1 Introduction
The theory of cyclotron auto resonance maser (CARM) and undulator-based free electron laser (U-FEL) devices is well established. The underlying physics is satisfactorily well described by the theoretical formulations dating back more than thirty years. The numerical codes originated by these analyses are powerful working tools, useful to analyse experimental data and to fix the design details of new devices.
Even though different, in many constructive details, the physical mechanism underlying the operation of the CARM (Nusinovich Reference Nusinovich2004) and U-FEL (Saldin, Schneidmiller & Yurkov Reference Saldin, Schneidmiller and Yurkov2000) traces back to the power transfer from the electron beam (e-beam) to a high-frequency electromagnetic field, through a klystron-like mechanism, which is the common ancestor for these families of free electron coherent generators.
The possibility of establishing a common thread between these devices has been discussed in many authoritative papers in the past see e.g. Gapanov et al. (Reference Gapanov, Petelin and Yulpatov1967), Bratman, Ginzburg & Petelin (Reference Bratman, Ginzburg and Petelin1979), Fliflet (Reference Fliflet1986). More recently (Ceccuzzi et al. Reference Ceccuzzi, Dattoli, Di Palma, Doria, Sabia and Spassovsky2015) some of the present authors used the theoretical analyses developed by different authors (Gapanov et al. Reference Gapanov, Petelin and Yulpatov1967; Bratman et al. Reference Bratman, Ginzburg and Petelin1979, Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981; Fliflet Reference Fliflet1986; Nusinovich, Latham & Li Reference Nusinovich, Latham and Li1994) to provide the translation of the CARM into U-FEL formalism by means of the analytical formulae summarized in Dattoli, Ottaviani & Pagnutti (Reference Dattoli, Ottaviani and Pagnutti2007), providing the back-bone of the code named PARSIFEL (Artioli et al. Reference Artioli, Dattoli, Ottaviani and Pagnutti2012).
The pathway leading to PARSIFEL is the result of an effort aimed at merging analytical formulae derived from U-FEL theory and scaling relations benchmarked through the available numerical codes, to get an accurate, reliable and semi-analytical physical model of the code devoted to preliminary design of FEL operating in different configurations.
In this paper we take a step further towards an extension of a PARSIFEL-like concept to CARM devices.
Regarding CARM we keep, as reference papers, those listed in Gapanov et al. (Reference Gapanov, Petelin and Yulpatov1967), Bratman et al. (Reference Bratman, Ginzburg and Petelin1979), Bratman et al. (Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981), Fliflet (Reference Fliflet1986), Nusinovich et al. (Reference Nusinovich, Latham and Li1994) and for U-FEL (Saldin et al. Reference Saldin, Schneidmiller and Yurkov2000) we have referred to the lines developed in the past by different authors and summarized in books and review articles, a partial list of which is reported in Marshall (Reference Marshall1985), Luchini & Motz (Reference Luchini and Motz1990), Colson (Reference Colson, Colson, Pellegrini and Renieri1990), Dattoli, Renieri & Torre (Reference Dattoli, Renieri and Torre1993).
In this introductory section, we will provide a quick review of the results obtained in Ceccuzzi et al. (Reference Ceccuzzi, Dattoli, Di Palma, Doria, Sabia and Spassovsky2015) and present the formalism we will adopt in the forthcoming parts of the paper.
We note that in a CARM–FEL a moderately relativistic continuous waveform e-beam, moves, inside a waveguide under the influence of an axial magnetic field, executing a helical path with a cyclotron frequency
$\unicode[STIX]{x1D6FA}_{0}=e\,B/m_{e}$
, ‘
$e$
’ being the electron charge, ‘
$me$
’ the electron mass and ‘
$B$
’ the magnetic field.
The kinematical variables of the electron beam are specified by the longitudinal (
$v_{z}$
) and transverse (
$v_{\bot }$
) velocity components, linked to the relativistic factor
$\unicode[STIX]{x1D6FE}$
by:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FD}_{z}^{2}+\unicode[STIX]{x1D6FD}_{\bot }^{2}=1-{\displaystyle \frac{1}{\unicode[STIX]{x1D6FE}^{2}}},\\ \unicode[STIX]{x1D6FD}_{z,\bot }={\displaystyle \frac{v_{z,\bot }}{c}},\\ \unicode[STIX]{x1D6FC}={\displaystyle \frac{v_{\bot }}{v_{z}}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FC}$
is the so called pitch factor.
The electrons, with longitudinal velocity
$v_{z}$
, interact with a co-propagating electromagnetic field characterized by a wavevector
$k_{z}$
, linked to the wavephase velocity
$v_{p}$
with the operational angular frequency (omega) by:
$$\begin{eqnarray}\displaystyle k_{z}=\frac{\unicode[STIX]{x1D714}}{v_{p}}. & & \displaystyle\end{eqnarray}$$
In the case of magnetic undulators the electrons (with the same aforementioned kinematical attributes) enter into an alternating (transverse static) magnetic field with period
$\unicode[STIX]{x1D706}_{u}$
, where execute transverse oscillations.
In this case the emission process is not constrained by any waveguide condition, therefore no complications arise with the wavephase velocity.
The two devices can be considered ‘topologically equivalent’, therefore useful analogies can be defined after an appropriate analysis of the emission mechanisms.
The derivation of the wavelength, characterizing the emission process inside the undulator, can be obtained using a fairly simple argument. The difference in velocities is such that, after one undulator period, the radiation has slipped ahead of the electron beam by the so called slippage length:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}=\left(c-v_{z}\right){\displaystyle \frac{\unicode[STIX]{x1D706}_{u}}{c}}=\left(1-\unicode[STIX]{x1D6FD}_{z}\right)\,\unicode[STIX]{x1D706}_{u}. & & \displaystyle\end{eqnarray}$$
Since
$\unicode[STIX]{x1D6FF}$
is linked to the phase advance of the electromagnetic wave with respect to the electrons, constructive interference of the wavefront of the emitted radiation at the next undulator period is ensured if:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D706}, & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D706}$
is the wavelength of the co-propagating field.
