Non-linear inverse Compton scattering

Non-linear inverse Compton scattering

See also: Compton 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 in a high-energy one (X-ray or gamma ray photon) during the interation 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 high-energy photon in output instead of receiving energy from a high-energy photon in input. Furthermore, differently from Compton scattering, this process is explicitly non-linear because the conditions for multiphoton absorption by the charged particle are reached in presence of a very intense electromagnetic field (non-linear regime of light-particle interaction), for example the one produced by high-intensity lasers.

Description

Non-linear inverse Compton scattering is a scattering process belonging to the category of light-matter interaction. It occurs when one moving charged particle emits in presence of an intense electromagnetic field. The process can be visualized by the absorption of multiple photons of this field with 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 ${a_{0}=eA/m_{e}c^{2}}$ helps to isolate the regime in which non-linear inverse Compton scattering occurs. If $a_{0}\ll 1$ 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 $a_{0}\gg 1$, NICS occurs and the probability amplitudes of emission have non-linear dependencies on the field. For this reason $a_{0}$ is called classical non-linearity parameter in the description of non-linear inverse Compton scattering.

Classical limit

The classical limit of this phenomenon, also called non-linear Thomson scattering or multiphoton Compton scattering, is a special case of synchrotron emission driven by the force exerted on a charged particle by intense electric and magnetic fields. Practically, a moving charge experiencing the Lorentz force induced by the presence of these electromagnetic fields emits electromagnetic radiation. In the following, the charged particles will be electroons for simplicity.

The component of the Lorentz force perpendicular to the particle velocity is the component responsible of the radial acceleration and thus of the relevant part of the radiation emission by a relativistic electron of charge $e$, mass $m$ and velocity $\mathbf {v}$.$$\mathbf {F} _{\perp }=-e\left(\mathbf {E} +{\dfrac {\mathbf {v} }{c}}\times \mathbf {B} \right)-\left[-e\left(\mathbf {E} +{\dfrac {\mathbf {v} }{c}}\times \mathbf {B} \right)\cdot {\dfrac {\mathbf {v} }{v}}\right]{\dfrac {\mathbf {v} }{v}}=-e\left(\mathbf {E} +{\dfrac {\mathbf {v} }{c}}\times \mathbf {B} \right)+e{\dfrac {\mathbf {E} \cdot \mathbf {v} }{v^{2}}}\mathbf {v}$$$\mathbf {E}$ and $\mathbf {B}$ are the electric and magnetic fields respectively e $v$ is the magnitude of the electron velocity The relativistic centripetal force acting on the particle is equal to the magnitude of $\mathbf {F} _{\perp }$:$$\gamma {\dfrac {mv^{2}}{\rho }}=e{\sqrt {\left(\mathbf {E} +{\dfrac {\mathbf {v} }{c}}\times \mathbf {B} \right)^{2}-\left({\dfrac {\mathbf {E} \cdot \mathbf {v} }{v}}\right)^{2}}}$$$\gamma$ is the Lorentz factor $\left(1-v^{2}/c^{2}\right)^{-1/2}$ . This equation defines the dependence of the radius of curvature on the particle velocity and on the electromagnetic fields felt by the particle. Whereas the motion of the particle is relativistic the magnitude $v$ can be substituted with the speed of light, in order to simplify the expression for $\rho$. Given an expression for $\rho$, the model and formulas given in Example 1: bending magnet well describe the classical limit of non-linear inverse compton scattering. Thus, the power distribution in frequency for classical NICS is completely equivalent to the general case of synchrotron emission with the main parameters made explicit according to the physics of non-linear Thomson scattering.

Electron quantum parameter

Increasing the intensity of the electromagnetic field and the particle velocity NICS progressively differs more and more from the classical limit described above. A dimensionless parameter, called electron quantum parameter, can be introduced to describe how far we are from the classical limit and how much non-liearity and quantum effect matter. This parameter is given by the following expression:

$$\chi ={\dfrac {\gamma }{E_{s}}}{\sqrt {\left(\mathbf {E} +{\dfrac {\mathbf {v} }{c}}\times \mathbf {B} \right)^{2}-\left({\dfrac {\mathbf {E} \cdot \mathbf {v} }{c}}\right)^{2}}}$$ 1

