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Nonlinear propagation of incoherent photons in a radiation background
Physics Letters A 330 (2004) 131–136 www.elsevier.com/locate/pla
Nonlinear propagation of incoherent photons in a radiation background Padma K. Shukla a,b , Mattias Marklund a,c,∗ , Gert Brodin b , Lennart Stenflo b a Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany b Department of Physics, Umeå University, SE-901 87 Umeå, Sweden c Department of Electromagnetics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Received 15 July 2004; accepted 30 July 2004 Available online 19 August 2004 Communicated by V.M. Agranovich
Abstract The non-linear propagation of intense incoherent photons in a photon gas is considered. The photon–photon interactions are governed by a pair of equations comprising a wave-kinetic equation for the incoherent photons in the presence of the slowly varying energy density perturbations of sound-like waves, and an equation for the latter waves in a background where the photon coupling is caused by quantum electrodynamical effects. The coupled equations are used to derive a dispersion relation, which admits new classes of modulational instabilities of incoherent photons. The present instabilities can lead to fragmentation of broadband short photon pulses in astrophysical and laboratory settings. 2004 Published by Elsevier B.V. PACS: 12.20.Ds; 95.30.Cq
The advent of quantum electrodynamics (QED) is one of the major scientific achievements of the 20th century. It is an experimentally well confirmed theory, and it has predicted a number of phenomena which were not expected in previous studies, e.g., the Casimir effect. Moreover, as opposed to the classical theory
* Corresponding author.
E-mail address: [email protected] (M. Marklund). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.07.073
of Maxwell, electromagnetic waves can according to QED interact in the absence of a material mediator. Due to the possibility of exchanging virtual electron– positron pairs, there is the intriguing effect of photon– photon scattering, first discussed even before the interaction between light and matter was well understood [1,2], and later derived within QED by Schwinger [3]. The fact that strong electromagnetic fields can interact in vacuum, opens up for very interesting applications in extreme astrophysical environments, such as magnetars [4], where photon splitting and lensing
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may take place [5–8]. The prospect of direct detection of the effect has also been discussed in the literature, and a number of suggestions for experimental setups have been given, involving second harmonic generation [9], self-focusing [10], non-linear wave mixing in cavities [11,12] and waveguide propagation [13], respectively. Apart from the astrophysical regions where high fields exist, the rapid development of ultra-high laser fields [14,15] and related laser–plasma techniques [16], gives hope that the critical field strengths will be produced in laboratories. The advent of the aforementioned high-intensity laser systems, currently reaching power outputs of 1021 –1022 W/cm2 [14], promises many interesting applications ranging from medicine and material science to fundamental physics research topics. As these laser powers are combined with plasma systems, even greater power outputs are expected [17], due to the non-linear interaction between the high-intensity photons and the plasma particles [18]. Moreover, the free electron laser, currently being developed (see, e.g., [19]), will hopefully reach even more extreme parameter values, making it likely that electromagnetic field strengths close to the Schwinger limit 1016 V/cm will be reached in the near future. Thus, the investigation of non-linear QED effects, becoming important close to the Schwinger intensity, certainly merits attention. We shall below investigate the effect of incoherence of high-frequency photons propagating on a radiation fluid background. This will extend the work presented in Ref. [20], and investigated numerically in Ref. [21], to the random phase regime. For this purpose, a wave kinetic theory for high-frequency photons coupled to an acoustic wave equation