The last two equations yield the FEL resonance condition, which can also be cast in the form:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D714}={\displaystyle \frac{2\unicode[STIX]{x03C0}\,c}{\unicode[STIX]{x1D706}}}={\displaystyle \frac{\unicode[STIX]{x1D714}_{u}}{1-\unicode[STIX]{x1D6FD}_{z}}},\\ \unicode[STIX]{x1D714}_{u}={\displaystyle \frac{2\,\unicode[STIX]{x03C0}\,c}{\unicode[STIX]{x1D706}_{u}}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
To this aim we note that the electrons, with relativistic factor
$\unicode[STIX]{x1D6FE}$
, enter inside the undulator, where on account of the Lorenz force induced by the magnetic field they acquire a transverse velocity component
$\unicode[STIX]{x1D6FD}_{\bot }$
, the longitudinal velocity can accordingly be written, using (1.1), as
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FD}_{z}=\sqrt{1-{\displaystyle \frac{1}{\unicode[STIX]{x1D6FE}_{z}^{2}}}},\\ \unicode[STIX]{x1D6FE}_{z}={\displaystyle \frac{\unicode[STIX]{x1D6FE}}{\sqrt{1+\bar{\unicode[STIX]{x1D6FC}}^{2}}}},\\ \bar{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}_{\bot }.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
If the relativistic factor is large enough to allow a series expansion of the square root, in the first of the equations (1.6), at the lowest order in
$1/\unicode[STIX]{x1D6FE}_{z}^{2}$
, we find:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}\cong 2\,{\displaystyle \frac{\unicode[STIX]{x1D6FE}^{2}}{1+\bar{\unicode[STIX]{x1D6FC}}^{2}}}\unicode[STIX]{x1D714}_{u}. & & \displaystyle\end{eqnarray}$$
The previous derivation can be extended to CARM by noting that the relevant ‘resonance’ condition, can be determined by using the same argument as before about constructive interference, which occurs whenever the accumulated slippage between radiation and electrons, in a helix period, equals the wavelength
$\unicode[STIX]{x1D6EC}$
. By taking into account that the link between helix period and guiding magnetic field is provided by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}={\displaystyle \frac{2\unicode[STIX]{x03C0}c}{\unicode[STIX]{x1D6FA}}}. & & \displaystyle\end{eqnarray}$$
We impose the resonance condition as
$$\begin{eqnarray}\displaystyle (v_{p}-v_{z})\,{\displaystyle \frac{\unicode[STIX]{x1D6EC}}{c}}=\unicode[STIX]{x1D706}, & & \displaystyle\end{eqnarray}$$
where we have used the phase velocity
$v_{p}$
to determine the radiation electron slippage.
The previous equation (1.9) can also be written in the more familiar form:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}={\displaystyle \frac{\unicode[STIX]{x1D6FA}_{0}}{\unicode[STIX]{x1D6FE}}}+k_{z}v_{z}. & & \displaystyle\end{eqnarray}$$
The above equation has been derived by using a kinematic argument and the analogy with U-FEL has been the pivotal element of the discussion. The physical origin of the previous identity can however be understood on the basis of different arguments, involving e.g. momentum (electron and fields) conservation (Bratman et al. Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981).
We can further elaborate the previous identities, denoting by
$\unicode[STIX]{x1D714}_{R}$
the resonant frequency, we obtain, from (1.10)
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D714}_{R}={\displaystyle \frac{\unicode[STIX]{x1D6FA}}{1-{\displaystyle \frac{v_{z}}{v_{p}}}}},\\ \unicode[STIX]{x1D6FA}={\displaystyle \frac{\unicode[STIX]{x1D6FA}_{0}}{\unicode[STIX]{x1D6FE}}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
It is worth stressing that, since the phase velocity is dependent on the field frequency, (1.11) is not an explicit solution for
$\unicode[STIX]{x1D714}$
but only an approximation.
Before establishing further the analogy between U-FEL and CARM, we discuss the physical meaning of the previous equations.
The CARM resonance condition can also be derived by requiring the matching between (1.10) and the waveguide dispersion relation:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{2}=c^{2}(k_{\bot }^{2}+k_{z}^{2}), & & \displaystyle\end{eqnarray}$$
where
$k_{\bot }$
is the transverse mode wavenumber, associated with the cutoff frequency
$\unicode[STIX]{x1D714}_{c}=ck_{\bot }$
. It is easily checked that, from (1.10) and (1.11), one gets
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\pm }\cong \frac{\unicode[STIX]{x1D6FA}}{1\mp {\displaystyle \frac{\unicode[STIX]{x1D6FD}_{z}}{\unicode[STIX]{x1D6FD}_{p}}}}. & & \displaystyle\end{eqnarray}$$
The down shifted intersection, yielding the gyrotron mode (Nusinovich Reference Nusinovich2004), will not be considered in the following.
