D -π/2θ RFig. 3.6 The integration contour in Weyl picture of angular spectrum of plane waves. The verticalpart of the contour where the imaginaty part of the angle θ is nonzero cancels out in integration.plane wave components (see Fig. 3.5) [13]. This fact is due to the specific frequency spectrum(3.2) that leads to the closed-form FWM’s (see the overview in following chapter).In the consequent publications (see Ref. [18]) Shaarawi et. al. demonstrated, that the parametersof the spectrum can be chosen so that the predominant part of the energy of theFWM’s is in forward propagating plane wave components.In the context of our approach this problem has to be considered as ill-posed – asall the wave fields that share the support of the angular spectrum of plane waves (3.12)are propagation-invariant regardless of their frequency spectrum, we can just choose onewithout the acausal components.3.2.2 FWM’s and evanescent wavesThe second topic that is closely related to the backward propagating plane wave componentsof the original FWM’s is the one of evanescent waves [18, 20].From the practical point of view it may seem peculiar to introduce the evanescentwaves, the intensity of which decays exponentially, in the context of the propagationinvariantwave fields where the depth of the propagation usually extend over several meters.However, the evanescent waves appear indeed in a Weyl picture of the FWM’s.Indeed, from Eqs. (2.21) and (3.5) one can writeA wen (k, χ) =B n (k) δhk − p k 2 − χ 2 − 2βi(3.49)for the angular spectrum of plane waves of the FWM’s so that the field amplitude reads∞XZ ∞ Z ∞Ψ (ρ, z, ϕ, t) = exp [±inϕ] dk dχχ B n (k) (3.50)×J n (χρ) δn=0hk − p k 2 − χ 2 − 2β00iexph ³i z p ´ik 2 − χ 2 − ckt .In Eq. (3.50), for the ranges χ

χ>k, the wave vector of the plane waves is purely imaginary and the integration is overthe evanescent waves [18, 35]. The situation may be more apparent if we transform tovariables χ, k → θ, k and write (for cylindrically symmetric component only for brevity)orΨ (±) (ρ, z, t) =Z ∞0dk k 2 ZD ± dθ cos θ sin θ (3.51)×B 0 (k) δ [k ∓ k cos θ − 2β] J 0 (kρ sin θ)exp[±ikz cos θ − iωt]³ ´Z2β2β sin θB 0Ψ (±) γ(1−cos θ)(ρ, z, t) = dθD γ (1 − cos θ)µ ± ·2βρsin θ2β×J 0 exp ±iγ (1 − cos θ) γ (1 − cos θ)(z cos θ ∓ ct)¸.(3.52)Here ”+” stands for forward propagating plane wave components and ”−” stands forbackward propagating plane wave components and the integration is carried out alongthe contours D ± of complex θ plane, χ/k =sin(θ R + iθ I ) (see Fig. 3.6). Also, if theanalysis is carried out for wave fields the angular spectrum of plane waves of which hasforward and backward propagating components, the total wave field can be written as[18]Ψ = ¡ Ψ + h + ¢ ¡ Ψ+ ev + Ψ−h + ev¢ Ψ− , (3.53)where subscript ”h” denotes homogeneous component of Ψ + or Ψ − ,i.e.,0 ≤ θ R ≤ 2π,θ I =0and subscript ”ev” denotes evanescent components, i.e., θ R = π/2, θ I < 0. Ithas been shown [18], that for the evanescent components of a free field one hasΨ + ev = −Ψ − ev, (3.54)so that the Weyl forward and backward propagating components add up resulting in thesource-free solution in Eq. (3.14).Again, in our approach the frequency spectrum is chosen so that the wave fields donot have any backward propagating components. Consequently, the integration is onlyalong the real part of the D + . Also, it is quite clear that for the free-space wave fieldsthe presence of the evanescent waves in the integration (3.52) is rather a peculiarity of theWeyl type angular spectrum of plane waves. For example, if we write the Weyl picture ofa plane wave pulse propagating perpendicularly to z axis, the corresponding Weyl pictureobviously do contain evanescent components. However, there is no physical content inthose components.3.2.3 Energy content of scalar FWM’sAs already noted, the total energy content of FWM’s is infinite [2, 6, 31]. Indeed, as theenergy content is calculated asU tot =Z ∞dzZ ∞dρρZ 2π−∞ 0 0dϕ |Ψ F (z, ρ, ϕ,t)| 2 . (3.55)33

In the Fourier picture, the Parseval relation and the angular spectrum of plane waves inEq. (3.13) can be used to yieldU tot = X Z ∞ Z πdk dθ ¯ ˜B ¯n (k) δ [θ − θ F (k)] ¯2n 0 0= X Z ∞ Z πdk dθ ¯ ˜B ¯n (k) ¯2 δ 2 [θ − θ F (k)] (3.56)n 0 0so thatU tot = ∞ (3.57)due to the δ 2 in the integrand (here and hereafter the tilde on angular spectrum indicatesthat the factor k 2 sin θ is included into the spectrum).Obviously the relation (3.57) is validwhenever there is a delta function in the definition of the angular spectrum of plane waves.Also, it has been proved that any wave field that is strictly propagation-invariant has necessarilyinfinite total energy [3, 4].The second important energetic parameter of the LW’s is their energy flow over a crosssectionper unit time – obviously, any physically feasible wave field has to have a finiteenergy flow. In terms of the previous section and using the two-dimensional Parsevalrelation this quantity can be calculated asZ Z ∞Φ xy = dk x dk y |A z (k x ,k y )| 2 , (3.58)−∞where A z (k x ,k y ) is again the projection of the angular spectrum of plane waves ontok x k y plane. Obviously the quantity is necessarily infinite, if only the projection of theangular spectrum can be written in terms of delta function in k x k y plane. Otherwisethe energy flow is finite, provided the function A z (k x ,k y ) is square integrable. Thecomparison of Figs. 3.2 and 3.4 shows that the FWM’s generally have finite total energyflow.In literature the finite energy LW’s have been constructed for example by means ofsuperpositions of FWM’s [6,31] and by applying finite time windows [45–49]. In section.6.3 we will describe our approach to this problem as described in Ref. III of this thesis.3.3 Alternate derivations of scalar FWM’s3.3.1 FWM’s as cylindrically symmetric superpositions of tilted pulsesAs to demonstrate the efficiency of the integral transform representations in describingthe properties of FWM’s, we give yet another description of FWM’s (Ref. IV).Let us represent FWM’s as the cylindrically symmetric superpositions of the interferingpairs of certain tilted pulses (see also Ref. I). In this representation the spatial amplitude34

