On the similarity between Thomson scattering from high-intensity circularly polarised lasers and synchrotron radiation

The mathematical similarity between Thomson Scattering (TS) and Synchrotron Radiation (SR) phenomena is elucidated, and some physical consequences are extracted from it. Analytical computations are reported in the frequency- and the time-domains so as to make this relationship more compelling. Algebraic equations for the relevant parameters implementing that similarity are obtained in the time-domain in their final form. Subsequently, those algebraic equations are numerically solved so as to make the connection available for a wide range of parameters in both phenomena. Prospects for a compact vacuum ultraviolet (VUV) or even x-ray source based on this analogy and using the interaction of high-intensity circularly polarized laser radiation with essentially monoenergetic electron beams are discussed.


Introduction and motivation
The scattering of electromagnetic radiation by free electrons, i.e. Thomson Scattering (TS), is at the basis of one of the most powerful diagnostics in fusion plasmas, being routinely used to obtain electron temperature and density with both spatial and temporal resolution [1][2][3]. TS is also a key ingredient in new (W7X, with first plasmas in 2015) and future (ITER) plasma devices. In particular, electron temperatures in excess of 40keV are expected in ITER when fully operational, and that will pose substantial experimental challenges due (in part, but not only) to the large contribution of relativistic effects, see for example [4].
Linear, incoherent TS is well documented, see [5][6][7][8][9][10] and references therein. By linear it is meant that the electron trajectory is very weakly perturbed by the laser, while incoherent TS means the electrons scatter the electromagnetic radiation independently of one another, so collective effects of electron motion in the computation of the scattered spectrum can be disregarded.
However, if radiation from ultra-high intensity lasers impinges upon the electron(s), the scattering becomes nonlinear and interesting new effects (harmonic generation, for instance) appear that are beyond conventional treatments of this effect. The computation of nonlinear TS (and, in general, of nonlinear radiation emission) from accelerated electrons has been addressed by several authors, using a variety of methods: analytic or semianalytic computations, Particle-in-Cell codes (PIC), or by following electron trajectories and subsequent computation of the Liénard-Wiechert radiated fields, for applications ranging from the physics and diagnosis of relativistic beams to high-energy astrophysical phenomena. These developments point to the lively and timely interest of this problem, in the context of new and powerful radiation sources (ultra-intense lasers) and their interaction with relativistic electrons. A substantial body of literature has accumulated on these subjects, see for example [11][12][13][14][15][16][17][18][19][20][21][22][23].
In previous papers the authors have developed Monte Carlo and analytical methods for the ab-initio computation (in a classical framework, disregarding quantum effects) of TS spectra under very general conditions regarding the electron distribution function, as well as the intensity and the polarization of the incoming laser radiation, for plasma fusion diagnostic applications [24][25][26][27][28]. In the course of these investigations,

Towards the TSC-HSR similarity: frequency-domain analytical computations
In this section the mathematical formalism used to compare both phenomena in the frequency-domain is presented; it will provide the basis for the comparison in the time-domain in section 3. The steps to be taken closely follow the results for TS in [24][25][26][27]. Excellent monographs on classical electrodynamics and SR exist, which have been instrumental at the starting level of the present research: see, for example, [29,30,35,36].

Helical electron trajectory for HSR
In the computation of the radiated magnetic field for HSR, the following assumptions will be made. The electron trajectory is helical, and its projection in a plane perpendicular to the applied constant magnetic field B 0 is a circle of radius r 0 . The helix axis is defined by the unit vector For HSR, t′ will denote in what follows the 'particle time' (to be clearly distinguished from the 'detector time' or 'retarded time at detector' denoted by t when needed). The electron trajectory and its velocity are given by: < . In figure 1, the trajectory and scattering geometry considered in this paper for HSR are shown in sketch.