The upper shifted counterpart
$\unicode[STIX]{x1D714}_{+}$
is the resonant (CARM) frequency and to better understand its role it will be rewritten as:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D714}_{R}\cong {\displaystyle \frac{\unicode[STIX]{x1D6FA}}{1-{\displaystyle \frac{\unicode[STIX]{x1D6FD}_{z}}{\unicode[STIX]{x1D6FD}_{p}}}}}={\displaystyle \frac{\unicode[STIX]{x1D6FA}}{1-{\displaystyle \frac{1}{\unicode[STIX]{x1D6FD}_{p}}}\sqrt{1-{\displaystyle \frac{1}{\unicode[STIX]{x1D6FE}_{z}^{2}}}}}}\cong 2\unicode[STIX]{x1D6FD}_{p}\,\unicode[STIX]{x1D6FE}_{z}^{2}\unicode[STIX]{x1D6FA},\\ \unicode[STIX]{x1D6FE}_{z}={\displaystyle \frac{\unicode[STIX]{x1D6FE}}{\sqrt{1+\bar{\unicode[STIX]{x1D6FC}}^{2}}}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$
based upon the assumption that
$\unicode[STIX]{x1D6FE}_{z}$
be sufficiently large that
$\sqrt{1-(1/\unicode[STIX]{x1D6FE}_{z}^{2})}\cong 1-(1/2\,\unicode[STIX]{x1D6FE}_{z}^{2})$
and
$\unicode[STIX]{x1D6FD}_{p}\simeq 1$
.
Equation (1.14) accounts for the frequency Doppler up-shift mechanism, characterizing most free electron devices.
It is important to emphasize that, at least formally, we have established so far an important analogy between CARM and U-FEL, namely
$\unicode[STIX]{x1D6EC}\leftrightarrow \unicode[STIX]{x1D706}_{u}$
, which justifies the remark that the two devices are topologically equivalent.
The role of the transverse velocity needs a more accurate comment. In the case of the U-FEL the transverse component, induced by the Lorenz force, is the tool allowing the coupling with the co-propagating electromagnetic (transverse) field. The explicit dependence of
$\bar{\unicode[STIX]{x1D6FC}}$
on the undulator parameters is provided by:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\bar{\unicode[STIX]{x1D6FC}}={\displaystyle \frac{K}{\sqrt{2}}},\\ K\propto e\,B_{0}\unicode[STIX]{x1D706}_{u}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
In the case of the CARM the role of the transverse velocity component is the same as that of the undulator strength in the FEL undulator.
This velocity component should be induced during the electron beam preparatory phase, before the injection into the cavity.
What we have described so far are the physical conditions underlying the ‘spontaneous’ emission, which is the prerequisite for the onset of the coherent emission process. As is well known, it occurs via the bunching mechanism. The interaction of the electrons with the cavity mode electric field determines their energy modulation, which transforms into a density modulation, followed by a coherent radio frequency (RF) emission when the electrons are bunched on a scale comparable to the RF electric field wavelength.
This description encompasses all devices of FEL type, CARM is however made peculiar by the fact that the auto resonance is guaranteed even near saturation because any increase of
$\unicode[STIX]{x1D6FA}$
is balanced by a corresponding decrease of the longitudinal velocityFootnote
1
.
A further important quantity, characterizing U-FEL, is the number of undulator periods, which is associated with the oscillations executed by an electron while travelling inside the undulator. In the case of CARM it can be linked to the number of helical turns of the electrons inside the magnet. Accordingly we get
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FA}\,{\displaystyle \frac{L}{\unicode[STIX]{x1D6FD}_{z}c}}=2\,\unicode[STIX]{x03C0}\,N,\\ N\cong {\displaystyle \frac{L}{\unicode[STIX]{x1D6EC}}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$
where
$L$
is the interaction length.
So far we have explained the main elements of our approach and in the following section we will see how the correspondences we have established may provide an effective tool to evaluate the CARM operation.
2 The CARM small signal theory
In this section we will develop further the analogy with U-FEL by showing that the equation describing the CARM field evolution in the linear regime can be written by taking advantage of the simplified expression which is valid in the former case.
In the analysis of the previous section we did not include any consideration regarding the interaction of the wave with the e-beam. The dispersion relation in (1.10)–(1.11) is appropriate for the ‘cold’ waveguide condition, which merely applies to the kinematic of the mode propagation.
The CARM dynamics, associated with the radiation intensity growth in the waveguide, undergoes different phases characterized by the amount of the field power density.
The weak coupling regime is characterized by a power level well below the threshold of the saturation intensity (namely the power density halving the small signal gain) and the relevant theory can be treated using perturbative methods and, to some extent, useful information can be drawn using analytical means. A significant result from such a treatment is the derivation of a modified dispersion relation including the interaction of the electrons with the waveguide modes. According to Chen & Wurtele (Reference Chen and Wurtele1991), and Friedman et al. (Reference Friedman, Hammer, Manheimer and Sprangle2012) we find
$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x1D714}^{2}}{c^{2}}}=(k_{\bot }^{2}+k_{z}^{2})+\tilde{\unicode[STIX]{x1D700}}{\displaystyle \frac{2k_{\bot }^{2}\left(\unicode[STIX]{x1D714}-k_{z}v_{z}\right)}{\left(\unicode[STIX]{x1D714}-{\displaystyle \frac{\,\unicode[STIX]{x1D6FA}_{0}}{\unicode[STIX]{x1D6FE}}}-k_{z}v_{z}\right)}}-\tilde{\unicode[STIX]{x1D700}}{\displaystyle \frac{k_{\bot }^{2}\unicode[STIX]{x1D6FD}_{\bot }^{2}\left(\unicode[STIX]{x1D714}^{2}-c^{2}k_{z}^{2}\right)}{\left(\unicode[STIX]{x1D714}-{\displaystyle \frac{\,\unicode[STIX]{x1D6FA}_{0}}{\unicode[STIX]{x1D6FE}}}-k_{z}v_{z}\right)^{2}}}, & & \displaystyle\end{eqnarray}$$
where
$\tilde{\unicode[STIX]{x1D700}}$
plays the role of coupling parameter. It depends on the beam current and on the geometrical parameters of the waveguide itself and will be specified later in this section. In (2.1) the terms containing the coupling
$\unicode[STIX]{x1D700}$
are those ruling the field electron evolution, we simplify the analysis by neglecting the first because the second is dominating near the resonance. We are therefore left with
$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x1D714}^{2}}{c^{2}}}=(k_{\bot }^{2}+k_{z}^{2})-\unicode[STIX]{x1D700}{\displaystyle \frac{k_{\bot }^{2}\left(\unicode[STIX]{x1D714}^{2}-c^{2}k_{z}^{2}\right)}{\left(\unicode[STIX]{x1D714}-{\displaystyle \frac{\,\unicode[STIX]{x1D6FA}_{0}}{\unicode[STIX]{x1D6FE}}}-k_{z}v_{z}\right)^{2}}}, & & \displaystyle\end{eqnarray}$$
in which we have set
$\unicode[STIX]{x1D700}=\tilde{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D6FD}_{\bot }^{2}$
.
