Self-Trapping of Electromagnetic Beams in Vacuum Supported by QED Nonlinear Effects

PHYSICAL REVIEW A, VOLUME 62, 043817

Self-trapping of electromagnetic beams in vacuum supported by QED nonlinear effects

Marin Soljacˇic´ and Mordechai Segev* Princeton University, Princeton, New Jersey 08544 ͑Received 10 June 1999; revised manuscript received 10 April 2000; published 19 September 2000͒ At very high radiation field intensities in vacuum, Maxwell equations need to be modified to account for the QED photon-photon interaction. We show that such modified equations support nondiffracting spatial radiation solitons that propagate for very long distances without changing their shapes. These solitons, and the underlying self-focusing and instability effects, should be observable in the near future.

PACS number͑s͒: 42.50.Gy, 42.65.Tg

A localized wave packet propagating in linear homoge- self-action ͓12͔, vacuum birefrengence ͓13͔, etc. In this pa- neous media has a tendency to change its spatial width as it per, we show that Maxwell equations, when modified to ac- propagates. However, everything in nature is nonlinear, in- count for the lowest order photon-photon QED interaction, cluding the vacuum ͓1͔. More specifically, at very high elec- allow for the existence of bright spatial solitons. Such soli- tromagnetic field intensities, Maxwell equations need to be tons are wave packets of high-intensity light, propagating for modified by small nonlinear terms. These terms come from long distances in a true vacuum, without changing their di- the quantum field theory of photon-photon scattering medi- mensions. We believe that these solitons, or at least the un- ated through exchange of virtual electron-positron pairs, as derlying self-focusing, will be observed in a near future in discovered theoretically by Euler and Heisenberg in 1936 specifically designed experiments. ͓1͔. The Euler-Heisenberg approximation of the ͑linear plus Up to the lowest-order correction, the nonlinear effective nonlinear͒ polarization of a true vacuum holds when the Lagrangian density is given by wavelengths of the interacting photons are all much longer 1 ␰ than the Compton wavelength of the electron, and, at the ␮␯ ␮␯ 2 LϭϪ F␮␯F Ϫ 5͑F␮␯F ͒ same time, the field is much weaker than the QED critical 16␲ 64␲ † field 2m2c3/(eh). Importantly, under these conditions, there Ϫ ␯␬ ␭␮ ͑ ͒ is no absorption, i.e., no real electrons and positrons are gen- 14F␮␯F F␬␭F ], 1 erated in the process ͓1͔. ͑This is not surprising, because like ץϪ ץ␰ϭប 4 ␲ 4 7 ϭ all off-resonance interactions, the real part of the susceptibil- where e /45 m c , and F␮␯ ␮A␯ ␯A␮ is the ͑ ͒ ity drops off much more slowly than the imaginary part electromagnetic-field tensor. The first term in Eq. 1 is just when the carrier frequency moves away from the resonant the usual clasical term, and the rest is the so-called ͓ ͔ frequency.͒ Heisenberg-Euler correction 1 . There are also QED correc- The addition of nonlinear terms to the wave equation can tions to the Lagrangian above that include the spatial and ␮␯ ͓ ͔ sometimes influence the dynamics of the wave packet’s temporal derivatives of F 20 . However, we intend to shape, so that its dimensions do not change at all as the wave work with light of characteristic wavelengths much longer packet propagates. Such a wave packet, in which the diffrac- than the Compton wavelength of an electron, and also at very tion, which tends to expand the pulse, is exactly balanced by high intensities, so the leading terms are in our case given by ͑ ͒ the nonlinear effects that are trying to shrink the pulse, is Eq. 1 , and we can safely ignore the lowest-order corrections to the Lagrangian that include the spatial and temporal loosely refered to as a soliton. Ever since solitons were sci- ␮␯ entifically documented ͓2͔, they have fascinated scientists in derivatives of F . Physically, these corrections are impor- many different fields. A universal nonlinear phenomenon, tant when the wavelength of the light is comparable to the solitons have been found in many different forms in nature. Compton wavelength of the electron, while the Heisenberg- For example, they were described on surface of shallow wa- Euler correction is important when the electric field becomes ter ͓2͔, in deep sea water ͓3͔, in plasma ͓4͔, on the surface of comparable to the critical electric field. One can show that black holes ͓5͔, for torsional waves on DNA molecules ͓6͔, the derivative terms are much smaller than the ones we in- ␻ ␻ 2Ӷ ␲3 2 ͑ etc. Solitons have also been studied extensively in nonlinear cluded as long as ( / c) (4/ )(E/Ec) , please see the ͒ ␻ optical systems; both spatial ͓7,21͔, and temporal solitons ͓8͔ Appendix , where c is the Compton frequency of the elec- have been investigated. Solitons also appear in quantum sys- tron, and Ec is the critical field. We are interested in the tems; for example, nucleons are a kind of solitons ͓9͔. regime where the Heisenberg-Euler correction is much more Over the years, various authors have studied the feasibil- important than the correction that involves derivatives. Nev- ity of using high-intensity radiation in vacuum to observe ertheless, we have to be careful that all proposed experimen- some nonlinear effects ͓10͔, such as four-wave mixing ͓11͔, tal verifications of our concept satisfy this requirement. Extremizing the action with respect to the four-vector po- tential A␮, one finds *FAX: ͑609͒ 258-1954. Present address: Physics Department, Bץ 1 Technion, Israel Institute of Technology, Haifa 32000, Israel. Email Bϭ0, ϫEϩ ϭ0, tץ address: [email protected] “• “ c

