2. QUASIENERGY ELECTRON SPECTRUM. CONDITION OF RADIATION ABSORPTION BY AN ELECTRON

The relativistic quantum states, as well as the quasienergy spectrum of an electron in the field of the circularly polarized wave of an arbitrary intensity, propagating along a homogeneous magnetic field, based on the exact Dirac solutions in these fields, have been considered in (Redmond, Reference Redmond1965; Oleinik & Sinyak, Reference Oleinik and Sinyak1975; Abakarov & Oleinik, Reference Abakarov and Oleinik1973; Barber & Melikian, Reference Barber and Melikian1998). This allows to reveal the specifics and various aspects of this interaction. In particular, the quasienergy spectrum, which was found in Barber and Melikian (Reference Barber and Melikian1998) (without taking into account the anomalous magnetic moment of the electron), is given by the following expression:

(1)$$\eqalign{Q_{q\comma n\comma {\rm \zeta} }^{\rm \mu} & = q^{\rm \mu} + \displaystyle{{k^{\rm \mu} } \over {\lpar kQ\rpar }}\vert e\vert B\lpar 2n + 1 + {\rm \zeta} \rpar \cr & + k^{\rm \mu} \displaystyle{{m^2 {\rm \xi} ^2 } \over {2\left({\lpar kQ\rpar - g\vert e\vert B} \right)}}\comma \;}$$

We use here the following notations: Q ^μ_q,n,ζ is the 4-quasimomentum of the electron, i.e., the time average of the kinetic 4-momentum of the electron in the state described by the quantum numbers n, ζ, where n = 0,1,2,… is the level number of the discrete spectrum of the electron, in the direction perpendicular to the magnetic field $\vec{B}$, and ζ = ±1 is the projection of the electron spin onto the direction of $\vec{B}$. The components Q ⁰_q,n,ζ and Q ³_q,n,ζ are the quasienergy and quasimomentum correspondingly. We have also denoted by $k^{\rm \mu} \lpar {\rm \omega} \comma \; {\rm \omega} \vec{n}\rpar $ the 4-vector of the electromagnetic wave, the frequency of the latter is denoted by ω, and the unit vector in the direction of propagation of the wave is denoted by $\vec{n}$. The parameter ξ = eE/mω is the parameter of the wave intensity, and E is the amplitude of the electric field of the electromagnetic field. Finally, q ^μ is the 4-momentum of the free electron, which corresponds to the electron 4-momentum before entering to the region of interaction with the external fields. The values of g = ±1 determine the directions of the right or left circular polarization of the wave.Footnote ¹

Some comments are in order. The second term in (1) describes the interaction of an electron with the magnetic field. If one considers the limit ξ → 0, then, taking into account the relations q ² = m ² and k ² = 0, one obtains from (1) the well-known result for the electron energy in a constant magnetic field (Berestetskii et al., Reference Berestetskii, Lifshitz and Pitaevskii1982; Sokolov & Ternov, Reference Sokolov and Ternov1986):

(2)$$\lpar Q_{q\comma n\comma {\rm \zeta} }^0 \rpar ^2 = \lpar Q_{q\comma n\comma {\rm \zeta} }^3 \rpar ^2 + m^2 + m{\rm \omega} _c \lpar 2n + 1 + {\rm \zeta} \rpar \comma \;$$

where ω_c = eB/m.

The third therm in (1) describes the interaction of an electron with the electromagnetic wave and the magnetic field simultaneously. In particular, taking the limit B → 0 one obtains from (1) the well-known expression for the 4-quasimomentum of the electron in the electromagnetic field (Berestetskii et al., Reference Berestetskii, Lifshitz and Pitaevskii1982):

(3)$$Q^{\rm \mu} = q^{\rm \mu} + k^{\rm \mu} \displaystyle{{m^2 {\rm \xi} ^2 } \over {2\lpar kq\rpar }}.$$

Using the condition k ² = 0 one obtains from (1) the relation (kQ) = (kq) and the following expression for the quasienergy of the electron, in the state described by the quantum numbers n, ζ:

(4)$$Q_{q\comma n\comma {\rm \zeta}}^0 = q^0 + {\rm \gamma}_0 {\rm \omega} _c \lpar 2n + 1 + {\rm \zeta} \rpar + \displaystyle{{m{\rm \gamma}_0 {\rm \xi} ^2 } \over {\rm \delta}}\comma \;$$

where have denoted for brevity ${\rm \delta} = 1 - g{\textstyle{{2{\rm \gamma}_0 {\rm \omega} _c } \over {\rm \omega} }}$, and γ₀ = (q ⁰ + q ³)/2m.