Frequency-domain radiated fields at detector position for HSR
The radiated field is computed far away from the electron orbit, at a point/detector located at distance D 0 ?r 0 from the origin, and lying in the X-Z plane. The scattering angle is denoted 0 q ¢ , hence n 0 = i k sin cos 0 0 q q ¢ + ¢ˆi s the scattering vector, assumed to be constant under the above general assumptions, and D y n Only the radiation part of the Liénard-Wiechert emitted fields is computed according to (e and ò 0 are the electron charge and the dielectric permittivity of vacuum, respectively): A priori, there could be one subtlety, namely, one could argue that the contribution of c t p b ¢ in t x SR ¢ ( ) (in the exponential inside the integral in equation (7)) could be larger than D 0 , for suitably large t¢ | |, so that the reliability of equation (7) would be in question. However, as n 1 0 , the contributions due to for large positive T, due to cancellations arising from oscillations in Figure 1. Helical trajectory is depicted in red, and its projections orthogonal to the constant magnetic field (chosen here along the Zaxis, in purple) are the circular dotted green curves. Blue lines represent X and Y axis respectively, while the black line represents a scattering (n 0 ) vector in the X-Z plane.
the integrations over t′, can be regarded as negligible compared to other contributions, thereby supporting a posteriori the physical validity of equation (7). The spectral amplitude is clearly perpendicular to the n 0 vector. The vector product n On the other hand, by using equations (1), (2), (3) and (4), we have: where we have introduced the new (Doppler) frequency Ω D for HSR: and a SR;c , a SR;s , a SR and the angle j sr : a r c a n n cos sin sin 11 SR c SR sr ; a r c a n n sin cos cos 12 SR s SR sr ; Alternative expressions for Ω D , a SR;c and a SR;c and the modulus of B 0 are given in appendix A. Let us introduce the unit vector: u 0 lies in the X-Z plane and is perpendicular to ĵ and to the scattering vector n 0 . Since the radiated magnetic field is perpendicular to n 0 , it can be projected onto two orthogonal quadratures along (among other choices) the unit orthogonal vectors ĵ and u 0 . Accordingly: The scattering unit vector n indicates the direction at which a detector is located at y: n i k sin cos 0 0 q q = +ˆand R n y 1 = -. In turn: x x = ¢ -( ) , t′ being the 'radiation' time for the electron; p(ξ) is the momentum of the radiating electron, also following from those dynamical equations [24,25]. x 1 (ξ), x 2 (ξ), x 3 (ξ) and p(ξ) are collected in appendix C. The quantities γ 1 , f 1 and f 2 , employed below, are integration constants of the dynamical equations of motion [24,25]. Λ (ξ) C can be recast as: sin 18 Like for HSR, we introduce the unit vector in the X-Z plane, perpendicular to both ĵ and n: We shall project the spectral amplitude B y, TS w ( ) on the orthonormal vectors ĵ and u: Notice that ω 0 ξ is dimensionless and that we have approximated exp[iω((R/c)+(g 0,C /ω 0 ))];exp[iω(R/c)] in the quadratures given in equations (25) and (26) We have scaled all ωdependent terms (for the sake of the HSR-TSC comparison), so that they depend on g C ) . In order to overcome the latter limitation, we could introduce two new orthonormal unit vectors j¢ and u¢ which are orthonormal to each other and to n. We shall omit the specific expressions for j¢ and u′, which would require a suitable rotation angle in the ĵ-u plane. As before, we could project the spectral amplitude B y, TS w ( ) on the orthonormal vectors j¢ and u′: where one now finds a contribution accompanying the sin 0 w x ( )term in u B y, TS w ¢ ·˜( ) . We shall omit for brevity the specific expressions for the ξ-independent amplitudes A TS j o , ; are given in appendix D. In the following developments, we shall make use of B j T S w ¢ ( ) and B u T S w ¢ ( ) , as they are more general. However, upon proceeding to numerical computations (in the time-domain), we shall restrict to

The mathematical basis of the TSC-HSR similarity: time-domain analytical computations and comparison
In this section the mathematical formalism used to compare both phenomena in the time-domain is presented. Time-domain analytical computations of fields for HSR and TSC have been operational to first suggest the connection investigated in this paper.
Using the standard expansion [37] ix il J x exp sin exp , 28 Next, we shall apply the inverse Fourier transform so as to proceed to the time-domain, namely, The resulting formulae for HSR and for TSC in the time-domain are given in the following two subsections, omitting intermediate computations.

