Propagation of localized optical waves in media 8 0 with dispersion, in dispersionless media and in 0 2 vacuum. Low diffractive regime n a J Lubomir M. Kovachev 4 Institute of Electronics, Bulgarian Academy of Sciences, 1 Tzarigradcko shossee 72,1784 Sofia, Bulgaria ] s February 2, 2008 c i t p o Abstract . s Wepresentasystematicstudyonlinearpropagationofultrashortlaser c pulses in media with dispersion, dispersionless media and vacuum. The i s applied methodofamplitudeenvelopesgivestheopportunitytoestimate y the limits of slowly warring amplitude approximation and to describe an h amplitudeintegro-differentialequation,governingthepropagationofopti- p calpulsesinsinglecycleregime. Thewellknownslowlyvaryingamplitude [ equationandtheamplitudeequationforvacuumarewrittenindimension- 1 lessform. Threeparametersareobtaineddefiningdifferentlinearregimes v oftheopticalpulsesevolution. Incontrasttopreviousstudieswedemon- 1 strate that in femtosecond region the nonparaxial terms are not small 8 and can dominate over transverse Laplacian. The normalized amplitude 0 nonparaxialequationsaresolvedusingthemethodofFouriertransforms. 2 FundamentalsolutionswithspectralkernelsdifferentfromFresneloneare . 1 found. One unexpected new result is the relative stability of light pulses 0 with sphericalandspheroidalspatial form,whenwecompare theirtrans- 8 verse enlargement with the paraxial diffraction of lights beam in air. It 0 is important to emphasize here the case of light disks, i.e. pulses whose : v longitudinalsizeissmallwithrespecttothetransverseone,whichinsome i partialcasesarepracticallydiffractionlessoverdistancesofthousandkilo- X meters. A new formula which calculates the diffraction length of optical r pulses is suggested. a 1 Introduction For long time few picosecond or femtosecond (fs) optical pulses with approxi- mately equaldurationin the x, y andz directions(Light Bullets orLB), andfs opticalpulses withrelativelylargetransverseandsmalllongitudinalsize (Light DisksorLD)areusedintheexperiments. TheevolutionofsogeneratedLBand LD in linear or nonlinear regime is quite different from the propagationof light 1 beams and they have drawn the researchers’ attention with their unexpected dynamical behavior. For example, self-channeling of femtosecond pulses with power little above the critical for self-focusing [1] and also below the nonlinear collapse threshold [2] (linear regime) in air, was observed. This is in contra- diction with the well known self-focusing and diffraction of an optical beam in the frame of paraxial optics. Various unidirectional propagation equations have been suggested to be found stable pulse propagation mainly in nonlinear regime(seee.g. MoloneyandKolesik[3],CouaironandMysyrowicz[4],Chinat all. [5], for a review). The basic studies in this field started with the so called spatio-temporal nonlinear Schr¨odinger equation (NSE) which is one compila- tionbetween paraxialapproximation,the groupvelocity dispersion(GVD) and nonlinearity[6,7,8,9]. Theinfluenceofadditionalphysicaleffectswerestudied by adding different terms to this scalar model as small nonparaxiality [15, 18], plasma defocussing, multiphoton ionization and vectorial generalizations. It is nothardtoseethatforpulseswithlowintensity(linearregime)inairandgases the additionalterms asGVD andothersbecome smallandthe basicmodelcan bereducedtoparaxialequation. Thisisthereasondiffractionofalowintensity