The previous identity is the crucial element of the forthcoming discussion and, for later convenience, we set
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\tilde{k}_{z}=k_{z}+\unicode[STIX]{x1D6FF}_{k_{z}},\\ k_{z}=\left({\displaystyle \frac{\unicode[STIX]{x1D714}^{2}}{c^{2}}}-k_{\bot }^{2}\right)^{1/2}={\displaystyle \frac{\unicode[STIX]{x1D6FE}\,\unicode[STIX]{x1D714}-\unicode[STIX]{x1D6FA}_{0}}{\unicode[STIX]{x1D6FE}\,v_{z}}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$
with
$\unicode[STIX]{x1D6FF}_{k_{z}}$
representing the deviation of the field longitudinal wavevector induced by the coupling with the electrons. Inserting (2.3) into (2.2) we find that
$\unicode[STIX]{x1D6FF}_{k_{z}}$
is specified by the following fourth degree algebraic equation
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{z}^{2}\unicode[STIX]{x1D6FF}_{k_{z}}^{4}+2\,\unicode[STIX]{x1D6FD}_{z}^{2}k_{z}\unicode[STIX]{x1D6FF}_{k_{z}}^{3}+\unicode[STIX]{x1D700}k_{\bot }^{2}\unicode[STIX]{x1D6FF}_{k_{z}}^{2}+2\,\unicode[STIX]{x1D700}\,k_{\bot }^{2}\,k_{z}\unicode[STIX]{x1D6FF}_{k_{z}}-\unicode[STIX]{x1D700}\,k_{\bot }^{4}=0. & & \displaystyle\end{eqnarray}$$
The roots of the above equation specifies the evolution of the CARM field amplitude along the coordinate
$z$
, according to
$$\begin{eqnarray}\displaystyle E\left(z\right)\propto \mathop{\sum }_{j=1}^{4}e_{j}e^{(\unicode[STIX]{x1D6FF}_{k_{z}})_{j}z}, & & \displaystyle\end{eqnarray}$$
where
$j$
refers to the roots of (2.4) and
$e_{j}$
are integration constants, fixed by the conditions
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}E\left(0\right)=1,\\ \left(\left({\displaystyle \frac{\text{d}}{\text{d}z}}\right)^{k}E(z)\right)_{z=0}=0,\\ k=1,2,3.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
The linearized field growth along the longitudinal coordinate can accordingly be obtained by plotting
$|E(z)|^{2}$
as shown in figure 1.
Comparison between U-FEL (blue) and CARM (green) intensity growth curves in a small signal regime with the result reported in Chen & Wurtele (Reference Chen and Wurtele1991).
The evolution curve exhibits the well-known shape, characterizing also the high gain U-FEL operating mode, namely an initial phase during which the electrons acquire the bunching, allowing the second phase, characterized by a linear (in logarithmic scale) growth with a characteristic gain length
$l_{g}$
. In the case of CARM such a quantity is specified by (Chen & Wurtele Reference Chen and Wurtele1991)
$$\begin{eqnarray}\displaystyle l_{g}^{-1}=2\unicode[STIX]{x1D6E4}=\sqrt{3}\left[\frac{\unicode[STIX]{x1D700}\,k_{\bot }^{4}}{16\,k_{z}\unicode[STIX]{x1D6FD}_{z}^{2}}\right]^{1/3}. & & \displaystyle\end{eqnarray}$$
Let us now invoke the analogy with the U-FEL, whose gain length is defined as (Dattoli et al. Reference Dattoli, Ottaviani and Pagnutti2007)
$$\begin{eqnarray}\displaystyle l_{g}=\frac{\unicode[STIX]{x1D706}_{u}}{4\,\unicode[STIX]{x03C0}\,\sqrt{3}\,\unicode[STIX]{x1D70C}}, & & \displaystyle\end{eqnarray}$$
with
$\unicode[STIX]{x1D70C}$
being the Pierce parameter linked to the small signal gain coefficient
$g_{0}$
by the identity (Dattoli et al.
Reference Dattoli, Ottaviani and Pagnutti2007)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}=\frac{(\unicode[STIX]{x03C0}\,g_{0})^{1/3}}{4\,\unicode[STIX]{x03C0}\,N}. & & \displaystyle\end{eqnarray}$$
The use of the correspondences established in the previous section and the comparison between (2.7), (2.8) allows the following identification
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}=\frac{\unicode[STIX]{x1D6EC}\,\unicode[STIX]{x1D6E4}}{4\,\unicode[STIX]{x03C0}\,\sqrt{3}}. & & \displaystyle\end{eqnarray}$$
The dependence of the U-FEL field amplitude on the longitudinal coordinate has been shown to be provided by (Dattoli et al. Reference Dattoli, Ottaviani and Pagnutti2007)Footnote 2 .