After injecting the electrons into a magnetic field, they will occupy some interval of the quasienergy levels Q ⁰_q,n,ζ, according to the spread of the values corresponding to q ⁰ and the quantum number n of the electrons. It is clear from (1) that the spread on the quantum number n can be determined according to the spread of the electron velocity V _e on angles ϕ, where ϕ is the angle between the electron velocity $\vec{V}_e$ and the z-axis.

From the expression (4) it follows that the quasienergy of the electron in external fields depends essentially on the polarization g of the electromagnetic field. It is clear from (4) that the corrections to the quasienergy of the electron, which are proportional to ξ², will always be positive if the polarization g = −1. Thus, by choosing appropriately the direction of the polarized wave, the growth of the electron energy will be independent on the initial phase of the electromagnetic wave. This is an essential circumstance from the practical point of view, when implementing the acceleration process.

3. ELECTRON ENERGY GROWTH. PHOTON ABSORPTION LENGTH FORMATION

From the formulas (1) and (4) one sees that during the interaction of the electron with the electromagnetic wave and the magnetic field the energy growth may happen as a result of two effects. First, it may happen due to consecutive multiple resonance absorption of photons from the wave, during the transition between the quasienergies Q ^μ_q,n,ζ → Q ^μ_{q ^′,n′,ζ}. Secondly, it may happen due to absorption of the s-photons from the wave, as a result of the correction to the electron energy, proportional to ξ².

The resonance absorption of a photon from the wave, with the 4-momentum $k^{\rm \mu} \lpar {\rm \omega} \comma \; \vec{k}\rpar $, may occur during the transition between the quasienergies Q ^μ_q,n,ζ → Q ^μ_{q ^′,n′,ζ}, if the length of the interaction (see Fig. 1) between the electron and the laser beam in the presence of a magnetic field is greater than the formation length l _a of the absorption of a photon.

Fig. 1.

Schematic diagram of the electron acceleration by laser radiation in the presence of a magnetic field. L _M is the length of the magnet.

Due to the energy-momentum conservation, one obtains:

(5)$$Q_{q\comma n\comma {\rm \zeta} }^{\rm \mu} + k^{\rm \mu} = Q_{q^{\prime}\comma n^{\prime}\comma {\rm \zeta} }^{\rm \mu}.$$

The changes of the quasienergy ΔQ ⁰ and the quasimomentum ΔQ ³ of an electron are related to the energy $\hbar\! {\rm \omega}$ and the momentum $\hbar\! k_z$ of the absorbed photon. Using the conservation of the quasienergy-quasimomentum:

(6)$$Q_{q\comma n\comma {\rm \zeta} }^0 + {\rm \omega} = Q_{q^{\prime}\comma n^{\prime}\comma {\rm \zeta} }^0$$

(7)$$Q_{q\comma n\comma {\rm \zeta} }^3 + {\rm \omega} \;{\rm cos}\;{\rm \theta} = Q_{q^{\prime}\comma n^{\prime}\comma {\rm \zeta} }^3 \comma \;$$

and the relation, following from (1):

(8)$$\eqalign{\left({Q_{q\comma n\comma {\rm \zeta} }^0 } \right)^2 =& \mathop {\left({Q_{q\comma n\comma {\rm \zeta} }^3 } \right)}\nolimits^2 + \;m^2 + m{\rm \omega} _c \lpar 2n + 1 + {\rm \zeta} \rpar \cr & + \displaystyle{{m^2 {\rm \xi} ^2 } \over {\rm \delta} }\comma \;}$$

we find the condition of the absorption of a photon by an electron:

(9)$$Q_{q\comma n\comma {\rm \zeta} }^0 - \lpar {\rm cos}{\rm \theta} \rpar Q_{q\comma n\comma {\rm \zeta} }^3 + \displaystyle{{{\rm \omega} {\rm sin}^2 {\rm \theta} } \over 2} = \displaystyle{{m{\rm \omega} _c \lpar n^{\prime} - n\rpar } \over {\rm \omega}}.$$

Here, Q ⁰_q,n,ζ and Q ³_q,n,ζ are the initial quasienergy and the z-component of the quasimomentum of the electron, and θ is the angle of the laser beam propagation, relative to the direction of the magnetic field $\vec{B}$.