opticalpulsegovernedby thismodelonseveraldiffractionlengthtobe equalto diffractionof a laserbeam. Onother hand, the experimentalists have discussed for a long time that in their measurements the diffraction length of an optical pulse is not equal to this of a laser beam zbeam = k r2, even when additional diff 0 ⊥ phase effects of lens and other optical devices can be reduced. Here k denotes 0 laser wave-number and r2 denotes the beam waist. Thus exist one deep differ- ⊥ ence between the existing models in linear regime, predicting paraxialbehavior in gases, and the real experiments. Thepurposeofthisworkistoperformasystematicstudyoflinearpropaga- tion of ultrashort optical pulses in media with dispersion, dispersionless media and vacuum and to suggest a model which is more close to the experimental results. In addition, there are several particular problems under consideration in this paper. The first one is to obtain (not slowly varying) amplitude envelope equation inmediawithdispersiongoverningtheevolutionofopticalpulsesinsingle-cycle regime. This problemis naturalinfemtosecondregionwherethe opticalperiod of a pulse is of order 2 3 fsec. The earliest model for pulses in single-cycle − regime suggestedby Brabec and Krausz [20] is obtained after Taylorexpansion of wave vector k2(ω) about ω . It is easy to show [21] that this expansion 0 divergein solidsfor single-cyclepulses. The higher orderdispersionterms start to dominate and the series can not cut off. This is the reason more carefully and accurately to derive the envelope equation before using Taylor series. In this way we obtain an integro-differential envelope equation where no Taylor expansion of the wave vector k2(ω), governing evolution of single cycle pulses in solids. The second problem is to investigate more precisely the slowly varying en- velope equations governing the evolution of optical pulses with high number of harmonics under the envelope. The slowly varying scalar Nonlinear Envelope Equation(NEE)isderivedinmanybooksandpapers[11,12,13,14,15,16,17]. 2 After the deriving of the NEE, most of the authors use a standard procedure to neglect the nonparaxial terms as small ones. Only some partial nonparaxial approximationsinfree space[10, 15,18]andopticalfibers [19]werestudied. In [22]werewritetheNEEindimensionlessformandestimatetheinfluence ofthe differentlinearandnonlineartermsontheevolutionofopticalpulses. Wefound thatbothnonparaxialtermsinNEE,secondderivativeinpropagationdirection and second derivative in time with 1/v2 coefficient, are not small corrections. InfsregiontheyareofsameorderastransverseLaplacianorstarttodominate. These equations with (not small) nonparaxialterms are solved in linear regime [22] and investigated numerically in nonlinear [23]. In this paper we include GVD term in the nonparaxial model and study also the envelope equation of electricalfieldin vacuumanddispersionlessmedia. Itis importantto note that the Vacuum Linear Amplitude Equation (VLAE) is obtained without any ex- pansion of the wave vector. That is why it work also for pulses in single-cycle regime (subfemto and attosecond pulses). Last but not least the nonparaxialequations for media with dispersion, dis- persionlessmedia andvacuumaresolvedinlinearregimeandnew fundamental solutions, including the GVD, are found. The solutions of these equations pre- dict new diffraction length for optical pulses zpulse = k2r4/z , where z is the diff 0 ⊥ 0 0 longitudinal spatial size of the pulse (the spatial analog of the time duration t ; z = vt ; v is group velocity). In case of fs propagation in gases and vac- 0 0 0 uum we demonstrate by these analytical and numerical solutions a significant decreasingofthediffractionenlargementinrespecttoparaxialbeammodeland a possibility to reach practically diffraction-free regime. 