![NICS picture](https://upload.wikimedia.org/wikipedia/commons/8/88/NICS_picture.png)
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
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...
or
gamma ray
A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation arising from the radioactive decay of atomic nuclei. It consists of the shortest wavelength electromagnetic waves, typically ...
) during the interaction with a
charged particle
In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary pa ...
, in many cases an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
.
This process is an inverted variant of
Compton scattering
Compton scattering, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a charged particle, usually an electron. If it results in a decrease in energy (increase in wavelength) of the photon ...
since, contrary to it, the charged particle transfers its
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
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
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
.
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 invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
.
The normalized
vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a ''vecto ...
helps to isolate the regime in which non-linear inverse Compton scattering occurs (
is the electron charge,
is the electron mass,
the speed of light and
the vector potential). If
, the emission phenomenon can be reduced to the scattering of a single photon by an electron, which is the case of
inverse Compton scattering
Compton scattering, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a charged particle, usually an electron. If it results in a decrease in energy (increase in wavelength) of the photon ...
. While, if
, 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,
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
keV electron beam colliding with a
Q-switched Nd:YAG laser delivering an intensity of
W/cm
2 (
), photons of frequency two times the one of the laser were produced, then in 1995 with a CPA laser of peak intensity around
W/cm
2 interacting with neon gas, and in 1998 in the interaction of a mode-locked Nd:YAG laser (
W/cm
2,
) 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
SLAC National Accelerator Laboratory, originally named the Stanford Linear Accelerator Center,
is a United States Department of Energy National Laboratory operated by Stanford University under the programmatic direction of the U.S. Departme ...
National Accelerator Laboratory at Stanford University, USA. In this experiment, the collision of an ultra-relativistic electron beam, with energy of about
GeV, with a terawatt
Nd:glass laser, with an intensity of
W/cm
2 (
,
), 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
Plasma acceleration is a technique for accelerating charged particles, such as electrons, positrons, and ions, using the electric field associated with electron plasma wave or other high-gradient plasma structures (like shock and sheath fields). ...
, 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
to
keV with a Ti:Sa laser beam (
W/cm
2).
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
Synchrotron radiation (also known as magnetobremsstrahlung radiation) is the electromagnetic radiation emitted when relativistic charged particles are subject to an acceleration perpendicular to their velocity (). It is produced artificially in ...
driven by the force exerted on a charged particle by intense
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic fields.
Practically, a moving charge emits
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
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
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
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
, mass
and velocity
.
In a simplified picture, one can suppose a local circular trajectory for a
relativistic particle
A relativistic particle is a particle which moves with a relativistic speed; that is, a speed comparable to the speed of light. This is achieved by photons to the extent that effects described by special relativity are able to describe those of su ...
and can assume a
relativistic centripetal force equal to the magnitude of the perpendicular Lorentz force acting on the particle:
and
are the electric and magnetic fields respectively,
is the magnitude of the electron velocity and
is the
Lorentz factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
.
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
can be substituted with the speed of light to simplify the expression for
.
Given an expression for
, the model given in
Example 1: bending magnet 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:where
V/m is the
Schwinger field. In scientific literature,
is also called
.
The Schwinger field
, appearing in this definition, is a critical field capable of performing on electrons a work of
over a reduced
Compton length , where
is the reduced
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
.
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
W/cm
2.
Consequently,
represents the work, in units of
, performed by the field over the Compton length
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
, however, they can be observed also well below this limit since ultra-relativistic particles with Lorentz factor equal to
see fields of the order of
in their rest frame.
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
In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
:
The four-force acting on the particle is equal to the derivative of the
four-momentum
In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
with respect to
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
.
Using this fact in the classical limit, the radiated power according to the relativistic generalization of the
Larmor formula
In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.
