Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth q

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1 Optics Communications 226 (23) Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth q M. Zamboni-Rached a, *, K.Z. Nobrega a, H.E. Hernandez-Figueroa a, E. Recami b,c a DMO-FEEC, State University at Campinas, Campinas, S.P., Brazil b Facolta di Ingegneria, Universita statale di Bergamo, Dalmine (BG), Italy c INFN-Sezione di Milano, Milan, Italy Received 7 February 23; received in revised form 18 July 23; accepted 2 July 23 Abstract In this paper we set forth new eact analytical superluminal localized solutions to the wave equation for arbitrary frequencies and adjustable bandwidth. The formulation presented here is rather simple and its results can be epressed in terms of the ordinary, so-called X-shaped waves. Moreover, by the present formalism we obtain the first analytical localized superluminal approimate solutions which represent pulses propagating in dispersive media. Our solutions may find application in different fields, like optics, microwaves, radio waves, and so on. Ó 23 Elsevier B.V. All rights reserved. PACS: 3.5.De; 41.2.Jb; 83.5.Vr; 91.3.Fn; 4.3.Nk; Bs Keywords: Wave equation; Wave propagation; Localized beams; Superluminal waves; Bessel beams; X-shaped waves; Optics; Dispersion compensation 1. Introduction q Work supported by FAPESP (Brazil), and by MIUR, INFN (Italy). This paper did first appear as e-print physics/ * Corresponding author. Tel./fa: address: mzamboni@dmo.fee.unicamp.br (M. Zamboni-Rached). For many years it has been known that localized (non-dispersive) solutions eist to the (homogeneous) wave equation [1 7], endowed with subluminal or superluminal [8 13] velocities. These solutions propagate without distortion for long distances in vacuum. Particular attention has been paid to the superluminal localized solutions (SLS) like the socalled X-waves [9,1,12] and their finite energy generalizations [11,12]. It is well known that such (SLS) have been eperimentally produced in acoustics [14], optics [15] and more recently microwave physics [16,17] /$ - see front matter Ó 23 Elsevier B.V. All rights reserved. doi:1.116/j.optcom

2 16 M. Zamboni-Rached et al. / Optics Communications 226 (23) As is well known, the standard X-wave has a broad band frequency spectrum, starting from zero [12,13] (it being therefore appropriate for low frequency applications). This fact can be viewed as a problem, because it is difficult or even impossible to define a carrier frequency for that solution, as well as to use it in high frequency applications. Therefore, it would be very interesting to obtain eact SLSs to the wave equations with spectra localized at higher (and arbitrary) frequencies and with an adjustable bandwidth (in other words, with a well-defined carrier frequency). To the best of our knowledge, only three attempts were made in this direction: one by Zamboni-Rached et al. [12,18], another by Salo et al. [19,2], and the last one by S~onajalg et al. [21,22]. The first two authors showed how to shift the spectrum to higher frequencies without dealing with its bandwidth, while the latter worked out computer simulations [21] (not analytically) and eperiments [22] on optical X pulses in a dispersive medium, obtaining very important results which confirm the efficacy of these superluminal localized waves. In this work, however, we are presenting analytical and eact superluminal localized solutions in vacuum, whose spectra can be localized inside any range of frequency with adjustable bandwidths, and therefore with the possibility of choosing a well-defined carrier frequency. In this way, we can get (without any approimation) radio, microwave, optical, etc., localized superluminal waves. Taking advantage of our methodology, it means, etending the solution found for the vacuum, we obtain the first analytical approimations to the SLSs in dispersive media (i.e., in media with a frequency dependent refractive inde). One of the interesting points of this work, let us stress, is that all results are obtained from simple mathematical operations on the standard Xwave, using simple analytical epressions to do it. 2. Superluminal localized waves in dispersionless media for arbitrary frequencies and adjustable bandwidths Let us start by dealing with SLSs in dispersionless media. From the aially symmetric solution to the wave equation in vacuum (n ¼ n ), in cylindrical coordinates, one can easily find that wðq; z; tþ ¼J ðk q qþe þikzz e it with kq 2 ¼ n2 ð2 = Þ kz 2; k2 q P, where J is the zeroth-order ordinary Bessel function, k z and k q are the aial and the transverse wavenumber, respectively, is the angular frequency and c is the light velocity. Using the following transformation: k q ¼ n c sin h; k z ¼ n ð1þ c cos h; wðq; z; tþ can be rewritten as the well-known Bessel beam wðq; fþ ¼J n c q sin h e þin c f cos h ; ð2þ where f z Vt while V ¼ c=ðn cos hþ is the phase velocity, quantity hð < h < p=2þ being the cone angle of the Bessel beam. As it is well known [14,21], a SLS (with aial symmetry) to the wave equation in a dispersionless medium can be written as Z r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 wðq; fþ ¼ SðÞJ V q V n 2 2 c 1 e þi 2 V f d; ð3þ where SðÞ is the frequency spectrum. In fact, such solutions are pulses propagating in free space without distortion and with the superluminal velocity V ¼ c=ðn cos hþ. The most popular spectrum SðÞ is that one given by SðÞ ¼e a, which provides the ordinary X-shaped waves V X wðq; fþ ¼q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð4þ ðav ifþ 2 þ q 2 ðn 2 V 2 1Þ Because of its non-dispersive properties and its low frequency spectrum, 1 the X-wave is being particularly applied in fields like acoustics [9]. To find analytical epression for Eq. (3) is a difficult task, being possible or not. Its numerical solutions usually brings some inconveniences for further analysis, uncertainties concerning the fast 1 It is easy to see that this spectrum starts from zero, it being suitable for low frequency applications, and has the bandwidth D ¼ 1=a.

3 M. Zamboni-Rached et al. / Optics Communications 226 (23) oscillating field components, etc.; besides implying a loss in the physical interpretation of the results. Thus, it is always worth looking for analytical epressions. In this way, one of our main objectives is finding out a spectrum which can preserve the integrability of Eq. (3) for any frequency range. In order to be able to shift our spectrum towards the desired frequency, let us locate it around a central frequency, c, with an arbitrary bandwidth D (denoting its full standard deviation, i.e., the spectral function full-width when its height equals 1=e of its maimum value) The SðÞ spectrum Fig. 1. Behavior of the derivative number, m, as a function of the normalized bandwidth frequency, D = c. Given a central frequency, c, and a bandwidth, D, one finds the eact value of m by substituting these values into the first epression of Eq. (6). Let us choose the spectrum SðÞ ¼ me a ; ð5þ V where V is the wave velocity, while m and a are free parameters. For m ¼ ; SðÞ ¼epð aþ, and one gets the (standard) X-wave spectrum. After some mathematical manipulations, one can easily find the following relations, valid for m 6¼ : 1 m ¼ ; ðd = c Þ lnð1þðd = c ÞÞ c ¼ m ; ð6þ a where D D þ or D. Because of the nonsymmetric character of spectrum (5), let us call D þ ð> ) the bandwidth to the right, and D (< ) the bandwidth to the left of the spectrum central frequency c ; so that D ¼ D þ D.It should be noted, however, that, already for small values of m (typically, for m P 1), one has D þ D. Once defined c and D, one can determine m from the first equation and then, using the second epression, to find a. Fig. 1 illustrates the behavior of the first relation of Eq. (6). From this figure, one can observe that the smaller D= c is, the higher m must be. Thus, one can notice that m plays the fundamental role of controlling the spectrum bandwidth. From the X-wave spectrum, it is known that a is related to the (negative) slope of the spectrum. Contrarily to a, quantity m has the effect of rising the spectrum. In this way, one parameter compensates the other, producing the localization of the spectrum inside a certain frequency range. At the same time, this fact also eplains (because of the second epression of Eq. (6)) why an increase of both m and a is necessary to keep the same c. This can be inferred from Fig. 2. In Fig. 2, both spectra have the same central value c. Taking the narrow spectrum as a reference, one can observe that, to get such a result, both quantities m and a have to increase. More- Fig. 2. Normalized spectra for c ¼ 23: Hz and different bandwidths. The first with m ¼ 27 (solid line) and the second spectrum with m ¼ 45 (dotted line). See the tet.