Time-domain radiated fields at detector position for HSR
One obtains the following explicitly real expansions into harmonics ( a D c ,

Time-domain radiated fields at detector position for TSC
We shall present the time-domain radiated field, for the general choice j j ¢ 1ˆand u u ¢ ¹ , as a spin-off to the calculations in the frequency domain. One gets the following explicitly real expansions into harmonics being the Doppler frequency for TSC): 3. Time-domain: comparison of radiated fields for HSR and TSC in the general case (j ĵ¢ ¹ and u u ¢ ¹ ) We shall now impose that the above projections of the spectral HSR and TSC fields have entirely similar functional forms and variations in the time-domain and we shall find the conditions that warrant those similarities. Specifically, we shall impose that the time-domain radiated field for HSR, as a function of tΩ D , and the one for TSC, as a function of t An inspection of the arguments of the various Bessel functions leads to the necessary condition (equivalent to ζ=ζ′): The two phase angles j sr and j c differ from each other ( sr c j j ¹ ) in general, although in special cases it could well be that sr , ; , ; We emphasize the role of the properties, summarized in appendix E, of the Bessel functions: they have led to the structures in the various equations above and, in particular, to the fact that the conditions warranting the TSC-HSR connection be independent on the J l (x)ʼs and hold for any harmonic (l).
We shall also impose that the ratio in equation (34) has the same value as the ratio in equation (35) for the general case j j ¢ 1ˆand u u ¢ ¹ ,as it is physically natural that those ratios be the same for both projections upon formulating the TSC-HSR relationship. And, once the later condition has been imposed, it will be useful to compute the common value of those ratios, which will give rise to another equation. All in all, we are left with a total of five equations as the basic framework for the TSC-HSR relationship.
In section 3.5, we shall treat (in addition to equation (33)) equations (34) and (35) And, in so doing, we shall also write explicitly the condition that the ratio in equation (34) has the same value as the ratio in equation (35) as well as the equation which determines the latter ratio (denoted below as 1/g 2 ). See equations (36) and (41).

Comparison of frequency-domain radiated fields for HSR and TSC: a short discussion
Let the conditions that warrant the similarities of the projections of the spectral HSR and TSC fields in the timedomain be imposed, namely, equations (33)- (35) and the additional equation expressing the equality between (34) and (35). Then, one can apply the Fourier transform to proceed back from the time-domain to the frequency one (namely, ), that is, to equations (29)- (32). Then, j B y , SR 0 ŵ ·˜( ) as a function of ω/ Ω D has the same functional structure as j B y, TS w ¢ ·˜( ) has as a function of g C 1, 0 w w , and the same holds for ) . We emphasize here the difficulties involved in trying to arrive at the above conditions (equations (33)- (35) and so on) directly in the frequency-domain. In fact, inspections of j B y , SR (p 0 is the modulus of the initial momentum p(ξ=0) of the electron, from appendix C), and after a little algebra equations (34) and (35) can be recast into the following explicit dimensionless equations that connect TSC and HSR: cos cos cos cos sin sin sin sin cos 38 α is the so-called laser parameter for TSC, given in appendix D, together with θ and f (for the initial momentum p(ξ=0) of the electron). Notice that we have imposed directly that the ratio in equation (36) has the same value (namely, 1/g 2 ) as the ratio in equation (41), for the actual case j j ¢ =ˆand u u ¢ = . The numerical computations will be based upon equations (33), (36) and (41). By starting out from the parameters characterizing one phenomenon, these five equations enable, in principle, to find the corresponding parameters in the other phenomenon giving a similar pattern of radiated fields (in the sense previously discussed), and conversely.