When any charged ...
becomes:
As a result, emission is improved by higher values of
and, therefore, some considerations can be done on which are the conditions for prolific emission, further evaluating the definition (). The electron quantum parameter increases with the energy of the electron (direct proportionality to
) 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:
where
is the
wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
of the plane wave and
the wavevector magnitude. Inserting this expression in the formula of
:
where the vectorial identity
was used. Elaborating the expression:
Since
for a plane wave and the last two terms under the square root compensate each other,
reduces to:
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.
* Inclusion of the discretization of the emitted radiation, i.e. the introduction of photons with respect to the continuous description of the classical limit.
This effect does not change quantitatively the emission features but changes how the emitted radiation is interpreted.
A parameter equivalent to
can be introduced for the photon of frequency
and it is called photon quantum parameter:
where
is the photon
four-wavevector and
is the three-dimensional wavevector. In the limit in which the particle approaches the speed of light, the ratio between
and
is equal to:
From the
Frequency distribution of radiated energy one can get a rate of high-energy photon emission distributed in
as a function of
and
but still valid in the classical limit:
where
stands for the
McDonald functions. The mean energy of the emitted photon is given by
. Consequently, a large Lorentz factor and intense fields increase the chance of producing high-energy photons.
goes as
because of this formula.
* The effect of
radiation reaction
In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes:
* ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visi ...
, due to photon recoil''.
''
The electron energy after the interaction process reduces because part of it is delivered to the emitted photon and the maximum energy achievable by the emitted photon cannot be higher than the electron
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
. This effect is not taken into account in non-linear Thomson scattering in which the electron energy is supposed to remain almost unaltered in energy such as in
elastic scattering
Elastic scattering is a form of particle scattering in scattering theory, nuclear physics and particle physics. In this process, the kinetic energy of a particle is conserved in the center-of-mass frame, but its direction of propagation is modif ...
. Quantum radiation reaction effects become important when the emitted photon energy approaches the electron energy. Since
, if
the classical limit of NICS is a valid description, while for
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 and spin effects''.
''
An accurate description of non-linear inverse Compton scattering is made considering the electron dynamics described with the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
in presence of an electromagnetic field.
Emission description when and
![Fplot](https://upload.wikimedia.org/wikipedia/commons/f/f3/Fplot.png)
When the incoming field is very intense
, 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
and
, is:
where the parameter
, is now defined as:
The result is similar to the classical one except for the different expression of
. For
it reduces to the classical spectrum (). Note that if
(
or
) 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
.
The total power emitted in radiation is given by the integration in
of the spectrum ():
where the result of the integration of
is contained in the last term:
This expression is equal to the classical one if
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:
A related quantity is the rate of photon emission:
where it is made explicit that the integration is limited by the condition that if
no photons can be produced.
This rate of photon emission depends explicitly on electron quantum parameter and on the Lorentz factor for the electron.
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
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,
nuclear reactions
In nuclear physics and nuclear chemistry, a nuclear reaction is a process in which two nuclei, or a nucleus and an external subatomic particle, collide to produce one or more new nuclides. Thus, a nuclear reaction must cause a transformation o ...
.
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 exploitation of non-linear inverse Compton scattering for high-energy photon production, for diagnostic of electron motion, and for probing non-linear quantum effects and non-linear QED.
Because of this reason, several numerical tools have been introduced to study non-linear inverse Compton scattering.
For example,
particle-in-cell
In plasma physics, the particle-in-cell (PIC) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles (or fluid elements) in a Lagrangian frame are tracked in continuous ph ...
codes for the study of laser-plasma acceleration have been developed with the capabilities of simulating non-linear inverse Compton scattering with
Monte Carlo methods
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
.
These tools are used to explore the different regimes of NICS in the context of laser-plasma interaction.
See also
*
Compton scattering
Compton scattering, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a charged particle, usually an electron. If it results in a decrease in energy (increase in wavelength) of the photon ...
*
Synchrotron radiation
*
Breit–Wheeler process
*
Quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
*
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
References
External links
High-energy photon emission & radiation reaction in the PIC code SMILEI- Example of particle-in-cell code with a module for NICS simulation.
- Example of research activity on NICS.
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