Dץ 1 Dϭ0, ϫHϪ ϭ0, ͑2͒ tץ c “ •“

where D and H are given by

Lץ Dϭ ϭEϩ4␲P, Eץ

␰ Pϭ ͓2͑E2ϪB2͒Eϩ7͑E B͒B͔, 4␲ • ͑3͒ Lץ HϭϪ ϭBϪ4␲M, FIG. 1. The QED soliton is a slowly varying amplitude modu- B lating two carrier plane waves. The figure shows the k vectors ofץ these carrier waves along with their polarization vectors nˆ. Solid ␰ vectors refer to one of the carrier waves, and the dashed vectors MϭϪ ͓2͑E2ϪB2͒BϪ7͑E B͒E͔. 4␲ • refer to the other. The polarization vector of each wave is perpen- dicular to the k vector to which it corresponds. The k vectors of The reason that the nonlinearities include precisely the both waves lie in the yz plane, and are mirror images of each other E2ϪB2 and E B terms is that these are the only Lorentz- with respect to the xy plane. The polarization vectors of the two • carrier waves are also mirror images of each other with respect to invariant scalar combinations of E and B fields. However, the xy plane. this implies that the nonlinear effects on a beam that is a plane wave modulated by some slowly varying envelope are 2͒. Without nonlinear effects, the beam’s width in x would only of the second order; a plane wave has E and B perpen- 2ϭ 2 grow as a function of z; the beam would diffract. Our goal is dicular to each other, and E B for plane waves in to construct a beam ͑a ‘‘beam’’ in space͒ whose shape does vacuum. Consequently, the lowest-order nonlinear effects ͑ ͒ not change in the x direction as the beam propagates. In the vanish identically in Eqs. 3 . One could, in principle, look slowly varying envelope approximation, we take ␣ӷ␭, into cases of very large beam divergence ͑beyond the ␭ ͒ where is the wavelength of the carrier. Then one of the paraxial approximation , or include corrections to P and M simplest E fields that satisfies all of these requirements is that include spatial derivatives, but all of these options will given to the lowest order by require field intensities that are many orders of magnitude higher than the sources available today, or else require wave- A͑x͒ lengths that are unaccessible to the laser technology. There- E͑x,y,z,t͒ϭ ͕nˆ cos͑k rϪ␻t͒ϩnˆ cos͑k rϪ␻t͖͒, 2 • • fore, in order to decrease the radiation intensity required to ͑5͒ observe self-trapped beams supported by the QED nonlinearities in vacuum, we explore slowly varying envelopes su- where nˆ is any unit vector, and a means the mirror inverted perimposed on top of two crossed plane waves, so that the image of a around the xy plane; for example, if a nonlinearities in Eq. ͑3͒ do not vanish to lowest order. ϭ ϭ Ϫ (ax ,ay ,az) is a vector, then a (ax ,ay , az). A(x) is the We proceed by forming the wave equations starting from slowly varying amplitude; the characteristic length scale of Eqs. ͑2͒. Although the original motivation for going to the A(x) is given by ␣. From the geometry shown in Fig. 1, it is ͑ equations with higher-order derivatives namely, decoupling clear that k has no component in the x direction; also k nˆ ͒ • E and B , appears to be lacking, equations with higher-order ϭ0 for lowest order fields in vacuum. Consequently, speci- derivatives are easier to analyze, and fields actually do de- ␻ fying nˆ, , the dispersion relation, and the sign of ky , speci- couple for certain relative polarizations. The wave equations fies k to the lowest order uniquely. are We substitute Eq. ͑5͒ into the second equation in Eq. ͑2͒ to find B to the lowest order, assuming 1/␣Ӷ1/␭: ץ 2D 1ץ 1 2DϪ ϭϪ4␲ͭ ϫ͑ ϫP͒Ϫ ͑ ϫM͒ͮ , t “ A͑x͒ץ t2 “ “ cץ c2 “ B͑x,y,z,t͒ϭ ͕kˆϫnˆ cos͑k rϪ␻t͒ ͑4͒ 2 • 6͒͑ ץ 2B 1ץ 1 2 Ϫ ϭ ␲ͭ ϫ͑ ϫ ͒Ϫ ͑ ϫ ͒ͮ ͖͒ P . Ϫ ˆϫ ͑ Ϫ␻ ץ M 4 2 ץ B 2 “ c t “ “ c t “ k nˆ cos k•r t . ϭ The beam we intend to study is a superposition of two Although E•B 0, the other scalar combination of E and B ͑ ͒ 2Ϫ 2ϭ 2 ͓ 2ϩ 2 2͔ plane waves, modified by a slowly varying envelope in the x appearing in Eq. 3 , E B A (x) ny nx (kz /k) ϫ ϩ Ϫ␻ Þ direction. The two k vectors of the two carrier waves lie in ͕cos(2kzz) cos͓2(kyy t)͔͖/2 0; as one might expect, the yz plane, and they are mirror images of each other with the dc term vanishes identically. More importantly, in our respect to the xy plane, as illustrated in Fig. 1. Our ‘‘beam’’ configuration of two beams propagating under an angle with is therefore infinite in the z direction, propagating in the y respect to each other, there is a symmetry with respect to the direction, and has a finite width ␣ in the x direction ͑see Fig. xy plane. Because of this symmetry, there can be no terms