Numerical computations backing up the analytical results
In this section the approach to the numerical solution of the algebraic equations in section 3.5 is presented. Most runs have as a target a TSC case (i.e., a set-up consisting of an electron of given momentum interacting with a large intensity laser), and then find the corresponding parameters in HSR giving the same pattern of radiated fields. The equations work in either direction however, and it is possible to begin with a set of HSR parameters to find the corresponding ones in TSC; some examples will be provided in section 5. Hundreds of different runs have been performed, varying the initial kinetic energy of the electron, the orientation of its momentum with respect to the laser propagation direction, and the laser parameter over wide ranges. The conclusion is that those algebraic equations have indeed a solution that can be efficiently computed to a high degree of accuracy. In some cases, the numerical computation has suggested an Ansatz for exact analytical solutions to the equations. We will also address here this aspect of the problem.
Before presenting a selection of cases backing up the connection between TSC-HSR, let us point out that if we just solve equations (33), (36) and (41) (five equations in total) one can find multiple solutions to them, differing (among others) in the relative Doppler frequencies at detector, namely, in Ω D /Ω 0 for HSR and ω D /ω 0 for TSC. This freedom/indeterminacy of solutions will allow to extend the potential physical applications, because it would be possible to connect both phenomena over largely different frequency regimes. Those aspects will also be addressed in section 5, and summarized in the Conclusions.
The algebraic equations have been numerically solved using the Mathematica® package, in particular the intrinsic function/procedure Nminimize [38]. Hence, the problem of finding a numerical solution to the algebraic equations is transformed into a minimization problem, namely, finding the minimum of a sum of square terms, each of them of the form w g N , , , ) . λ j are the target values (the denominators of equations (36) and (41)) and N , , , ) are the numerators of the corresponding equations as functions of the parameters to be found (n p denoting the number of parameters). In the case of TSC target values, the HSR parameters we are looking for would be the parallel and perpendicular velocities β p , β 0 , the angles ψ 0 , f 0 giving the orientation of helix axis (magnetic field) and the initial position of electron with respect to it (that is, r 0 and δ 0 ), and the scattering angle θ 0 '. An equivalent set of HSR parameters turns out to be the kinetic energy E SR,kin of the electron, the fraction of parallel to perpendicular velocities, Frac=β p /β 0 , and the set of angles, together with r 0 , mentioned above. The 'extra' parameter g 2 in the minimination is always found to be numerically equal to the initial Lorentz γ factor of the electron (namely, ) and hence is not an independent one.
The problem is reduced to finding the minimum of w g N , , , vary over given ranges. w j are weights that control the relative importance of the different terms in the minimization process. The weights associated to equations (33), (36) and (41) (five equations in total) have always been set to 1.0, while the weight associated to the relative Doppler frequency at detector has been set either to 0.0 or 1.0 (a value of 0.0 for a weight means of course that this particular condition is not included in the minimization for that particular run). In the large majority of cases that have been tested, the final value for the weigthed sum is found to be ≈10 −15 or even less, meaning that minimization conditions have simultaneously been met to better than seven significant digits.
Case-study number 1: The general conditions for the comparison TSC-HSR are summarised in table 1. The value for the weighted sum of squares after minimization is 6.8×10 −16 . In this run the equality of normalised Doppler frequencies at detector has been imposed, i.e., the corresponding weight has been set to 1.0. Figure 2 gives the corresponding radiated fields in (normalised) time domain, and the plot of j-quadrature versus uquadrature. Taking into account that 'standard' synchrotron radiation set-up (with β p =0.0) gives necessarily , a substantial β p is needed to meet a target value g 1 1.576 04 . Figures 3, 4 give the corresponding radiated fields in time ) is possible by a (simultaneous) change of electron energy, helical trajectory parameters, share of parallel to perpendicular velocity and scattering angle, and this large freedom in fitting different observed Doppler frequencies opens the possibility of modeling some phenomena associated with synchrotron radiation at other frequency range(s) using the connection TSC-HSR. Also in this case, the match between both phenomena (figures 3, 4) is excellent; 16 harmonics have been added to compute the quadratures.