2 From Maxwell’s equations of a source-free, dis- persive, nonlinear Kerr type medium to the amplitude equation The propagation of ultra-short laser pulses in isotropic media, can be charac- terized by the following dependence of the polarization of first P~ and third lin P~ order on the electrical field E~: nl t P~ = δ(τ t)+4πχ(1)(τ t) E~ (τ,r)dτ = lin − − −Z∞ (cid:16) (cid:17) t ε(τ t)E~ (τ,x,y,z)dτ, (1) − −Z∞ 3 t t t P~(3) =3π χ(3)(τ t,τ t,τ t) nl 1− 2− 3− −Z∞−Z∞−Z∞ E~(τ ,r) E~∗(τ ,r) E~(τ ,r)dτ dτ dτ , (2) 1 2 3 1 2 3 × · (cid:16) (cid:17) whereχ(1) andεarethelinearelectricsusceptibilityandthedielectricconstant, χ(3) is the nonlinear susceptibility of third order, and we denote r = (x,y,z). We use the expression of the nonlinear polarization (2), as we will investigate only linearly or only circularly polarized light and in addition we neglect the third harmonics term. The Maxwell’s equations in this case becomes: 1∂B~ E~ = , (3) ∇× −c ∂t 1∂D~ H~ = , (4) ∇× c ∂t D~ =0, (5) ∇· B~ = H~ =0, (6) ∇· ∇· B~ =H~, D~ =P~ +P~ , (7) lin nl where E~ and H~ are the electric and magnetic fields strengths, D~ and B~ are the electric and magnetic inductions. We should point out here that these equations are valid when the time duration of the optical pulses t is greater 0 thanthe characteristicresponsetime ofthe mediaτ (t >>τ ), andalsowhen 0 0 0 the time duration of the pulses is of the order of time response of the media (t τ ). Taking the curl of equation (3) and using (4) and (7), we obtain: 0 0 ≤ 1 ∂2D~ E~ ∆E~ = , (8) ∇ ∇· − −c2 ∂t2 (cid:16) (cid:17) where ∆ 2 is the Laplace operator. Equation (8) is derived without using ≡ ∇ the third Maxwell’s equation. Using equation (5) and the expression for the linearand nonlinearpolarizations(1) and(2), we canestimate the secondterm in equation (8) for arbitrary localized vector function of the electrical field. It is not difficult to show that for localized functions in nonlinear media with and without dispersion E~ =0 and we can write equation (8) as follows: ∇· ∼ 1 ∂2D~ ∆E~ = . (9) c2 ∂t2 4 We will now replace the electrical field in linear and nonlinear polarization on the right-hand side of (9) with it’s Fourier integral: +∞ E~ (r,t)= Eˆ~ (r,ω)exp( iωt)dω, (10) − −Z∞ where with Eˆ~ (r,ω) we denote the time Fourier transformof the electrical field. We thus obtain: t ∞ ∆E~ = 1 ∂2 ε(τ t)Eˆ~ (r,ω)exp( iωτ)dωdτ + c2∂t2  − −  −Z∞−Z∞   (11) t t t ∞ 3π ∂2 χ(3)(τ t,τ t,τ t) Eˆ~ (r,ω) 2Eˆ~ (r,ω) c2 ∂t2 1− 2− 3− −Z∞−Z∞−Z∞−Z∞ (cid:12) (cid:12) (cid:12) (cid:12) exp( i(ω(τ τ +(cid:12)τ )))dωd(cid:12)τ dτ dτ . 