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

xv g tθ F(k 0)ϑphasefrontszλ 0v g t 1v p tk 0x(a)FWMz(b)xt = 0v p t 1 t = t 1z-v g t 1v p t 1(c)Fig. 3.7 (a) On the spatial amplitude of tilted pulses; (b) comparison of spatial amplitudes of thesuperposition of a pair of tilted pulses (in left) and of the corresponding FWM (in right); (c) on thedifference of phase and group velocities of the FWM’s (see text).36

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

xv p t =v g tλ 0zθ 0v p t 1ctk 0x(a)Bessel-Xzx(b)t = 0 v p t v g 1 = t 1t = t1zv g t 1(c)Fig. 3.9 (a) On the phase and group velocity of a plane wave pulse propagating at angle θ 0 relativeto z-axis; (b) comparison of spatial amplitudes of the superposition of a pair of plane wave pulses (inleft) and of their corresponding cylindrically symmetric superposition – Bessel-X pulse (in right);(c) on the group velocity of Bessel-X pulses.38

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

Re(Ψ F)arb. unitsz5×10 -3 mx4×10 -5 mFig. 3.10A numerical example of the spatial amplitude of the original FWM’s.misleading in some respects. First of all, the constant β is by no means the carrier wavenumber of the FWM’s as one might expect from the corresponding monochromatic expression– in the following Chapter 4 we will see that the convenient choice of parameterfor optically feasible FWM’s with the carrier wave number k 0 ≈ 1×10 7 radmthe parameteris of the order of magnitude β . 100 radm. Secondly, the requirement of optical feasibilityalso implies that a 1 ¿ 1 (see Sec. 4.2) and with this condition the typical spatialamplitude distribution of the original FWM’s (see Fig. 3.10) do not resemble that of theGaussian beam as they appear in the textbook examples. The reason for the "abnormal"behaviour is obvious – with the above conditions the direct analogy to the monochromaticcase, where β =2π/λ, yields for the beam waist in Eq. (3.72) W 0 ¿ λ. Sothatwehavea limiting case where the waist of the Gaussian beam is much less than its wavelength –clearly here the different physical nature of the FWM’s show up.Next we would like to discuss the claim, often encountered in literature, that the originalFWM’s are carrier free wave fields. First of all, in lights of the general physicalconsiderations in section 3.1.5 it should be evident that the non-oscillating shape of thecentral peak in Fig. 3.10 is a direct consequence of the ultra-wide frequency spectrum ofthe wave field – if the pulse length of the corresponding source plane wave pulse is lessthan the central wavelength, the resulting FWM is effectively an half-cycle pulse and inthis condition the concept of carrier wavelength is rather meaningless of course. However,in above sections it was shown that the general FWM’s are not confined to the one particularfrequency spectrum. Correspondingly, we can choose a feasible frequency spectrumand the carrier free behaviour of the original FWM’s should certainly not be mentionedas the defining property of the original FWM’s, this is just a mathematical peculiarity ofa particular integral transform table entry.The issue can be given an alternate description if we note that, using the analogy to the40

arb. unitsarb. unitsarb. unitsz = 0mm-202-6ζ (10 )secz = 20mm-202-6ζ (10 )secz = 40mm-202-6ζ (10 )secFig. 3.11 On the character of the Gouy phase shift term in the closed form expression of theFWM’s: The real part of the original FWM (dotted blue line) is depicted for three z-coordinatevalues together with the modulus (solid green line) and real part (solid black line) of an FWM withnarrower bandwidth (see text).41

monochromatic Gaussian beams the termµ µ ζG (z, ζ) =i arctan +2βza 1(3.73)in the expression Eq. (3.70) could be interpreted as the Gouy phase shift [161] of theFWM’s. In previous section 3.3.1 we described the FWM’s as the cylindrically symmetricsuperpositions of certain tilted pulses. Now, the original FWM’s differ from those, depictedin Fig. 3.7b only by the ultra-wide bandwidths. In Fig. (3.11) we have depicted theon-axis spatial evolution of an FWM as described by Eq. (3.70) and of one of reasonablebandwidth, calculated h from ³ ´i the Eq. (3.14). The comparison of the two waveforms showsthat term exp i arctan ζa 1of the phase term G can be interpreted as the remnants ofthe sinusoidal waveform, lost due to the ultra-wide bandwidth and the term exp [i2βz] isadded as the monothonically growing phase factor that is due to the difference betweenthe group and phase velocities of FWM’s. The latter term is characteristic to the FWM’sonly – instead of having a single focus with accompanying Gouy phase shift or a "frozen"Gouy phase shift as the X-type pulses, FWM’s have periodically evolving phase shiftterm.The idea of Gouy phase shift, initially introduced in the Fresnel approximation of thediffraction theory of monochromatic focused beams, has attracted a renewed interest recentlyin the context of propagation of subcycle Gaussian pulses (see Refs. [126–128,133,135,139,141]). We believe, that the simple physical interpretation of the term (3.73)in the context of FWM’s, as being the result of the difference between phase and groupvelocities of the wave field, might add to the general understanding of the phenomenon.3.3.3 FWM’s as the Lorentz transforms of focused monochromatic beamsAn interesting interpretation to the FWM’s can be given in terms of special theory of relativity.Namely, in Ref. [5] Bélanger demonstrated that Gaussian monochromatic beamsappear as FWM’s (Gaussian packetlike beams) when observed in an inertial system movingat relativistic speeds relative to the focused wave. In this short note we would like togive an another mathematical representation to this claim.Suppose we take a focused monochromatic wave of the formΨ (ρ, z, t) =Z πwith angular spectrum of plane waves0dθK (θ)exp[ik 0 (z cos θ − ct)] (3.74)A 0 (k, θ) =K (θ) δ (k − k 0 ) . (3.75)If we observe the beam from a moving inertial system, the plane wave components ofthe field suffer from the relativistic Doppler shift. As the result, their wave vectors andfrequencies transform as described by Lorentz transformations. Specifically, the wavenumber of the wave vector and its longitudinal and transversal components in the inertial42