We stress that we have considered here only the transitions between the quasienergies of the electrons without taking into account the changes in spin direction, since the probability associated with such changes is ignorable (Berestetskii et al., Reference Berestetskii, Lifshitz and Pitaevskii1982; Sokolov & Ternov, Reference Sokolov and Ternov1986).

For the radiation absorption at optical or lower frequencies, and for the relevant to our case values of the angle ${\rm \theta} \ll 1$, one can ignore in (9) the quantum corrections, since:

(10)$$\hbar\! {\rm \omega} \sin^2 \lpar {\rm \theta} \rpar \ll \displaystyle{{mc^2 {\rm \omega} _c \lpar n^{\prime} - n\rpar } \over {\rm \omega} }.$$

Thus, using the equation (4), and substituting Q _q,n,ζ³ from (8) into (9), one can write the condition of the photon absorption in the form:

(11)$${\rm \omega} \simeq {{\rm \omega} _c \lpar n^{\prime} - n\rpar \over \displaystyle{\rm \gamma} \left(1 - \cos\lpar {\rm \theta}\rpar \sqrt{1-{1\over {\rm \gamma}^2}-{{\rm \omega} _c \lpar 2n + 1 + {\rm \zeta} \rpar \over {\rm \gamma}^2 m}-{{\rm \xi}^2\over {\rm \gamma}^{2}{\rm \delta}}} \right)}\comma \;$$

where we have introduced the notation:

(12)$${\rm \gamma} = \displaystyle{{Q_{q\comma n\comma {\rm \zeta} }^0 } \over m} = {\rm \gamma}_0 \left[{1 + \displaystyle{1 \over {4{\rm \gamma}_0^2 }} + \displaystyle{{{\rm \omega} _c \lpar 2n + 1 + {\rm \zeta} \rpar } \over m} + \displaystyle{{{\rm \xi} ^2 } \over {\rm \delta} }} \right].$$

From the formula (11) one can easily obtain the classical condition for the radiation absorption. Indeed, considering the transitions for the main harmonics corresponding to n′ − n = 1, disregarding the quantum correction $2n\hbar {\rm \omega} _c / {\rm \gamma} ^2 mc^2 $, and considering ξ²/γ²δ → 0, one obtains the well-known classical condition for the radiation absorption (Kolomenskii & Lebedev, Reference Kolomenskii and Lebedev1962; Reference Kolomenski and Lebedev1963; Reference Kolomenski and Lebedev1966; Milantiev, Reference Milantiev1997).

Since we are interested in the case for which ${\rm \theta} \ll 1$ and ${\rm \gamma}_0 \gg 1$, one can use the relations (kQ) = (kq), as well as the formulas (9) and (11), to find the approximate value of the difference ΔQ ⁰ = Q ⁰_q′,n′,ζ − Q _q,n,ζ⁰ between the quasienergies of the electron:

(13)$$\Delta Q^0 \simeq \displaystyle{{2{\rm \gamma}_0 {\rm \omega} _c \lpar n^{\prime} - n\rpar } \over {1 + {\rm \theta} ^2 {\rm \gamma}_0^2 \left[{1 + {\displaystyle{{{\rm \xi} ^2 } \over {\rm \delta} }} + {\displaystyle{{{\rm \omega} _c \lpar 2n + 1 + {\rm \zeta} \rpar } \over m}}} \right]}}.$$

It is clear from this formula, that the difference ΔQ ⁰ between the quasienergies is not equidistant, and is decreasing with the increase of the quantum number n. Due to this behavior we see that for the quantum numbers n > n _cr, where n _cr is some critical value, one will have $\Delta Q ^0 \lt \hbar {\rm \!\omega} $. The latter means that the condition of the resonance absorption of a photon breaks down, and, as a result, the process of electron acceleration stops. We stress that ΔQ ⁰, according to the formula (13), does depend on the intensity parameter ξ, unless θ = 0.