$$\begin{eqnarray}\displaystyle a(\unicode[STIX]{x1D70F}) & = & \displaystyle {\displaystyle \frac{a_{0}}{3\,(\unicode[STIX]{x1D708}+p+q)}}\;\text{e}^{-(2/3)\;\text{i}\unicode[STIX]{x1D708}\,\unicode[STIX]{x1D70F}}\left\{\vphantom{\left({\displaystyle \frac{\sqrt{3}}{6}}\right)}(-\unicode[STIX]{x1D708}+p+q)\,\text{e}^{-\text{i}/3\;(p+q)\,\unicode[STIX]{x1D70F}}+\,2\,(2\,\unicode[STIX]{x1D708}+p+q)\;\text{e}^{\text{i}/6\;(p+q)\,\unicode[STIX]{x1D70F}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.\times \,\left[\cosh \left({\displaystyle \frac{\sqrt{3}}{6}}\left(p-q\right)\unicode[STIX]{x1D70F}\right)+\text{i}{\displaystyle \frac{\sqrt{3}\,\unicode[STIX]{x1D708}}{p-q}}\sinh \left({\displaystyle \frac{\sqrt{3}}{6}}\left(p-q\right)\unicode[STIX]{x1D70F}\right)\right]\right\},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle p & = & \displaystyle \left[{\textstyle \frac{1}{2}}(r+\sqrt{d})\right]^{1/3},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle q & = & \displaystyle \left[{\textstyle \frac{1}{2}}(r-\sqrt{d})\right]^{1/3},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle r & = & \displaystyle 27\,\unicode[STIX]{x03C0}\,g_{0}-2\,\unicode[STIX]{x1D708}^{3},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle d & = & \displaystyle 27\,\unicode[STIX]{x03C0}\,g_{0}[27\,\unicode[STIX]{x03C0}\,g_{0}-4\,\unicode[STIX]{x1D708}^{3}].\end{eqnarray}$$
The various parameters entering the above expression are recognized as
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D708}\equiv \text{detuning parameter}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle z\equiv \text{longitudinal coordinate}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle L\equiv N\,\unicode[STIX]{x1D706}_{u}\equiv \text{interaction length}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}\equiv \text{dimensionless time}. & \displaystyle\end{eqnarray}$$
The correspondence with the CARM variables is obtained by defining the normalized detuning
$\bar{\unicode[STIX]{x1D708}}$
parameter as
$$\begin{eqnarray}\displaystyle \bar{\unicode[STIX]{x1D708}}={\displaystyle \frac{\unicode[STIX]{x1D708}}{(27\unicode[STIX]{x03C0}\;g_{0})^{1/3}}}, & & \displaystyle\end{eqnarray}$$
and then by casting, using the relations (2.7)–(2.9), the dimensionless time in the form
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}={\displaystyle \frac{z}{L}}=\frac{z}{Nl_{g}4\unicode[STIX]{x03C0}\sqrt{3}\unicode[STIX]{x1D70C}}=\frac{2\unicode[STIX]{x1D6E4}z}{\sqrt{3}(\unicode[STIX]{x03C0}g_{0})^{1/3}}, & & \displaystyle\end{eqnarray}$$
thus finally ending up with
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D708}\unicode[STIX]{x1D70F}=2\sqrt{3}\unicode[STIX]{x1D6E4}\bar{\unicode[STIX]{x1D708}}z. & & \displaystyle\end{eqnarray}$$
The complex amplitude (2.11) can now be assumed to be a function of the normalized detuning parameter
$\bar{\unicode[STIX]{x1D708}}$
and of the inverse gain length,
$\unicode[STIX]{x1D6E4}$
, which will be exploited to describe the small signal growth of the radiation field amplitude, namely
$$\begin{eqnarray}\displaystyle a(z,\unicode[STIX]{x1D6E4},\bar{\unicode[STIX]{x1D708}}) & = & \displaystyle \frac{a_{0}}{3}\text{e}^{-\text{i}(4\unicode[STIX]{x1D6E4}\bar{\unicode[STIX]{x1D708}}z/\sqrt{3}\unicode[STIX]{x1D6FD}_{z})}\cdot \left\{A_{1}\text{e}^{-\text{i}(2\unicode[STIX]{x1D6E4}zA_{3}^{(+)}/\sqrt{3})}+2A_{2}\text{e}^{\text{i}(\unicode[STIX]{x1D6E4}zA_{3}^{(+)}/\sqrt{3})}\right.\nonumber\\ \displaystyle & & \displaystyle \cdot \,\left[\left.\cosh (\unicode[STIX]{x1D6E4}zA_{3}^{(-)})+\imath \frac{\sqrt{3}\bar{\unicode[STIX]{x1D708}}}{A_{3}^{(+)}}\sinh (\unicode[STIX]{x1D6E4}zA_{3}^{(-)})\right]\right\},\end{eqnarray}$$
with
$$\begin{eqnarray}\displaystyle A_{1} & := & \displaystyle A_{1}(\bar{\unicode[STIX]{x1D708}})={\displaystyle \frac{(-\unicode[STIX]{x1D708}+p+q)}{\unicode[STIX]{x1D708}+p+q}}\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{\left(\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}+\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}+\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}-\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}-\sqrt[3]{2}\bar{\unicode[STIX]{x1D708}}\right)}{\left(\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}+\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}+\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}-\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}+\sqrt[3]{2}\bar{\unicode[STIX]{x1D708}}\right)}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle A_{2} & := & \displaystyle A_{2}(\bar{\unicode[STIX]{x1D708}})=\frac{(2\unicode[STIX]{x1D708}+p+q)}{\unicode[STIX]{x1D708}+p+q}\nonumber\\ \displaystyle & = & \displaystyle \frac{\left(\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}+\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}+\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}-\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}+2\sqrt[3]{2}\bar{\unicode[STIX]{x1D708}}\right)}{\left(\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}+\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}+\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}-\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}+\sqrt[3]{2}\bar{\unicode[STIX]{x1D708}}\right)},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle A_{3}^{(\pm )} & := & \displaystyle A_{3}^{(\pm )}(\bar{\unicode[STIX]{x1D708}})=\frac{1}{\sqrt[3]{2}}\left(\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}+\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}\pm \sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}-\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}\right),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle (p\pm q)\unicode[STIX]{x1D70F} & = & \displaystyle \frac{\sqrt{3}}{\sqrt[3]{2}}2\unicode[STIX]{x1D6E4}z\left(\sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}+\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}\pm \sqrt[3]{1-2\bar{\unicode[STIX]{x1D708}}^{3}-\sqrt{1-4\bar{\unicode[STIX]{x1D708}}^{3}}}\right)\nonumber\\ \displaystyle & = & \displaystyle 2\sqrt{3}\unicode[STIX]{x1D6E4}zA_{3}^{(\pm )}.\end{eqnarray}$$
In figure 1 we have provided a comparison between the prediction of the CARM theory and of the U-FEL scaling equations, given in (2.11). The agreement is satisfactory and further comments will be given below.