for a radiation fluid is presented. It is shown that modulational instabilities are inherent in the system of equations. Moreover, in the limit of the slow time-variation approximation, we obtain a Vlasov equation with self-interaction for the high-frequency photons. We consider an incoherent non-thermal high-frequency spectrum of photons. As will be shown, this spectrum will be able to interact with low-frequency acoustic-like perturbations. The high-frequency part is treated by means of a wave kinetic description, whereas the low-frequency part is described by an acoustic wave equation with a driver [20] which follows from a radiation fluid description. Let Nk (t, r) denote the high frequency photon distribution func-
tion, normalized such that the corresponding number density is given by n = Nk d 3 k. Then Nk will satisfy the wave kinetic equation [22] ∂Nk ∂ωk ∂Nk ∂Nk + vg · − · = 0, ∂t ∂r ∂r ∂k where v g = ∂ωk /∂k is the group velocity, and 2 ωk = ck 1 − λE , 3
(1)
(2)
where E is the radiation fluid density [20,23], and λ = 8κ or 14κ, depending on the polarization state of the photon. Here κ ≡ 2α 2 h¯ 3 /45m4e c5 ≈ 1.63 × 10−30 ms2 /kg, α is the fine-structure constant, 2π h¯ the Planck constant, me the electron mass, and c the velocity of light in vacuum. The dispersion relation (2) is valid as long as there is no pair creation and the field strength is smaller than the QED critical field, i.e., ω me c2 /h¯
and |E| Ecrit ≡ me c2 /eλc ,
(3)
respectively. Here e is the elementary charge, λc is the Compton wave length, and Ecrit 1018 V/m. The high-frequency photons drive low-frequency acoustic perturbations according to [20] 2 c2 2 ∂ ∇ − E ∂t 2 3 2λE0 ∂ 2 2 2 +c ∇ h¯ ωk Nk d 3 k, =− (4) 3 ∂t 2 where the constant E0 is the background radiation fluid energy density. This hybrid description, where the high-frequency part is treated kinetically, and the low-frequency part is described within a fluid theory, applies when the mean-free path between photon– photon collisions is shorter than the wavelengths of the low-frequency perturbations. We note that the inEq. (1), and is nortensity Ik = h¯ ωk Nk /0 satisfies malized such that |E|2 = Ik d 3 k, where E is the high-frequency electric field strength, and 0 is the dielectric constant of vacuum. The equations presented here resemble the photon–electron system in Ref. [24], where the interaction between random phase photons and sound waves in an electron–positron plasma has been investigated. Next we consider a small low-frequency long wavelength perturbation of a homogeneous background spectrum, i.e., Nk = Nk0 + Nk1 exp[i(Kz − Ωt)],
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Nk1 Nk0 , and E = E1 exp[i(Kz − Ωt)] and linearize our equations. We thus obtain (using expression (2) for ωk , and introducing kˆ = k/k) Nk1 =
Kc 2λkE1 ∂Nk0 zˆ · 3 Ω − Kckˆ · zˆ ∂k
(5a)
and
2λch¯ E0 Ω 2 + K 2 c2 kNk1 d 3 k, (5b) 3 Ω 2 − K 2 c2 /3 which, when combined, give the non-linear dispersion relation
E1 = −
µK Ω 2 + K 2 c2 3 Ω 2 − K 2 c2 /3 k2 ∂Nk0 3 × d k, zˆ · ˆ ∂k Ω − Kck · zˆ
1=−
(6)
Fig. 1. The transverse instability for the mono-energetic case. Γ /Kc plotted as a function of vT /c, as given in Eq. (8).
where we have introduced the constant µ = 43 λ2 c2 h¯ E0 . (a) For a mono-energetic high frequency background, we have Nk0 = n0 δ(k − k 0 ). The non-linear dispersion relation (6) then reduces to 2 2 Ω − K 2 c2 /3 Ω − Kc cos θ0 1 = µn0 k0 K Ω 2 + K 2 c2 3 × Kc + (2Ω − 3Kc cos θ0 ) cos θ0 , (7) where we have introduced cos θ0 ≡ kˆ0 · zˆ . This monoenergetic background has a transverse instability when θ0 = π/2, with the growth rate
4 2 1 vT vT Γ = √ Kc + 14 +1 c c 6 (8) 2 1/2 vT − −1 , c where Γ ≡ −iΩ, and vT2 ≡ µn0 k0 c, and vT is a characteristic speed of the system. The expression in the square bracket is positive definite. In Fig. 1 we have plotted Γ /Kc as a function of vT /c. In fact, under most circumstances, vT c. Using the expression (7), we then have √ two branches. The branch corresponding to Ω Kc/ 3 is always stable for small vT , while for the branch corresponding to Ω Kc cos θ0 we obtain the growth rate
1 − cos θ0 Γ = KvT (9) , 1 − 3 cos θ0
Fig. 2. (Γ /KvT )2 , according to Eq. (9), plotted as a function of α0 = cos θ0 in the mono-energetic case.
