Propagation of localized optical waves in media with dispersion, in dispersionless media and in vacuum. Low diffractive regime
aa r X i v : . [ phy s i c s . op ti c s ] J a n Propagation of localized optical waves in mediawith dispersion, in dispersionless media and invacuum. Low diffractive regime
Lubomir M. KovachevInstitute of Electronics, Bulgarian Academy of Sciences,Tzarigradcko shossee 72,1784 Sofia, BulgariaOctober 23, 2018
Abstract
We present a systematic study on linear propagation of ultrashort laserpulses in media with dispersion, dispersionless media and vacuum. Theapplied method of amplitude envelopes gives the opportunity to estimatethe limits of slowly warring amplitude approximation and to describe anamplitude integro-differential equation, governing the propagation of opti-cal pulses in single cycle regime. The well known slowly varying amplitudeequation and the amplitude equation for vacuum are written in dimension-less form. Three parameters are obtained defining different linear regimesof the optical pulses evolution. In contrast to previous studies we demon-strate that in femtosecond region the nonparaxial terms are not smalland can dominate over transverse Laplacian. The normalized amplitudenonparaxial equations are solved using the method of Fourier transforms.Fundamental solutions with spectral kernels different from Fresnel one arefound. One unexpected new result is the relative stability of light pulseswith spherical and spheroidal spatial form, when we compare their trans-verse enlargement with the paraxial diffraction of lights beam in air. Itis important to emphasize here the case of light disks, i.e. pulses whoselongitudinal size is small with respect to the transverse one, which in somepartial cases are practically diffractionless over distances of thousand kilo-meters. A new formula which calculates the diffraction length of opticalpulses is suggested.
For long time few picosecond or femtosecond (fs) optical pulses with approxi-mately equal duration in the x , y and z directions (Light Bullets or LB), and fsoptical pulses with relatively large transverse and small longitudinal size (LightDisks or LD) are used in the experiments. The evolution of so generated LB andLD in linear or nonlinear regime is quite different from the propagation of light1eams and they have drawn the researchers’ attention with their unexpecteddynamical behavior. For example, self-channeling of femtosecond pulses withpower little above the critical for self-focusing [1] and also below the nonlinearcollapse threshold [2] (linear regime) in air, was observed. This is in contra-diction with the well known self-focusing and diffraction of an optical beamin the frame of paraxial optics. Various unidirectional propagation equationshave been suggested to be found stable pulse propagation mainly in nonlinearregime (see e.g. Moloney and Kolesik [3], Couairon and Mysyrowicz [4], Chin atall. [5], for a review). The basic studies in this field started with the so calledspatio-temporal nonlinear Schr¨odinger equation (NSE) which is one compila-tion between paraxial approximation, the group velocity dispersion (GVD) andnonlinearity [6, 7, 8, 9]. The influence of additional physical effects were studiedby adding different terms to this scalar model as small nonparaxiality [15, 18],plasma defocussing, multiphoton ionization and vectorial generalizations. It isnot hard to see that for pulses with low intensity (linear regime) in air and gasesthe additional terms as GVD and others become small and the basic model canbe reduced to paraxial equation. This is the reason diffraction of a low intensityoptical pulse governed by this model on several diffraction length to be equal todiffraction of a laser beam. On other hand, the experimentalists have discussedfor a long time that in their measurements the diffraction length of an opticalpulse is not equal to this of a laser beam z beamdiff = k r ⊥ , even when additionalphase effects of lens and other optical devices can be reduced. Here k denoteslaser wave-number and r ⊥ denotes the beam waist. Thus exist one deep differ-ence between the existing models in linear regime, predicting paraxial behaviorin gases, and the real experiments.The purpose of this work is to perform a systematic study of linear propaga-tion of ultrashort optical pulses in media with dispersion, dispersionless mediaand vacuum and to suggest a model which is more close to the experimentalresults. In addition, there are several particular problems under considerationin this paper.The first one is to obtain (not slowly varying) amplitude envelope equationin media with dispersion governing the evolution of optical pulses in single-cycleregime. This problem is natural in femtosecond region where the optical periodof a pulse is of order 2 − k ( ω ) about ω . It is easy to show [21] that this expansiondiverge in solids for single-cycle pulses. The higher order dispersion terms startto dominate and the series can not cut off. This is the reason more carefullyand accurately to derive the envelope equation before using Taylor series. Inthis way we obtain an integro-differential envelope equation where no Taylorexpansion of the wave vector k ( ω ), governing evolution of single cycle pulsesin solids.The second problem is to investigate more precisely the slowly varying en-velope equations governing the evolution of optical pulses with high number ofharmonics under the envelope. The slowly varying scalar Nonlinear EnvelopeEquation (NEE) is derived in many books and papers [11, 12, 13, 14, 15, 16, 17].2fter the deriving of the NEE, most of the authors use a standard procedureto neglect the nonparaxial terms as small ones. Only some partial nonparaxialapproximations in free space [10, 15, 18] and optical fibers [19] were studied. In[22] we rewrite the NEE in dimensionless form and estimate the influence of thedifferent linear and nonlinear terms on the evolution of optical pulses. We foundthat both nonparaxial terms in NEE, second derivative in propagation directionand second derivative in time with 1 /v coefficient, are not small corrections.In fs region they are of same order as transverse Laplacian or start to dominate.These equations with (not small) nonparaxial terms are solved in linear regime[22] and investigated numerically in nonlinear [23]. In this paper we includeGVD term in the nonparaxial model and study also the envelope equation ofelectrical field in vacuum and dispersionless media. It is important to note thatthe 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-cycleregime (subfemto and attosecond pulses).Last but not least the nonparaxial equations for media with dispersion, dis-persionless media