We should put in evidence that the linear solution obtained solving the dispersion relation (2.2) has been regularized neglecting the oscillating root of the (2.4) as reported in figure 2 comparing the amplitude signal with and without the oscillating solution .
Small signal CARM growing intensity, with and without the imaginary contributions from (2.4).
The following two remarks are in order to complete the previous discussion:
-
(i) The dispersion relations for CARM and U-FEL lead to a fourth and third degree algebraic equations respectively. This is a consequence of the fact that the CARM field equations have been derived without the assumption of paraxial approximation, while in the case of U-FEL the small signal problem is solved by the approximation of a slowly varying envelope (SVE). This assumption leads to a treatment involving algebraic equations of one degree lower. In adapting U-FEL to CARM theory, according to the prescription of this paper and of those reported in Ceccuzzi et al. (Reference Ceccuzzi, Dattoli, Di Palma, Doria, Sabia and Spassovsky2015), we did not find particular differences, except for the initial part, where the SVE approximation smoothen the field oscillations. We must underline that the approximation of the dispersion equation with the elimination of the quasi resonant term is a further cause of the fluctuation at low intensities.
-
(ii) The equations (2.2)–(2.7) have been written without fixing the waveguide mode structure, we can however factorize the
$\unicode[STIX]{x1D700}$
coupling parameter as the product of two terms, namely (2.18)where
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D700}=\unicode[STIX]{x1D6EF}\,\text{f},\\ \text{f}={\displaystyle \frac{4\,\unicode[STIX]{x03C0}\,\unicode[STIX]{x1D6FD}_{\bot }^{2}}{\unicode[STIX]{x1D6FE}\,\unicode[STIX]{x1D6FD}_{z}}}\left({\displaystyle \frac{I_{b}}{I_{A}}}\right),\end{array}\right\} & & \displaystyle\end{eqnarray}$$
$I_{b,A}$
denote the beam and Alfvén current, the parameter
$\unicode[STIX]{x1D6EF}$
summarizes the details of the cavity mode and the effect due to the geometrical overlapping between electrons and waveguide modes.
This is not a secondary issue, in particular when the operation on higher-order transverse modes is foreseen. In this case the relevant spatial distribution is not provided by a Gaussian, covering smoothly the transverse surface, but by a kind of circular corona which demands an appropriate shaping of the transverse structure of the e-beam to optimize the coupling.
In figure 3 we have reported the gain length for different higher-order modes, along with the transverse mode distribution.
Transverse mode pattern and associated gain length. The circle in the middle represents the electron beam transverse distribution, which is assumed to stay well inside the mode itself thus maximizing the filling factor.
3 Numerical analysis, saturation and final comments
In the previous section we have developed quite a straightforward formalism to prove that most of the scaling formulae developed within the framework of FEL theory can be adapted to the study and design as well of CARM devices, at least for the case of the small signal regime.
In this section we include the nonlinear contributions and show that the logistic curve model (Artioli et al. Reference Artioli, Dattoli, Ottaviani and Pagnutti2012; Ceccuzzi et al. Reference Ceccuzzi, Dattoli, Di Palma, Doria, Sabia and Spassovsky2015) is an effective tool to study the evolution of the system up to the saturation.
We will provide a comparison of this analysis with the logistic growth model based on a family of S-shaped curves, the model has been shown to be very effective in reproducing the evolution of any system undergoing a dynamical behaviour ruled by an equation of the type
$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}}{\text{d}\,z}}P={\displaystyle \frac{P}{l_{g}}}\left[1-{\displaystyle \frac{P}{P_{S}}}\right], & & \displaystyle\end{eqnarray}$$
even though either CARM and U-FEL satisfy more complicated nonlinear equations for the growth of the power density. Equation (3.1) captures the essential physics of the problem, namely a linear growth followed by a quadratic nonlinearity when the power approaches
$P_{S}$
which denotes the saturated power. The solution of (3.1) can be written as
$$\begin{eqnarray}\displaystyle P\left(z\right)=P_{0}{\displaystyle \frac{\text{e}^{z/l_{g}}}{1+{\displaystyle \frac{P_{0}}{P_{S}}}\left[\text{e}^{z/l_{g}}-1\right]}}, & & \displaystyle\end{eqnarray}$$
where
$P_{0}$
is the input radiation seed.
The definition of the CARM saturated power
$P_{S}$
is easily given by just following the prescriptions reported in Bratman et al. (Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981), we set therefore
$$\begin{eqnarray}\displaystyle P_{S}\cong \unicode[STIX]{x1D702}\,P_{E}, & & \displaystyle\end{eqnarray}$$
where
$P_{E}$
is the electron beam power and
$\unicode[STIX]{x1D702}$
the efficiency of the device in turn provided by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}_{sp}\unicode[STIX]{x1D702}_{C}, & & \displaystyle\end{eqnarray}$$
where we have denoted by
$\unicode[STIX]{x1D702}_{sp}$
the single particle efficiency, (Bratman et al.
Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981) which can be written using the analogy in terms of the Pierce parameter (Saldin et al.
Reference Saldin, Schneidmiller and Yurkov2000; Dattoli et al.
Reference Dattoli, Pagnutti, Ottaviani and Asgekar2012) as
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{sp}\cong \sqrt{2}\unicode[STIX]{x1D70C}=\frac{\sqrt{2}\unicode[STIX]{x1D6EC}\,\unicode[STIX]{x1D6E4}}{4\,\unicode[STIX]{x03C0}\,\sqrt{3}}, & & \displaystyle\end{eqnarray}$$
while the collective efficiency
$\unicode[STIX]{x1D702}_{C}$
is specified by (Bratman et al.
Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}_{C}\cong \frac{1}{(1-\unicode[STIX]{x1D6FD}_{p}^{-2})\,(1-\unicode[STIX]{x1D6FE}^{-1})}\frac{\unicode[STIX]{x1D6FD}_{\bot }^{2}}{b},\quad \text{with }b={\displaystyle \frac{\unicode[STIX]{x1D6FD}_{\bot }^{2}}{2\,\unicode[STIX]{x1D6FD}_{z}\unicode[STIX]{x1D6FD}_{p}\left(1-{\displaystyle \frac{\unicode[STIX]{x1D6FD}_{z}}{\unicode[STIX]{x1D6FD}_{p}}}\right)}}, & & \displaystyle\end{eqnarray}$$
where, using the terminology of Bratman et al. (Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981),
$b$
denotes the electron recoil parameter. It accounts for the auto resonance contribution, including the effect of axial momentum and velocity change with the electron energy loss (Bratman et al.
Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981). Regarding the analogy with U-FEL it can be associated with the undulator tapering parameter (Kroll, Morton & Rosenbluth Reference Kroll, Morton and Rosenbluth1981; Dattoli et al.
Reference Dattoli, Pagnutti, Ottaviani and Asgekar2012).
According to the previous identity the saturated power can be cast in the form
$$\begin{eqnarray}\displaystyle P_{S}\cong {\displaystyle \frac{\sqrt{2}}{4\unicode[STIX]{x03C0}\sqrt{3}}}{\displaystyle \frac{\unicode[STIX]{x1D6EC}\,\unicode[STIX]{x1D6E4}}{(1-\unicode[STIX]{x1D6FD}_{p}^{-2})\,(1-\unicode[STIX]{x1D6FE}^{-1})}}{\displaystyle \frac{\unicode[STIX]{x1D6FD}_{\bot }^{2}}{b}}P_{E}. & & \displaystyle\end{eqnarray}$$
We have recovered all the crucial parameters (gain length and saturated power) to draw the CARM power growth curve using the logistic equation. However, equation (3.2) accounts only for the exponential growth prior to the saturation and does not contain any lethargic phase. To overcome this problem we replace the exponential term in (3.2) with the square modulus of the small signal amplitude derived in the previous section, thus writing
$$\begin{eqnarray}\displaystyle P(z)\cong P_{0}{\displaystyle \frac{\left|\unicode[STIX]{x1D6FD}(z)\right|^{2}}{1+{\displaystyle \frac{P_{0}}{P_{S}}}\left(\left|\unicode[STIX]{x1D6FD}(z)\right|^{2}-1\right)}}\quad \text{and}\quad \unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D70F})=\frac{a(\unicode[STIX]{x1D70F}(z))}{a_{0}}, & & \displaystyle\end{eqnarray}$$
with
$a_{0}$
initial amplitude.
To check the validity of the previous formula we have developed an ad hoc numerical code to integrate the CARM equations (Gapanov et al. Reference Gapanov, Petelin and Yulpatov1967; Bratman et al. Reference Bratman, Ginzburg and Petelin1979, Reference Bratman, Ginzburg, Nusinovich, Petelin and Strklkov1981; Fliflet Reference Fliflet1986; Chen & Wurtele Reference Chen and Wurtele1991; Nusinovich et al. Reference Nusinovich, Latham and Li1994).
The dynamical systems accounting for the evolution of CARM devices are described by a set of equations coupling electrons and field. From the mathematical point of view the problem is that of solving a system of nonlinear ordinary differential equation, consisting of four differential equations three of which account for the electron motion and the other for the complex field amplitude evolution inside the cavity, namely:
$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}u}{\text{d}\unicode[STIX]{x1D701}}}={\displaystyle \frac{[1-u]^{s/2}}{1-bu}}Re(F_{s}\text{e}^{-\text{i}\unicode[STIX]{x1D703}}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D703}}{\text{d}\unicode[STIX]{x1D701}}}={\displaystyle \frac{1}{1-bu}}\left[\unicode[STIX]{x1D6E5}-u-b\unicode[STIX]{x1D6F6}(\unicode[STIX]{x1D701})+{\displaystyle \frac{s}{2}}[1-u]^{(s/2)-1}Re(\text{i}F_{s}\text{e}^{-\text{i}\unicode[STIX]{x1D703}})\right], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}\unicode[STIX]{x1D6F6}}{\text{d}\unicode[STIX]{x1D701}}}=-{\displaystyle \frac{[1-u]^{s/2}}{1-bu}}Re\left(\text{i}{\displaystyle \frac{\text{d}F_{s}}{\text{d}\unicode[STIX]{x1D701}}}\text{e}^{-\text{i}\unicode[STIX]{x1D703}}\right), & \displaystyle\end{eqnarray}$$
$u,\unicode[STIX]{x1D703},\unicode[STIX]{x1D6F6}$
variables are associated with the electron energy, the electron wavephase and the axial momentum correction respectively,
$s$
is the order of the harmonics,
$\unicode[STIX]{x1D701}$
is the normalized space variable and
$F_{s}$
accounts for the complex mode field amplitude, whose evolution is fixed by the equation 
$$\begin{eqnarray}\displaystyle \frac{\text{d}F_{s}}{\text{d}\unicode[STIX]{x1D701}}=I_{g}\left\langle \frac{[1-u]^{s/2}}{1-bu}\text{e}^{\text{i}\unicode[STIX]{x1D703}}\right\rangle , & & \displaystyle\end{eqnarray}$$
with
$\langle \cdots \rangle _{\unicode[STIX]{x1D703}_{0}}$
denoting the average taken on the initial phase distribution (Fliflet Reference Fliflet1986; Dattoli et al.