Case-study number 3: Let us now present two examples of numerical computations which lead to obtain much simpler solutions to the algebraic equations. This is interesting at least for two reasons: on the one hand an explicit analytical solution to the full TSC-HSR connection can be offered at least for some combinations of parameters, and on the other hand, the solutions obtained can benchmark the numerical techniques used, increasing our confidence on their correctdness under more challenging conditions. The cases to be studied now require that f=3π/2, and a sufficiently high initial kinetic energy such that the condition p 2 0 y 0 a p + = (equivalent to j c =−π/2) is fulfilled. In such a case, a HSR solution exists at exactly the same energy and scattering angle, and with ψ 0 =f 0 =δ 0 =0. The additional condition that the arguments inside Bessel functions be the same in TSC and HSR, fully solves the problem by providing the fraction of parallel to It is also possible to prove that in those cases, a scattering angle exists θ * , given by the condition p tan 2 z 0 * * q a pz q = ( ( ) ), at which the j-quadratures of the emitted B-field exactly vanishes, and then the radiated field is fully linearly polarised as is standard SR when measured in the plane of the electron(s) orbit. See appendix F for details. The target parameters for TSC illustrating the above mentioned results are given in table 3; figures 5, 6 show the field quadratures in time domain when 32 harmonics are added. It is perhaps interesting to mention that if 3 2 f p ¹ , but the condition j c =−π/2 can still be met,  Case-study number 4: To end this section, let us present some numerical computations that are inspired in, and connect with [34]. In the latter work, the authors report on high-order nonlinear TS for a set-up in which high energy electrons (up to 200MeV) counterpropagate against a ultrahigh-intensity laser, and the corresponding scattering is also measured in a (basically) backscattering geometry. We shall show here 3 examples where the electron is counterpropagating against the laser (hence θ=π, f=0), with scattering angle θ 0 =π, laser parameter α=4π and initial kinetic energies for the electron successively equal to 250.0 keV, 2.5 MeV and 25.0 MeV. The results are summarised in table 4 and figure 7.   Figure 5. Comparison between the radiated TS fields (a) and SR fields (b). Apart from a dilation/contraction in time axis, readily accounted for through the corresponding Doppler effects, the temporal shape of both quadratures in HSR is exactly the same as in TSC.

Physical applications suggested by the relationship between TSC-HSR
The relationship between nonlinear TS from circularly polarised high-intensity lasers and HSR suggests some physical applications that we shall briefly mention below. It is not the aim of this paper to fully develop them, but the tools needed to do so, both in frequency and time domains, are at hand. For completeness, a brief comment on the interaction of high-intensity circularly polarized lasers with fusion plasmas is included, although in that case the emphasis is not placed specifically on the TSC-HSR relationship.
(a) The laser synchrotron light source. When the condition p 2 0 y 0 ), the field quadratures radiated by TSC have the expected symmetry of SR for an electron orbit lying in the X-Y plane, the radiation being measured at a point out of the orbital plane in general. Usually F 0 ) is needed to obtain quadrature matching, but the analogy can be made closer by explicitly looking for solutions of the algebraic equations having F=0, see table 5 and figure 8 for a typical example. If, in addition, f=3π/2 holds then an angle θ * exists, given by the condition p tan 2 z 0 * * q a pz q = ( ( ) ), at which the j-quadratures of the emitted B-field exactly vanishes ( p 2 0 y 0 for a certain range of f angles, but the condition to obtain the vanishing of the j-quadrature is more complicated). The radiated field is then fully linearly polarised as is the standard SR measured in the plane of the electron(s) orbit. The spectral content of the emitted fields can be tuned by a combination of laser wavelength/intensity tuning and/or electron energy tuning: see, for instance, the case reported in table 6 and figure 9. In summary, varying E SR kin , , α, θ, f, 0 * q , a tunable radiation source is feasible, based on circularly polarised lasers interacting with (essentially) monoenergetic electrons, having the same properties as regards to its spectrum, spatial distribution of power, polarization, etc., as SR, provided that the magnitude and orientation of the initial momentum be adequately chosen.