where $E_{s}=m^{2}c^{3}/(\hbar e)\simeq 1.3\cdot 10^{18}$ V/m is the Schwinger field. In literature, $\chi$ is also called $\eta$. It controls in general the importance of quantum effects as will be explained later. The Schwinger field $E_{s}$, appearing in its definition, is a field capable of performing on electrons a work of $mc^{2}$ over a reduced Compton length $\hbar /(mc)$. The presence of such a strong field implies the instability of vacuum and 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 $10^{29}$ W/cm². Consequently, $\chi$ represents the work, in units of $mc^{2}$ performed by the field over the Compton length $\hbar /(mc)$ and in this way measures also the importance of quantum non-linear effects. The electron quantum parameter is linked with the magnitude of the Lorentz four-force acting on the particle due to the electromagnetic field:$$\chi ={\dfrac {e\hbar }{m^{3}c^{4}}}|F_{\alpha \beta }p^{\alpha }|$$The 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:$$P={\dfrac {2}{3}}{\dfrac {e^{2}m^{2}c^{3}}{\hbar ^{2}}}\chi ^{2}$$As a result emission is improved by higher values of $\chi$, some considerations can be done on which are the conditions for prolific emission. Definition \ref{chi}(1) can be further evaluated. The electron quantum parameter increases with the energy of the electron (direct proportionality to $\gamma$) and is larger when the force exerted by the field perpendicularly to the particle velocity increases. Moreover, considering approximately the pulse as a plane wave the parameter can be rewritten using the fact that:$$\mathbf {B} ={\dfrac {\mathbf {k} \times \mathbf {E} }{k}}$$where $\mathbf {k}$ is the wavevector of the plane wave whose magnitude is $k$. Inserting this expression in the formula of $\chi$:$$\chi ={\dfrac {\gamma }{E_{s}}}{\sqrt {\left(\mathbf {E} +{\dfrac {(\mathbf {E} \cdot \mathbf {v} )}{c}}{\dfrac {\mathbf {k} }{k}}-{\dfrac {(\mathbf {v} \cdot \mathbf {k} )}{kc}}\mathbf {E} \right)^{2}-\left({\dfrac {\mathbf {E} \cdot \mathbf {v} }{c}}\right)^{2}}}$$where the vectorial identity $\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )=(\mathbf {A} \cdot \mathbf {C} )\mathbf {B} -(\mathbf {A} \cdot \mathbf {B} )\mathbf {C}$ was used. Elaborating the expression:$$\chi ={\dfrac {\gamma }{E_{s}}}{\sqrt {\left[\mathbf {E} \left(1-{\dfrac {\mathbf {v} \cdot \mathbf {k} }{kc}}\right)\right]^{2}-2\left(1-{\dfrac {\mathbf {v} \cdot \mathbf {k} }{kc}}\right)\left({\dfrac {\mathbf {E} \cdot \mathbf {v} }{kc}}\right)\mathbf {k} \cdot \mathbf {E} +\left({\dfrac {(\mathbf {E} \cdot \mathbf {v} )}{c}}{\dfrac {\mathbf {k} }{k}}\right)^{2}-\left({\dfrac {\mathbf {E} \cdot \mathbf {v} }{c}}\right)^{2}}}$$Since $\mathbf {k} \cdot \mathbf {E} =0$ for a plane wave and the last two terms under the square root compensate each other, $\chi$ reduces to:

$$\chi ={\dfrac {\gamma |\mathbf {E} |}{E_{s}}}{\sqrt {\left(1-{\dfrac {\mathbf {v} \cdot \mathbf {k} }{kc}}\right)^{2}}}$$

In the simplified configuration of a plane wave impinging on the electron, higher values of the electron quantum parameter are obtained when the laser is counterpropagating with respect to the electron velocity.

Quantum Effects

A full description of non-linear inverse Compton scattering must include some effects releted to the quantization of light and matter. They principal ones are listed below.

• Inclusion of the discretization of the emitted radiation, i.e. the introduction of photons with respect to the classical limit. This effect does not change quantitatively the emission features but changes the way in which the emtted radiation is interpreted. A parameter equivalent to $\chi$ can be introduced for the photon of frequency $\omega$, it is called photon quantum parameter:$$\eta ={\dfrac {e\hbar ^{2}}{m^{3}c^{4}}}|F_{\alpha \beta }k^{\alpha }|$$where $k^{\alpha }=(\omega /c,\mathbf {k} )$ is the photon four-wavevector, $\mathbf {k}$ is the three-dimensional wavevector. In the limit in which the particle approaches the speed of light the ratio between $\eta$ and $\chi$ is equal to:$$\zeta ={\dfrac {\eta }{\chi }}\simeq {\dfrac {\hbar \omega }{\gamma mc^{2}}}$$From the Frequency distribution of radiated energy one can get the rate of high-energy photon emission distributed in $\eta$ as a function of $\chi$ and $\eta$ but still valid in the classical limit:

$${\dfrac {d^{2}N}{d\eta dt}}={\dfrac {1}{\zeta \hbar \chi }}{\dfrac {dP}{d\omega }}={\dfrac {\sqrt {3}}{2\pi }}{\dfrac {q^{2}mc}{\hbar ^{2}}}{\dfrac {\chi }{\gamma \eta }}F(y)\quad {\text{where}}\quad F(y)=y\int _{y}^{\infty }K_{\frac {5}{3}}(x)dx\quad {\text{and}}\quad y={\dfrac {2\eta }{3\chi ^{2}}}$$ 2

