Electron-many photon scattering
Non-linear inverse Compton scattering (NICS), also known as non-linear Compton scattering and multiphoton Compton scattering, is the scattering of multiple low-energy photons, given by an intense electromagnetic field, in a high-energy photon (X-ray or gamma ray) during the interaction with a charged particle, in many cases an electron. This process is an inverted variant of Compton scattering since, contrary to it, the charged particle transfers its energy to the outgoing high-energy photon instead of receiving energy from an incoming high-energy photon. Furthermore, differently from Compton scattering, this process is explicitly non-linear because the conditions for multiphoton absorption by the charged particle are reached in the presence of a very intense electromagnetic field, for example, the one produced by high-intensity lasers.
Non-linear inverse Compton scattering is a scattering process belonging to the category of light-matter interaction phenomena. The absorption of multiple photons of the electromagnetic field by the charged particle causes the consequent emission of an X-ray or a gamma ray with energy comparable or higher with respect to the charged particle rest energy.
The normalized vector potential formula_1 helps to isolate the regime in which non-linear inverse Compton scattering occurs (formula_2 is the electron charge, formula_3 is the electron mass, formula_4 the speed of light and formula_5 the vector potential). If formula_6, the emission phenomenon can be reduced to the scattering of a single photon by an electron, which is the case of inverse Compton scattering. While, if formula_7, NICS occurs and the probability amplitudes of emission have non-linear dependencies on the field. For this reason, in the description of non-linear inverse Compton scattering, formula_8 is called classical non-linearity parameter.
History.
The physical process of non-linear inverse Compton scattering has been first introduced theoretically in different scientific articles starting from 1964. Before this date, some seminal works had emerged dealing with the description of the classical limit of NICS, called non-linear Thomson scattering or multiphoton Thomson scattering. In 1964, different papers were published on the topic of electron scattering in intense electromagnetic fields by L. S. Brown and T. W. B. Kibble, and by A. I. Nikishov and V. I. Ritus, among the others. The development of the high-intensity laser systems required to study the phenomenon has motivated the continuous advancements in the theoretical and experimental studies of NICS. At the time of the first theoretical studies, the terms non-linear (inverse) Compton scattering and multiphoton Compton scattering were not in use yet and they progressively emerged in later works. The case of an electron scattering off high-energy photons in the field of a monochromatic background plane wave with either circular or linear polarization was one of the most studied topics at the beginning. Then, some groups have studied more complicated non-linear inverse Compton scattering scenario, considering complex electromagnetic fields of finite spatial and temporal extension, typical of laser pulses.
The advent of laser amplification techniques and in particular of chirped pulse amplification (CPA) has allowed to reach sufficiently high-laser intensities to study new regimes of light-matter interaction and to significantly observe non-linear inverse Compton scattering and its peculiar effects. Non-linear Thomson scattering was first observed in 1983 with formula_9 keV electron beam colliding with a Q-switched delivering an intensity of formula_10 W/cm2 (formula_11), photons of frequency two times the one of the laser were produced, then in 1995 with a CPA laser of peak intensity around formula_12 W/cm2 interacting with neon gas, and in 1998 in the interaction of a mode-locked Nd:YAG laser (formula_13 W/cm2, formula_14) with plasma electrons from a helium gas jet, producing multiple harmonics of the laser frequency. NICS was detected for the first time in a pioneering experiment at the SLAC National Accelerator Laboratory at Stanford University, USA. In this experiment, the collision of an ultra-relativistic electron beam, with energy of about formula_15 GeV, with a terawatt , with an intensity of formula_12 W/cm2 (formula_17, formula_18), produced NICS photons which were observed indirectly via a nonlinear energy shift in the spectrum of electrons in output; consequent positron generation was also observed in this experiment.
