\mathrsfs K + ( ℓ ) of the Volkov state in the same pulse (right panel). Makes a clear identification of individual Zel’dovich levels difficult.įigure 4: The light-front momentum component π − along the classical trajectory of an electron in a short laser pulse as aįunction of the laser phase ϕ = ω x + (left panel) is compared to the spectral component In the right panel a large number of spectral components contribute the width of each level is larger than their separation which One can make a clear connection between these Zel’dovich levels and the peaks in the Volkov spectrum Zel’dovich levels at ℓ = n + m a 2 0 8 ω γ in the case of an infinite monochromatic plane wave. In the left panel the black vertical lines indicate the positions of the įigure 3: Spectral components of the Volkov state with a small (large) range of spectral components ℓ Modifications of this level structure due to radiative corrections, i.e. the electron self-energy have been calculated as well. Or the resonant singularities in second-order strong-field scattering processes can be seen as transitions betweenĭifferent Zel’dovich of the incident and final state Volkov electrons. The level structure furnishes an easy interpretation of strong-field phenomena.įor instance, the appearance of harmonics in the non-linear Compton spectra = ∞ ∑ n = − ∞ e − i ( q + n k ) ⋅ x E I P W p ( n ) , They correspond to the ⧸ k ⧸ A term in the Volkov states, i.e. the Pauli interaction term. 1 are non-zero only in regions where the laser pulse is present. The tensor projection of the Ritus matrices ![]() This behaviour corresponds to the build-up of an intensity dependent ponderomotive quasi-momentum inside the laser pulse. In the case of the scalar projection of the Ritus matrices theĮffect of the laser pulse is a local tilt of the electron wave fronts, see left panel of Fig. With the upcoming generation of multi-petawatt lasers one expects to reach intensities up toġ0 23 … 10 25 \watt \per \centi \metre 2 and even beyondĢ 2 2For instance, the Vulcan laser at RAL-CLF (UK) Which corresponds to a 0 ∼ 100 at λ = \unit 800 \nano \metre typical for Ti:Sa laser systems. I = \unit 2 × 10 22 \watt \per \centi \metre 2, With present laser technology one is able to reach light intensities on the order of The classical nonlinearity parameter a 0 represents the laser energy density seen by the probe electron andĬan be related to the laser intensity viaĪ 2 0 = 7.309 × 10 − 19 I λ 2. Laser peak intensity and the laser bandwidth a 2 0 Δ ϕ ∝ I / ( Δ ω / ω ). ![]() (iii) the quantum efficiency parameter χ e = e √ ( F ⋅ p ) 2 / m 3 = a 0 b 0.įor a pulsed laser field one has in addition the dephasing parameter a 2 0 Δ ϕ, which represents the ratio of the (ii) the quantum energy parameter b 0 = ( k p ) / m 2, and ![]() (i) the classical nonlinearity parameter a 0 = e A 0 / m , ![]() The interaction of a probe electron with four-momentum p μ with the plane wave field can be characterized by the followingĭimensionless gauge and Lorentz invariant parameters: With a complex polarization vector ε, and an envelope function g with pulse duration Δ ϕ. The field can be represented by a transverse vector potential, A μ = ( 0, A ⊥ ( k ⋅ x ), 0 ), with k ⋅ A = 0,Īnd parametrized as A ⊥ = A 0 g ( ϕ / Δ ϕ ) R e In addition, Maxwell’s equations in vacuum require that F μ ν is transverse, k μ F μ ν = 0. In this case, ϕ = k ⋅ x = ω x +, which tells us that the field depends solely on the light-front variable x + = t + z (cf. Appendix A). The field strength is a univariate function, F μ ν = F μ ν ( k ⋅ x ). Their wave vector is a light-like four-vector, k 2 = ω 2 − k 2 = 0, and Let us first recall some of the properties of (transverse) plane waves, which we assume to propagate along the negative z axis,
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