The mean energy of the emitted photon is given by:$$\langle \hbar \omega \rangle ={\frac {4}{5{\sqrt {3}}}}\chi \gamma mc^{2}$$which means that having relativistic particles ($\gamma \gg 1$) and intense fields increases the chance of producing high-energy photons. $\zeta \sim \chi$ because of this formula.

• The effect of radiation reaction, due to photon recoil . The electron energy after the interaction process reduces because of the energy delivered to the emitted photon, this effect is not taken into account in non-linear Thomson scattering in which the electron is supposed to remain more or less unaltered in energy such as in elastic scattering. Quantum radiation reaction effects become important when the emitted photon energy approaches the electron energy. Since $\chi \sim \zeta \sim \hbar \omega /(\gamma mc^{2})$ , if $\chi ,\zeta \ll 1$ the classical limit of NICS is a valid description, while for $\chi ,\zeta \sim 1$ the energy of the emitted photon is of the order of the electron energy and photon recoil is very relevant.
• The quantization of the motion of the electron . The quantization of the electron motion can be roughly estimated by $\hbar \omega _{0}/\gamma mc^{2}$ where $\omega _{0}=c/\rho$, $\hbar \omega _{0}$ expresses the interval between adjacent levels for an electron in circular motion . This ratio goes as $\chi /\gamma ^{3}$ and is very small for a relativistic electron, meaning that the electron motion is quasi-classical. However, increasing $\chi$ the effects of quantization start to be more relevant.
• Non-linear quantum effects, like the production of an electron-positron pair in vacuum. They occur above the critical field $E_{s}$, however, they can be observed also well below that limit since ultra-relativistic particles, with gamma factor equal to $E_{s}/|\mathbf {E} |$, see, in their rest frame, fields of the order of $E_{s}$. $\chi$, as a measure of the magnitude of non-linear quantum effects, is called also non-linear quantum parameter.

An accurate description of non-linear inverse Compton scattering is made considering the electron dynamics described with Dirac equation in presence of an electromagnetic field. Since the incoming field is intense enough, this description is completely equivalent to necessary to the interaction of electron with multiple photons, with no need of explicitly quantize the electromagnetic field of the incoming low-energy radiation. Then the interaction with the radiation field, 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 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 is quite general. The spectrum of non-linear inverse Compton scattering, obtained with this approach, is:

$${\dfrac {d^{2}N}{d\eta dt}}={\dfrac {\sqrt {3}}{2\pi }}{\dfrac {q^{2}mc}{\hbar ^{2}}}{\dfrac {\chi }{\gamma \eta }}F(\chi ,\eta )\quad {\text{where}}\quad F(\chi ,\eta )={\dfrac {\eta ^{2}}{\chi ^{2}}}yK_{\frac {2}{3}}(y)+\left(1-{\dfrac {\eta }{\chi }}\right)y\int _{y}^{\infty }K_{\frac {5}{3}}(x)dx$$ 3

where the parameter $y$, is now defined as:$$y={\dfrac {2\eta }{3\chi (\chi -\eta )}}={\dfrac {2\zeta }{3\chi (1-\zeta )}}$$The result is very similar to the classical one except for the different expression of $F$. For $\chi ,\zeta \to 0$ it reduces to the classical spectrum (2) , see Figure . Note that if $\zeta \geq 1$ ($\eta \geq \chi$ or $y<0$) 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 $(\gamma -1)mc^{2}$.