which is consistent with (8) in the limit θ0 → π/2. In Fig. 2, the behavior of the growth rate (9) is depicted. (b) The high-frequency photons have generally a spread in momentum space. For simplicity we here choose the background intensity distribution as a shifted Gaussian, i.e. [25] I0 (k − k 0 )2 exp − , Ik0 = (10) 3 2 kW π 3/2 kW where I0 = |E0 |2 is the (constant) background intensity and kW is the width of the distribution around
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k 0 . Then the dispersion relation is b 2 η2 + 1 k(k0 cos θ0 − k cos θ ) 1=− 5 2 η − 1/3 η − cos θ kW 2 k − 2k · k 0 d 3 k, × exp − 2 kW
(11)
2 ) and where b2 = (4/9π 3/2 )λ2 0 E0 I0 exp(−k02 /kW η ≡ Ω/Kc. Assuming that the deviation of k 0 from the zˆ -axis is small, and that δ ≡ k0 /kW 1, we can integrate Eq. (11), keeping terms linear in δ, to obtain √ 2 3 π 2 η +1 1 −πb 2 + 8δη cos θ0 2 η − 1/3 √ 3 π η − 4δη cos θ0 + δ cos θ0 − 4 × (2 arctanh η − iπ) , (12)
for 0 < η < 1. Thus, we see that the non-zero width of the distribution complicates the characteristic behavior of the dispersion relation by a considerable amount. It is clear though, that the width will introduce a reduction of the growth rate, as compared to the monoenergetic case. We may also look at the case when the timedependence is weak, i.e., ∂ 2 E /∂t 2 c2 ∇ 2 E , such that Eq. (4) yields E = 2λE0 h¯ ωk Nk d 3 k. (13) Upon using Eq. (13) in (2), we find ∂ωk ∂ = −µk k Nk d 3 k . ∂r ∂r
(14)
Hence Eq. (1) becomes ∂Nk ∂Nk + vg · ∂t ∂r ∂Nk ∂ = 0, + µk k Nk d 3 k · ∂r ∂k
(15)
which in the one-dimensional case reduces to ∂Nk ∂Nk + vg ∂t ∂x ∂ ∂Nk = 0. + µk k Nk dk ∂x ∂k
(16)
A similar equation may of course be derived for the intensity Ik . Eq. (16) gives the evolution of high-frequency photons on a slowly varying background radiation fluid, and it may be used to analyze the long term behavior of amplitude modulated intense short incoherent laser pulses. Some astrophysical environments exhibit extremely large energy scales. Ordinary neutron stars have surface magnetic field strengths of the order of 1010 – 1013 G. The even more extreme magnetars [26] can reach 1014–1015 G, which is actually close to the Schwinger limit, where the vacuum becomes fully non-linear. In fact, it is believed that single-particle QED effects due to photon–photon scattering, such as photon splitting, play a significant role in the physics of these extreme systems [27,28]. It has become a standard fact that neutron stars and magnetars experience quakes, due to the extreme magnetic field strengths. These will cause increasing tension in the star crust over long periods of time, releasing sudden bursts of energy as the surface of the star can no longer withstand the strains induced by the fields. It is expected that large quantities of low-frequency photons can be ejected from such an event, and this would then form an incoherent spectrum of electromagnetic waves [29]. This photon gas could reach energy densities E0 ∼ 1017 –1026 Jm−3 . A short, incoherent high frequency electromagnetic pulse, with wavelengths from the UV to gamma range, could be modeled by Eq. (1), while the low-frequency photon gas would be governed by the acoustic-like wave equation (4). Because of the high field strengths in the neutron star atmosphere during these quakes, such events could be of importance for gamma-ray bursts, i.e., short ( tens of second) emissions of photons in the gamma range [30]. From pulse propagation in air, it is known that there will be a significant blueshift of the electromagnetic pulse, due to the formation of steep intensity gradients and air ionization [31]. A blueshift can also occur within the system presented in this Letter, thus making the frequency up-shifting to the gamma-range photons possible even in a ‘vacuum’ environment. The novel modulational instability of photon systems presented in this Letter, might in this sense play a role for the formation of localized gamma-ray bursts, shedding new light on such events. In order to gain insight into this problem, the formation of intense photon pulses
P.K. Shukla et al. / Physics Letters A 330 (2004) 131–136
with frequencies up to the gamma regime, due to the incoherent photon gas created by neutron star and magnetar quakes, could be analyzed non-pertubatively using the system of Eqs. (1) and (4). The full exploration of this system will, however, require extensive numerical simulations. Although the astrophysical environment is very likely to offer conditions at which non-linear vacuum effects, such as the ones presented here, are important, perhaps even more exciting is the laboratory possibilities that has surfaced during the last twenty years. During this period, the development of state-of-the-art lasers has displayed an immense progress. Currently, the achievable laser powers reaches 1021 –1022 W/cm2 , and in the near future one may reach 1023–1024 W/cm2 [14]. The free electron laser (FEL) is under heavy development as well, and the planned systems will reach intensities making laboratory astrophysics a reality (see, e.g., [19,32]). It has even been suggested that the FEL could reach intensities of the order 1024–1028 W/cm2 , although this would require the development of new X-ray optical techniques [33]. Apart from the development of these high intensity laser systems, there is also rapid progress in the area of laser–plasma systems [17]. Since plasmas can sustain enormous field strengths, the use of plasmas as guiding and focusing material for lasers will make it possible to attain power outputs well above the laser limit [14,17]. As high intensity laser pulses propagate trough plasmas, the charges will become highly relativistic due to the large ponderomotive force. Due to this, there will be an almost complete evacuation of the plasma at the high intensity regions, creating an environment where the pulse will selffocus, thus reaching intensities close to the Schwinger critical field [16,34,35]. It has been suggested that within the environment created by the next generation laser–plasma systems, non-linear QED effects such as photon–photon scattering will indeed be important [36], and that this will yield a facility for experimental detection of elastic collisions between photons in vacuum [37]. Moreover, the fact that the imaginary part of the full photon QED Lagrangian corresponds to pair creation in vacuum, indicates that within the high intensity systems that will be operational in the near future one may even create anti-matter by suitable direct laser irradiation [38]. Thus, the results presented in this Letter will play a significant role in understand-
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ing the full dynamics of laser–plasma interaction in the near future, since photon–photon scattering of partially coherent laser pulses with the trapped photons gives rise to completely new types of instabilities, altering the evolution of the plasma channels and the guided intense fields. We have considered the non-linear propagation of randomly distributed intense short photon pulses in a photon gas. The photon–photon interactions are described by means of a QED model, in which an ensemble of incoherent photon pulses is governed by a wave kinetic equation where the coupling between the intense photon pulses and the sound-like waves of the photon gas is due to slow variations of the sound wave energy distribution. The intense photon pressure, in turn, modifies the sound wave propagation. The wave kinetic and the sound wave equations form a closed system, which has been used to derive a dispersion relation. By choosing appropriate spectra for short pulse photons, we analyze the dispersion relation to show the existence of new classes of modulational instabilities. The latter can cause fragmentation of incoherent photon pulses in astrophysical contexts and in forthcoming experiments using very intense short laser pulses.