1 2 3 1 2 3 × − − The causality principle imposes the following conditions on the response func- tions: ε(τ t)=0; χ(3)(τ t,τ t,τ t)=0, 1 2 3 − − − − τ t>0; τ t>0; i=1,2,3. (12) i − − That is why we can extend the upper integralboundary to infinity and use the standard Fourier transform [13]: t +∞ ε(τ t)exp( iωτ)dτ = ε(τ t)exp( iωτ)dτ, (13) − − − − −Z∞ −Z∞ t t t χ(3)(τ t,τ t,τ t)dτ dτ dτ = 1 2 3 1 2 3 − − − −Z∞−Z∞−Z∞ +∞+∞+∞ χ(3)(τ t,τ t,τ t)dτ dτ dτ . (14) 1 2 3 1 2 3 − − − −Z∞−Z∞−Z∞ Thespectralrepresentationofthelinearopticalsusceptibilityεˆ(ω)isconnected 0 to the non-stationary optical response function by the following Fourier trans- form: 5 +∞ εˆ(ω)exp( iωt)= ε(τ t)exp( iωτ)dτ. (15) − − − −Z∞ The expression for the spectral representation of the non-stationary nonlinear optical susceptibility χˆ(3) is similar : +∞+∞+∞ χˆ(3)(ω)exp( iωt)= χ(3)(τ t,τ t,τ t) 1 2 3 − − − − −Z∞−Z∞−Z∞ exp( i(ω(τ τ +τ )))dτ dτ dτ . (16) 1 2 3 1 2 3 × − − Thus, after brief calculations, equation (11) can be represented as ∞ ∆E~ = ω2εˆ(ω)Eˆ~ (r,ω)exp( iωt)dω − c2 − −Z∞ ∞ + ω2χˆ(3)(ω) Eˆ~ (r,ω) 2Eˆ~ (r,ω)exp( i(ωt))dω. (17) c2 − −Z∞ (cid:12) (cid:12) (cid:12) (cid:12) We now define the square o(cid:12)f the lin(cid:12)ear k2 and the generalized nonlinear kˆ2 nl wave vectors, as well as the nonlinear refractive index n with the expressions: 2 ω2εˆ(ω) k2 = , (18) c2 3πω2χˆ(3)(ω) kˆ2 = =k2n , (19) nl c2 2 where 3πχˆ(3)(ω) n (ω)= . (20) 2 εˆ(ω) The connection between the usual dimensionless nonlinear wave vector k2 and nl the generalized one (19) is: k2 = kˆ2 Eˆ~ (r,ω) 2. In terms of these quantities, nl nl equation (17) can be expressed by: (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ∞ ∆E~ = k2(ω)Eˆ~ (r,ω)exp( iωt)dω − − −Z∞ ∞ k2(ω)n (ω) Eˆ~ (r,ω) 2Eˆ~ (r,ω)exp( i(ωt))dω. (21) 2 − − −Z∞ (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 6 Let us introduce here the amplitude function A~(r,t) for the electrical field E~(r,t): E~ (x,y,z,t)=A~(x,y,z,t)exp(i(k z ω t)), (22) 0 0 − where ω and k are the carrier frequency and the carrier wave number of the 0 0 wavepacket. The writing of the amplitude function in this formmeans thatwe considerpropagationonlyin+z -directionandneglectthe oppositeone. Letus write here also the Fourier transform of the amplitude function Aˆ~(r,ω ω ): 0 − +∞ A~(r,t)= Aˆ~(r,ω ω )exp( i(ω ω )t)dω, (23) 0 0 − − − −Z∞ and the following relation between the Fourier transform of the electrical field and the Fourier transform of the amplitude function: Eˆ~ (r,ω)exp( iωt)= − exp( i(k z ω t))Aˆ~(r,ω ω )exp(i(ω ω )t), (24) 0 0 0 0 − − − − Since we investigate optical pulses, we assume that the amplitude function and its Fourierexpressionaretime -andfrequency-localized. Substituting (33),(23) and (24) into equation (21) we finally obtain the following nonlinear integro- differential amplitude equation: ∂A~(r,t) ∆A~(r,t)+2ik k2A~(r,t)= 0 ∂z − 0 (25) ∞ k2(ω) 1+n (ω) Aˆ~(r,ω ω ) 2 Aˆ~(r,ω ω )exp( i(ω ω )t)dω 2 0 0 0 − − − − − −Z∞ (cid:18) (cid:12) (cid:12) (cid:19) (cid:12) (cid:12) (cid:12) (cid:12) Equation(25)wasderivedwithonlyonerestriction,namely,thattheamplitude function and its Fourier expression are localized functions. That is why, if we know the analytical expression of k2(ω) and n (ω), the Fourier integral 2 on the right- hand side of (25) is a finite integral away from resonances. In this way we can also investigate optical pulses with time duration t of the 0 order of the optical period T =2π/ω . Generally, using the nonlinear integro- 0 0 differential amplitude equation (25) we can also investigate wave packets with time duration of the order of the optical period, as well as wave packets with a large number of harmonics under the pulse. The nonlinear integro-differential amplitude equation (25) can be written as a nonlinear differential equation for 7 the Fourier transform of the amplitude function Aˆ~, after we apply the time Fourier transformation (23) to the left-hand side of (25) : ∆Aˆ~(r,ω ω )+2ik ∂Aˆ~(r,ω−ω0) 0 0 − ∂z + 1+n (ω) Aˆ~(r,ω ω ) 2 k2(ω) k2(ω ) Aˆ~(r,ω ω )=0. (26) 2 − 0 − 0 0 − 0 (cid:18)(cid:18) (cid:12) (cid:12) (cid:19) (cid:19) (cid:12) (cid:12) We should note he(cid:12)re the well-kn(cid:12)own fact that the Fourier component of the amplitude function in equation (26) depends on the spectral difference ω = △ ω ω , rather than on the frequency, as is the case for the electrical field. 