4 18 M. Zamboni-Rached et al. / Optics Communications 226 (23) over, this figure shows the important role of m for generating a wider, or narrower, spectrum. Let us notice that the spectrum (5) was first considered in three relevant works by Friberg and co-workers [19,2,23], as well as in another paper by Zamboni-Rached et al. [12]. However, in those previous works only the spectrum shifting was discussed (second epression of Eq. (6)), meanwhile nothing was mentioned about its bandwidth (first epression of Eq. (6)). To our knowledge, the present work is the first effort to completely describe the relationship between the central frequency and its bandwidth of the spectrum given by Eq. (5) X-type waves in a dispersionless medium Applying our spectrum epressed by Eq. (5), Eq. (3) can be written as Z 1 m wðq; fþ ¼V V r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! J V q V n 2 2 c 1 e ðav ifþ=v d : 2 V ð7þ Therefore, we have seen that the use of a spectrum like (5) allows shifting it towards any frequency and confining it within the desired frequency range. In fact, this is one of its most important characteristics. It can be noticed that, using SðÞ ¼epð aþ, Eqs. (3) and (4) are equivalent. Thus, differentiating Eq. (3) with respect to ðav ifþ, a multiplicative factor ð =V Þ is each time produced (an obvious property of Laplace transforms). In this way, it is possible to write Eq. (7) as wðq; fþ ¼ð 1Þ m o m X oðav ifþ m : ð8þ A different epression for Eq. (7), without any need of calculating the mth differentiation of the X- wave, can be found by using identity (6.621) of [24] mþ1 Cðm þ 1ÞX m þ 1 wðq; fþ ¼ F ; m V m 2 2 ; 1; n 2 V 2 1 q 2 X 2 ; ð8 Þ V 2 Fig. 3. The real part of an X-shaped beam for microwave frequencies in a dispersionless medium. where X is the ordinary X-wave given in Eq. (4), and F is a GaussÕ hypergeometric function. Eq. (8 ) can be useful in the cases of large values of m. Fig. 3 shows an eample of an X-shaped wave for microwave frequencies. To that aim, it was chosen c ¼ 6 GHz, D ¼ :9 c, V ¼ 5c and the values of m and a were calculated by using Eq. (6): thus obtaining m ¼ 1 and a ¼ 1: s. As one can see, the resulting wave has really the same shape and the same properties as the classical X-waves: namely, both a longitudinal and a transverse localization. 3. Superluminal localized waves in dispersive media In Section 2, Eq. (1) was written for a dispersionless medium (n ¼ constant, independent of the frequency). However, for a typical medium, when the refractive inde depends on the wave frequency, nðþ, those equations become [21] k q ðþ ¼ nðþ sin h; c k z ðþ ¼ ð9þ nðþ cos h: c The above equations describe one of the basic points of this work. In Section 2 it was mentioned that h determines the wave velocity: a fact that can be eploited when one looks for a localized wave that does not suffer dispersion. In other words, one can choose a particular frequency dependence of h to compensate the ( material ) dispersion due to

5 M. Zamboni-Rached et al. / Optics Communications 226 (23) the variation with the frequency of the refractive inde [21]. If the frequency dependence of the refractive inde in a medium is known, within a certain frequency range, let us see how the consequent dispersion can be compensated for. When a dispersionless pulse is desired, the constraint k z ¼ d þ b must be satisfied (in other words the group velocity, o=ok z, does not depend on the frequency). And, by using the last term in Eq. (9), one infers that such a constraint is forwarded by the following relationship between h and : cðd þ bþ cos hðþ ¼ ; ð1þ nðþ where d and b are arbitrary constants (and b is related to the wave velocity: b ¼ 1=V ). For convenience, we shall consider d ¼. Then, Eq. (3) can be rewritten as wðq;fþ¼ Z 1 SðÞJ V q r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! n 2 ðþv 2 1 e þibf d: ð11þ Let us stress that this equation is a priori suited for many kinds of applications. In fact, whatever its frequency be (in the optical, acoustic, microwave, etc., range), it constitutes the integral formula representing a wave which propagates without dispersion in a dispersive medium. Let us recall, now, that there are at least three different ways (see, e.g. [21,22]) for