(b) Learning the basics of SR from table-top laser experiments. A compact laser/electron beam set-up can be used to learn the basic physics of SR. Synchrotrons radiation properties can be (partly) modeled and characterised in more comfortable optical/UV range(s), provided that the laser wavelength, its intensity, and the energy of the electron beam be suitably chosen. Of course, that set-up would not be an alternative to those critical and very important modern applications of synchrotrons making use of very energetic electrons (in the range of a few GeV to ≈10 GeV). Figure 6. Comparison between the radiated TS fields (a) and SR fields (b). Apart from a dilation/contraction in time axis, readily accounted for through the corresponding Doppler effects, the temporal shape of both quadratures in HSR is exactly the same as in TSC. Table 4. HSR parameters fitting a given TSC run, see main text for details.
TSC/HSR parameters (target/found)    field, and the theory developed in this paper should apply as to the computation and basic properties of the emitted radiation. Other subtle effects in gyrotron physics, as the role of the resonant cavity, or the co-ordinated movement of charges, which gives rise to the output coherent radiation, lie beyond our scope here. Figures 10, 11 give an example of electrons of 75 keV kinetic energy, with F(≡(β p /β 0 ))=0.5, and moving in a magnetic field B 0 =1 T. Larmor radius and Larmor frequency are computed according to the well-known formulae ( In this particular case, and for 2 0 q p ¢ = , two TSC configurations giving rise to quadrature matching are given. The difference among the two cases is that in figure 10, the relative Doppler frequencies are matched (to   ) . The interacting configuration of electron beam/laser in this case could be called an 'optical gyrotron' since the quadratures follow the same pattern as in its microwave counterpart. Obviously, the frequency ranges in which energy is emitted are quite different in one case from the other, but due to the freedom in choosing the matching relative frequency, the output frequency of the optical gyrotron can be (widely) tuned, an interesting feature to explore in more detail. The capability of tuning the gyrotron frequency shot to shot or in real time would be a possible break-through in electron cyclotron resonance heating in fusion plasmas.
(d) The interaction of high-intensity circularly polarised laser radiation with fusion plasmas. Fusion plasmas are currently probed by a host of active and passive diagnostics in order to ascertain their properties. One of the most important diagnostics to measure electron temperature and density is TS from powerful lasers, probing the electron distribution function through the measurement of the scattered spectrum. To the best of our knowledge, no attempt has been made to probe fusion plasmas with ultra-intense circularly polarised lasers, either with diagnostic purposes, or to obtain from them a light source with the special properties predicted from the nonlinear effects. Numerical work presented by the authors in [24,25,27] suggest that using an ultra-intense laser could shift the emitted spectrum to the red, and hence spectral zones that are normally unaccessible due to practical considerations, like the one around the laser wavelength, could now be directly observed, increasing the diagnostic capability. The predicted harmonics to the (bulk) scattered spectrum would also be of potential diagnostic use. Experiments like the one reported in [34] make fusion plasmas a natural candidate for further study.
(e) Geophysical/astrophysical applications. Short portions of the trajectories of charged particles on the Earth or planetary magnetospheres are well modeled by the helicoidal trajectories considered in this paper. For suitable energy ranges of the corresponding particles and local magnetic fields in which they are moving, it could be possible to find the corresponding parameters of a TSC laser experiment that would model that behaviour, and predict their electromagnetic emission.
Also, the emission from high-energy electrons in pulsars having magnetic fields substantially larger than those achieved under laboratory conditions seems a good candidate for study. In fact, SR from charges spiraling in the magnetic fields of pulsars has been considered as a possible mechanism explaining (part of) their rich phenomenology [31,32,39]. For those objects, the far-field approximation made in this paper for the emitted radiation would be a very good one, even for particles that travel along the magnetic field for substantial Figure 9. Comparison between the radiated TS fields (a, c) and SR fields (b, d) for the parameters referred to in table 6. It is apparent from the figure that harmonic content and signal amplitude can be tuned/changed by changing either the α parameter and/or the initial kinetic energy. 128 harmonics (panels (a), (b)) or 256 harmonics (panels (c), (d)) have been added to produce the B-field quadratures in real time.
distances. Depending on the share of parallel to perpendicular velocity of charged particles in the pulsar magnetosphere, some of the most energetic radiation processes taking part in them could be modeled by TSC experiments under conditions that could be achieved with current laser technology. The theoretical analogy developed in this paper can be easily extended to other charged particles: it could be applied to the emission of, say, ultra-energetic protons in pulsar magnetospheres, and one could speculate about a corresponding TSC experiment capturing the characteristics of such emission.