043817-2 SELF-TRAPPING OF ELECTROMAGNETIC BEAMS IN . . . PHYSICAL REVIEW A 62 043817

ϩ Ϫ␻ 2Ϫ 2 ͑ like cos͓2(kyy kzz t)͔ appearing in E B . This will by interaction between those small terms and the carier have profound consequences below. The second-harmonic beam͒, one expects the efficiency of this process to be very terms that survived combine with E and B in Eqs. ͑3͒ for P significantly suppressed; therefore, it should be unobservable and M, respectively, to produce the first-harmonic terms, and for fairly long propagation distances, compared to the self- also third-harmonic terms. After a few lines of algebra, we trapping length. Substituting E, B, P, and M from Eqs. ͑5͒, obtain nonvanishing P and M to the lowest order: ͑6͒, and ͑7͒, into the wave equations ͑4͒, and keeping only the lowest order terms, yields ␰ 2 2 2ץ kz ϭ 2͑ ͒ͫ 2ϩ 2ͩ ͪ ͬ ϩ͑ ͒ 1 E kz ,P ␲ A x ny nx E TOH , ٌ2EϪ ϭϪ2␰A2͑x͒ͫn2ϩn2ͩ ͪ ͬk2E t2 y x k zץ k c2 4 ͑ ͒ 7 ͑8͒ 2 2ץ ␰ 2 kz 1 B kz ,Mϭ A2͑x͒ͫn2ϩn2ͩ ͪ ͬBϩ͑TOH͒, ٌ2BϪ ϭϪ2␰A2͑x͒ͫn2ϩn2ͩ ͪ ͬk2B =t2 y x k yץ 4␲ y x k c2 where Bϭ(ϪB ,B ,B )ifBϭ(B ,B ,B ). where n and n are components of the unit vector nˆ. In Eqs. = x y z x y z x y Unlike any other case in nonlinear optics, where we have 7, ͑TOH͒ stands for ‘‘third-order harmonics,’’ such as those equations for the electric field only, here we have two equa- proportional to cos(3k z), etc. Specifically, these terms are z tions, one for E and one for B, which represent the same proportional to cos(3k z)cos(k yϪ␻t), and cos(k z)cos͓3(k y z y z y electromagnetic field. This is a manifestation of the full sym- Ϫ␻t)͔. One might worry that these terms might lead to sig- metry between E and B in QED: the nonlinearity here is nificant energy drainage from the initial carrier beam. How- driven neither by electric dipoles of the medium nor by mag- ever, as we will see below, k is of the same order as k . y z netic dipoles of the medium, but by the interaction of elec- Thus these third-order harmonics do not satisfy the disper- tromagnetic field with itself in vacuum, for which the sym- 2 2ϩ 2 ϭ␻2 sion relation c (ky kz ) . Thereby, in any equilibrium metry between E and B is complete. Thus one must find a ͑ ͒ self-trapped shape, the corrections to Eq. 5 that they induce way to make these two equations consistent with each other. have to be tiny in order to be well behaved when plugged The only way to do this, and to keep Eqs. ͑8͒ self-consistent into Eqs. ͑4͒; in particular, the corrections to E and B have to ͑ ͒ ͑ ͒ 2 after substituting Eqs. 5 and 6 into them, is to have ky be O(⌫/␻ ) smaller than the fields in Eqs. ͑5͒ and ͑6͒. ͑Here ϭ ϭ ⌫ kz , and nˆ xˆ. Therefore, in the case of the ansatz of Eqs. is the nonlinear correction to the dispersion relation of (5) and (6), the two carrier waves have to intersect perpen- vacuum ͓to be defined in Eq. ͑9͒ below͔; as one would ex- ͒ dicular to one another, and the radiation has to be polarized pect, and as we show below, this correction is indeed tiny. in the x direction. After setting k ϭk and nˆϭxˆ, both of In the language of nonlinear