frame, moving at speed V along the z axis, obey equationsk 0 = γ l k 0 (1 − β l cos θ) (3.76a)k 0 z = γ l k 0 (cos θ − β l ) (3.76b)k 0 x = k 0 sin θ = k x (3.76c)k 0 y = k 0 sin θ = k y (3.76d)(Eq. (11.29) of Ref. [159]) where k 0 is the wave number of the wave field in rest frameandβ l (V ) = V cγ l (V ) =1p1 − β2(3.77a)(3.77b)We can use Eq. (3.76a) to eliminate θ from Eq. (3.76b) to getkz 0 = − 1µβ l (V ) k0 − γ l (V ) k 0 β l (V ) − 1 β l (V )and if we define the parameters asγ (V ) =1β l (V )(3.78)(3.79a)β (V,k 0 ) = γ l (V ) k 02, (3.79b)we can write for the z component of the wave vectorµ 1kz 0 = −γ (V ) k 0 − 2β (V,k 0 )γ (V ) − γ (V ) . (3.80)Thus, if the velocity of the moving frame is close to the speed of light, the angular spectrumof plane waves of the wave field in the moving frame is the one of the FWM thatmoves in negative direction of z axis – for the FWM’s we have in Eq. (3.5)k z = γk − 2βγ (3.81)and using both k 0 and V as parameters we can model every possible support of angularspectrum of plane waves of FWM’s. In Fig. 3.12 the support of angular spectrum of planewaves of the beam as seen from the moving reference system is depicted for variousvalues of the speed V and fixed value for k 0 .Note that an alternate approach to describe the LW’s in terms of generalized Lorentztransforms can be found in Ref. [35] – in this work it was shown that the superluminaland subluminal Lorentz transformations can be used to derive LW solutions to the scalarwave equation by boosting known solutions of the wave equation.43

6k x(10 rad/m)V = 0.999c3V = 0.9c V = 0.5cV = 0V = 0.999999c-1 - 0.50.5 17k z(10 rad/m)-3Fig. 3.12 The support of angular spectrum of plane waves of a focused monochromatic beam asseen from inertial reference systems moving at different velocities relative to the rest system of themonochromatic beam (k 0 =1× 10 7 rad ). Due to the relativistiv Doppler shift the direction ofmpropagation and the frequency of the monochromatic components of the focused beam transformso that the beam is seen as the FWM in the moving reference system.3.3.4 FWM’s as a construction of generalized functions in the FourierdomainIn Refs. [15,16] Donnelly and Ziolkowski realized, that various separable and non-separablesolutions to the wave equation can be constructed in spatial and temporal Fourier domainby choosing the Fourier transform of the solution of the differential equation so that, whenmultiplied by the transform of the particular differential operator, it gives zero in the senseof generalized functions. In the special case of scalar homogeneous wave equation thecorresponding relation readsµχ 2 + kz 2 − ω2c 2 ψ (k,ω)=0, (3.82)where ψ (k,ω) (2.9) is (3+1) dimensional Fourier transform of the solution of the waveequation (2.8). It can be shown that the function of the general type·¸ ·¸ψ (k,ω)=Ξ (χ, β) δ k z −µβ − χ2δ ω − cµβ + χ2(3.83)4β4βsatisfies (3.82) and yields all the known FWM’s (in the sense defined in this thesis). Forexample the choice [15]· ¸Ξ (χ, β) = π2iβ exp − χ2 a 1(3.84)4βleads to the original FWM’s.One can notice, that if we eliminate the term χ 2 /4β from the delta functions we get thecondition (3.5) and thus the Eq. (3.83) is yet another transcription of the support of theangular spectrum of plane waves, derived in section 3.1.1.44

4 AN OUTLINE OF SCALAR LOCALIZEDWAVES STUDIED IN LITERATURE SO FAR4.1 IntroductionOver the years a considerable effort has been made to find closed-form localized solutionsto the homogeneous scalar wave equations. The main aim of this work is to study thefeasibility of LW’s in optical domain. Without debasing the value of those solutions itappears, that this approach often leads to the source schemes that are difficult to realizeeven in radio frequency domain.Though there has been several publications that provide an unified approach for thedescription of LW’s [15, 31, 35], to our best knowledge, the optical feasibility of thosewave fields has not been estimated in literature. Moreover, the analysis of the numericalexamples that have been published in literature show, that authors have often choose theparameters of the LW’s so that the frequency spectrum is in the radio frequency domain.In our opinion in optical domain the best representation for the analysis is the Whittakertype plane wave decomposition. First of all, the mental picture of the Fourier lens thatproduces the two-dimensional Fourier transform of monochromatic wave field betweenits focal planes is often very useful in modeling the optical setups – we precisely know,how and in what approximations the elementary components of the Fourier picture, theplane waves and Bessel beams, can be generated. Secondly, the approach of the section3.1.5 allows us easily estimate the spatial shape of the wave fields under the discussion.In the following overview we define the term "optically feasible" by two rather obviousrestrictions:1. The frequency spectrum of an optically feasible wave field should be in optical domain;2. The plane wave spectrum of an optically feasible wave field should not contain planewaves propagating at non-paraxial angles relative to optical axis.The latter requirement can be justified by a very simple geometrical estimate, describedin Fig. 6.29 – if the FWM’s has to propagate over distances that exceed the diameter ofthe source more than, say, five times, the maximum angle of the plane wave componentsin the wave field has to be less than 5 degrees.Note, that the energy content of most of the wave fields discussed in this outline isinfinite, thus, they are not physically realizable as such. However, as we will see inchapters that follow, in optical implementations the finite energy approximations of theLW’s follow naturally from the finite aperture of the setups and this approximation do notchange the general properties of the LW’s, so that the two conditions for optical feasibility,posed here, are also valid for LW’s with finite energy content.In our numerical examples we try to optimize the parameters of each LW so that (i) the45