Multiple experiments have been then performed by crossing a high-energy laser pulse with a relativistic electron beam from a conventional linear electron accelerator, but a further achievement in the study of non-linear inverse Compton scattering has been achieved with the realization of all-optical setups. In these cases, a laser pulse is both responsible for the electron acceleration, through the mechanisms of plasma acceleration, and for the non-linear inverse Compton scattering occurring in the interaction of accelerated electrons with a laser pulse (possibly counter-propagating with respect to electrons). One of the first experiment of this type was made in 2006 producing photons of energy from formula_19 to formula_20 keV with a Ti:Sa laser beam (formula_21W/cm2). Research is still ongoing and active in this field as attested by the numerous theoretical and experimental publications.
Classical limit.
The classical limit of non-linear inverse Compton scattering, also called non-linear Thomson scattering and multiphoton Thomson scattering, is a special case of classical synchrotron emission driven by the force exerted on a charged particle by intense electric and magnetic fields. Practically, a moving charge emits electromagnetic radiation while experiencing the Lorentz force induced by the presence of these electromagnetic fields. The calculation of the emitted spectrum in this classical case is based on the solution of the Lorentz equation for the particle and the substitution of the corresponding particle trajectory in the Liénard-Wiechert fields. In the following, the considered charged particles will be electrons, and gaussian units will be used.
The component of the Lorentz force perpendicular to the particle velocity is the component responsible for the local radial acceleration and thus of the relevant part of the radiation emission by a relativistic electron of charge formula_2, mass formula_3 and velocity formula_24. In a simplified picture, one can suppose a local circular trajectory for a relativistic particle and can assume a relativistic centripetal force equal to the magnitude of the perpendicular Lorentz force acting on the particle: formula_25formula_26 and formula_27 are the electric and magnetic fields respectively, formula_28 is the magnitude of the electron velocity and formula_29 is the Lorentz factor formula_30. This equation defines a simple dependence of the local radius of curvature on the particle velocity and on the electromagnetic fields felt by the particle. Since the motion of the particle is relativistic, the magnitude formula_28 can be substituted with the speed of light to simplify the expression for formula_32. Given an expression for formula_32, the model given in can be used to approximately describe the classical limit of non-linear inverse Compton scattering. Thus, the power distribution in frequency of non-linear Thomson scattering by a relativistic charged particle can be seen as equivalent to the general case of synchrotron emission with the main parameters made explicitly dependent on the particle velocity and on the electromagnetic fields.
Electron quantum parameter.
Increasing the intensity of the electromagnetic field and the particle velocity, the emission of photons with energy comparable to the electron one becomes more probable and non-linear inverse Compton scattering starts to progressively differ from the classical limit because of quantum effects such as photon recoil. A dimensionless parameter, called electron quantum parameter, can be introduced to describe how far the physical condition are from the classical limit and how much non-linear and quantum effects matter. This parameter is given by the following expression: ;templatestyles src="Numbered block/styles.css" / ;{c}\times\mathbf{B}\right)^2-\left(\dfrac{\mathbf{E}\cdot\mathbf{v}}{c}\right)^2} 
 ;/math ;|1}}where formula_34 V/m is the Schwinger field. In scientific literature, formula_35 is also called formula_36. The Schwinger field formula_37, appearing in this definition, is a critical field capable of performing on electrons a work of formula_38 over a reduced Compton length formula_39, where formula_40 is the reduced Planck constant. The presence of such a strong field implies the instability of vacuum and it is necessary to explore non-linear QED effects, such as the production of pairs from vacuum. The Schwinger field corresponds to an intensity of nearly formula_41 W/cm2. Consequently, formula_42 represents the work, in units of formula_38, performed by the field over the Compton length formula_39 and in this way it also measures the importance of quantum non-linear effects since it compares the field strength in the rest frame of the electron with that of the critical field. Non-linear quantum effects, like the production of an electron-positron pair in vacuum, occur above the critical field formula_37, however, they can be observed also well below this limit since ultra-relativistic particles with Lorentz factor equal to formula_46 see fields of the order of formula_37 in their rest frame. formula_42 is called also non-linear quantum parameter whereas it is a measure of the magnitude of non-linear quantum effects. The electron quantum parameter is linked to the magnitude of the Lorentz four-force acting on the particle due to the electromagnetic field and it is a Lorentz-invariant:formula_49The four-force acting on the particle is equal to the derivative of the four-momentum with respect to proper time. Using this fact in the classical limit, the radiated power according to the relativistic generalization of the Larmor formula becomes:formula_50As a result, emission is improved by higher values of formula_42 and, therefore, some considerations can be done on which are the conditions for prolific emission, further evaluating the definition (1). The electron quantum parameter increases with the energy of the electron (direct proportionality to formula_29) and it is larger when the force exerted by the field perpendicularly to the particle velocity increases.