Let us also emphasize, that according to the formulas (6) and (7):

(14)$$Q_{q^{\prime}\comma n^{\prime}\comma {\rm \zeta} }^0 - Q_{q\comma n\comma {\rm \zeta} }^0 = \displaystyle{{Q_{q^{\prime}\comma n^{\prime}\comma {\rm \zeta} }^3 - Q_{q\comma n\comma {\rm \zeta} }^3 } \over {\cos\lpar {\rm \theta} \rpar }}\comma \;$$

and, therefore, in the case when ${\rm \theta} \ll 1$, the growth of the electron energy mainly occurs along the z-axis. This is in complete agreement with the fact that during the photon absorption the electron receives not only the energy, but also the momentum of the photon, along the direction of the z-axis.

From the condition for the photon absorption (13) it follows that besides the main harmonics n′ − n = 1, which corresponds to the classical physics, there are also possible the transitions with $n^{\prime} - n \gg 1$. The probabilities of the quantum transitions with $n^{\prime} - n \gg 1$ has been obtained and studied in (Abakarov & Oleinik, Reference Abakarov and Oleinik1973). It has been shown there, that if the quantum transition occurs under the influence of the pulsed electromagnetic field with the vector potential of the form A = aexp(−τ²/τ₀²)cos(ωt) (where τ₀ is the duration of the wave pulse), then the probability of transitions n → n′ is determined by the parameter:

(15)$${\rm \zeta} _{tr} \simeq \sqrt {\rm \pi} {\rm \xi} {\rm \gamma}_0 \displaystyle{{\sqrt {{\rm \omega} _c m} } \over {\rm \omega} }{\rm \tau} _0 {\rm \omega} \exp\left[{ - \displaystyle{{{\rm \tau} _0^2 } \over 4}\lpar 2{\rm \gamma}_0 {\rm \omega} _c - {\rm \omega} \rpar ^2 } \right]$$

In the paper (Abakarov & Oleinik, Reference Abakarov and Oleinik1973) it was shown that if ${\rm \zeta} _{tr} \gg 1$ then the probability of transitions n → n′ has the maximum value when $\vert n^{\prime} - n\vert \sim {\rm \zeta} _{tr}^2/ 2 \gg 1$.

From the equation (13) it follows that if the energy of photons is equal to ω = 2γ₀ω_c, then during the trasition between the quasienergies Q _q,n,ζ^μ → Q ^μ_{q ^′,n′,ζ′} there occurs n′ − n-tuple photon absorption by electrons. Moreover, one can estimate the value of |n′ − n|~ζ_tr²/2 using the formula (15) for ζ_tr, depending on the values of γ₀, ξ and τ₀. If $\vert 1 - 2{\rm \gamma}_0 {\rm \omega} _c / {\rm \omega} \vert \ll 1$, then one approximately obtains that n′ − n~ζ_tr²/2 ≃ω/(π| ω − 2γ₀ω_c|) when the probability of transitions Q _q,n,ζ^μ → Q ^μ_{q ^′,n′,ζ′} reaches maximum.

In this paper we consider the accelration of electrons for g = −1. However, according to the formula (4), for the case g = +1, the electrons energy growth is equal to $m{\rm \gamma}_0 {\rm \xi} ^2 /\lpar 1 - {2{\rm \gamma}_0 {\rm \omega} _c \over {\rm \omega}}\rpar $ and becomes stronger, in comparison with the case g =−1, at the resonance condition ω → 2γ₀ω_c. To understand and explain the peculiarities of the electron acceleration for the case g = +1 one needs to perform a more detailed analysis, which will be considered in a future publication.

As we have mentioned above, the absorption of a photon from the wave by an electron may occur only in the case when the interaction length of the electron with the laser beam in a magnetic field is greater the formation length l _a of the photon absorption by an electron. To estimate the value of l _a we note, that the process of photon absorption by an electron is formed on some definite length l _a of the electron trajectory, when at least the wave with the length λ may be absorbed during the time Δt = l _a/V _e (Landau & Lifshitz, Reference Landau and Lifshitz1975; Berestetskii et al., Reference Berestetskii, Lifshitz and Pitaevskii1982; Sokolov & Ternov, Reference Sokolov and Ternov1986). The length l _a of the photon absorption is determined from the formula (Landau & Lifshitz, Reference Landau and Lifshitz1975; Sokolov & Ternov, Reference Sokolov and Ternov1986):