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

each of any realistic light source. In fact, due to this extraordinary large bandwidth theoriginal FWM’s in Eq. (4.1) are essentially half-cycle pulses, as already noted in section3.3.2.As deduced in section 3.3.2 (and also in terms of the section 3.1.5), the parameter a 1determines the waist of the wave field – in our numerical example a 1 =1.4 × 10 −7 ,sothat the Eq. (3.72) givesra1W 0 =β ∼ 6 × 10−5 m.In literature it has been argued, that the FWM’s determined by Eq. (4.1) are nonphysicalas the wave field contain acausal components. In the discussion of Ref. [18] it has beenshown that the acausality can be eliminated by proper choice of parameters a 1 and β –it has been shown that if βa 1 < 1, the predominant contribution to the spectrum comesfrom the plane waves moving in positive z axis direction. In Fig 4.13b it can be seen,that this is indeed the case, however the field is still far from convenient for any opticalimplementation due to ultra-wide bandwidth.Note, that various closed-form sub- and superluminal FWM’s (γ 6= 1) have been derivedfor example in Ref. [15].4.3 Bessel-Gauss pulsesThe Bessel-Gauss pulses were introduced by Overfelt in Ref. [33] (see also Refs. [15,50,51]). In this publication it was shown, that the scalar wave fieldµ a 1Ψ BG (ρ, µ, ζ) =a 1 + iζ J κa1 ρ0exp [iβµ]a 1 + iζ¸ ·× exp·− βρ2κ 2 ¸a 1 ζexp −i, (4.6)a 1 + iζ 4β [a 1 + iζ]where the physical meaning of the parameters a 1 , β, alsoζ and µ is consistent with theprevious discussion. The expression has an additional free parameter κ as compared tothe FWM’s in (4.1), in fact, the latter is the special case of the former in the limitingcase κ → 0. The Bessel-Gauss pulses were further investigated in Ref. [15] where it wasshown, that in Fourier picture as in Eq. (2.9) the spatiotemporal Fourier transform of thefield can be written as·¸ ·¸ψ BG (k,ω)=Ξ BG (χ, β) δ k z −µβ − χ2δ ω + cµβ + χ2, (4.7)4β4βwhereΞ BG (χ, β) = a 14π 3 µ · ¸β I κa1 χ0 exp − κ2 a 1exp·− a 1χ 2 ¸(4.8)2β4β 4β(see section 3.3.4 for the notation). From Eq. (4.7) it can be seen that the support of theplane wave spectrum of the Bessel-Gauss pulses is the same as described by Eq. (3.6) [or48

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 )BG5×10 -3 m(c)xz4×10 -5 mFig. 4.14 A numerical example of a Bessel-Gauss pulse optimized for optical generation with theparameters σ = 40000 2π , m a1 =5× 10−6 m, β =40 radm, γ =1: (a) The angular spectrumof plane waves in two perspectives; (b) The frequency spectrum of the Bessel-Gauss pulse (blackline), the frequency spectrum of an optically feasible wave field (green line), the angle θ F (k) asthe function of the wave number (dashed blue line); (c) The spatial amplitude of the pulse.49

Eq. (3.9)]. The change of variables in (4.8) yields for the frequency spectrumB (k) = a 14π 3 µ β I κa1 k sin θ F (k)02β·× exp − a 1¡ ¢¸κ 2 + k 2 sin 2 θ F (k) , (4.9)4βwhere κ>0, a 1 > 0 and β>0.In the original paper the Bessel-Gauss pulses were introduced as the wave fields thatare "more highly localized than the fundamental Gaussian solutions because of its extraspectral degree of freedom". The additional free parameter is indeed advantageous, however,in our opinion not in the sense proposed in this publication – the spatial localizationof any wideband free-space wave field is directly proportional to its bandwidth and thelatter is inappropriately large even for the original FWM’s (see Ref. [15] for a related discussion).It may be the consequence of this general emphasis of the original paper thatit is not generally recognized that the extra parameter κ in Eqs. (4.6) – (4.9) gives onethe necessary degree of freedom to fit an arbitrary bandlimited Gaussian-like spectrum –from Eq. (4.9) it can be seen, that the central frequency and bandwidth of the spectra ofthe pulse are independently adjustable by the parameters κ and a 1 respectively.Analogously to the discussion in section 3.3.2 the Bessel-Gauss pulses can be given theform that, in some respect, resembles that of the monochromatic Gaussian beam:· µ ¸W 01Ψ BG (ρ, z, ζ) =exp[−iβζ]a 1 (ζ) J 0 κa 1 ρβa 2 1 (ζ) − i 1R (ζ)¸× exp·− ρ2a 2 1 (ζ) − κ2 a 1 ζ(4.10)4βR(ζ)· µ κ 2 µ µ a 1 ζ βρ2ζ× exp −i4β 2 − − i arctan − 2βz + π ¸,a 2 1 (ζ) R (ζ)a 1 2here againµ # 12 2ζa 1 (ζ) = W 0"1+(4.11a)a 1"R (ζ) = ζ 1+µ # 2 a1ζ(4.11b)ra1W 0 =β .(4.11c)The general form of the Bessel-Gauss pulses (4.10) is very advantageous in the sense thathere we can actually write out its carrier wave number – often this quantity is elusive forthe wideband wave fields. Indeed, around the point ζ =0, along the optical axis (ρ =0)with (4.11a) we can write for the z axis component of the carrier wave numberk 0z = β + σ24β − 1 (4.12)a 1This result is actually quite significant, if we once more remind that in literature the50

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

4k x(10 rad/m)12 3 4 57k z(10 rad/m)~B(k)(a)unitsarb. units8arb.12 3 4 5(b)7k (10 rad/m)Re(Ψ )B5×10 -3 m(c)xz4×10 -5 mFig. 4.15 A numerical example of a monochromatic Bessel beam with the parametersθ 0 =0.223 deg, k 0 ∼ 1 × 10 7 : (a) The angular spectrum of its plane waves in two perspectives;(b) The delta-shaped frequency spectrum; (c) The spatial amplitude.52