and vacuum are solved in linear regime and new fundamentalsolutions, including the GVD, are found. The solutions of these equations pre-dict new diffraction length for optical pulses z pulsediff = k r ⊥ /z , where z is thelongitudinal 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-uum we demonstrate by these analytical and numerical solutions a significantdecreasing of the diffraction enlargement in respect to paraxial beam model anda possibility to reach practically diffraction-free regime. The propagation of ultra-short laser pulses in isotropic media, can be charac-terized by the following dependence of the polarization of first ~P lin and third ~P nl order on the electrical field ~E : ~P lin = t Z −∞ (cid:16) δ ( τ − t ) + 4 πχ (1) ( τ − t ) (cid:17) ~E ( τ, r ) dτ = t Z −∞ ε ( τ − t ) ~E ( τ, x, y, z ) dτ, (1)3 P (3) nl = 3 π t Z −∞ t Z −∞ t Z −∞ χ (3) ( τ − t, τ − t, τ − t ) × (cid:16) ~E ( τ , r ) · ~E ∗ ( τ , r ) (cid:17) ~E ( τ , r ) dτ dτ dτ , (2)where χ (1) and ε are the linear electric susceptibility and the dielectric constant, 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 investigateonly linearly or only circularly polarized light and in addition we neglect thethird harmonics term. The Maxwell’s equations in this case becomes: ∇ × ~E = − c ∂ ~B∂t , (3) ∇ × ~H = 1 c ∂ ~D∂t , (4) ∇ · ~D = 0 , (5) ∇ · ~B = ∇ · ~H = 0 , (6) ~B = ~H, ~D = ~P lin + ~P nl , (7)where ~E and ~H are the electric and magnetic fields strengths, ~D and ~B arethe electric and magnetic inductions. We should point out here that theseequations are valid when the time duration of the optical pulses t is greaterthan the characteristic response time of the media τ ( t >> τ ), and also whenthe 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: ∇ (cid:16) ∇ · ~E (cid:17) − ∆ ~E = − c ∂ ~D∂t , (8)where ∆ ≡ ∇ is the Laplace operator. Equation (8) is derived without usingthe third Maxwell’s equation. Using equation (5) and the expression for thelinear and nonlinear polarizations (1) and (2), we can estimate the second termin equation (8) for arbitrary localized vector function of the electrical field. Itis not difficult to show that for localized functions in nonlinear media with andwithout dispersion ∇ · ~E ∼ = 0 and we can write equation (8) as follows:∆ ~E = 1 c ∂ ~D∂t . (9)4e will now replace the electrical field in linear and nonlinear polarization onthe right-hand side of (9) with it’s Fourier integral: ~E ( r, t ) = + ∞ Z −∞ ˆ ~E ( r, ω ) exp ( − iωt ) dω, (10)where with ˆ ~E ( r, ω ) we denote the time Fourier transform of the electrical field.We thus obtain:∆ ~E = 1 c ∂ ∂t t Z −∞ ∞ Z −∞ ε ( τ − t ) ˆ ~E ( r, ω ) exp ( − iωτ ) dωdτ + (11)3 πc ∂ ∂t t Z −∞ t Z −∞ t Z −∞ ∞ Z −∞ χ (3) ( τ − t, τ − t, τ − t ) (cid:12)(cid:12)(cid:12) ˆ ~E ( r, ω ) (cid:12)(cid:12)(cid:12) ˆ ~E ( r, ω ) × exp ( − i ( ω ( τ − τ + τ ))) dωdτ dτ dτ . 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 ,τ − t > τ i − t > i = 1 , , . (12)That is why we can extend the upper integral boundary to infinity and use thestandard Fourier transform [13]: t Z −∞ ε ( τ − t ) exp ( − iωτ ) dτ = + ∞ Z −∞ ε ( τ − t ) exp ( − iωτ ) dτ , (13) t Z −∞ t Z −∞ t Z −∞ χ (3) ( τ − t, τ − t, τ − t ) dτ dτ dτ = + ∞ Z −∞ + ∞ Z −∞ + ∞ Z −∞ χ (3) ( τ − t, τ − t, τ − t ) dτ dτ dτ . (14)The spectral representation of the linear optical susceptibility ˆ ε ( ω ) is connectedto the non-stationary optical response function by the following Fourier trans-form: 5 ε ( ω ) exp ( − iωt ) = + ∞ Z −∞ ε ( τ − t ) exp ( − iωτ ) dτ . (15)The expression for the spectral representation of the non-stationary nonlinearoptical susceptibility ˆ χ (3) is similar :ˆ χ (3) ( ω ) exp ( − iωt ) = + ∞ Z −∞ + ∞ Z −∞ + ∞ Z −∞ χ (3) ( τ − t, τ − t, τ − t ) × exp ( − i ( ω ( τ − τ + τ ))) dτ dτ dτ . (16)Thus, after brief calculations, equation (11) can be represented as∆ ~E = − ∞ Z −∞ ω ˆ ε ( ω ) c ˆ ~E ( r, ω ) exp ( − iωt ) dω + ∞ Z −∞ ω ˆ χ (3) ( ω ) c (cid:12)(cid:12)(cid:12) ˆ ~E ( r, ω ) (cid:12)(cid:12)(cid:12) ˆ ~E ( r, ω ) exp ( − i ( ωt )) dω. (17)We now define the square of the linear k and the generalized nonlinear ˆ k nl wave vectors, as well as the nonlinear refractive index n with the expressions: k = ω ˆ ε ( ω ) c , (18)ˆ k nl = 3 πω ˆ χ (3) ( ω ) c = k n , (19)where n ( ω ) = 3 π ˆ χ (3) ( ω )ˆ ε ( ω ) . (20)The connection between the usual dimensionless nonlinear wave vector k nl andthe generalized one (19) is: k nl = ˆ k nl (cid:12)(cid:12)(cid:12) ˆ ~E ( r, ω ) (cid:12)(cid:12)(cid:12) . In terms of these quantities,equation (17) can be expressed by:∆ ~E = − ∞ Z −∞ k ( ω ) ˆ ~E ( r, ω ) exp ( − iωt ) dω − ∞ Z −∞ k ( ω ) n ( ω ) (cid:12)(cid:12)(cid:12) ˆ ~E ( r, ω ) (cid:12)(cid:12)(cid:12) ˆ ~E ( r, ω ) exp ( − i ( ωt )) dω. (21)6et 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)where ω and k are the carrier frequency and the carrier wave number of thewave packet. The writing of the amplitude function in this form means that weconsider propagation only in + z -direction and neglect the opposite one. Let uswrite here also the Fourier transform of the amplitude function ˆ ~A ( r, ω − ω ): ~A ( r, t ) = + ∞ Z −∞ ˆ ~A ( r, ω − ω ) exp ( − i ( ω − ω ) t ) dω, (23)and the following relation between the Fourier transform of the electrical fieldand the Fourier transform of the amplitude function:ˆ ~E ( r, ω ) exp( − iωt ) =exp ( − i ( k z − ω t )) ˆ ~A ( r, ω − ω ) exp ( i ( ω − ω ) t ) , (24)Since we investigate optical pulses, we assume that the amplitude function andits Fourier expression are time - and frequency-localized. Substituting (33),(23)and (24) into equation (21) we finally obtain the following nonlinear integro-differential amplitude equation: ∆ ~A ( r, t ) + 2 ik ∂ ~A ( r, t ) ∂z − k ~A ( r, t ) = (25) − ∞ Z −∞ k ( ω ) (cid:18) n ( ω ) (cid:12)(cid:12)(cid:12) ˆ ~A ( r, ω − ω ) (cid:12)(cid:12)(cid:12) (cid:19) ˆ ~A ( r, ω − ω ) exp ( − i ( ω − ω ) t ) dω Equation (25) was derived with only one restriction, namely, that the amplitudefunction and its Fourier expression are localized functions. That is why, ifwe know the analytical expression of k ( ω ) and n ( ω ), the Fourier integralon the right- hand side of (25) is a finite integral away from resonances. Inthis way we can also investigate optical pulses with time duration t of theorder of the optical period T = 2 π/ω . Generally, using the nonlinear integro-differential amplitude equation (25) we can also investigate wave packets withtime duration of the order of the optical period, as well as wave packets with alarge number of harmonics under the pulse. The nonlinear integro-differentialamplitude equation (25) can be written as a nonlinear differential equation for7he Fourier transform of the amplitude