generating these dispersionless localized superluminal pulses, whose wave vectors and frequencies obey Eq. (9) and whose cone angle is given by Eq. (1). Such different approaches may be found discussed with some details in [21,22,25 29], and here we shall describe them briefly. The first one uses the classical [1] set-up of a multiannular aperture plus a lens, where each frequency component of the pulse illuminates however a different (but specific) annular slit [25], in such a way that relations (9) and (1) get satisfied. In the known eperiments by Durnin et al. [2,3] a single annular slit and a lens were adopted to generate a Bessel beam. A second, simple approach, which works for a medium with low dispersion, consists in using an Aicon (see [21,22], and references therein). When illuminated by a pulse, this conical lens gets relation Eq. (9) satisfied. Moreover, for small values of the angle c between the conical and the flat surface, the cone angle h can be epressed by the simple epression hð; cþ ¼ðn A ðþ 1Þc, where n A ðþ is the refractive inde of the aicon material [21,22,26]. In the case of a low dispersion medium and of group-velocities V > c, one can therefore look for the best value of c to fit also Eq. (1) by means of the previous relation h ¼ hð; cþ, which is linear in c. The last approach consists in using holographic elements, which make possible to deal satisfactorily also with stronger dispersion media. Elements of this kind have been considered in papers as [27 29], whose transmission function can correspond to two successive optical elements (an aicon and a thin lens). In general, one can follow a procedure similar to the one illustrated in Fig. 4. In such a case, there is a different deviation of the wave vector for each spectral component in passing through the setup chosen (one of the three listed above). Such deviation, associated with the dispersion due to the medium, makes the phase velocity equal for each frequency. This corresponds to no dispersion for the group velocity. More details about the physics under consideration can be found in [21,22]. Now, let us consider a nearly gaussian spectrum as that one given by Eq. (5), and assume the presence of a dispersive medium whose refractive inde (for the frequency range of interest) can be written in the form nðþ ¼n þ d; ð12þ where n is a constant, while d is a free parameter that makes it possible a linear behavior of nðþ: something that is actually realizable for frequencies far from the resonances associated with the used material. Notice that the linear relationship between the refractive inde and the wave frequency assumed in Eq. (12) is not necessary: but its eistence gets our calculations simplified. In this way, on substituting Eq. (12) into Eq. (11), and considering the spectrum, shifted towards optical frequencies, given by Eq. (5), a relation similar to Eq. (13) is found

6 2 M. Zamboni-Rached et al. / Optics Communications 226 (23) Fig. 4. Sketch of a generic setup (aicon, hologram, etc.) suited to properly deviating the wave vector of each spectral component. Z 1 mj wðq;f;dþ¼v V r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! V q V 2 c ðn 2 þ dþ 2 1 e ðav ifþ=v d : V ð13þ To the purpose of evaluating Eq. (13), let us make a Taylor epansion and rewrite it as wðq; f; dþ ¼wðq; f; Þþd ow od þ d2 o 2 w d¼ 2! od 2 d¼ þ d3 o 3 w 3! od 3 þ: ð14þ d¼ For the above equation it is known that, if d is small enough, it is possible to truncate the series at its first derivative. For the time being, let us assume this is the case and that there is no problem on truncating Eq. (14). One can check Fig. 5, which shows typical values of d for SiO 2, a typical raw-material in fiber optics. Looking at Eq. (14), one can notice that its first term wðq; f; Þ is already known to us, because it coincides with the solution given by our Eq. (7). To complete the epansion (14), one must find ow j od d¼. After some simple mathematical manipulations, one gets that Z ow V 4 qn 1 mþ2e ðav ifþ=v od ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d¼ n 2 V 2 1 V J 1 q r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! V n 2 2 V c 1 d : ð15þ 2 V This integral can be easily evaluated by using identity of [24], so to obtain ow od ¼ð 1Þ mþ4 V 3 n d¼ n 2 V 2 1 o mþ2 ½ oðav ifþ ðav ifþx Š: ð16þ mþ2 As in the case of Eqs. (8), (8 ), another form for epressing Eq. (15) can be found by having recourse once more to the identity (6.621) of [24] ow od ¼ n q 2 Cðm þ 4ÞX mþ4 d¼ 2 V m m þ 4 m 1 F ; ; 2; 2 2 n 2 V 2 1 q 2 X 2 ; ð16 Þ V 2 where, as before, X is the ordinary X-wave given by Eq. (4) and F is again a GaussÕ hypergeometric