0 − 3 From amplitude equation to the slowly vary- ing envelope approximation (SVEA) Equation (25) is obtained without imposing any restrictions on the square of the linear k2(ω) and generalized nonlinear kˆ2 = k2(ω)n (ω) wave vectors. To nl 2 obtainSVEA,wewillrestrictourinvestigationtothecaseswhenitispossibleto approximatek2andkˆ2 asapowerserieswithrespecttothefrequencydifference nl ω ω as: 0 − ω2εˆ (ω) ∂ k2(ω ) k2(ω)= 0 =k2(ω )+ 0 (ω ω ) c2 0 ∂ω − 0 (cid:0) 0 (cid:1) +1∂2 k2(ω0) (ω ω )2+..., (27) 2 ∂ω2 − 0 (cid:0) 0 (cid:1) kˆ2 (ω)= ω2χˆ(3)(ω) =kˆ2 (ω )+ ∂ kˆn2l(ω0) (ω ω )+... (28) nl c2 nl 0 (cid:16) ∂ω (cid:17) − 0 0 To obtain SVEA in second approximation to the linear dispersion and in first approximation to the nonlinear dispersion, we must cut off these series to the second derivative term for the linear wave vector and to the first derivative term for the nonlinear wave vector. This is possible only if the series (27) and (28) are strongly convergent. Then, the main value in the Fourier integrals in equation (25) yields the first and second derivative terms in (27), and the zero and first derivative terms in (28). The first term in (27) cancels the last term on the left-hand side of equation (25). The convergence of the series (27) and (28) for spectrally limited pulses propagating in the transparent UV and optical regions of solids materials, liquids and gases, depends mainly on the number of harmonics under the pulses [21]. For wave packets with more than 10 harmonics under the envelope, the series (27) is strongly convergent, and the third derivative term (third order of dispersion) is smaller than the 8 second derivative term (second order of dispersion) by three to four orders of magnitude for all materials. In this case we can cut the series to the second derivative term in (27), as the next terms in the series contribute very little to the Fourier integral in equation (25). When there are 2 6 harmonics under − the pulse, the series (27) is weakly convergent for solids and continue to be strongly convergent for gases. Then for solids we must take into account the dispersion terms of higher orders as small parameters. In the case of wave packets with only one or two harmonics under the envelope, propagating in solids, the series (27) is divergent. This is the reason why the SVEA does notgovernthe dynamics of wavepacketswith time durationofthe orderof the opticalperiodinsolids. Substitutingtheseries(27)and(28)in(25)andbearing in mind the expressions for the time-derivative of the amplitude function, the SVEA of secondorder with respect to the linear dispersionand first order with respect to the nonlinear dispersion is expressed in the following form: ∂A~ ∂A~ ∆A~+2ik +2ik k′ = 0 0 ∂z ∂t 2 ∂2A~ 2 ∂ A~ A~ k k”+k′2 kˆ2 A~ A~ 2ikˆ kˆ′ , (29) 0 ∂t2 − 0nl − 0nl 0nl (cid:12)∂(cid:12)t (cid:12) (cid:12) where k0 =(cid:0) k(ω0) and(cid:1) kˆ02nl = kˆn2l(cid:12)(cid:12)(cid:12)(ω0(cid:12)(cid:12)(cid:12)). We