References [1] W. Heisenberg, H. Euler, Z. Phys. 98 (1936) 714. [2] V.S. Weisskopf, K. Dan, Vidensk. Selsk. Mat. Fys. Medd. 14 (1936) 1. [3] J. Schwinger, Phys. Rev. 82 (1951) 664. [4] C. Kouveliotou, et al., Nature 393 (1998) 235. [5] J.S. Heyl, L. Hernquist, J. Phys. A: Math. Gen. 30 (1997) 6485. [6] J.S. Heyl, L. Hernquist, Phys. Rev. D 55 (1997) 2449. [7] N.J. Shaviv, J.S. Heyl, Y. Lithwick, Mon. Not. R. Astron. Soc. 306 (1999) 333. [8] V.I. Denisov, S.I. Svertilov, Astron. Astrophys. 399 (2003) L39. [9] Y.J. Ding, A.E. Kaplan, Phys. Rev. Lett. 63 (1989) 2725. [10] M. Soljaˇci´c, M. Segev, Phys. Rev. A 62 (2000) 043817. [11] G. Brodin, M. Marklund, L. Stenflo, Phys. Rev. Lett. 87 (2001) 171801. [12] G. Brodin, M. Marklund, L. Stenflo, Phys. Scripta T 98 (2002) 127. [13] G. Brodin, L. Stenflo, D. Anderson, M. Lisak, M. Marklund, P. Johannisson, Phys. Lett. A 306 (2003) 206. [14] G.A. Mourou, C.P.J. Barty, M.D. Perry, Phys. Today 51 (1998) 22. [15] A. Pukhov, Rep. Prog. Phys. 66 (2003) 47.
136
P.K. Shukla et al. / Physics Letters A 330 (2004) 131–136
[16] S.V. Bulanov, T. Esirkepov, T. Tajima, Phys. Rev. Lett. 91 (2003) 085001. [17] R. Bingham, Nature 424 (2003) 258. [18] R.A. Cairns, A. Reitsma, R. Bingham, Phys. Plasmas 11 (2004) 766. [19] http://www-hasylab.desy.de/facility/fel. [20] M. Marklund, G. Brodin, L. Stenflo, Phys. Rev. Lett. 91 (2003) 163601. [21] P.K. Shukla, B. Eliasson, Phys. Rev. Lett. 92 (2004) 073601. [22] J.T. Mendonça, Theory of Photon Acceleration, Institute of Physics Publishing, Bristol, 2001. [23] Z. Bialynicka-Birula, I. Bialynicki-Birula, Phys. Rev. D 2 (1970) 2341. [24] P.K. Shukla, L. Stenflo, Phys. Plasmas 5 (1998) 1554. [25] P.K. Shukla, L. Stenflo, Phys. Lett. A 237 (1998) 385. [26] C. Kouveliotou, et al., Nature 393 (1998) 235.
[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
M.G. Baring, A.K. Harding, Astrophys. J. 507 (1998) L55. A.K. Harding, Science 251 (1991) 1033. V.N. Kondratyev, Phys. Rev. Lett. 88 (2002) 221101. T. Piran, Phys. Rep. 314 (1999) 575. A.L. Gaeta, Phys. Rev. Lett. 84 (2000) 3582. T. Tajima, Plasma Phys. Rep. 29 (2003) 207. P.S. Chen, C. Pellegrini, Quantum Aspects of Beam Physics, World Scientific, Singapore, 1999. M.Y. Yu, P.K. Shukla, N.L. Tsintsadze, Phys. Fluids 25 (1982) 1049. M. Marklund, B. Eliasson, P.K. Shukla, JETP Lett. 79 (2004) 208. B. Shen, M.Y. Yu, Phys. Rev. E 68 (2003) 026501. B. Shen, M.Y. Yu, X. Wang, Phys. Plasmas 10 (2003) 4570. S.S. Bulanov, Phys. Rev. E 69 (2004) 036408.