function ˆ ~A , after we apply the timeFourier transformation (23) to the left-hand side of (25) :∆ ˆ ~A ( r, ω − ω ) + 2 ik ∂ ˆ ~A ( r, ω − ω ) ∂z + (cid:18)(cid:18) n ( ω ) (cid:12)(cid:12)(cid:12) ˆ ~A ( r, ω − ω ) (cid:12)(cid:12)(cid:12) (cid:19) k ( ω ) − k ( ω ) (cid:19) ˆ ~A ( r, ω − ω ) = 0 . (26)We should note here the well-known fact that the Fourier component of theamplitude 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. Equation (25) is obtained without imposing any restrictions on the square ofthe linear k ( ω ) and generalized nonlinear ˆ k nl = k ( ω ) n ( ω ) wave vectors. Toobtain SVEA, we will restrict our investigation to the cases when it is possible toapproximate k and ˆ k nl as a power series with respect to the frequency difference ω − ω as: k ( ω ) = ω ˆ ε ( ω ) c = k ( ω ) + ∂ (cid:0) k ( ω ) (cid:1) ∂ω ( ω − ω )+ 12 ∂ (cid:0) k ( ω ) (cid:1) ∂ω ( ω − ω ) + ..., (27)ˆ k nl ( ω ) = ω ˆ χ (3) ( ω ) c = ˆ k nl ( ω ) + ∂ (cid:16) ˆ k nl ( ω ) (cid:17) ∂ω ( ω − ω ) + ... (28)To obtain SVEA in second approximation to the linear dispersion and in firstapproximation to the nonlinear dispersion, we must cut off these series to thesecond derivative term for the linear wave vector and to the first derivativeterm 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 integralsin equation (25) yields the first and second derivative terms in (27), and thezero and first derivative terms in (28). The first term in (27) cancels the lastterm on the left-hand side of equation (25). The convergence of the series(27) and (28) for spectrally limited pulses propagating in the transparent UVand optical regions of solids materials, liquids and gases, depends mainly onthe number of harmonics under the pulses [21]. For wave packets with morethan 10 harmonics under the envelope, the series (27) is strongly convergent,and the third derivative term (third order of dispersion) is smaller than the8econd derivative term (second order of dispersion) by three to four orders ofmagnitude for all materials. In this case we can cut the series to the secondderivative term in (27), as the next terms in the series contribute very little tothe Fourier integral in equation (25). When there are 2 − ~A + 2 ik ∂ ~A∂z + 2 ik k ′ ∂ ~A∂t = (cid:0) k k ” + k ′ (cid:1) ∂ ~A∂t − ˆ k nl (cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A − i ˆ k nl ˆ k ′ nl ∂ (cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A∂t , (29)where k = k ( ω ) and ˆ k nl = ˆ k nl ( ω ). We will now define other importantconstants connected with the wave packets carrier frequency: linear wave vector k ≡ k ( ω ) = ω p ε ( ω ) /c ; linear refractive index n ( ω ) = p ε ( ω ); nonlinearrefractive index n ( ω ) = 3 πχ (3) ( ω ) /ε ( ω ); group velocity: v ( ω ) = 1 k ′ = c p ε ( ω ) + ω q ε ∂ε∂ω , (30)nonlinear addition to the group velocity (ˆ k nl ) ′ : (cid:16) ˆ k nl (cid:17) ′ = 2 k n v + k ∂n ∂ω , (31)and dispersion of the group velocity k ”( ω ) = ∂ k/∂ω ω = ω . All these quantitiesallow a direct physical interpretation and we will therefore rewrite equation (29)in a form consistent with these constants: − i ∂ ~A∂t + v ∂ ~A∂z + (cid:18) n + k v ∂n ∂ω (cid:19) ∂ (cid:18)(cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A (cid:19) ∂t = v k ∆ ~A − v (cid:18) k ” + 1 k v (cid:19) ∂ ~A∂t + k vn (cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A. (32)9his equation can be considered to be SVEA of second approximation withrespect to the linear dispersion and of first approximation to the nonlineardispersion (nonlinear addition to the group velocity). It includes the effectsof translation in z direction with group velocity v , self-steepening, diffraction,dispersion of second order and self-action terms. The equations (32) and (29),with and without the self-steepening term, are derived in many books and papers[11, 12, 14, 15, 16]. From equations (32), (29), after neglecting some of thedifferential terms and using a special ”moving in time” coordinate system, it isnot hard to obtain the well known spatio-temporal model. Our intention in thispaper is another: before canceling some of the differential terms in SVEA (32),we must write (32) in dimensionless form. Then we can estimate and neglectthe small terms, depending on the media parameters, the carrier frequency andwave vector, and also on the different initial shape of the pulses. This approachwe will apply in Section 5. As a result we will obtain equations quite differentfrom the spatio-temporal ones in the femtosecond region. The theory of light envelopes is not restricted only to the cases of non-stationaryoptical (and magnetic) response. Even in vacuum, where ε = 1 and ~P nl = 0, wecan write an amplitude equation by applying solutions of the kind (33) to thewave equation (8). We denote here by ~V ( x, y, z, t ) the amplitude function forthe electrical field ~E ( r, t ) in vacuum: ~E ( x, y, z, t ) = ~V ( x, y, z, t ) exp ( i ( k z − ω t )) , (33)where ω and k again are the carrier frequency and the carrier wave number ofthe wave packet. We thus obtain the following linear equation for the amplitudeenvelope of the electrical field: − i ∂ ~V∂t + c ∂ ~V∂z ! = c k ∆ ~V − k c ∂ ~V∂t . (34)The vacuum linear amplitude equation (VLAE) (34) is obtained directly fromthe wave equation without any restrictions. This is in contrast to the case ofdispersive medium, where we use the series of the square of the wave vector andwe 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 thatthe 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 opticalresponse ε = const . In this case, the propagating constant will be v = c/ √ εµ .10 SVEA and VLAE in a normalized form
Starting from Maxwell’s equations for media with non-stationary linear andnonlinear response, we obtained an amplitude equation and a SVEA using onlytwo restrictions, which are physically acceptable for ultra-short pulses. Havingadopted the first restriction, namely, investigation of localized in time and spaceamplitude functions only, we introduced the amplitude equation (25). Followingthe second restriction, i.e., limiting ourselves with the case of a large numberof harmonics under the localized envelopes, we obtained the SVEA (32). Asit was pointed out in the previous section, the second restriction do not affectthe VLAE (34). The next step is writing SVEA (32) and VLAE (34) in di-mensionless variables and estimating the influence of the different differentialterms. In this case, the coefficients in front of the differential operators in (32)and (34) will be numbers of different orders, depending on the medium n and n , the spectral region of propagation k and ω , the field intensity | A | , andthe initial shape of the pulses, namely, light