will now d(cid:12)efi(cid:12)ne other important constantsconnectedwiththewavepacketscarrierfrequency: linearwavevector k k(ω ) = ω ε(ω )/c; linear refractive index n(ω ) = ε(ω ); nonlinear 0 0 0 0 0 0 ≡ refractive index n (ω )=3πχ(3)(ω )/ε(ω ); group velocity: p2 0 0 0 p 1 c v(ω )= = , (30) 0 k′ ε(ω )+ ω0 1 ∂ε 0 2 ε∂ω p q nonlinear addition to the group velocity (kˆ2 )′: 0nl ′ 2k n ∂n kˆ2 = 0 2 +k2 2, (31) 0nl v 0 ∂ω (cid:16) (cid:17) anddispersionof the groupvelocityk”(ω )=∂2k/∂ω2 .All these quantities 0 ω=ω0 allowadirectphysicalinterpretationandwewillthereforerewriteequation(29) in a form consistent with these constants: 2 ∂ A~ A~ ∂A~ ∂A~ k v∂n i +v + n2+ 0 2 (cid:18)(cid:12) (cid:12) (cid:19)= − ∂t ∂z 2 ∂ω (cid:12)∂(cid:12)t (cid:18) (cid:19) (cid:12) (cid:12)      v v 1 ∂2A~ k vn 2 ∆A~ k”+ + 0 2 A~ A~. (32) 2k − 2 k v2 ∂t2 2 0 (cid:18) 0 (cid:19) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 9 This equation can be considered to be SVEA of second approximation with respect to the linear dispersion and of first approximation to the nonlinear dispersion (nonlinear addition to the group velocity). It includes the effects of translation in z direction with group velocity v, self-steepening, diffraction, dispersion of second order and self-action terms. The equations (32) and (29), withandwithouttheself-steepeningterm,arederivedinmanybooksandpapers [11, 12, 14, 15, 16]. From equations (32), (29), after neglecting some of the differential terms and using a special ”moving in time” coordinate system, it is nothardtoobtainthewellknownspatio-temporalmodel. Ourintentioninthis paper is another: before canceling some of the differential terms in SVEA (32), we must write (32) in dimensionless form. Then we can estimate and neglect the smallterms, depending onthe media parameters,the carrierfrequency and wavevector,andalsoonthe differentinitial shapeofthe pulses. This approach we will apply in Section 5. As a result we will obtain equations quite different from the spatio-temporal ones in the femtosecond region. 4 Propagation of optical pulses in vacuum and dispersionless media Thetheoryoflightenvelopesisnotrestrictedonlytothecasesofnon-stationary optical(andmagnetic)response. Eveninvacuum,whereε=1 andP~ =0,we nl can write an amplitude equation by applying solutions of the kind (33) to the wave equation (8). We denote here by V~(x,y,z,t) the amplitude function for the electrical field E~(r,t) in vacuum: E~ (x,y,z,t)=V~ (x,y,z,t)exp(i(k z ω t)), (33) 0 0 − whereω andk againarethe carrierfrequencyandthe carrierwavenumberof 0 0 thewavepacket. Wethusobtainthefollowinglinearequationfortheamplitude envelope of the electrical field: ∂V~ ∂V~ c 1 ∂2V~ i +c = ∆V~ . (34) − ∂t ∂z ! 2k0 − 2k0c ∂t2 The vacuum linear amplitude equation (VLAE) (34) is obtained directly from the wave equation without any restrictions. This is in contrast to the case of dispersivemedium,whereweusetheseriesofthesquareofthe wavevectorand we require the series (27) to be strongly convergent. That is why equation(34) describes both amplitudes with many harmonics under the pulse, and ampli- tudes with only one or a few harmonics under the envelope. It is obvious that the envelope V~ in equation (34) will propagate with the speed of light c in vac- uum. Equation (34) is valid also for transparent media with stationary optical response ε=const. In this case, the propagating constant will be v =c/√εµ. 10