Nonlinear propagation of incoherent photons in a radiation background Padma K. Shukla a,b , Mattias Marklund a,c,∗ , Gert Brodin b , Lennart Stenflo b a Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany b Department of Physics, Umeå University, SE-901 87 Umeå, Sweden c Department of Electromagnetics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Received 15 July 2004; accepted 30 July 2004 Available online 19 August 2004 Communicated by V.M. Agranovich
Abstract The non-linear propagation of intense incoherent photons in a photon gas is considered. The photon–photon interactions are governed by a pair of equations comprising a wave-kinetic equation for the incoherent photons in the presence of the slowly varying energy density perturbations of sound-like waves, and an equation for the latter waves in a background where the photon coupling is caused by quantum electrodynamical effects. The coupled equations are used to derive a dispersion relation, which admits new classes of modulational instabilities of incoherent photons. The present instabilities can lead to fragmentation of broadband short photon pulses in astrophysical and laboratory settings. 2004 Published by Elsevier B.V. PACS: 12.20.Ds; 95.30.Cq
The advent of quantum electrodynamics (QED) is one of the major scientific achievements of the 20th century. It is an experimentally well confirmed theory, and it has predicted a number of phenomena which were not expected in previous studies, e.g., the Casimir effect. Moreover, as opposed to the classical theory
* Corresponding author.
E-mail address: [email protected] (M. Marklund). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.07.073
of Maxwell, electromagnetic waves can according to QED interact in the absence of a material mediator. Due to the possibility of exchanging virtual electron– positron pairs, there is the intriguing effect of photon– photon scattering, first discussed even before the interaction between light and matter was well understood [1,2], and later derived within QED by Schwinger [3]. The fact that strong electromagnetic fields can interact in vacuum, opens up for very interesting applications in extreme astrophysical environments, such as magnetars [4], where photon splitting and lensing
132
P.K. Shukla et al. / Physics Letters A 330 (2004) 131–136
may take place [5–8]. The prospect of direct detection of the effect has also been discussed in the literature, and a number of suggestions for experimental setups have been given, involving second harmonic generation [9], self-focusing [10], non-linear wave mixing in cavities [11,12] and waveguide propagation [13], respectively. Apart from the astrophysical regions where high fields exist, the rapid development of ultra-high laser fields [14,15] and related laser–plasma techniques [16], gives hope that the critical field strengths will be produced in laboratories. The advent of the aforementioned high-intensity laser systems, currently reaching power outputs of 1021 –1022 W/cm2 [14], promises many interesting applications ranging from medicine and material science to fundamental physics research topics. As these laser powers are combined with plasma systems, even greater power outputs are expected [17], due to the non-linear interaction between the high-intensity photons and the plasma particles [18]. Moreover, the free electron laser, currently being developed (see, e.g., [19]), will hopefully reach even more extreme parameter values, making it likely that electromagnetic field strengths close to the Schwinger limit 1016 V/cm will be reached in the near future. Thus, the investigation of non-linear QED effects, becoming important close to the Schwinger intensity, certainly merits attention. We shall below investigate the effect of incoherence of high-frequency photons propagating on a radiation fluid background. This will extend the work presented in Ref. [20], and investigated numerically in Ref. [21], to the random phase regime. For this purpose, a wave kinetic theory for high-frequency photons coupled to an acoustic wave equation for a radiation fluid is presented. It is shown that modulational instabilities are inherent