(16)$$l_a \simeq \displaystyle{{\rm \lambda} \over {\lpar 1 - \vec {\rm \beta} \vec{n}\rpar }}\comma \;$$

where $\vec {\rm \beta} = {{\mathop {\vec V}\nolimits_e } / c}$, and ${1 / {\lpar 1 - \vec {\rm \beta} \vec{n}\rpar }}$ is the Doppler factor. Clearly, in order the process of the photon absorption by an electron to be possible, the condition l _a < L _M, where L _M is the length of the magnet, must be imposed. Representing $\vec {\rm \beta} \vec{n}$ in the component form $\vec {\rm \beta} \vec{n} = {\rm \beta} _z n_z + {\rm \beta} _ \bot n_ \bot = {\rm \beta} _z \cos{\rm \theta} + {\rm \beta} \sin{\rm \phi} \sin{\rm \theta} $, and taking into account the formula (16), one obtains:

(17)$${\rm \beta} _z \cos{\rm \theta} \simeq 1 - \displaystyle{{\rm \lambda} \over {l_a }} - {\rm \beta} \sin{\rm \phi} \sin{\rm \theta} .$$

Using the relation γ₀ = q ⁰ (1 + β_z)/2m, one can write the condition of the photon absorption (13), for the case θ = 0, in the following form:

(18)$${\rm \omega} = \displaystyle{{q^0 } \over m}{\rm \omega} _c \lpar 1 + {\rm \beta} _z \rpar.$$

Substituting β_z, for the case θ = 0, from the equation (17) into the above expression (18), we find:

(19)$$l_a \simeq {\rm \lambda}\displaystyle{{q^0 {\rm \omega} _c } \over {2q^0 {\rm \omega} _c - m{\rm \omega} }}.$$

From the above expressions, using the limitation 0 < β_z < 1, one finds the following restriction on ω_c:

(20)$$\displaystyle{{m{\rm \omega} } \over {2q^0 }} \lt {\rm \omega} _c \lt \displaystyle{{m{\rm \omega} } \over {q^0 }}.$$

It is clear from (19) that for the given values of ω and q ⁰, by changing the values of ω_c within the limits determined in (20), the length l _a will be varying within the interval (λ,νλ), where ν = τ₀c/λ = 1,2,… is determined from the duration τ₀ of the laser pulse.

Let us note, that in the formulas (16), (17) and (19) the length l _a represents the length of formation only of one photon. One can find the length l _a of formation of photon absorption for ξ ≥ 1, when the electron energy growth occurs because of absorption from the wave of s-photons, and the energy absorbed by the electron (on the wave length λ) is comparable or greater than the initial energy of the electron (Landau & Lifshitz, Reference Landau and Lifshitz1975). The overview of quantum processes in the field of a strong electromagnetic plane wave is given in (Nikishev & Ritus Reference Nikishov and Ritus1979).

From (4), (6) and (7) one finds that for case when ξ ≥ 1, due to the interaction with the electromagnetic wave and magnetic field, the electron acquires the energy:

(21)$$\Delta {\rm \epsilon} = Q_{q\comma n\comma {\rm \zeta} }^0 - q^0 = s{\rm \omega} \simeq \displaystyle{{m{\rm \gamma}_0 {\rm \xi} ^2 } \over {1 + {\textstyle{{2{\rm \gamma}_0 {\rm \omega} _c } \over {\rm \omega} }}}}.$$

From this formula it is clear that, for ξ ≥ 1, the energy Δε absorbed by an electron may take values much greater than the initial electron energy q ⁰.