optics, this is very similar to y z Ϫ␻ these equations reduce to saying that terms like cos(3kzz)cos(kyy t) are not phase 2 ͑ ͒ ␰ matched with our beam, and terms like cos(kzz)cos͓3(kyy d A x ϩ⌫A͑x͒ϭϪ A3͑x͒/k2, ͑9͒ Ϫ␻t)͔ are asynchroneous with our beam. Therefore, both dx2 2 types of third-harmonic terms can be neglected. The lack of Ϫ␻ ͑ ͒ ⌫ϵ ␻ 2Ϫ 2ϩ 2 terms like cos(3kzz)cos͓3(kyy t)͔ in Eq. 7 differs mark- where ( /c) (kz ky) is the correction to the disper- edly from the usual nonlinear optics systems. Algebraically, sion relation due to the nonlinear effects. One can think of ⌫ it is a direct consequence of the fact that there are no terms as an eigenvalue that has to be specified together with the ϩ Ϫ␻ 2Ϫ 2 ͑ ͒ like cos͓2(kyy kzz t)͔ in E B , which in turn is caused eigenfunction A(x) in order to solve Eq. 9 self- by the fact that our ‘‘carrier wave’’ is not a plane wave but consistently. From Eq. ͑9͒ we see that ͉⌫͉ϭO(1/␣2)Ӷk2,so an intersection of two plane waves. Physically, one can think this correction is small. of our configuration as being equivalent ͑apart for the bound- Equation ͑9͒ is the famous (1ϩ1)D cubic nonlinear ary conditions͒ to a beam propagating between two infinite Schro¨dinger equation ͑NLSE͓͒14,21͔. This equation is inte- conducting sheets parallel with the xy plane. In such a con- grable, so all of its solutions can be written in an analytical figuration, there is an effective dispersion created by the ge- form. The equation supports bright solitons, and solitons of ometry rather than the presence of some absorption reso- all orders can be found. The lowest order soliton is given by nance. This is the intuitive reason why our system displays A(x)ϭ2 sech(x/␣)/(␣kͱ␰), and its corresponding ⌫ϭ no third-harmonic generation ͑THG͒, although one might ex- Ϫ1/␣2. pect THG to occur since one normally thinks of vacuum as Having found nondiffracting solutions, the question about being essentially dispersionless. their stability arises naturally. Stability of (1ϩ1)D solitons The only thing we have to be careful about is that the of Eq. ͑9͒ has been studied extensively ͓14͔, and there is no ͑TOH͒ terms from Eq. ͑7͒ can in principle ͑through the non- doubt that these solitons are very stable. Nevertheless, one ͒ linearity induce terms proportional to cos(3kzz)cos͓3(kyy should also think about the stability of the solutions with Ϫ␻t)͔, which are phase matched with the starting beam, and respect to the underlying two wave equations. For the ansatz thus can drain the energy from the original beam. However, of Eqs. ͑5͒ and ͑6͒, we have found a self-consistent solution since this is only an indirect process, that requires drainage only if the two carrier beams are propagating under a 90° of energy from the first into the third harmonic ͑through the angle with respect to each other. Thus one should ask non-phase-matched and/or asynchroneous terms neglected whether our solution is also stable under small deviations of above͒, and then back from the third harmonic into the first this angle. Furthermore, one should also question the stabil-