in the system of equations. Moreover, in the limit of the slow time-variation approximation, we obtain a Vlasov equation with self-interaction for the high-frequency photons. We consider an incoherent non-thermal high-frequency spectrum of photons. As will be shown, this spectrum will be able to interact with low-frequency acoustic-like perturbations. The high-frequency part is treated by means of a wave kinetic description, whereas the low-frequency part is described by an acoustic wave equation with a driver [20] which follows from a radiation fluid description. Let Nk (t, r) denote the high frequency photon distribution func-
tion, normalized such that the corresponding number density is given by n = Nk d 3 k. Then Nk will satisfy the wave kinetic equation [22] ∂Nk ∂ωk ∂Nk ∂Nk + vg · − · = 0, ∂t ∂r ∂r ∂k where v g = ∂ωk /∂k is the group velocity, and 2 ωk = ck 1 − λE , 3
(1)
(2)
where E is the radiation fluid density [20,23], and λ = 8κ or 14κ, depending on the polarization state of the photon. Here κ ≡ 2α 2 h¯ 3 /45m4e c5 ≈ 1.63 × 10−30 ms2 /kg, α is the fine-structure constant, 2π h¯ the Planck constant, me the electron mass, and c the velocity of light in vacuum. The dispersion relation (2) is valid as long as there is no pair creation and the field strength is smaller than the QED critical field, i.e., ω me c2 /h¯
and |E| Ecrit ≡ me c2 /eλc ,
(3)
respectively. Here e is the elementary charge, λc is the Compton wave length, and Ecrit 1018 V/m. The high-frequency photons drive low-frequency acoustic perturbations according to [20] 2 c2 2 ∂ ∇ − E ∂t 2 3 2λE0 ∂ 2 2 2 +c ∇ h¯ ωk Nk d 3 k, =− (4) 3 ∂t 2 where the constant E0 is the background radiation fluid energy density. This hybrid description, where the high-frequency part is treated kinetically, and the low-frequency part is described within a fluid theory, applies when the mean-free path between photon– photon collisions is shorter than the wavelengths of the low-frequency perturbations. We note that the inEq. (1), and is nortensity Ik = h¯ ωk Nk /0 satisfies malized such that |E|2 = Ik d 3 k, where E is the high-frequency electric field strength, and 0 is the dielectric constant of vacuum. The equations presented here resemble the photon–electron system in Ref. [24], where the interaction between random phase photons and sound waves in an electron–positron plasma has been investigated. Next we consider a small low-frequency long wavelength perturbation of a homogeneous background spectrum, i.e., Nk = Nk0 + Nk1 exp[i(Kz − Ωt)],
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Nk1 Nk0 , and E = E1 exp[i(Kz − Ωt)] and linearize our equations. We thus obtain (using expression (2) for ωk , and introducing kˆ = k/k) Nk1 =
Kc 2λkE1 ∂Nk0 zˆ · 3 Ω − Kckˆ · zˆ ∂k
(5a)
and
2λch¯ E0 Ω 2 + K 2 c2 kNk1 d 3 k, (5b) 3 Ω 2 − K 2 c2 /3 which, when combined, give the non-linear dispersion relation
E1 = −
µK Ω 2 + K 2 c2 3 Ω 2 − K 2 c2 /3 k2 ∂Nk0 3 × d k, zˆ · ˆ ∂k Ω − Kck · zˆ
1=−
(6)
Fig. 1. The transverse instability for the mono-energetic case. Γ /Kc plotted as a function of vT /c, as given in Eq. (8).
where we have introduced the constant µ = 43 λ2 c2 h¯ E0 . (a) For a mono-energetic high frequency background, we have Nk0 = n0 δ(k − k 0 ). The non-linear dispersion relation (6) then reduces to 2 2 Ω − K 2 c2 /3 Ω − Kc cos θ0 1 = µn0 k0 K Ω 2 + K 2 c2 3 × Kc + (2Ω − 3Kc cos θ0 ) cos θ0 , (7) where we have introduced cos θ0 ≡ kˆ0 · zˆ . This monoenergetic background has a transverse instability when θ0 = π/2, with the growth rate
4 2 1 vT vT Γ = √ Kc + 14 +1 c c 6 (8) 2 1/2 vT − −1 , c where Γ ≡ −iΩ, and vT2 ≡ µn0 k0 c, and vT is a characteristic speed of the system. The expression in the square bracket is positive definite. In Fig. 1 we have plotted Γ /Kc as a function of vT /c. In fact, under most circumstances, vT c. Using the expression (7), we then have √ two branches. The branch corresponding to Ω Kc/ 3 is always stable for small vT , while for the branch corresponding to Ω Kc cos θ0 we obtain the growth rate
1 − cos θ0 Γ = KvT (9) , 1 − 3 cos θ0
Fig. 2. (Γ /KvT )2 , according to Eq. (9), plotted as a function of α0 = cos θ0 in the mono-energetic case.