filament r ⊥ << z (LF), LightBullets (LB) r ⊥ ≈ z or Light disks (LD) r ⊥ >> z . With r ⊥ we denote herethe initial transverse dimension, ”the spot” of the pulse, and with z we de-note the initial longitudinal dimension, which is simply the spatial analog ofthe initial time duration t , determined by the relation z = vt or z = ct inthe vacuum case. The SVEA (32) and VLAE (34) are written in a Cartesianlaboratory coordinate system. To investigate the dynamics of optical pulses atlong distances, it is convenient to rewrite these equations in a Galilean coordi-nate system, where the new reference frame moves with the group velocity forequation (32), t ′ = t ; z ′ = z − vt : − i ∂ ~A∂t ′ + (cid:18) n + k v ∂n ∂ω (cid:19) ∂ (cid:18)(cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A (cid:19) ∂t ′ − ∂ (cid:18)(cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A (cid:19) ∂z ′ = v k ∆ ⊥ ~A − (35) v k ”0 ∂ ~A∂z ′ − v (cid:18) k ” + 1 k v (cid:19) ∂ ~A∂t ′ − v ∂ ~A∂t ′ ∂z ′ ! + n k v (cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A, and with the velocity of light for equation (34), t ′ = t ; z ′ = z − ct : − i ∂ ~V∂t ′ = c k ∆ ⊥ ~V − k c ∂ ~V∂t ′ + 1 k ∂ ~V∂t ′ ∂z ′ . (36)With ∆ ⊥ = ∂ ∂x + ∂ ∂y we denote the transverse Laplacian. We define thefollowing dimensionless variables connected with the initial amplitude and withthe spatial and temporal dimensions of the pulses through the relations: ~A = A ~A ”; ~V = V ~V ”; x = r ⊥ x ”; y = r ⊥ y ”; z ′ = z z ”;11 ′ = t t ”; z = z z ”; t = t t ” . (37)After the substitution of these variables in (32), (35), (34) and (36) and makinguse of the expressions for the diffraction z dif = k r ⊥ and dispersion z disp = t /k ” lengths, we obtain the following five dimensionless parameters in front ofthe differential terms in the equations (32), (35) (34) and (36): α = k z ; δ = r ⊥ z ; β = z dif z disp ; γ = k r n | A | ; γ = | A | (cid:18) n + k v ∂n ∂ω (cid:19) . (38)Omitting the seconds in the new dimensionless variables and constants, theequations (32), (35), (34) and (36) can be represented as follows:Case a. SVEA (32) in a laboratory frame (”Laboratory”) − iαδ ∂ ~A∂t + ∂ ~A∂z + γ ∂ (cid:18)(cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A (cid:19) ∂t = ∆ ⊥ ~A + δ ∂ ~A∂z − ∂ ~A∂t ! − (39) β ∂ ~A∂t + γ (cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A. Case b. SVEA (35) in a frame moving with the group velocity: − iαδ ∂ ~A∂t ′ + γ ∂ (cid:18)(cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A (cid:19) ∂t ′ − ∂ (cid:18)(cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A (cid:19) ∂z ′ = ∆ ⊥ ~A − β ∂ ~A∂z ′ − (40) (cid:0) β + δ (cid:1) ∂ ~A∂t ′ − ∂ ~A∂t ′ ∂z ′ ! + γ (cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A, Case c. VLAE (34) in a laboratory frame: − iαδ ∂ ~V∂t + ∂ ~V∂z ! = ∆ ⊥ ~V + δ ∂ ~V∂z − ∂ ~V∂t ! . (41)Case d. VLAE (36) in a Galilean frame: − iαδ ∂ ~V∂t ′ = ∆ ⊥ ~V − δ ∂ ~V∂t ′ − ∂ ~V∂t ′ ∂z ′ ! . (42)12t should be noted here that equal dimensionless constants in front of the dif-ferential terms in both the ”Laboratory” and ”Galilean” frames are obtained.This gives us the possibility to investigate and estimate simultaneously the dif-ferent terms in the normalized equations (39), (40), (41) and (42). We will nowdiscuss these constants in detail, as they play a significant role in determiningthe different pulse propagation regimes.- The first constant α = k z = 2 πz /λ determines with precision 2 π the”number of harmonics” on a FWHM level of the pulses. Since we use the slowlyvarying amplitude approximation, α is always a large number ( α >> δ = r ⊥ /z determines the relation between the initialtransverse and longitudinal size of the optical pulses. This parameter distin-guishes the case of light filaments (LF) δ = r ⊥ /z << δ = r ⊥ /z ∼ = 1 and the case of light disks LD δ = r ⊥ /z >>
1. For lightfilaments δ << δ . It is not difficult to see that in this case the SVEA (39), (40) and VLAE(40), (41) can be transformed to the standard paraxial approximation of thelinear and nonlinear optics. If we set the possible values of the optical pulsestransverse dimensions at 3 − mm > r ⊥ > µm , we can directly obtain theabove distinction in dependence on the time duration of the pulses. For lightpulses with time duration ns > t > − ps we obtain δ = r ⊥ /z << t ≈ − ps up to 500 − f s it is possible to reach δ = r ⊥ /z ∼ = 1 and we are in the regime of LB. For pulses in the time range300 f s − f s we can prepare the initial shape of the pulses to satisfy the rela-tion δ = r ⊥ /z >> t ≥ f s contain more than 10-15 optical harmonics under the pulse, so thatwe are still in SVEA approximation. In the last two cases (LB and LD) thedifferential terms with δ cannot be ignored and the equations (39), (40) and(41) governing the propagation of pulses with initial form of LB and LD arequite different from the paraxial approximation.-The third parameter is β = k r ⊥ /z dis , where z dis = t /k ” determinesthe relation between the diffraction and dispersion lengths. The dispersionparameter k ” in the visible and UV transparency region of dielectrics has valuesfrom k ” ∼ − s /cm for gases and metal vapors up to k ” ∼ − s /cm forsolid materials. It is convenient to express this parameter using the product ofthe second constant δ and the parameter β = k v k ” by the relation β = β δ .For typical values of the dispersion k ” in the visible and UV region listed above,the dimensionless parameter β is very small ( β << β ∝
1. Theparameter β can be also negative and may reach β ≃ − β ≃ − γ = k r ⊥ n | A | and αδ γ ≃ αδ n | A | α >> δ ≥ γ >> αδ γ . For optical pulses with power near the criticalthreshold for self-focusing γ ∼ = 1 and less (linear regime) γ <<
1, the nonlinearaddition to the group velocity is very small ( αδ γ <<
1) and from here to theend of this paper we will neglect the terms with the first addition to the nonlineardispersion. The analysis of the dimensionless constants performed above leadsus to the following conclusion: Dynamics of wave packets with power near tocritical for self-focusing γ ∝ − iαδ ∂ ~A∂t + ∂ ~A∂z ! = ∆ ⊥ ~A + δ ∂ ~A∂z − δ ( β + 1) ∂ ~A∂t + γ (cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A. (43)Case b. SVEA in frame moving with group velocity (”Galilean”): − iαδ ∂ ~A∂t ′ = ∆ ⊥ ~A − β δ ∂ ~A∂z ′ − δ ( β + 1) ∂ ~A∂t ′ − ∂ ~A∂t ′ ∂z ′ ! + γ (cid:12)(cid:12)(cid:12) ~A (cid:12)(cid:12)(cid:12) ~A. (44)Equations (43) and (44) are quite different from the well known paraxial spatio-temporal evolution equations. Here are included also the second derivativealong the z direction, a mixed term and additional second derivative in timeterm. This leads to dynamics of the ultrashort fs pulses different from spatio-temporal model. In this paper we will investigate the propagation in linearregime, when γ << The behavior of long pulses is similar to that of optical beams, since theirpropagation is governed by a equation where the nonparaxial terms becomesmall. That is why we can expect the diffraction enlargement of long pulsesto be of the same order as are the optical beams. The situation regarding LBand LD is different. Their propagation is governed by equations in media withnon-stationary optical response - SVEA (43) and (44), and by VLAE (34) and(36) in media with linear stationary optical response (or vacuum), where thenonparaxial terms are of same order and bigger than transverse Laplacian. Inthis section we will solve the equations (43), (44) in linear regime ( γ <<