Conclusions
It is proven that Thomson Scattering from ultrahigh-intensity circularly polarised lasers (TSC) interacting with (essentially) monoenergetic electrons, and helical synchrotron radiation (HSR) have the same mathematical form. To achieve this connection, the circular polarization of the laser is essential; for other states of polarization (linear, say) some similarities still remain, but no longer an exact one. Figure 10. Comparison of first-harmonic and full-field quadratures for HSR (top panels) and TSC (middle panels). Bottom panel shows B u versus B j for both processes (TSC and HSR). As it is apparent, quadrature matching is excellent. See main text for details.
The TSC-HSR similarity is not due to the 'trivial' reason that both HSR and TSC trajectories be helices in real space-time ( t x -¢), because that is not true in a strict sense. Although the HSR trajectories are indeed helices in t x -¢, the TSC ones are not, as shown by extensive numerical computation of the implicit equations t x c 3 x x = ¢ -( ) , etc in appendix C. TSC trajectories in t x -¢ show in general highly non-harmonic (although periodic) velocity components in all co-ordinate axis, and hence they substantially differ from the standard helix considered for HSR. It is after the change of variables t t x c 3 x x ¢  = ¢ -( ) that the mathematical similarity between TSC-HSR is displayed in full generality, and so it plays an absolutely central role in providing a basis for the TSC-HSR connection. The kinematics of TSC trajectories, and its role for a deeper physical understanding of the TSC-HSR similarity, would deserve further investigation.
The TSC-HSR similarity has been studied both in the frequency and time domains. As a by-product, exact and general expressions in the time domain giving the radiated magnetic fields for general helical trajectories and scattering geometries are obtained. Figure 11. Comparison of first-harmonic and full-field quadratures for HSR (top panels) and TSC (middle panels). Bottom panel shows B u versus B j for both processes (TSC and HSR). As it is apparent, quadrature matching is excellent. See main text for details.
The conditions leading to this similarity are fully clarified, and the problem is reduced to solving a set of nonlinear algebraic equations connecting both phenomena. Those equations have been numerically solved to provide compelling examples of the above mentioned connection. In some specific cases (Appendix F), they are also amenable to large analytical simplifications. In general the solution to the set of algebraic equations is not unique, but can be made so if further conditions are imposed, as for example a definite value for the target normalised frequency. This opens prospects for the study of synchrotron radiation processes in spectral ranges that can perhaps be more readily accessible from an experimental point of view (visible or UV, for example) using current ultrahigh laser technology.
Physical applications derived from the mathematical analogy, ranging from the all-optical production of radiation having exactly the same properties as regards to polarization, harmonic content, etc as synchrotron radiation, to the experimental simulation of geophysical or astrophysical processes under laboratory conditions exploiting current laser technology, have been discussed. This paper has developed what could be called a 'dictionary' in which every process in TSC has a precise translation (although possibly not a single one) into a HSR one, and viceversa. Whenever a mathematical connection between different physical phenomena has been established, the corresponding physical realms have benefited from insights coming from each other. In this sense, it is hoped that our work provides a connection that allows to see TS and synchrotron processes from an unified point of view.
sin cos cos cos sin sin cos cos cos cos cos sin sin sin A.1 sin sin cos sin cos sin A.3 The expressions for Ω D , a SR;c and a SR; s are: The modulus of the applied magnetic field B 0 is (compare with equation (46)): B mc er 1  For HSR, we shall suppose that ψ 0 =f 0 =δ 0 =0 and j sr =π/2. Then, from section 3.5, it follows that: The interesting outcome, so far, in this appendix is that for the special configurations for HSR and TSC considered above the four equations (36) and (41)  and since the left-hand side takes all values from +¥ to -¥ when 0 * q runs from 0 to π while the right-hand side remains finite, there is always at least one value of 0 * q fulfilling it. The particular choice made above implies that ORCID iDs I Pastor https:/ /orcid.org/0000-0003-0891-0941