which is consistent with (8) in the limit θ0 → π/2. In Fig. 2, the behavior of the growth rate (9) is depicted. (b) The high-frequency photons have generally a spread in momentum space. For simplicity we here choose the background intensity distribution as a shifted Gaussian, i.e. [25] I0 (k − k 0 )2 exp − , Ik0 = (10) 3 2 kW π 3/2 kW where I0 = |E0 |2 is the (constant) background intensity and kW is the width of the distribution around
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k 0 . Then the dispersion relation is b 2 η2 + 1 k(k0 cos θ0 − k cos θ ) 1=− 5 2 η − 1/3 η − cos θ kW 2 k − 2k · k 0 d 3 k, × exp − 2 kW
(11)
2 ) and where b2 = (4/9π 3/2 )λ2 0 E0 I0 exp(−k02 /kW η ≡ Ω/Kc. Assuming that the deviation of k 0 from the zˆ -axis is small, and that δ ≡ k0 /kW 1, we can integrate Eq. (11), keeping terms linear in δ, to obtain √ 2 3 π 2 η +1 1 −πb 2 + 8δη cos θ0 2 η − 1/3 √ 3 π η − 4δη cos θ0 + δ cos θ0 − 4 × (2 arctanh η − iπ) , (12)
for 0 < η < 1. Thus, we see that the non-zero width of the distribution complicates the characteristic behavior of the dispersion relation by a considerable amount. It is clear though, that the width will introduce a reduction of the growth rate, as compared to the monoenergetic case. We may also look at the case when the timedependence is weak, i.e., ∂ 2 E /∂t 2 c2 ∇ 2 E , such that Eq. (4) yields E = 2λE0 h¯ ωk Nk d 3 k. (13) Upon using Eq. (13) in (2), we find ∂ωk ∂ = −µk k Nk d 3 k . ∂r ∂r
(14)
Hence Eq. (1) becomes ∂Nk ∂Nk + vg · ∂t ∂r ∂Nk ∂ = 0, + µk k Nk d 3 k · ∂r ∂k
(15)
which in the one-dimensional case reduces to ∂Nk ∂Nk + vg ∂t ∂x ∂ ∂Nk = 0. + µk k Nk dk ∂x ∂k
(16)
A similar equation may of course be derived for the intensity Ik . Eq. (16) gives the evolution of high-frequency photons on a slowly varying background radiation fluid, and it may be used to analyze the long term behavior of amplitude modulated intense short incoherent laser pulses. Some astrophysical environments exhibit extremely large energy scales. Ordinary neutron stars have surface magnetic field strengths of the order of 1010 – 1013 G. The even more extreme magnetars [26] can reach 1014–1015 G, which is actually close to the Schwinger limit, where the vacuum becomes fully non-linear. In fact, it is believed that single-particle QED effects due to photon–photon scattering, such as photon splitting, play a significant role in the physics of these extreme systems [27,28]. It has become a standard fact that neutron stars and magnetars experience quakes, due to the extreme magnetic field strengths. These will cause increasing tension in the star crust over long periods of time, releasing sudden bursts of energy as the surface of the star can no longer withstand the strains induced by the fields. It is expected that large quantities of low-frequency photons can be ejected from such an event, and this would then form an incoherent spectrum of electromagnetic waves [29]. This photon gas could reach energy densities E0 ∼ 1017 –1026 Jm−3 . A short, incoherent high frequency electromagnetic pulse, with wavelengths from the UV to gamma range, could be modeled by Eq. (1), while the low-frequency photon gas would be governed by the acoustic-like wave equation (4). Because of the high field strengths in the neutron star atmosphere during these quakes, such events could be of importance for gamma-ray bursts, i.e., short ( tens of second) emissions of photons in the gamma range [30]. From pulse propagation in air, it is known that there will be a significant blueshift of the electromagnetic pulse, due to the formation of steep intensity gradients and air ionization [31]. A blueshift can also occur within the system presented in this Letter, thus making the frequency up-shifting to the gamma-range photons possible even in a ‘vacuum’ environment. The novel modulational instability of photon systems presented in this Letter, might in this sense play a role for the formation of localized gamma-ray bursts, shedding new light on such events. In order to gain insight into this problem, the formation of intense photon pulses
P.K. Shukla et al. / Physics Letters A 330 (2004) 131–136
with frequencies up to the gamma regime, due to the incoherent photon gas created by neutron star and magnetar quakes, could be analyzed non-pertubatively using the system of Eqs. (1) and (4). The full exploration of this system will, however, require extensive numerical simulations. Although the astrophysical environment is very likely to offer conditions at which non-linear vacuum effects, such as the ones presented here, are important, perhaps even more exciting is the laboratory possibilities that has surfaced during the last twenty years. During this period, the development of state-of-the-art lasers has displayed an immense progress. Currently, the achievable laser powers reaches 1021 –1022 W/cm2 , and in the near future one may reach 1023–1024 W/cm2 [14]. The free electron laser (FEL) is under heavy development as well, and the planned systems will reach intensities making laboratory astrophysics a reality (see, e.g., [19,32]). It has even been suggested that the FEL could reach intensities of the order 1024–1028 W/cm2 , although this would require the development of new X-ray optical techniques [33]. Apart from the development of these high intensity laser systems, there is also rapid progress in the area of laser–plasma systems [17]. Since plasmas can sustain enormous field strengths, the use of plasmas as guiding and focusing material for lasers will make it possible to attain power outputs well above the laser limit [14,17]. As high intensity laser pulses propagate trough plasmas, the charges will become highly relativistic due to the large ponderomotive force. Due to this, there will be an almost complete evacuation of the plasma at the high intensity regions, creating an environment where the pulse will selffocus, thus reaching intensities close to the Schwinger critical field [16,34,35]. It has been suggested that within the environment created by the next generation laser–plasma systems, non-linear QED effects such as photon–photon scattering will indeed be important [36], and that this will yield a facility for experimental detection of elastic collisions between photons in vacuum [37]. Moreover, the fact that the imaginary part of the full photon QED Lagrangian corresponds to pair creation in vacuum, indicates that within the high intensity systems that will be operational in the near future one may even create anti-matter by suitable direct laser irradiation [38]. Thus, the results presented in this Letter will play a significant role in understand-
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ing the full dynamics of laser–plasma interaction in the near future, since photon–photon scattering of partially coherent laser pulses with the trapped photons gives rise to completely new types of instabilities, altering the evolution of the plasma channels and the guided intense fields. We have considered the non-linear propagation of randomly distributed intense short photon pulses in a photon gas. The photon–photon interactions are described by means of a QED model, in which an ensemble of incoherent photon pulses is governed by a wave kinetic equation where the coupling between the intense photon pulses and the sound-like waves of the photon gas is due to slow variations of the sound wave energy distribution. The intense photon pressure, in turn, modifies the sound wave propagation. The wave kinetic and the sound wave equations form a closed system, which has been used to derive a dispersion relation. By choosing appropriate spectra for short pulse photons, we analyze the dispersion relation to show the existence of new classes of modulational instabilities. The latter can cause fragmentation of incoherent photon pulses in astrophysical contexts and in forthcoming experiments using very intense short laser pulses.