the Bessel beams propagates much further than the Rayleigh range predicts.Note, that though there has been numerous experiments on Bessel beams, they are notrealizable in experiment in the exact form (4.14) – indeed, the analysis of section 3.2.3immediately shows, that this wave field has both infinite total energy and energy flow overits cross-section. We will discuss this point in what follows.In this thesis the Bessel beams appear as the components of the Fourier decompositionin Eqs. (2.15) – (2.17) for example. Later in this thesis we will refer to their most importantproperties in some detail. At this point we just depict its angular spectrum of planewaves and the typical spatial amplitude (see Fig. 4.15).4.4.2 X-pulsesIn [99] Lu et al demonstrated that the choiceA (X)0 (k, θ) =B (k) δ (θ − θ 0 ) (4.16)in representation (2.18) with the frequency spectrumB (k) = 1 k 2 exp [−ka 0] (4.17)yields the propagation-invariant wave field with the following spatial amplitude:a 0Ψ X (ρ, z, t) =(4.18)q(ρ sin θ 0 ) 2 +[a 0 − i (z cos θ 0 − ct)] 2(see Ref. [102] for the description of higher order X-pulses). From the angular spectrumin Eq. (4.16) it can be seen, that the support of angular spectrum of plane waves of theX-pulses is a cone in k-space, i.e., all the plane wave components of the wave field propagateat the equal angle from the propagation axis. The frequency spectrum of X-pulses inEq. (4.16) is uniform (see Fig. 4.16a) – the immediate conclusion of the approach of section3.1.5 that the corresponding amplitude should have exponentially decaying behaviourin both z axis and xy plane is confirmed in Fig. 4.16c.X-wave fields have been further investigated in Refs. [100–103], recently the topichave been given an overview and general description in Ref [106]. We mention here theso called bowtie waves that are generally introduced as the derivatives of the X-waves:Ψ mX (ρ, z, t) = ∂m Ψ 0 (r,t)∂x m . (4.19)The derivatives of X-waves have been shown to possess non-symmetric nature and haveextended localization along a radial direction. In our terms the physical nature of suchwave fields can be interpreted by applying the derivation operation on the general angular53

84k x(10 rad/m)12 3 4 57k z(10 rad/m)~B(k)(a)θF(k)degarb. units0.212 3 4 5(b)7k (10 rad/m)Re(Ψ )Xarb. units5×10 -3 m(c)xz4×10 -5 mFig. 4.16 A numerical example of a X-pulse with the parameters θ 0 =0.223 deg, γ =0.99999:(a) The angular spectrum of plane waves in two perspectives; (b) The frequency spectrum of theX-pulse (black line), the frequency spectrum of an optically feasible wave field(greenline),theangle θ 0 as the (constant) function of the wave number (dashed blue line); (c) The spatial amplitudeof the pulse.54

spectrum representation of the free-space scalar wave fields in Eq. (2.15). We easily get∂ m Ψ 0 (r,t)∂x m = 1 Z ∞ Z πZ 2π(2π) 4 dk k 2 dθ (sin θ) m+1 dφ cos m φ000× A (k sin θ cos φ, k sin θ cos φ, k cos θ)× exp [ik (x sin θ cos φ + y sin θ cos φ + z cos θ − iωt)] , (4.20)so that the angular spectrum of plane waves of such wave fields is not cylindrically symmetric,correspondingly the wave field is a superposition of higher order monochromaticBessel beams as described by Eq. (2.19) for example.Due to the exponential shape of the frequency spectrum the X-waves are not appropriatefor optical implementation.4.4.3 Bessel-X pulsesThe Bessel-X pulses were introduced by Saari in Ref. [115, 116] as the bandlimited versionof X-pulses. Their angular spectrum of plane waves can be described aswhere˜B (k) =à (BX)0 (k, θ) = ˜B (k) δ (θ − θ 0 ) , (4.21)σ k√2πrkk 0exp"− σ2 k (k − k 0) 22#, (4.22)σ k being defined in (3.37) and k 0 being the carrier wave number, so that for the fieldamplitude one can writeΨ BX (ρ, z, t) =Z ∞0dk ˜B (k)×J 0 [kρ sin θ 0 ]exp[−ik (z cos θ 0 − ct)] . (4.23)The integration in Eq. (2.18) can be carried out to yield [116]whereandΨ BX (ρ, z, t) = p Z (d)·× exp − 1 ¡ 2¢¸ρ 22σ 2 sin 2 θ + d J 0 [Z (d) ρk 0 sin θ]exp[ik 0 d] , (4.24)kZ (d) =1+idk 0 σ 2 k(4.25)d = z cos θ − ct. (4.26)From the Eqs. (4.21) and (4.22) it can be seen that,again, the support of angular spectrumof plane waves is a cone in k-space (see Fig. 4.17a). However, unlike the X-pulses,the frequency spectrum of Bessel-X pulse is Gaussian and it can be optimized to approximatethat of our initial conditions. Thus, the Bessel-X pulses are optically feasible in the55

84k x(10 rad/m)12 3 4 57k z(10 rad/m)~B(k)(a)θ(k)deg1arb. units0.512 3 4 5(b)7k (10 rad/m)Re(Ψ ) BXarb. units4×10 -3 m(c)xz4×10 -5 mFig. 4.17 A numerical example of a Bessel-X pulse with the parameters θ 0 = 0.223 deg,γ =0.99999: (a) The angular spectrum of plane waves in two perspectives; (b) The frequencyspectrum of the Bessel-X pulse (black line) as compared to the frequency spectrum of an opticallyfeasible wave field (green line), the angle θ 0 as the (constant) function of the wavel number (dashedblue line); (c) The spatial amplitude of the pulse.56

sense defined in this chapter.4.5 Two limiting cases of the propagation-invariance4.5.1 Pulsed wave fields with infinite group velocityConsider the special case γ =0of the support of angular spectrum of plane waves (3.8)that readsk z (k) =ξ = const. (4.27)From the general definition of group velocity in Eq. (3.4) it is obvious, that for this particularcase we have v g = ∞. In what follows we give a physical description to the wavefields that have such a peculiar property.A closed-form solution of the homogeneous scalar wave equation that obeys (4.27) canbe easily found. The angular spectrum of plane waves of the wave field reads· µ ¸A (fi)ξ0 (k, θ) =B (k) δ θ − arccos . (4.28)kThe substitution in Whittaker type superposition (2.18) yields⎛ s ⎞Z ∞µ 2Ψ fi (ρ, z, t) = dk ˜B ξ(k) J 0⎝kρ 1 − ⎠0k· µ ¸ ξ× exp ikk z − ct , (4.29)whereso thatΨ fi (ρ, z, t) =exp[iξz]µ 2 ξ˜B (k) =ks1 2 − B (k) , (4.30)kZ ∞0dk ˜B (k) J 0µρqk 2 − ξ 2 exp [−ikct] . (4.31)If we choose ˜B (k) =const =1and use the integral transforms [157]Z ∞0if 0