The total power emitted in radiation is given by the integration in $\eta$ of the spectrum (3) :$$P={\dfrac {2}{3}}{\dfrac {e^{2}m^{2}c^{3}}{\hbar ^{2}}}\chi ^{2}g(\chi )$$where the result of integration of $F(\chi ,\eta )$ is contained in the last term$$g(\chi )={\dfrac {3{\sqrt {3}}}{2\pi \chi ^{2}}}\int _{0}^{+\infty }F(\chi ,\eta )d\eta ={\dfrac {9{\sqrt {3}}}{8\pi }}\int _{0}^{+\infty }\left[{\dfrac {2y^{2}K_{\frac {5}{3}}(y)}{(2+3\chi y)^{2}}}+{\dfrac {36\chi ^{2}y^{3}K_{\frac {2}{3}}(y)}{2+3\chi y)^{4}}}\right]dy$$The expression is equal to the classical one if $g(\chi )$ is equal to one and it can be expanded in two limiting cases, near the classical limit and when quantum effects are of huge importance:$${\begin{cases}P\approx {\dfrac {2}{3}}{\dfrac {e^{2}m^{2}c^{3}}{\hbar ^{2}}}\left(1-{\dfrac {55{\sqrt {3}}}{16}}\chi +48\chi ^{2}\right),&{\text{for }}\chi \ll 1\\P\approx 0.37{\dfrac {e^{2}m^{2}c^{3}}{\hbar ^{2}}}(3\chi )^{\frac {2}{3}},&{\text{for }}\chi \gg 1\end{cases}}$$A related quantity is the rate of photon emission:$${\dfrac {dN}{dt}}={\dfrac {\sqrt {3}}{2\pi }}{\dfrac {q^{2}mc}{\hbar ^{2}}}{\dfrac {\chi }{\gamma }}\int _{0}^{\chi }{\dfrac {F(\chi ,\eta )}{\eta }}d\eta$$where it is made explicit that the integration is limited by the condition that if $\eta \geq \chi$ no photons can be produced. This rate of photon emission depends explicity on electron quantum parameter and on the Lorentz factor for the electron.

History

The phenomenon was first studied theoretically around 1960-1970 in different papers by A.I. Nikishov and V.I.Ritus in the case of an electron scattering off high-energy photons in the field of an intense electromagnetic plane wave. At that time the names non-linear inverse Compton scattering and multiphoton Compton scattering were not in use yet since they emerged only later. After the work of Nikishov and Ritus, others studied the same process but considered more complex electromagnetic fields such as those of finite spatial and temporal extension that characterize a laser pulse.

The advent of chirped pulse amplification and other laser amplification techniques allowed to reach sufficiently high laser intensities to significantly observe the phenomenon and its most peculiar quantum effects.

In 1983, the classical and non-relativistic limit of NICS (multiphoton Thomson scattering) was first observed in the collision of a $1$ keV electron beam with a Q-switched Nd:YAG laser delivering an intensity of $1.7\cdot 10^{14}$ W/cm² ($a_{0}=0.01$), and producing a photon of frequency two times the one of the laser , and then in 1998 in the interaction of a mode-locked Nd:YAG laser ($4.4\cdot 10^{18}$ W/cm2, $a_{0}=1.88$) with plasma electrons from an helium gas jet producing multiple harmonics of the laser frequency . NICS with quantum effects was detected for the first time in a pioneering experiment at the SLAC National Accelerator Laboratory at Stanford University, USA. In this experiment an ultra-relativistic electron beam with energy of about $46.6$ GeV collided with a terawatt Nd:glass laser with an intensity of $10^{18}$ W/cm² ($a_{0}=0.8$, $\chi =0.3$) and NICS was observed indirectly via a nonlinear energy shift in the spectrum of the outgoing electrons evidencing the absorption of up to four laser photons, 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.

In 2000s the study of NICS in an all-optical setup has started. In this case, a laser pulse is both responsible for the electron acceleration, through the mechanism of plasma acceleration, and for the non-linear inverse Compton scattering occurring in the interaction of accelerated electrons with a laser pulse (possibly counterpropagating with respect to electrons).

Applications

Non-linear inverse Compton scattering is an interesting phenomenon for all applications requiring high-energy photons since NICS is capable of producing photons with energy comparable to $mc^{2}$ and higher. In the case of electrons, this means that it is possible to produce photons with MeV energy that can consequently trigger other phenomena such as pair production, Breit-Wheeler pair production, Compton scattering, vacuum polarization.

In the context of laser-plasma acceleration, both relativistic electrons and laser pulses of ultra-high intensity can be present setting favourable conditions for the observation and the exploitment of non-linear inverse Compton scattering for diagnostic purpose, for high-energy photon production and for probing non-linear quantum effects and and non-linear QED. Because of this reason in the 2010s, particle-in-cell codes used to study laser-plasma acceleration have been developed with the capabilities of simulating non-linear inverse Compton scattering with Monte Carlo methods. These tools are used to explore the different regimes of NICS in the context of laser-plasma interaction.

See Also

• Compton scattering
• Synchrotron radiation
• Breit–Wheeler process
• Quantum electrodynamics
• Laser