Using the relation (13) for the case n′ − n = 1 and cosθ = 0, valid for all values ξ, one obtains from (21):

(22)$$\displaystyle{{q^0 \lpar 1 + {\rm \beta} _z \rpar {\rm \omega} _c } \over {m{\rm \omega} }} = \displaystyle{{m{\rm \gamma}_0 {\rm \xi} ^2 } \over {\Delta {\rm \epsilon}}} - 1$$

Using the relation γ₀ = q ₀(1 + β_z)/2m, and taking ${\rm \beta} _z \simeq 1 - {\textstyle{{\rm \lambda} \over {l_a }}}$, which follows from the formula (17) for the case ${\rm \theta} = 0$, one obtains from (21):

(23)$$\displaystyle{{2\Delta {\rm \epsilon}} \over {{\rm \xi} ^2 q^0 }} - 1 \simeq {\rm \beta} _z \simeq 1 - \displaystyle{{\rm \lambda} \over {l_a }}.$$

From the formula (23) we obtain the restrictions ξ²q ⁰/2 < Δε < ξ²q ⁰ and l _a ≥ l _a,min = λ. In addition, we obtain the relation:

(24)$$l_a \simeq \displaystyle{{\rm \lambda} \over 2}\displaystyle{{{\rm \xi} ^2 q^0 } \over {{\rm \xi} ^2 q^0 - \Delta {\rm \epsilon}}}.$$

The maximal value of the length l _a is limited by the value of the laser pulse duration τ, i.e., l _a,max = νλ. The dependence of Δε on l _a, according to the formula (23), has the form:

(25)$$\Delta {\rm \epsilon} \simeq {\rm \xi} ^2 q^0 \left({1 - \displaystyle{{\rm \lambda} \over {2l_a }}} \right).$$

In particular, in the case when the magnetic field B → 0, one obtains from the formula (1) the famous Dirac solution for an electron in the field of the plane electromagnetic wave (Berestetskii et al., Reference Berestetskii, Lifshitz and Pitaevskii1982). In this case, according to the expression (21), the energy growth is equal to mc ²γ₀ξ². In the paper (Stupakov & Zolotorev, Reference Stupakov and Zolotorev2001) the authors had considered the acceleration of electrons interacting with a focused high intensity short duration laser beam in vacuum, by means of ponderomotive light forces. It was shown that the maximal energy gain of electrons can reach up to the value mc ²γ₀ξ² on the Rayleigh length of the order γ₀ξƛ.

To estimate the necessary laser intensity I _las we use the relation between I _las and the amplitude E of the electromagnetic wave's electric field (Landau & Lifshitz, Reference Landau and Lifshitz1975):

(26)$$E\left[{\displaystyle{{\rm V} \over {{\rm cm}}}} \right]\simeq 19\comma \; 46\sqrt {I_{{\rm las}} \left[{\displaystyle{{\rm W} \over {{\rm cm}^2 }}} \right]}.$$

Using (25) and (26) we see that the energy growth Δε increases proportionally with the increase of the laser intensity I _las and the initial electron energy q ⁰. To give a concrete numeric estimate, we take, for example, γ₀ = 10², λ = 1,06 · 10⁻³ cm, I _las ≃ 10¹⁸W/cm², τ = 1ps, we find that the energy growth has, according to the formula (25), the value Δε≃ 2,1 GeV (see Fig. 2)

Fig. 2.

Dependence of the electron energy growth Δε on the number ν of periods in the wave pulse according to the formula (25), for the parameters γ₀ = 10², I _las ≃ 10¹⁸ W/cm² (or ξ ≃ 6.42), λ = 1,06 · 10⁻³ cm.

4. RELATIVE PRECISION OF THE ACCELERATED ELECTRONS ENERGY

For the case we consider in this paper, the parameters are ξ ≥ 1, g =− 1, and the electron energy growth is determined from the formula (25). The relative precision of an accelerated electron beam can be found from (25), using the following approximate formula:

(27)$${{\rm \delta} {\rm \epsilon} \over {\rm \epsilon}}\simeq 2{{\rm \delta} {\rm \xi} \over {\rm \xi} } + {{\rm \delta} q^0 \over q^0}+{1\over 2{\rm \nu} - 1}{{\rm \delta} {\rm \tau} _0 \over {\rm \tau} _0}.$$

We see from this formula, that Δε/ε depends on the relative precision of the initial electron beam energy, the intensity, as well as the duration of the laser pulse. For example, taking ${{{\rm \delta} q^0 } / {q^0 }} \le 10^{ - 3}\comma \; {{\rm \delta} {\rm \omega} / {\rm \omega} } \le 10^{ - 3} $, and δξ/ξ ≤ 0,5 · 10⁻², one obtains the order of the relative precision of the accelerated electron beam: δε/ε ≤ 10⁻².