time84k x(10 rad/m)12 3 4 57k z(10 rad/m)~B(k)(a)θ(k)deg80arb. units4012 3 4 5(b)7k (10 rad/m)Re(Ψ )fit = -1fsarb. unitst = 0fszt = 1fs2×10 -6 m(c)x1.2×10 -5 m(d)t = 2fsFig. 4.18 A numerical example of a the wave fieldwithinfinite group velocity with the parametersξ =6.7 × 10 6 m, γ = ∞: (a) The angular spectrum of plane waves in two perspectives; (b) Thefrequency spectrum of the pulse (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 pulse; (d) Three snapshots of the temporal evolution of the pulse.58

the integral (4.31) can be evaluated explicitly to yield⎧⎪⎨Ψ fi (ρ, z, t) =⎪⎩exphiξz−iξ √ iρ 2 −c 2 t√ 2 if 0

4k x(10 rad/m)12 3 4 57k z(10 rad/m)8arb. units~B(k)(a)θ(k)deg1arb. units0.512 3 4 5(b)7k (10 rad/m)Re(Ψ ) b5×10 -3 m(c)xz4×10 -5 mFig. 4.19 A numerical example of a the pulsed wave field with frequency independent beamwidthwith the parameters α 0 =4× 10 4 rad , γ ∼ 0: (a) The angular spectrum of plane waves in twomperspectives; (b) The frequency spectrum of the pulse (black line), the frequency spectrum of anoptically feasible wave field (green line), the cone angle of the component Bessel beams as thefunction of the wave number (dashed blue line); (c) The spatial amplitude of the pulse.60

The support of the angular spectrum of plane waves of the wave field Eq. (4.37) isdepicted in Fig. 4.19 (in the numerical example the frequency spectrum B (k) is Gaussianwith the bandwidth corresponding to ∼ 6fs pulse). Using the approach of section 3.1.5one can immediately tell the general spatial shape of such wave fields. Indeed, in thiscase we have a simple special case, where the projection of the angular spectrum of planewaves onto the k x k y -plane is delta-ring, correspondingly, the transversal amplitude atz =0, t =0should be of the shape of the Bessel function. As for longitudinal shape, itsenvelope is determined by the bandwidth by it Fourier transform, i.e., we should have aslice of a Bessel beam. The numerical simulation in Fig. 4.19d shows that this estimate istrue. Also, one can see that the wave field has generally infinite energy flow.Comparing the support in Eq. (4.36) to that of the propagation-invariant pulsed wavefield in Eq. (3.6) and (3.12) one can see, that the wave field (4.37) is not propagationinvariant.Consequently, the localized part of the wave field spreads as it propagates. Wecan also suggest the best condition for limited propagation-invariance – the comparisonof the support in Fig. 4.19a to those of FWM’s in Fig. 3.2 implies that for restrictedbandwidths the support (4.35) could be optimized to approximate the "horizontal" part ofthe ellipsoidal supports of the subluminal FWM’s (γ >1).4.6 Physically realizable approximations to FWM’sAs it was explained in section 3.1.5, the presence of the delta function in the support of theangular spectrum of plane waves of the free-space scalar wave fields necessarily resultsin infinite total energy content of the wave field. Consequently, for all the above reviewedwave fields the total energy content is infinite,U tot =Z ∞−∞dzZ ∞0dρρZ 2π0dϕ |Ψ F (ρ, z, ϕ, t)| 2 = ∞. (4.39)Here we proceed by reviewing the approaches used in literature to overcome this difficulty.In later chapters we introduce the approach that is especially useful for analyzingoptical experiments.4.6.1 Electromagnetic directed-energy pulse trains (EDEPT)One approach has been to construct various continuous superpositions of FWM’s (4.1)over the parameter β (see Refs. [6, 14, 15, 31] and references therein), in this case onewritesΨ LW (z,ρ,t) ==Z ∞0dβΛ (β) Ψ F (z,ρ,t; β)a 14πi (a 1 + iζ)Z ∞0dβΛ (β)exp[s (z, ρ, t)] , (4.40)61

k xk zΛ(β)k xk z(a)k xk zk xΛ(β)kz(b)Fig. 4.20 On the effect of integrating over the parameter β on the support of the angular spectrumof plane waves of the FWM’s: (a) The special case of the "mean" support of the angular spectrumof plane waves where γ =1(v g = c), β 6= 0; (b) The special case where γ>0 (v g

wheres (z,ρ,t) =−βρ2 + iβ (z + ct) (4.41)a 1 + iζΛ (β) is a weighting function and the subscript LW means "localized wave". As the supportsof the angular spectrum of the FWM’s for different values of parameter β generallydo not overlap and change smoothly in k-space, the integration indeed eliminates the deltafunction in the expression for angular spectrum of plane waves (see Fig. 4.20). It can beshown [31] that Eq. (4.40) yields finite total energy wave field if only the function Λ (β)satisfies conditionZ1 ∞dβ |Λ (β)| 2 12a 1 0β < ∞ (4.42)(see Eq. 2.8 of Ref. [31]), i.e., if only β −1/2 Λ (β) is square integrable. The LW’s of thegeneral form (4.40) have been called EDEPT solutions of the scalar wave equation.From the discussion of previous chapters it is obvious, that the wave field of the generalform (4.40) are not strictly propagation-invariant. At first glance it may seem surprisingbecause (i) FWM solutions with different values of parameter β do travel without anyspread and (ii) all the FWM’s overlap in every space-time point as their group velocitiesare equal. However, the effect can be easily understood if we recollect from section 3.3.1that the phase velocities of the pulses are different leading to the z axis position dependentinterference and spread of the superposition of the component pulses (see Ref. [3, 4] foralternate proofs of this claim).4.6.1.1 Modified power spectrum pulse (MPS)The modified power spectrum pulses [31] have been introduced by the following bidirectionalplane wave spectrum (see Eq. (3.3) of Ref. [31] and Eq. (3.32) of Ref. [14])⎧h ³ ³ ´³C (m)0 ˜α, ˜β,χ´ ⎨ p(p˜β−b) q−1= 2πΓ(q)exp − ˜αa 1 + p˜β − b a 2´i, if ˜β > b p⎩ 0 ,if b p > ˜β. (4.43)≥ 0Here a 2 , b, q and p are new parameters and Γ denotes the gamma function. Using therelations (2.24a) and (2.24b) the corresponding Whittaker type plane wave spectrum canbe written as (see also Eq. (3.13a) and (3.13b) of Ref. [31])Ã (MPS)0 (χ, k z )=⎧⎪⎨⎪⎩p[ p 2 (k−kz)−b]q−12πΓ(q)exph³− (k+kz)a12+,ifk z < p b´ ib − (k−kz)p a 22χ 24 − b pχ 24 − b p0 ,ifk z > p b(4.44)where k = p χ 2 + kz 2 and the relation (2.26) has been used. The spatial amplitude distri-,63