1) andwill compare the solutions with the solutions of the linear VLAE (34) and (36).Neglecting the small nonlinear terms in (43), (44) we obtain:14. Linear SVEA in a laboratory coordinate frame: − iαδ ∂ ~A∂t + ∂ ~A∂z ! = ∆ ⊥ ~A + δ ∂ ~A∂z − δ ( β + 1) ∂ ~A∂t . (45)b. Linear SVEA in a Galilean coordinate frame: − iαδ ∂ ~A∂t ′ = ∆ ⊥ ~A − δ ( β + 1) ∂ ~A∂t ′ − ∂ ~A∂t ′ ∂z ′ ! − δ β ∂ ~A∂z ′ . (46)For comparison we will rewrite here the corresponding linear VLAE:c. Linear VLAE in a laboratory frame: − iαδ ∂ ~V∂t + ∂ ~V∂z ! = ∆ ⊥ ~V + δ ∂ ~V∂z − ∂ ~V∂t ! . (47)d. Linear VLAE in a Galilean frame: − iαδ ∂ ~V∂t ′ = ∆ ⊥ ~V − δ ∂ ~V∂t ′ − ∂ ~V∂t ′ ∂z ′ ! . (48)As can be expected, the equations for ultra-short optical pulses in vacuumand dispersionless media (47), (48) become identical with the equations withdispersion (45), (46) when the dispersion parameter β and δ β are small. Inequations (45), (46) there are three dimensionless parameters, α , δ and β ,while in (47), (48) there are only two α and δ . These parameters can bechanged considerably in fs region, and this leads, as we can see later, to quitedifferent dynamics in the particular cases.In the general case we apply the Fourier method to solve the linear SVEA(45), (46) which describe the propagation of ultrashort optical pulses in mediumwith dispersion and linear VLAE (47), (48) which govern the propagation of lightpulses in vacuum and dispersionless media. We mark the Fourier transform ofthe amplitude functions of SVEA in Galilean frame (45) with ~A G ( k x , k y , k z , t ) = F ( ~A ( x, y, z, t )), and in Lab frame (46) with ~A L ( k x , k y , k z , t ) = F ( ~A ( x, y, z, t )).The Fourier transform of the amplitude functions of VLAE in Galilean framebecome (47), ~B G ( k x , k y , k z , t ) = F ( ~V ( x, y, z, t )), while in Laboratory frame (48)we write ~B L ( k x , k y , k z , t ) = F ( ~V ( x, y, z, t )). Applying spatial Fourier transfor-mation to the components of the amplitude vector functions ~A and ~V , thefollowing ordinary linear differential equations in k x , k y , k z space for SVEA:a. Laboratory: 15 iαδ ∂ ~A L ∂t = − (cid:0) k x + k y + δ ( k z − αk z ) (cid:1) ~A L − δ ( β + 1) ∂ ~A L ∂t , (49)b. Galilean: − iδ ( α − ( β + 1) k z ) ∂ ~A G ∂t = − (cid:0) k x + k y − δ β k z (cid:1) ~A G − δ ( β + 1) ∂ ~A G ∂t , (50)and the following equations for VLAE:c. Laboratory: − iαδ ∂ ~B L ∂t = − (cid:0) k x + k y + δ ( k z − αk z ) (cid:1) ~B L − δ ∂ ~B L ∂t , (51)d. Galilean: − iδ ( α − k z ) ∂ ~B G ∂t = − (cid:0) k x + k y (cid:1) ~B G − δ ∂ ~B G ∂t , (52)are obtained. We look for solutions of the kind of ~A L = ~A L ( k x , k y , k z ) exp( i Ω L t )and ~A G = ~A G ( k x , k y , k z ) exp( i Ω G t ) for the equations (49), (50), and for solu-tions ~B L = ~B L ( k x , k y , k z ) exp( i Φ L t ) and ~B G = ~B G ( k x , k y , k z ) exp( i Φ G t ) for theequations (51) and (52) correspondingly. Let us denote the square of the sumof the wave vectors as: ˆ k = k x + k y + δ ( k z − αk z ). The solutions existwhen Ω L , Ω G , Φ L and Φ G satisfy the following quadratic equations:Ω L − αβ + 1 Ω L − ˆ k δ ( β + 1) = 0 , (53)Ω G − α − ( β + 1) k z ) β + 1 Ω G − k x + k y − δ β k z δ ( β + 1) = 0 . (54)Φ L − α Φ L − ˆ k δ = 0 , (55)Φ G − α − k z )Φ G − k x + k y δ = 0 . (56)16he solutions of (53), (54) for media with dispersion are:Ω L , = αβ + 1 ± s α ( β + 1) + ˆ k δ ( β + 1) , (57)Ω G , = α − ( β + 1) k z β + 1 ± s ( α − ( β + 1) k z ) ( β + 1) + k x + k y − δ β k z δ ( β + 1) , (58)while the solutions of (55), and (56) for dispersionless media and vacuumbecome: Φ L , = α ± q α + ˆ k /δ , (59)Φ G , = ( α − k z ) ± q α + ˆ k /δ . (60)Now the necessity of a parallel investigation of the propagation of optical pulsesin media with dispersion, in dispersionless media and in vacuum becomes obvi-ous. Further, we will introduce here the concept of weak and strong dispersionmedia depending on the value of dimensionless dispersion parameter β . When β << β << β can reach the values β ≃ − β + 1 withrespect to the solutions without dispersion. We consider here the regime ofpropagation far away from electronic resonances and the Langmuir frequencyin electronic plasmas, where it is possible to obtain a strongly negative disper-sion parameter β ∝ −
1. We point out here again that in the case of LB, when β ∼ = −
1, the amplitude equations (32) can be transformed into the 3D+1 linearand nonlinear vector Schrodinger equations [17]. Generally said, the dispersionparameter β varies slowly from the visible to the UV transparency region ofthe materials from very small values up to β ≃ − α and δ change significantly. For example α varies from 10 to 10 , while δ varies from 10 − − − for LF to 10 for LBand 10 − for LD. This is the reason to investigate more precisely in thenext paragraph the solutions of the equations for media with weak dispersionas air where β <<
1, (58), as we expect that the solutions for media withstrong dispersion (UV transparency region of solids and liquids) will be onlyslightly modified by the factor β + 1; β ≤