043817-3 MARIN SOLJACˇ IC´ AND MORDECHAI SEGEV PHYSICAL REVIEW A 62 043817

ciple, these solitons, or at least the underlying self-focusing effects, can therefore be observed in vacuum, in space, or perhaps even in a laboratory environment, as creatures of stationary, nonvarying, and nonpropagating shapes that are made solely of photons. These creatures would be supported by the mutual interaction between the photons that make up the soliton. Finally, it is instructive to discuss possible experimental realizations. The solitons we propose are not observable with the laser technology available today. Nevertheless, given the rate at which the laser technology has been advancing in the recent decades, the solitons proposed here should probably FIG. 2. The geometry of the proposed configuration. The energy be observable within a decade or two. The peak intensity is transported along the y direction, and the beam is self-trapped in needed to support the soliton is given by approximately the x direction; the characteristic length of the beam in the x direc- ␧ c(␭/␣)2␰Ϫ1/␲2Ϸ1037 W/m2. Although 1037 W/m2 is an tion is ␣. The two carrier waves interfere, forming interference 0 enormous intensity, we have a great flexibility in decreasing fringes. the intensity requirements for an experimental observation by making the size of the soliton large compared to the car- ity regarding the particular polarization of the E field neces- rier wavelength. The largest laser intensities available today sary for the solution in Eqs. ͑5͒ and ͑6͒ to be valid. These are O(1024 W/m2), and they are in the near-infrared ͓17͔.On kinds of stability are much more complicated to study than the other hand, the shortest wavelength lasers demonstrated the ‘‘conventional’’ stability analyses of solitons of the cubic thus far are x-ray lasers, with a carrier wavelength of NLSE, because of the complicated underlying interplay of O(10 nm) ͓18͔. When designing an experiment, the first con- the vectorial E and B fields. Thus far, even after an intensive sideration is the available propagation length required to effort, we were not able to prove or disprove their stability. confirm that a beam forms a soliton ͑or at least to see con- Further study of the stability of these QED solitons will have siderable self-focusing effects͒: the diffraction length. We to address these questions. envision that such experiments will be done with ultrashort Our initial reason for investigating QED solitons in the lasers pulses, so this will enormously reduce the requirement (1ϩ1)D topology proposed above were the facts that the on the total pulse energy required, to be within reach of calculation is fully analytic, and more importantly, that the near-future technology. A possible realization is under labo- solutions are known to be stable, at least in the context of a ratory conditions: the idea is to superimpose two sheets of single nonlinear equation ͓14͔ as given by Eq. ͑9͒. Of course, light propagating at a 90° angle with respect to each other. If it would be much nicer to have solitons of a (2ϩ1)D topol- the length of the