8arb. units4k x(10 rad/m)12 3 4 57k z(10 rad/m)(a)C0(0,β, χ)Re(Ψ MPS)40(b)β (rad/m)80z5×10 -3 m(c)x4×10 -5 mFig. 4.21 A numerical example of a MPS with the parameters a 1 =1.4 × 10 −7 m, a 2 =4000m,q =10, p =0.0001, b =0.002, (γ =1, β 0 =40 rad ): (a) The angular spectrum of plane wavesmin two perspectives; (b) The β -distribution in bidirectional plane wave spectrum for α =0;(c)Thespatial amplitude of the MPS for t =0.64

ution of the MPS’s is described by equation (see Eqs. (3.34) and (1.4) of Ref. [14])⎡³ ´ ⎤Ψ MPS (ρ, ζ, η) = ⎣1 exp − bs p³ ⎦ ´q , (4.45)4π (a 1 + iζ)a 2 + s pwhereρ 2s =− iη. (4.46)4π (a 1 + iζ)The comparison of the bidirectional plane wave spectra of MPS (4.43) with that of theFWM’s (4.2) one can see, that the latter is a special case a 2 =0, q =1of the former.Consequently, the parameter a 1 in (4.43) has the same interpretation as in case of FWM’s– it determines the frequency spectra of the wave field. From (4.44) it is also obvious thatthe parameter a 2 determines the width of the β distribution and parameter b determinesthe central value of β. As for parameter q, it can be used to optimize the shape of the βdistribution.A numerical example of the MPS is depicted in Fig. 4.21. In this example we tried tooptimize the parameters so as to satisfy the conditions for optical feasibility as stated inthe introduction of this overview. From the angular spectrum of plane waves in Fig. 4.21aone can see, that the MPS’s generally have the same inconvenience as FWM’s – there nofreedom to choose the frequency spectrum as to optimize for any convenient light sourceand they are generally half-cycle pulses.For an interpretation of MPS’s as being the field generated by a combined point-likesource and a sink placed at a complex-number coordinate see Refs. [139, 141]It is not our aim at this point to study the temporal behaviour of the EDEPT solutions,thus, the spatial amplitude of the wave fields is given only for the time t =0.4.6.2 Splash pulsesSplash pulses [6] appear if one chooses the bidirectional plane wave spectrum as (Eq. (3.13)of Ref. [14])³C (SP)0 ˜α, ˜β,χ´= π 2 ˜βhq−1exp −³˜αa 1 + 2´i˜βa . (4.47)One can see, that the bidirectional spectrum is similar in the structure as the one of theMPS (4.43). Here again the term exp [−˜αa 1 ] can be interpreted as the spectra of the"central" FWM and the parameters a 2 and q determine the distribution function over theparameter ˜β. The integration in the bidirectional plane wave decomposition (2.22) can becarried out to yield (Eq. (3.19) of Ref. [14], Eq. (17) of Ref. [6])·¸Γ (q)1Ψ SP (ρ, ζ, η) =(a 2 − iη)+(4.48)4π (a 1 + iζ)(a 1 + iζ)The wave field has been called as "splash pulse" in Ref. [6] as for its characteristicspatial shape. However, in our numerical example we tried once more to find a set ofparameters suitable for optical generation. It appeared (see Fig. 4.22), that in this case the65

4k x(10rad/m)12 3 4 57k z(10 rad/m)(a)C (0,β, χ)08arb. unitsRe(Ψ )SP40(c)β (rad/m)80z5×10 -3 m(c)x4×10 -5 mFig. 4.22 A numerical example of a Splash mode with the parameters a 1 = 1.4 × 10 −7 m,a 2 =0.4m, q =16,(γ =1, β 0 =40 rad ): (a) The angular spectrum of plane waves in twomperspectives; (b) The β -distribution in bidirectional plane wave spectrum for ˜α =0;(c)Thespatialamplitude of the Splash mode for t =0.66

angular spectrum of plane waves is very similar to that of the MPS’s as in Fig. 4.21.4.7 Several more LW’sTo date, the literature on LW’s and on propagation of ultrashort electromagnetic pulses isoverwhelming and this overview is by no means complete. Our aim was to demonstratethe applicability of our approach on most important special cases.As already mentioned, in Ref. [35] Besieris et al derived several closed-form superluminaland subluminal LW solutions to the scalar wave equation by "boosting" knownsolutions of other Lorentz invariant equations.In section 3.3.4, we already reviewed the approach of solving the homogeneous scalarwave equation and Klein-Gordon equation, introduced in Refs. [15, 16] by Donnelly andZiolkowski. In those works, they also deduced various closed-form separable and nonseparablesolutions to the wave equation.In Ref. [32] Overfelt found a continua of localized wave solutions to the scalar homogeneouswave, damped wave, and Klein-Gordon equations by means of a complexsimilarity transform technic.The numerous publications that are involved with ultrashort-pulse solutions of the timedependentparaxial wave equation (e.g. isodiffracting pulses) should be mentioned here(see Refs. [127–141] and references therein).4.8 On the transition to the vector theoryEven though the scalar theory is often used in description of propagation of electromagneticwave fields, generally the solutions to the Maxwell equations have to be used. However,the latter approach is generally much more involved.In the context of this thesis, we investigate the free-space wave fields and mostly usethe angular spectrum representation of the wave fields. In this context the limitations ofthe scalar theory can be easily formulated – the scalar theory is reasonably accurate if onlythe plane wave components of the wave field propagate at small (paraxial) angles relativeto optical axis (see Wolf and Mandel [158], for example). In this thesis we investigatethe possibilities of optical generation of FWM’s (and LW’s), correspondingly, the aboveformulated restriction is satisfied in all practical cases and we can restrict ourselves toscalar theory.(Of course, this is not the case with the original FWM’s and LW’s published in literature– the plane wave components of those wave fields propagate even perpendicularly to thedirection of propagation and in their exact description the transition to the vector theoryis obligatory.)The vector theory of FWM’s and LW’s has been formulated and used in several publications[1–3,8,31,104,123,125,126]. The preferred approach has been the use of the Hertzvectors as formulated in Eqs. (2.7a) and (2.7b). One can refer to the theory expounded by67