Reference Dattoli, Renieri and Torre1993; Nusinovich Reference Nusinovich2004; Ceccuzzi et al.
Reference Ceccuzzi, Dattoli, Di Palma, Doria, Sabia and Spassovsky2015).
The pivotal parameters characterizing the previous equations system related to the CARM dynamics are summarized by three dimensionless quantities:
$b$
, accounting for the auto resonance,
$\unicode[STIX]{x1D6E5}$
, normalized detuning
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}={\displaystyle \frac{2\left(1-{\displaystyle \frac{\unicode[STIX]{x1D6FD}_{z}}{\unicode[STIX]{x1D6FD}_{p}}}\right)^{2}\left(1-{\displaystyle \frac{\unicode[STIX]{x1D714}_{R}}{\unicode[STIX]{x1D714}}}\right)}{\unicode[STIX]{x1D6FD}_{\bot }^{2}(1-\unicode[STIX]{x1D6FD}_{p}^{-2})}}, & & \displaystyle\end{eqnarray}$$
and
$I_{g}$
normalized beam current, proportional to the beam current
$I_{b}$
and expressible in terms of
$\unicode[STIX]{x1D70C}$
parameter as
$$\begin{eqnarray}\displaystyle I_{g}=\left[{\displaystyle \frac{4\sqrt[3]{2}\left(1-{\displaystyle \frac{\unicode[STIX]{x1D6FD}_{z}}{\unicode[STIX]{x1D6FD}_{p}}}\right)}{\unicode[STIX]{x1D6FD}_{\bot }^{2}\left(1-\unicode[STIX]{x1D6FD}_{p}^{-2}\right)}}\;\unicode[STIX]{x1D70C}\right]^{3}. & & \displaystyle\end{eqnarray}$$
We will solve the previous set of equations valid, as already underscored, in the linear and nonlinear regimes and compare the result with the small signal analysis of the previous sections.
The revisited semi-analytical formula from FEL compared with numerical code.
The adopted numerical procedure uses a Runge–Kutta scheme for the electron dynamics, with the field amplitude kept constant during one discretization step. Furthermore, a finite difference method has been applied to evaluate the differential equation concerning the amplitude wave evolution, in which the crucial step is the careful average of the electron phase distribution and the transverse velocity distribution in order to include correctly the effect of the beam qualities.
To study the effect of the particles velocity spread, starting form a fixed
$\unicode[STIX]{x1D6FE}_{0}$
beam energy and an
$\unicode[STIX]{x1D6FC}_{0}$
pitch, a Gaussian distribution of the transverse velocity has been generated centred at the initial value of
$\unicode[STIX]{x1D6FD}_{\bot 0}$
.
For each particle we considered an ODE system characterized by a
$b(\unicode[STIX]{x1D6FD}_{\bot 0}^{i})$
and
$\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FD}_{\bot 0}^{i})$
parameters and the integral average of the electron phase and velocity distribution have been evaluated by the use of a standard trapezoidal scheme.
Furthermore, the orbital efficiency has been obtained by averaging the electron motion on the electron phase and velocity distribution, allowing to evaluate the CARM power growth versus
$z$
as reported in figure 4.
The analytical growth linear rate of the signal compared with the numerical simulation.
Efficiency behaviour (for optimal detuning
$\unicode[STIX]{x1D6E5}$
) versus transverse velocity spreadfor different values of
$I_{g}$
(dotted line numerical computation, red line Lorentzian fit).
The comparison between (3.8) and the power evolution obtained via the numerical implementation are shown in figures 4 and 5. The two curves compare fairly well; the use of these formulae for fixing the working points of a CARM device is therefore justified. The aim of the paper is to explore the analogy between CARM and U-FEL on the basis of a unifying common theoretical framework. This attempt is far from being exhaustive and the analogies should be exploited with care, for example we have not clarified whether the effects of beam quality on CARM performance can be accounted for by the ‘universal’ scaling formula as happens in the case of the U-FEL (Dattoli et al. Reference Dattoli, Ottaviani and Pagnutti2007; Artioli et al. Reference Artioli, Dattoli, Ottaviani and Pagnutti2012).
The impact of the beam qualities on the CARM output power is shown in figure 6 where we have reported the power versus the root mean square of the transverse velocity (
$\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}_{\bot }}$
). The curves are nicely fitted by a Lorentzian (
$\hat{\unicode[STIX]{x1D702}}_{sp}\propto (1/(1+a\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FD}_{\bot }}^{2}))$
, with
$\hat{\unicode[STIX]{x1D702}}_{sp}$
the numerical single particle efficiency and
$a$
is a fit parameter), but a ‘universal parameter’ embedding inhomogeneous broadening effects and
$I_{g}$
(or
$\unicode[STIX]{x1D70C}$
as well) has not been formulated yet. According to the suggestion put forward in Nusinovich et al. (Reference Nusinovich, Latham and Li1994) and later in Ceccuzzi et al. (Reference Ceccuzzi, Dattoli, Di Palma, Doria, Sabia and Spassovsky2015) this seems to be possible, but further numerical analysis is needed to provide a more sound conclusion and is left for future investigations.