References [1] W. Heisenberg, H. Euler, Z. Phys. 98 (1936) 714. [2] V.S. Weisskopf, K. Dan, Vidensk. Selsk. Mat. Fys. Medd. 14 (1936) 1. [3] J. Schwinger, Phys. Rev. 82 (1951) 664. [4] C. Kouveliotou, et al., Nature 393 (1998) 235. [5] J.S. Heyl, L. Hernquist, J. Phys. A: Math. Gen. 30 (1997) 6485. [6] J.S. Heyl, L. Hernquist, Phys. Rev. D 55 (1997) 2449. [7] N.J. Shaviv, J.S. Heyl, Y. Lithwick, Mon. Not. R. Astron. Soc. 306 (1999) 333. [8] V.I. Denisov, S.I. Svertilov, Astron. Astrophys. 399 (2003) L39. [9] Y.J. Ding, A.E. Kaplan, Phys. Rev. Lett. 63 (1989) 2725. [10] M. Soljaˇci´c, M. Segev, Phys. Rev. A 62 (2000) 043817. [11] G. Brodin, M. Marklund, L. Stenflo, Phys. Rev. Lett. 87 (2001) 171801. [12] G. Brodin, M. Marklund, L. Stenflo, Phys. Scripta T 98 (2002) 127. [13] G. Brodin, L. Stenflo, D. Anderson, M. Lisak, M. Marklund, P. Johannisson, Phys. Lett. A 306 (2003) 206. [14] G.A. Mourou, C.P.J. Barty, M.D. Perry, Phys. Today 51 (1998) 22. [15] A. Pukhov, Rep. Prog. Phys. 66 (2003) 47.
136
P.K. Shukla et al. / Physics Letters A 330 (2004) 131–136
[16] S.V. Bulanov, T. Esirkepov, T. Tajima, Phys. Rev. Lett. 91 (2003) 085001. [17] R. Bingham, Nature 424 (2003) 258. [18] R.A. Cairns, A. Reitsma, R. Bingham, Phys. Plasmas 11 (2004) 766. [19] http://www-hasylab.desy.de/facility/fel. [20] M. Marklund, G. Brodin, L. Stenflo, Phys. Rev. Lett. 91 (2003) 163601. [21] P.K. Shukla, B. Eliasson, Phys. Rev. Lett. 92 (2004) 073601. [22] J.T. Mendonça, Theory of Photon Acceleration, Institute of Physics Publishing, Bristol, 2001. [23] Z. Bialynicka-Birula, I. Bialynicki-Birula, Phys. Rev. D 2 (1970) 2341. [24] P.K. Shukla, L. Stenflo, Phys. Plasmas 5 (1998) 1554. [25] P.K. Shukla, L. Stenflo, Phys. Lett. A 237 (1998) 385. [26] C. Kouveliotou, et al., Nature 393 (1998) 235.
[27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
M.G. Baring, A.K. Harding, Astrophys. J. 507 (1998) L55. A.K. Harding, Science 251 (1991) 1033. V.N. Kondratyev, Phys. Rev. Lett. 88 (2002) 221101. T. Piran, Phys. Rep. 314 (1999) 575. A.L. Gaeta, Phys. Rev. Lett. 84 (2000) 3582. T. Tajima, Plasma Phys. Rep. 29 (2003) 207. P.S. Chen, C. Pellegrini, Quantum Aspects of Beam Physics, World Scientific, Singapore, 1999. M.Y. Yu, P.K. Shukla, N.L. Tsintsadze, Phys. Fluids 25 (1982) 1049. M. Marklund, B. Eliasson, P.K. Shukla, JETP Lett. 79 (2004) 208. B. Shen, M.Y. Yu, Phys. Rev. E 68 (2003) 026501. B. Shen, M.Y. Yu, X. Wang, Phys. Plasmas 10 (2003) 4570. S.S. Bulanov, Phys. Rev. E 69 (2004) 036408.