7 M. Zamboni-Rached et al. / Optics Communications 226 (23) Fig. 5. Variation of the refractive inde nðþ with frequency for fused silica. The solid line is its behaviour, according to SellmeierÕs formulae. The open circles and squares are the linear approimations for m ¼ 45 and m ¼ 27, respectively. function. Once more, Eq. (16 ) can be useful in the cases of large values of m. Finally, from our basic solution (8) and its first derivative (16), one can write the desired solution of Eq. (11) as wðq; fþ ¼ð 1Þ m o m X oðav ifþ m o mþ2 þð 1Þ mþ4 V 3 n n 2 V 2 1 c oðav ifþ mþ2 2 ðav ½ ifþx Šd: ð17þ However, if one wants to use Eqs. (8 ) and (16 ), instead of Eqs. (8) and (16), the solution (17) can be written in the form mþ1 Cðmþ1ÞX wðq;fþ¼ F mþ1 V m 2 ; m 2 ;1; V 2 1 q 2X d 2 n q 2 Cðmþ4ÞX mþ4 n 2 V 2 2 V m F mþ4 2 ; m 1 ;2; n 2 V 2 1 q 2X 2 ; 2 V 2 ð17 Þ It is also interesting to notice that the approimate superluminal localized solution (17) for a dispersive medium has been obtained from simple mathematical operations (derivatives) applied to the standard X-wave. 4. Optical applications To illustrate what was said before, two practical eamples will be considered, both in optical frequencies. When mentioning optics, it is natural to refer ourselves to optical fibers. Then, let us suppose the bulk of the dispersive medium under consideration to be fused silica (SiO 2 ). Far from the medium resonances (which is our case), the refractive inde can be approimated by the well-known Sellmeier equation [3] n 2 ðþ ¼1 þ XN j¼1 B j 2 j 2 j 2 ; ð18þ where j is the resonance frequency, B j is the strength of the jth resonance, and N is the total number of the material resonances that appear in the frequency range of interest. For typical frequencies of long-haul transmission in optics, it is necessary to choose N ¼ 3, which leads us [29] to the values B 1 ¼ : , B 2 ¼ :479426, B 3 ¼

8 22 M. Zamboni-Rached et al. / Optics Communications 226 (23) : , k 1 ¼ :68443 lm, k 2 ¼ : lm andk 3 ¼ 9: lm. Fig. 5 illustrates the variation of the refractive inde nðþ with frequency for fused silica. The solid line is its behaviour, according to SellmeierÕs formulae. The open circles and squares are the linear approimations for m ¼ 45 and m ¼ 27, respectively, and specifies the ranges that will be adopted here. In the two eamples, the spectra are localized around the angular frequency c ¼ 23: Hz (which corresponds to the wavelength k c ¼ :8 lm), with two different bandwidth D 1 ¼ :6 c and D 2 ¼ :45 c. The values of a and m corresponding to these two situations are a ¼ 1: , m ¼ 27, and a ¼ 1: , m ¼ 45, respectively. We also have kept 2 the same velocity, b ¼ 1=V ¼ 1=ð5cÞ. Looking at these windows, one can notice that silica does not suffer strong variations of its refractive inde. As a matter of fact, a linear approimation to n ¼ nðþ is quite satisfactory in these cases. Moreover, for both situations, and for their respective n values, the value of parameter d results to be very small, verifying condition (11): which means that it is quite acceptable our truncation of the Taylor epansion. The beam intensity profiles for both bandwidths are shown in Figs. 6 and 7. In the first figure, one can see a pattern similar to the fundamental X-wave, but here, of course, the pulse is much more localized spatially and temporally (typically, it is a femtosecond pulse). In the second figure, one can observe some little differences with respect to the first one, mainly in the spatial oscillations inside the wave envelope [18]. This may be eplained by taking into account that, for certain values of the bandwidth, the carrier wavelength become shorter than the width of the spatial envelope; so that one meets a well-defined carrier frequency. Let us point out that both these waves are transversally and longitudinally