Plane wave case.
Considering a plane wave the electron quantum parameter can be rewritten using this relation between electric and magnetic fields:formula_53where formula_54 is the wavevector of the plane wave and formula_55 the wavevector magnitude. Inserting this expression in the formula of formula_42:formula_57where the vectorial identity formula_58 was used. Elaborating the expression:formula_59Since formula_60 for a plane wave and the last two terms under the square root compensate each other, formula_42 reduces to:
formula_62
In the simplified configuration of a plane wave impinging on the electron, higher values of the electron quantum parameter are obtained when the plane wave is counter-propagating with respect to the electron velocity.
Quantum effects.
A full description of non-linear inverse Compton scattering must include some effects related to the quantization of light and matter. The principal ones are listed below.
 ;templatestyles src="Numbered block/styles.css" / ;{2\pi}\dfrac{e^2 m c}{\hbar^2 }\dfrac{\chi}{\gamma\eta} F(y) \quad \text{where}\quad F(y)=y\int_y^{\infty}K_{\frac{5}{3}}(x)dx\quad\text{and}\quad y=\dfrac{2\eta}{3\chi^2} ;/math ;|2}}where formula_74 stands for the McDonald functions. The mean energy of the emitted photon is given by formula_75. Consequently, a large Lorentz factor and intense fields increase the chance of producing high-energy photons. formula_76 goes as formula_42 because of this formula.
Emission description when formula_7 and formula_82.
When the incoming field is very intense formula_7, the interaction of the electron with the electromagnetic field is completely equivalent to the interaction of the electron with multiple photons, with no need of explicitly quantize the electromagnetic field of the incoming low-energy radiation. While the interaction with the radiation field, i.e. the emitted photon, is treated with perturbation theory: the probability of photon emission is evaluated considering the transition between the states of the electron in presence of the electromagnetic field. This problem has been solved primarily in the case in which electric and magnetic fields are orthogonal and equal in magnitude (crossed field); in particular, the case of a plane electromagnetic wave has been considered. Crossed fields represent in good approximation many existing fields so the found solution can be considered quite general. The spectrum of non-linear inverse Compton scattering, obtained with this approach and valid for formula_7 and formula_82, is:
 ;templatestyles src="Numbered block/styles.css" / ;{2\pi}\dfrac{q^2 m c}{\hbar^2 }\dfrac{\chi}{\gamma\eta} F(\chi,\eta) \quad \text{where} \quad F(\chi,\eta)=\dfrac{\eta^2}{\chi^2} y K_{\frac{2}{3}}(y)+\left(1-\dfrac{\eta}{\chi}\right)y\int_y^{\infty}K_{\frac{5}{3}}(x)dx
 ;/math ;|3}}where the parameter formula_86, is now defined as:formula_87The result is similar to the classical one except for the different expression of formula_88. For formula_89 it reduces to the classical spectrum (2). Note that if formula_90 (formula_91 or formula_92) the spectrum must be zero because the energy of the emitted photon cannot be higher than the electron energy, in particular could not be higher than the electron kinetic energy formula_93.
The total power emitted in radiation is given by the integration in formula_68 of the spectrum (3):formula_95where the result of the integration of formula_96 is contained in the last term:
formula_97This expression is equal to the classical one if formula_98 is equal to one and it can be expanded in two limiting cases, near the classical limit and when quantum effects are of major importance:formula_99A related quantity is the rate of photon emission:formula_100where it is made explicit that