1. We obtained solutions of the17haracteristic equations (57), (58), (59) and (60). The solutions of the corre-sponding linear differential equations SVEA (49), (50) and VLAE (51), (52) inthe k -space become:a. Solution of SVEA in the k-space and laboratory coordinate frame: ~A L = ~A L ( k x , k y , k z , t = 0) × exp i αβ + 1 ± s α ( β + 1) + ˆ k δ ( β + 1) t . (61)b. Solution of SVEA in the k-space and Galilean coordinate frame: ~A G = ~A G ( k x , k y , k z , t = 0) × (62)exp i α − ( β + 1) k z β + 1 ± s ( α − ( β + 1) k z ) ( β + 1) + k x + k y − δ β k z δ ( β + 1) t . c. Solution of VLAE in k-space and laboratory coordinate frame: ~B L = ~B L ( k x , k y , k z , t = 0) exp (cid:18) i (cid:18) α ± q α + ˆ k /δ (cid:19) t (cid:19) . (63)d. Solution of VLAE in the k-space and Galilean coordinate frame: ~B G = ~B G ( k x , k y , k z , t = 0) exp (cid:18) i (cid:18) ( α − k z ) ± q α + ˆ k /δ (cid:19) t (cid:19) . (64)It is obvious that the solutions (63) and (64) of equations (51) and (52) shouldbe equal with accuracy - a wave number in z direction. This follows from theFourier transform of such evolution equations and leads to only one differencebetween the solutions in the real space - the motion of the pulse in the z directionin a Laboratory frame and its stationarity in a Galilean frame. As it waspointed out at the beginning, we investigate here only localized in space and timeinitial functions of the amplitude envelopes. Thus, the images of these functionsafter Fourier transform in the k x , k y , k z space are also localized functions. Thesolutions of our amplitude equations in k space (61), (62),(63) and (64) are theproduct of the initial localized in k x , k y , k z -space functions and the new spectralkernels which are periodic (different from the Fresnel’s one). The product ofa localized function and a periodic function is also a function localized in the k x , k y , k z space. Therefore, the solutions of our amplitude equations in k space(63), (64),(61) and (62) are also localized functions in this space and we canapply the inverse Fourier transform to obtain again the fundamental localized18olutions in the x, y, z, t space. More precisely, we use the convolution theoremto present our fundamental solutions in the real space as a convolution of inverseFourier transform of the initial pulse with the inverse Fourier transforms of thenew spectral kernels:a. Fundamental solution of SVEA (45) in laboratory coordinate frame: ~A ( x, y, z, t ) = F − (cid:16) ~A L ( k x , k y , k z , t = 0) (cid:17) ⊗ F − exp i αβ + 1 ± s α ( β + 1) + ˆ k δ ( β + 1) t . (65)b. Fundamental solution of SVEA (46) in Galilean coordinate frame: ~A ( x, y, z ′ , t ′ ) = F − (cid:16) ~A G ( k x , k y , k z , t = 0) (cid:17) ⊗ (66) F − exp i α − ( β + 1) k z β + 1 ± s ( α − ( β + 1) k z ) ( β + 1) + k x + k y − δ β k z δ ( β + 1) t . c. Fundamental solution of VLAE (47) in laboratory coordinate frame: ~V ( x, y, z, t ) = F − (cid:16) ~B L ( k x , k y , k z , t = 0) (cid:17) ⊗ F − (cid:18) exp (cid:18) i (cid:18) α ± q α + ˆ k /δ (cid:19) t (cid:19)(cid:19) . (67)d. Fundamental solution of VLAE (48) in Galilean coordinate frame: ~V ( x, y, z ′ , t ′ ) = F − (cid:16) ~B G ( k x , k y , k z , t = 0) (cid:17) ⊗ F − (cid:18) exp (cid:18) i (cid:18) ( α − k z ) ± q α + ˆ k /δ (cid:19) t (cid:19)(cid:19) , (68)where with F − we denote the spatial three-dimensional inverse Fourier trans-form and with ⊗ we denote the convolution symbol. The difference between theFresnel’s integrals, describing propagation of optical beams and long pulses inlinear regime, and the new integrals (65), (66), (67) and (68), which are solutionsof the linear evolution equations (45), (46), (47) and (48) is quite obvious. Inaddition, in the new spectral kernels there are three dimensionless parameters: α , δ and β . Let us fix α to be always large, i.e. α >>
1. As pointed out above,the condition α >> δ and β , on the evolution of the initial pulse, wewill rewrite the expression for the spectral kernel (57) of the solutions (65) ofequation (45) in the following form: Ω L , = αβ + 1 ± s α ( β + 1) + 1 δ ( β + 1) ( k x + k y ) + 1 β + 1 ( k z − αk z ) . (69)As α >> β ≤
1, the diffraction widening will be determined by thesecond term under the square root in (69):1 δ ( β + 1) ( k x + k y ) , (70)which determines the transverse diffraction and dispersion widening of the pulses.We pointed out above that the dispersion parameter varies very slowly withinthe limits 0 ≤ β < , while the relations between the transverse and lon-gitudinal part varies significantly 10 − < δ < . This is why we estimatemainly the influence of the different values of δ on the diffraction widening.We investigate the following basic cases:a/ Long pulses, when δ <<
1. It is easy to estimate from (70), that thetransverse enlargement k x + k y will dominate significantly as:1 δ ( β + 1) >> . (71)In the case of long pulses we have also αδ ∼ but not equal ) diffraction length tothat of optical beam ( z beamdiff = k r ⊥ ). The difference is only in the factors αδ ∼
1. When pulse propagate in optical transparency region of the air andgases β <<
1, normalized dispersion parameter is to small that the mainfactor which determinate the diffraction widening become z pulsediff = αδ z beamdiff = k r ⊥ /z . The validity of this new diffraction formula for pulses will be studiedmore precisely in the next paragraph not only for long pulses but also for LBand LD.b/ LB: δ ≃
1. In the case of optical pulses with approximately equal trans-verse and longitudinal size, we obtain the following coefficient of the transverse k x + k y diffraction terms: 1 δ ( β + 1) ∼ = 1 / . (72)Hence, the diffraction and dispersion transverse enlargement will be reduced bythe factor δ ( β + 1) with respect to the diffraction of long pulses and Fresnel’sdiffraction. 20n addition, we will point out here an important asymptotic behavior ofLB: When α is small (pulses with only few harmonics under the envelope)and β << ∼ exp (cid:16) i ( q k x + k y + ( k z − α ) ) t (cid:17) ∼ = exp ( i ( | k | t )), which is actually the spectralkernel of the 3D wave equation. For this reason we can expect for opticalpulses with only one or two harmonics under the envelope (subfemtosecond andattosecond pulses) diffraction similar to the typical diffraction of the 3D waveequation, whose dynamics is characterized by internal and external fronts anda significant widening of the pulse.c/ Light disks: This is the case when the longitudinal size z is mush shorterthan the transverse size r ⊥ and δ >>
1. As indicated above, the typical timeregion for such pulses is 30 − f s < t < − f s . We determine thelower limits of this relation from the condition α >>
