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Mendonc,a J.T. Theory of photon acceleration (plasma) (IOP 2000)(232s).pdf |
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Size 1.0Mb Date Jul 15, 2004 |
Acknowledgments 1 Introduction 1.1 Definition of the concept 1.2 Historical background 1.3 Description of the contents Photon ray theory 2.1 Geometric optics 2.2 Space and time refraction 2.2.1 Refraction 2.2.2 Time refraction 2.2.3 Space–time refraction 2.3 Generalized Snell’s law 2.4 Photon effective mass 2.5 Covariant formulation Photon dynamics 3.1 Ionization fronts 3.2 Accelerated fronts 3.3 Photon trapping 3.3.1 Generation of laser wakefields 3.3.2 Nonlinear photon resonance 3.3.3 Covariant formulation 3.4 Stochastic photon acceleration 3.4.1 Motion in two wakefields 3.4.2 Photon discrete mapping 3.5 Photon Fermi acceleration 3.6 Magnetoplasmas and other optical media Photon kinetic theory 4.1 Klimontovich equation for photons 4.2 Wigner–Moyal equation for electromagnetic radiation 4.2.1 Non-dispersive medium 4.2.2 Dispersive medium...
1.1 Definition of the concept
The concept of photon acceleration appeared quite recently in plasma physics. It is a simple and general concept associated with electromagnetic wave propagation, and can be used to describe a large number of effects occurring not only in plasmas but also in other optical media. Photon acceleration is so simple that it could be considered a trivial concept, if it were not a subtle one. Let us first try to define the concept. The best way to do it is to establish a comparison between this and a few other well-known concepts, such as with refraction. For instance, photon acceleration can be seen as a space–time refraction. Everybody knows that refraction is the change of direction suffered by a light beam when it crosses the boundary between two optical media. In more technical terms we can say that the wavevector associated with this light beam changes, because the properties of the optical medium vary in space. We can imagine a symmetric situation where the properties of the optical medium are constant in space but vary in time. Now the light wavevector remains 1...
1.2 Historical background
Let us now give a short historical account of this concept. The basic equations necessary for the description of photon acceleration have been known for many years, even if their explicit meaning has only recently been understood. This is due to the existence of a kind of conceptual barrier, which prevented formally simple jumps in the theory to take place, which could provide a good example of what Bachelard would call an obstacle epistemologique [6]....
The above description of wave propagation is only valid for uniform and stationary media. Let us now assume that propagation is taking place in a nonuniform and non-stationary medium. If the space and time variations in the medium are slow enough (in such a way that, locally both in space and in time, the medium can still be considered as approximately uniform and constant), we can replace the wave electric field (2.1) by a similar expression: E (r , t ) = E 0 (r , t ) exp iψ (r , t ) (2.5)...
This is the velocity of the centroid of an electromagnetic wavepacket moving in the medium, vg = dr /dt . The above equation can be written as ∂ k ∂ ∂ω + vg · =− . (2.11) ∂t ∂r ∂r The differential operator on the left-hand side is nothing but the total time derivative d/dt . It means that we can rewrite the last two equations as dr ∂ω = , dt ∂k ∂ω dk =− . dt ∂r (2.12)...
where H ( y ) = 0 for y < 0 and H ( y ) = 1 for y > 0 is the well-known step function or Heaviside function. First of all, it should be noticed that from equations (2.18) and (2.19) we have dω = 0. (2.21) dt This means that the wave frequency is a constant of motion ω (r , k , t ) = ω0 = const. If the plane of incidence coincides with z = 0, the time variation of the wavevector components is determined by dk x =0 dt dk y ∂ ω0 n = ω0 ln n ( y ) = kb sech2 (kb y ). dt ∂y 2n ( y ) (2.22) (2.23)...
We see that the photon frequency is shifted as time evolves, following a law similar to that of the wavevector change during refraction. On the other hand, if...
This relation means that, when a photon interacts with a moving boundary layer, the wave frequency and wavevector are both shifted, in a kind of space– time refraction. We notice that in order to establish the Snell’s laws (2.24, 2.25), or their temporal counterparts (2.29, 2.30), we had to define some constant of motion. In the first case, the constants of motion were the photon frequency ω and the wavevector component parallel to the boundary layer k x . In the second case, the constant of motion was the photon wavevector. Now, for the case of refraction in a moving boundary, or space–time refraction, we need to find a new constant of motion because neither the frequency nor the wavevector are conserved. This new constant of motion can be directly obtained from equation (2.33), but it is useful to explore the Hamiltonian properties of the ray equations (2.12, 2.13). For such a purpose, let us then define a canonical transformation from the variables(x , k ) to the new pair of variables (x , k ), such that x
=...
This is nothing but the well-known formula for the relativistic mirror, which was derived here in a very simple way, without invoking relativity or using any Lorentz transformation. Such a result is possible because the photon equations are exactly relativistic by nature. Apart from providing a very simple and alternative way to derive the relativistic mirror effect, this result is also interesting because it shows that such an effect can be seen as a particular case of the more general space–time refraction or photon acceleration processes....
2.4 Photon effective mass
Let us now go deeper in the analysis of the photon dynamics and explore the analogies between the photon ray equations and the equations of motion of an arbitrary point particle with finite rest mass. From the above definitions of the local wavevector k and local frequency ω, it is obvious that the total variation of the phase ψ , as a function of the space and time coordinates, is dψ = k · dr − ω dt . (2.64)...
2 This reduces to equation (2.83) for χ = −ωp /ω2 . We will now make a final comment on the nature of this force. For a nonuniform but stationary medium, the force acting on the photon is responsible for the change in the photon velocity according to the usual laws of refraction: a gradient of the refractive index always leads to a change in the photon velocity, as stated by this expression for the force. We have then a variation in the relativistic γ factor: ∂ γ /∂ t = 0. However this is not equivalent to photon acceleration, because the total energy of the photon remains unchanged: ∂ ω /∂ t = 0. The reason is that the variation in velocity (or in kinetic energy) is exactly compensated by an equal and opposite variation in the effective mass (or in the rest energy)...
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