sheets in the z direction is of a size similar ogy. If nothing else, this might relax the power requirements to that of the propagation length in the y direction, one for experimental observation. Therefore, as an avenue to- should be able to observe solitonic effects before the sheets ward further research we propose forming necklace solitons ‘‘walk off’’ each other. So, for example, if ␭ϭ10 nm and ͓15,16͔ out of the QED solitons we just created, and studying ␣/␭ϭ104, and the peak intensity is 1029 W/m2 ͑which cor- their properties, and also stability. Intuitively, a necklace responds to the field Eϭ6.15ϫ1015 V/m), such soliton ef- soliton is the soliton solution of Eq. ͑9͒, wrapped around its fects should be observed after a propagation distance ͑dif- own tail, to form a ring. It is best if the radius of the ring L fraction length͒ of 1 m. Since the beams propagate at 90° is much larger than the thickness of the ring ␣. In the limit of with respect to each other, the cross section of the configu- L→ϱ, keeping everything else fixed, the necklace soliton ration has to be at least 1 m tall if we do not want the beams reduces to the soliton solution of Eq. ͑9͒. Thus, if the QED to stop overlapping before we observe the effect. Thus the solitons of Eq. ͑9͒ are stable, the necklace solitons should be peak power required is 1028 W. Finally, we have to check stable as well. Such necklace solitons offer a major advan- whether this experimental configuration satisfies our assump- tage over the (1ϩ1)D geometry offered above: the two su- tion that we can ignore the corrections to the Lagrangian in perimposed beams propagating at 90° with respect to each Eq. ͑1͒ that include the spatial and temporal derivatives of ␮␯ ␻ ␻ 2Ӷ ␲3 2 other will experimentally ‘‘walk off’’ each other, because in F : i.e., is ( / c) (4/ )(E/Ec) satisfied, as explained any such experiment the extent of each beam in its ‘‘uni- in the Appendix. This condition translates into 1Ӷ466 for form’’ direction will be finite. Therefore, in a (1ϩ1)D ge- the parameters proposed, so neglecting the corrections to the ometry the beams will intersect only over a finite distance, Lagrangian, due to the derivatives, is safe. which will limit the observation length, and also the possi- Observations and studying of QED solitons ͑or of the un- bilities of many applications, like shooting these solitons be- derlying effects of self-focusing and modulation instability͒ tween planets in outer space. In contrast, the necklace soli- could offer several fundamentally new aspects to physics in tons could experimentally exist in principle forever in a general. First, they would be the first experiments of photon- vacuum. photon scattering in vacuum, which involve the creation of We point out that the group velocity of all the solitons we virtual electron-positron pairs. Thus far, ‘‘nonlinear optics’’ described is c/&. This means that we can Lorentz boost to with QED nonlinearities in vacuum was demonstrated only the frame in which they are not propagating at all. In prin- for the creation of real electron-positron pairs ͓19͔, and the