1, i.e a large numberof harmonics under the envelope. This condition still holds true for pulses inthe visible and UV regions with time duration 30 − f s . The dimensionlessparameter in front of the transverse diffraction and dispersion (cid:0) k x + k y (cid:1) willbe of the order of: 1 δ ( β + 1) << . (73)We thus see that the transverse enlargement is of the order of δ ( β + 1), ornegligible as compared with LB, and smaller by a factor of about ( δ ( β +1)) than in the cases of long pulses and Fresnel’s diffraction. To summarizethe results of this section, we can expect that the transverse diffraction anddispersion enlargement of LB should be smaller by a factor δ ( β + 1) thanthose of LF, while the transverse diffraction and dispersion enlargement of LDshould be smaller by a factor of ( δ ( β + 1)) than those of long pulses andparaxial approximation. Practically no transverse enlargements of LD would beobserved over long distances, namely, more than tens and hundred of diffractionlength. In the beginning of this section we will discuss more widely the Galilean invari-ancy and connections between normalized equations written in different coordi-nate systems: Laboratory and Galilean. The Galilean invariancy of the SVEA(45) and VLAE (47) is not obvious. Indeed, after using the transformation t ′ = t and z ′ = z − vt where v = 1 the new equations in Galilean frame, (46) and (48)admit mixed z, t terms and look quite different. The Galilean invariancy can beseen only from the kind of the fundamental solutions of equations (65)-(68) inLaboratory and Galilean frames. The solutions are equal with precision wave21igure 1: Intensity profiles of a Gaussian beam with initial condition A x ( x, y, z = 0) = exp (cid:16) − x + y (cid:17) governed by the 2 D paraxial equation (74).The transverse size (the spot) grows by factor √ z = 1. This correspond to a real distance z = z beamdiff = 7 . cm for the selectedin the paper laser source on λ = 800 nm .number k z , which gives the stationarity in Galilean and translation in z directionin Laboratory frame. The numerical solutions with initial conditions - Gaus-sian pulses provided in [22] for both coordinate systems demonstrate again thatlocalized waves admit equal spatial and phase deformation in both coordinatesand there is only one difference - stationarity in Galilean and translation withnormalized velocity v=1 in Laboratory frame. Naturally, the best way is tosolve numerically the equations (45) or (47) in Laboratory system and to seeone spatial and phase transformation of the pulse as well as its translation. In-convenience in such one approach is, that at long distances the pulses will moveout of the grid. And here the Galilean invariancy of the normalized equationshelp us. It is well known that the times in Galilean and Laboratory systems areequal t ′ = t . As the normalized velocity is v = 1, the same translation z = 1 inLaboratory frame correspond to time evolution in both system t ′ = t = 1 . Thisis demonstrated in [22] but here it gives us one additional opportunity. We cansolve the equation (46) in Galilean frame for long time t’ without pulses to moveout of the grid and to connect the normalized time t’ in Galilean with normal-ized time t and translation z in Laboratory frame t ′ = t = z ; v = 1. After that,using normalized constants, we can obtain the real distance of propagation ofoptical pulses. It is not hard to see that normalized distance z = 1 correspondto a real distance z = k r ⊥ when αδ = 1. And this is the natural way to com-pare the dynamics of optical pulses governed by equation (46) in Galilean framewith the evolution of a laser beam in scalar paraxial approximation describedby normalized equation: − i ∂A∂z + 12 ∆ ⊥ A = 0 , (74)22here, as it is well known, z = 1 in normalized coordinates corresponds to areal distance z = k r ⊥ called diffraction length. This length determines thedistance where the laser beam increase its width on level e − from the maximumwith factor √
2. We investigate here only laser sources with spectrally limited,not phase modulated initial Gaussian profile. The optical lens and devices addadditional phase modulation on the initial pulse and influenced on the wideningin linear regime.Evolution of real laser pulse with the following characteristics is considered:light source form Ti:sapphire with width on level e − ; r ⊥ = 100 µm . Usuallysuch small spot of the pulse is made by focusing by lens. To obtain no modulatedin phase initial pulse the additional phase from the lens must be reduced to zeroby a system of lens. The solutions of linear SVEA (46) in Galilean frame arecarried out for optical wave on wave-length λ = 800 nm propagating in air andthe following constants: carrying wave number k = n b ω/c = 7 . × ; cm − ,where n b ≈ . k ” = 3 . × − sec /cm ; normalizedGVD coefficient β = k v k ” ∼ = 2 . × − . To find the difference in dynamics ofLF, LB, and LD we select different time duration of pulses for LF ( t = 260 ps ),LB ( t = 330 f s ) and LD ( t = 33 f s ). Using the above parameters of thelaser sources and the material constants we obtain the following dimmensionlessparameters in SVEA (46) for the particular cases:a) long pulse and αδ = 1 ( t = 260 ps ): α = 6 . × ; δ = 1 / . × − ; β = 2 . × − .b) light bullet (330 fs): α = 785 . δ = 1 . β = 2 . × − .c) light disk (33 fs): α = 78 , δ = 100 . β = 2 . × − .Other important parameter for comparing the pulse dynamics and paraxialevolution of a laser beam is the diffraction length z diff = k r ⊥ = 7 . cm . Inaddition, we should point out that all coming numerical computations are per-formed with pulses satisfying the boundary conditions lim x.y,z L/ A ( x, y, z, t ) ~x =0 and also lim x.y,z Λ / A ( k x , k y , k z , t ) = 0, where L and Λ are respectively thespatial and wave-number intervals for the calculations. The initial conditions for linearly polarized normalized Gaussian beam reads: ~A = A x ~x ; A x ( x, y, z = 0) = exp (cid:18) − x + y (cid:19) . (75)The evolution of the initial Gaussian beam (75) governed by the paraxial equa-tion (74) is described by the Fresnel’s integral or can be found by numericalcalculation of the inverse Fourier transform of the solution in the ( k x , k y )-space.The intensity profile of a solution A ( x, y, z ) of the paraxial equation (74) withinitial condition (75) on a normalized distance z = 1 is illustrated on Fig.1.23etting in ming the above real parameters of a laser system on 800 nm , thenormalized distance z = 1 corresponds to one diffraction length and real dis-tance of z = z diff = k r ⊥ = 7 . cm .Figure 2: Transverse intensity distribution of 260 ps pulse on carrying frequency800 nm (long Gaussian pulse). Numerical solutions of the linear SVEA (46) inGalilean frame is performed by the following particularly selected initial condi-tions to satisfy αδ = 1: A x ( x, y, z, t = 0) = exp (cid:16) − x + y + z (cid:17) , α = 6 . × ; δ = 1 / . × − ; β = 2 . × − . The surfaces | A ( x, y, z ′ = 0 , t ′ = 0; t ′ = 1) | are plotted. The transverse size (the spot) grows by factor √ t ′ = z = 1. This correspond to real distance z = z beamdiff = 7 . cm equal to the diffraction length of a laser beam (compare with Fig.1). The pointed above choice for the parameter αδ = 1 of a long pulse is usedparticularly to compare it’s diffraction with the diffraction length of a laserbeam. We mark also that in the general case, the real diffraction length of along pulse (ns or ps) is similar to the diffraction of laser beam and the differenceis in the factor αδ or: z pulsediff = αδ z beamdiff = k r ⊥ /z . (76)The validity of this expression is illustrated on the next two figures where thedynamics of an initial long pulse is governed by the linear SVEA (46) in Galileanframe with initial condition ~A = A x ~x ; A x ( x, y, z ′ , t ′ = 0) = exp (cid:16) − x + y + z ′ (cid:17) and dimensionless constants:a) long pulse on 260 ps : αδ = 1; α = 6 . × ; δ = 1 / . × − ; β = 2 . × − . 