Ziolkowski [31] where he used the Hertz vectors of the formΠ (e) = z Ψ F (4.49a)Π (m) = z Ψ F , (4.49b)where z is the unit vector along the propagation axis and Ψ F is the (localized) solution ofthe scalar wave equation. With (4.49a) and (4.49b) one get TE or TM field with respectto z respectively. A more general treatment can be found in Ref. [126], where the Hertzvectors are written as the superpositions of the solutions of scalar wave equation Ψ i asΠ (m) = x X pa p(m) Ψ p + y X qb q(m) Ψ q + z X sc (m)s Ψ s (4.50a)Π (e) = x X pa p(e) Ψ p + y X qb q(e) Ψ q + z X sc (e)s Ψ s . (4.50b)The computation of the field components using (2.7a) and (2.7b), although straightforward,results in very complex formulas.The intuitive analysis of the effect of the transfer to exact vector theory that is more inthe spirit of this thesis can be carried out in terms of the results that have been publishedon vector Bessel beams in Refs. [120,121,124,164]. For example, the result in Ref. [164]reveals, that for TE and TM fields the vector Bessel beams retain their paraxial-Besselbeam nature up to cone angles ∼ 14 ◦ and this result indeed amply justifies the use of thescalar theory in this thesis.4.8.1 The derivation of vector FWM’s by directly applying the Maxwell’sequationsTo finish this chapter we nevertheless advance in some extent the second approach mentionedin Sec. 2.1, where we gave the general expression for the plane wave decompositionof the solution of the free-space Maxwell equations.To find the vector form for the FWM’s as described in Eq. (3.14) we use the Eqs. (2.3a)– (2.5c). In correspondence with Eq. (3.7) we chooseE x (k,ω) = Ẽx (k, φ) δ [k z − k cos θ F (k)] (4.51)E y (k,ω) = Ẽy (k, φ) δ [k z − k cos θ F (k)] (4.52)and this choice yields from Eq. (2.3a) and (2.3b)E i (r,t) =14exp [−2iβγz] (4.53)(2π)Z 2π Z ∞× dφ dkk 2 sin θ F (k) Ẽi (k, φ) ×0 −∞× exp [ik (x sin θ F (k)cosφ + y sin θ F (k)sinφ + γz − ct)] ,68

H i (r,t) =14exp [−2iβγz] (4.54)(2π)Z 2π Z ∞× dφ dkk 2 sin θ F (k) ˜H i (k, φ)0 −∞× exp [ik (x sin θ F (k)cosφ + y sin θ F (k)sinφ + γz − ct)] ,where from Eqs. (2.4) – (2.5c)Ẽ z (k, φ) =− tan θ F (k)hicos φẼx (k, φ)+sinφẼy (k, φ)(4.55)and1h˜H x (k, φ) = −sin 2 θ F (k)sinφ cos φẼx (k, φ)+cµ 0 cos θ F (k)+ ¡ 1 − sin 2 θ F (k)cos 2 φ ¢ iẼ y (φ, k)(4.56)˜H y (k, φ) =kh ¡1− sin 2 θ F (k)sin 2 φ ¢ Ẽ x (k, φ)+cµ 0 cos θ F (k)i+sin 2 θ F (k)sinφ cos φẼy (k, φ)(4.57)˜H z (k, φ) = 1 hisin θ F (k) sin φẼx (k, φ) − cos φẼy (k, φ) . (4.58)cµ 0By expanding∞XẼ i (k, φ) = Ẽ i (k, n)exp[inφ] , (4.59)wheren=−∞Z 2πẼ i (k, n) = 1 dφẼi (k, φ)exp[−inφ] , (4.60)2π 0the integration over the φ can be carried out to yieldE i (r,t) =Z ∞1(2π) 2 exp [−2iβγz] X dkn −∞× exp [inφ] L E i (k, ρ, n)exp[ik (γz − ct)] (4.61)H i (r,t) =Z ∞1(2π) 2 exp [−2iβγz] X dkn −∞× exp [inφ] L H i (k, ρ, n)exp[ik (γz − ct)] , (4.62)whereL E x (k, ρ, n) =k2 sin θ F (k) Ẽ x (k, n) J n (kρ sin θ F (k)) (4.63)and L E z (k, n, ρ), L H iL E y (k, ρ, n) =k 2 sin θ F (k) Ẽy (k, n) J n (kρ sin θ F (k)) (4.64)(k, ρ, n) can be expressed as the linear combinations of Bessel func-69

tions of different order. (see Refs. [121, 164, 172] for relevant discussions).We also note, that in addition to the TM and TE wave fields azimuthally polarized,radially polarized and circularly polarized vector FWM’s can be derived [164].4.9 Conclusions.The main conclusion of this section are:1. At this point it should be clear, that all the possible closed-form FWM’s can be analyzedin a single framework where the support of the angular spectrum of plane waves(3.6) – (3.12) is the only definitive property for propagation-invariance. The questionof whether an integration over the support has or has not a closed-form result is thequestion of mathematical convenience only.2. With a proper choice of parameters some of the closed form FWM’s (Bessel-Gausspulses, Bessel-X) are well suited for use as the models for simulating the result ofoptical experiments. In contrary, the LW’s we reviewed here – the MPS’s, splashpulses and the original FWM’s – are not feasible in this context. Mostly it is becauseof the ultra-wide bandwidth and non-paraxial angular spectrum content of the pulses.3. In our opinion, the procedure of modeling finite-thickness supports for finite energyapproximations of FWM’s reviewed in this section lacks a convenient physical interpretationand to estimate its practical value this topic has to be addressed in the contextof a particular launching setup instead.70