043817-4 SELF-TRAPPING OF ELECTROMAGNETIC BEAMS IN . . . PHYSICAL REVIEW A 62 043817 demonstration could not provide a direct quantitative mea- APPENDIX surement of nonlinear QED effects in a vacuum. Further- In this appendix, we determine in which regime of the more, the solitons we described would provide a direct quan- parameters are the Heisenberg-Euler corrections, the domi- titative test of QED vacuum nonlinearities in the regime of nant lowest-order QED corrections to the classical Lagrang- much higher photon wavelengths, and much smaller peak ian, while the terms involving the derivatives can be ne- intensities than were tested so far. Moreover, in contrast to glected, and vice versa. To simplify the algebra we work in previously performed QED experiments, a macroscopic units where cϭបϭ1. In these units, the Heisenberg-Euler number of photons would participate in the effect. Second, correction ͓1͔ is given by exploration of the QED radiation soliton ideas might provide a means of communicating in space by line-of-sight: one might be able to launch a very narrow beam from a satellite e4 L ϭ ͕4F2ϩ7G2͖, ͑A1͒ orbiting the Earth to a nearby planet ͑or to an asteroid͒ and HE 360␲2m4 back, in order to investigate the reflection at a pin-point pre- cision. Furthermore, these ideas could be used to deliver very where Fϭ(B2ϪE2)/2, and GϭE B. Similarly, the correc- large powers to small, far-away objects, such as asteroids at • tions involving derivatives ͓20͔ are given by a trajectory too close to earth and deflect them. Finally, soliton-related mechanisms of self-focusing and catastrophic 2ץ collapse could offer ideas for explaining astronomical obser- e2 ͮ␤␣ ͪ ␯␤͒ϩ ͩ Ϫٌ2 ץ͑͒␣ ץϭ ͭ Ϫ͑ L ␣F ␯F F␣␤ F , t2ץ ␤ vations of bursts of EM radiation ͑gamma ray Bursts͒ that D 360␲m2 are localized in space in a narrow divergence angle ͑self- ͑A2͒ focused ‘‘jets’’?͒. It is possible that QED-related self- focusing might naturally occur near active stars, where the where F␣␤ is just the usual ͑real͒ electromagnetic field ten- EM fields are huge. All this, together with the possibility of sor. perhaps exciting applications, make the QED radiation soli- We would like to compare L and L for an arbitrary tons and their related effects well worth studying. HE D typical field configuration. Thus the best we can do is dimen- In conclusion, we have shown that modified Maxwell sional analysis. Unless we are dealing with a particularly equations in vacuum can give rise to spatial solitons ͑nondif- pathological field configuration, L Ϸe4E4/(90␲2m4), fracting solutions͒, which are supported by the nonlinearity HE while L Ϸe2E2␻2/(360 ␲m2), where E is the electric field that arises from QED photon-photon interactions at very D magnitude and ␻ is the carrier frequency. Thus, L ӶL is high radiation intensities. These solitons present an exciting D HE satisfied if and only if ␻2Ӷ4e2E2/(␲m2). To make this physical system to be studied experimentally but there are more easily transferable between the different systems of many questions left open. Some questions about the stability units, we note that the Compton frequency ␻ ϭm, while the of such (1ϩ1)D and (2ϩ1)D solutions are yet to be stud- c critical field E ϭm2/(␲e) in this system of units. Thus our ied theoretically. From an experimental standpoint, we be- c constraint can be written as: lieve that these solitons should be observable in the near future. ␻2 4 E2 We are grateful to Professor A.E. Kaplan from Johns Ӷ , ͑A3͒ ␻2 ␲3 E2 Hopkins University for useful discussions and for finding the c c error in our estimates for the peak intensity. In addition, we would like to thank Professor C. Callan Jr., Professor Eric which is the expression we use in the text. This expression Prebys of Princeton University, Amir Levinson of Tel Aviv compares two dimensionless ratios, thus it is the same in all University, Professor M. Moshe, and Professor M. Reznikoff systems of units. Just to be able to make the comparison ␻ ϭ ប of the Technion for useful discussions. This work was sup- quantitative, we note that in SI, c mc/ , while Ec ported by the U.S. Army Research Office. ϭm2c3/(e␲ប).

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