24igure 3: Intensity distribution profiles of 43 ps pulse on carrying frequency800 nm (long Gaussian pulse). Numerical solutions of the linear SVEA (46) inGalilean frame is performed with initial conditions satisfy αδ = 6; α = 1 . × ; δ = 6 . × − ; β = 2 . × − . The transverse size (the spot) grows by factor √ t ′ = z = 6. This correspond to a real distance z pulsediff = αδ z beamdiff = 47 . cm .b) long pulse on 43 ps : αδ = 6; α = 1 . × − ; δ = 6 . × − ; β =2 . × − .Case a) is illustrated on Fig. 2, where the spot (x,y size) of the pulse isplotted and the parameters are selected to satisfy the relation αδ = 1. That iswhy the pulse enlarge its spatial width by a factor √ t ′ = z = 1 as in the case of laser beam. Case b) is illustrated onFig. 3., where the important dimmensionless parameter is αδ = 6. As can beexpected the spot of the pulse grows by factor √ the same order to the one of a laser beam and this lengthcan be equal only in some partial cases, satisfy αδ = 1. The evolution of LB in media with dispersion, is governed by the same SVEA(46) as in the case of long pulses. The shape of the LB is symmetric in the x , y and z plane, so that the linearly polarized initial Gaussian profile can bewritten as: ~A = A x ~x ; α = 785; δ = r ⊥ z = 1 , β = 2 . × − A x ( x, y, z, t = 0) = exp (cid:18) − x + y + z (cid:19) . (77)25igure 4: Evolution of a Gaussian light bullet with 330 f s time dura-tion governed by linear SVEA (46) in Galilean frame under initial condition A x ( x, y, z, t = 0) = exp (cid:16) − x + y + z (cid:17) , α = 785 . δ = 1 . β = 2 . × − .The surfaces | A ( x, y, z ′ = 0; t ′ = 0; t ′ = 785 / t ′ = 785) | are plotted. Thetransverse size (the spot) grows by factor √ t ′ = z = 785. For the selected in the paper laser source this corresponds to areal distance z pulsediff = αδ z beamdiff ≃ m .From the qualitative analysis presented in the previous section, when δ = 1,the widening of the LB is expected to be α = 785; z bulletdiff = αz beamdiff = k r ⊥ .The surface (x,y plane) of the solution of VLAE (67) with initial conditionsof the kind of (77) on normalized time-distance t ′ = z = 785, calculated byexploit of FFT technique, is illustrated on Fig. 4. One can see that the pulseenlarge its spot by factor √ z bulletdiff = αz beamdiff =785 z beamdiff = 6162 . cm ∼ = 61 m . Let us remark once again that this resultis only correct if the number of harmonics under the pulse, multiplied by 2 π (dimensionless parameter α ), is large. As it was mentioned in the beginning, optical pulses with small longitudinaland large transverse size, while at the same time the large number of harmonicsunder the pulse remaines, can be obtained without significant experimentaldifficulties. This can easily be realized in the optical region for pulses withtime duration from 200 − f s up to 30 − f s . We consider again thepropagation of LD in the framework of the solutions (66) of the SVEA (46) inGalilean coordinates under initial conditions of the form:26igure 5: Transverse intensity distribution of 33 f s pulse on carrying frequency800 nm (light disk) governed by the same SVEA (46) and initial condition A x ( x, y, z, t = 0) = exp (cid:16) − x + y + z (cid:17) , α = 78 . δ = 100; β = 2 . × − . Thesurfaces | A ( x, y, z ′ = 0; t ′ = 0; t ′ = 7850 / t ′ = 7850) | are presented. TheLD enlarges its transverse size by factor √ t ′ = z = 7850. This correspond to 7850 diffraction lengths of a laser beam orfor the selected laser source: z pulsediff = αδ z beamdiff ≃ m . ~A = A x ~x ; α = 78 . δ = r ⊥ z = 100 ,A x ( x, y, z ′ , t ′ = 0) = exp (cid:18) − x + y + z ′ (cid:19) . (78)Results of calculations of solution (66) with initial conditions of kind (78), usingFFT and inverse FFT technique are presented on Fig. 5. The numerical solutionconfirm our expectation that the LD enlarges its shape by factor √ z diskdiff = αδ z beamdiff = k r ⊥ /z = 7850 z beamdiff ∼ = 616 m . The new formula for diffraction length of opticalpulses (76) gives remarkable opportunity to select the parameters of the laserpulse and to obtain pulses with negligible diffraction. From (76) it is seen that z pulsediff depends on the spot diameter of the pulse by four degree ( z diskdiff ∼ r ⊥ ).If we use pulse large enough in transverse dimension we can obtain practicallydiffractionless pulses. For example, using (76) and light disk with waist r ⊥ = 1 cm , time duration t = 33 f s ( z = 10 µm ), and k = 7 . × cm − ( λ = 800 nm ) we can obtain pulse diffraction length of order of z diskdiff ∼ km . SuchLD will be propagate in transparency region of gases or vacuum on severalthousand kilometers without practical diffraction enlargement.27 Conclusion
In this paper dynamics of ultrashort laser pulses in media with dispersion, dis-persionless media and vacuum are investigated in the frame of non-paraxial gen-eralization of the amplitude equation. In partial case of media with dispersion,we obtained an integro - differential nonlinear equation, governing propagationof optical pulses with time duration of order of the optical period. The slowlyvarying envelope approximation (many harmonics under the pulse) reduced thisamplitude integro - differential equation to the well known slowly-varying ampli-tude vector nonlinear differential equation with different orders of dispersion ofthe linear and nonlinear susceptibility. In case of propagation of optical pulsesin dispersionless media and vacuum, we obtained an nonparaxial amplitudeequation which is valid in both cases, namely, pulses with many harmonics andpulses with only one-two harmonics under the envelope. We normalized theseamplitude equations and obtained five dimensionless parameters determiningdifferent linear and nonlinear regimes. The nonparaxial envelope equations formedia with dispersion, dispersionless media and vacuum are solved in linearregime and new fundamental solutions, including the GVD, are found. In gasesand vacuum the solutions of these equations predict new diffraction length foroptical pulses z pulsediff = k r ⊥ /z . We demonstrate by these analytical and numer-ical solutions a significant decreasing of the diffraction enlargement of f s pulses(LB and LD) in respect to paraxial widening of a laser beam and a possibilityto reach diffraction-free regime. This work is partially supported by the Bulgarian Science Foundation undergrant F 1515/2005.
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