localized, and 2 Suitable ways for generating such pulses can be the first and especially the third one listed in Section 3, after Eq. (11). By contrast, the second approach (with an Aicon) is not appropriate, whenever the considered medium has non-negligible dispersive effects. Fig. 6. The real part of an X-shaped beam for optical frequencies in a dispersive medium, with m ¼ 27. It refers to the outside window in Fig. 5. Fig. 7. The real part of an X-shaped beam for optical frequencies in a dispersive medium, with m ¼ 45. It refers to the inner window in Fig. 5. that, since the dependence of w on z and t is given by f ¼ z Vt, they are free from dispersion, just like a classical X-shaped wave. 5. Conclusions In this paper we have first worked out analytical superluminal localized solutions to the wave equation for arbitrary frequencies and with adjustable bandwidth in vacuum. The same methodology has been then used to obtain new, analytical epressions representing X-shaped waves (with arbitrary frequencies and adjustable bandwidth)

9 M. Zamboni-Rached et al. / Optics Communications 226 (23) which propagate in dispersive media. Such epressions have been obtained, on one hand, by adopting the appropriate spectrum (which made possible to us both choosing the carrier frequency rather freely, and controlling the spectral bandwidth), and, on the other hand, by having recourse to simple mathematics. Finally, we have illustrated some eamples of our approach with applications in optics, considering fused silica as the dispersive medium. Acknowledgements The authors are grateful to the anonymous referees of this article, as well as to C.A. Dartora and Amr Shaarawi for continuous scientific collaboration; and to Jane M. Madureira for stimulating discussions. For useful discussions they also thank V. Abate, C. Becchi, M. Brambilla, C. Cocca, R. Collina, G.C. Costa, G. Degli Antoni, F. Fontana, M. Pernici, M. Villa and M.T. Vasconselos. References [1] J.N. Brittingham, J. Appl. Phys. 54 (1983) [2] J. Durnin, J.J. Miceli, J.H. Eberly, Phys. Rev. Lett. 58 (1987) [3] J. Durnin, J.J. Miceli Jr., J.H. Eberly, Opt. Lett. 13 (1988) [4] R.W. Ziolkowski, J. Math. Phys. 26 (1985) 861. [5] R.W. Ziolkowski, Phys. Rev. A 39 (1989) 25. [6] A. Shaarawi, I.M. Besieris, R.W. Ziolkowski, J. Math. Phys. 3 (1989) [7] A. Shaarawi, R.W. Ziolkowski, I.M. Besieris, J. Math. Phys. 36 (1985) [8] R. Donnelly, R.W. Ziolkowski, Proc. R. Soc. Lond. A 44 (1993) 541. [9] J.-y. Lu, J.F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39 (1992) 441. [1] E. Recami, Physica A 252 (1998) 586, and references therein. [11] I.M. Besieris, M. Abdel-Rahman, A. Shaarawi, A. Chatzipetros, Prog. Electromagn. Res. 19 (1998) 1. [12] M. Zamboni-Rached, E. Recami, H.E. Hernandez-Figueroa, Eur. Phys. J. D 21 (22) 217. [13] M. Zamboni-Rached, Localized solutions: structure and applications, M.Sc. Thesis, Physics Department, State University of Campinas, [14] J.-y. Lu, J.F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39 (1992) 19. [15] P. Saari, K. Reivelt, Phys. Rev. Lett. 79 (1997) [16] D. Mugnai, A. Ranfagni, R. Ruggeri, Phys. Rev. Lett. 84 (2) 483. [17] E. Recami, Found. Phys. 31 (21) [18] M. Zamboni-Rached, K.Z. Nobrega, E. Recami, H.E. Hernandez, Phys. Rev. E 66 (22), article [19] J. Salo, J. Fagerholm, A.T. Friberg, M.M. Salomaa, Phys. Rev. E 62 (2) [2] J. Salo, A.T. Friberg, M.M. Salomaa, J. Phys. A 34 (21) [21] H. S~onajalg, P. Saari, Opt. Lett. 21 (1996) [22] H. S~onajalg, M. Ratsep, P. Saari, Opt. Lett. 22 (1997) 31. [23] J. Fagerholm, A.T. Friberg, J. Huttunen, D.P. Morgan, M.M. Salomaa, Phys. Rev. E 54 (1996) [24] I.S. Gradshteyn, I.M. Ryzhik, Integrals, Series and Products, fourth ed., Academic Press, New York, [25] H. S~onajalg, A. Gorokhovskii, R. Kaarli, et al., Opt. Commun. 71 (1989) 377. [26] R.M. Herman, T.A. Wiggins, J. Opt. Soc. Am. A 8 (1991) 932. [27] A. Vasara, J. Turunen, A.T. Friberg, J. Opt. Soc. Am. A 6 (1989) [28] R.P. MacDonald, J. Chrostowski, S.A. Boothroyd, B.A. Syrett, Appl. Opt. 32 (1983) 647. [29] See, e.g., V.P. Koronkevich, I.A. Mikhaltsova, E.G. Churin, Y.I. Yurlov, Appl. Opt. 34 (1995) 5761, and references therein. [3] G.P. Agrawal, Nonlinear Fiber Optics, second ed., Academic Press, New York, 1995.