Modelling elliptically polarised free electron lasers

In the averaged FEL model the electron orbits are first calculated in the absence of any radiation field from the Lorentz force equation:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{{\boldsymbol{\beta }}}_{j}}{{\rm{d}}t}=-\displaystyle \frac{e}{{\gamma }_{j}m}{{\boldsymbol{\beta }}}_{j}\times {{\boldsymbol{B}}}_{u},\end{eqnarray} \tag{ 4 }$$

where ${{\boldsymbol{\beta }}}_{j}={{\bf{v}}}_{j}/c$ and ${\gamma }_{j}$ are the jth electron's velocity scaled with respect to the speed of light c, and the corresponding Lorentz factor. Substituting for the undulator field (1), and integrating the Lorentz equation (4), the scaled electron velocity components are obtained:

$$\begin{eqnarray}&&{\beta }_{{xj}}={\left(\displaystyle \frac{2{u}_{e}^{2}}{1+{u}_{e}^{2}}\right)}^{1/2}\displaystyle \frac{{\bar{a}}_{u}}{{\gamma }_{j}}\mathrm{sin}({k}_{u}z),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\beta }_{{yj}}=-{\left(\displaystyle \frac{2}{1+{u}_{e}^{2}}\right)}^{1/2}\displaystyle \frac{{\bar{a}}_{u}}{{\gamma }_{j}}\mathrm{cos}({k}_{u}z),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\beta }_{{zj}}={\left[{\bar{\beta }}_{z}^{2}-\left(\displaystyle \frac{1-{u}_{e}^{2}}{1+{u}_{e}^{2}}\right)\displaystyle \frac{{\bar{a}}_{u}^{2}}{{\gamma }_{j}^{2}}\mathrm{cos}(2{k}_{u}z)\right]}^{1/2},\end{eqnarray} \tag{ 7 }$$

where ${\bar{v}}_{z}=c{\bar{\beta }}_{z}$ is the average longitudinal electron velocity. The constants m and e take their usual meanings of rest mass and charge magnitude of the electron. Introducing the non-unit vector basis ${\bf{f}}=\tfrac{1}{\sqrt{2}}({u}_{e}\hat{{\bf{x}}}+{\rm{i}}\hat{{\bf{y}}})$, so that ${\bf{f}}\cdot {\bf{f}}=-(1-{u}_{e}^{2})/2$ and ${\bf{f}}\cdot {{\bf{f}}}^{* }=(1+{u}_{e}^{2})/2$, the perpendicular components may be written:

$$\begin{eqnarray}&&{{\boldsymbol{\beta }}}_{\perp j}=\displaystyle \frac{{\rm{i}}}{\sqrt{1+{u}_{e}^{2}}}\displaystyle \frac{{\bar{a}}_{u}}{{\gamma }_{j}}({\bf{f}}\mathrm{exp}(-{\rm{i}}{k}_{u}z)-{\rm{c.c.}}).\ \end{eqnarray} \tag{ 8 }$$

Integrating equation (7), the longitudinal electron trajectory in the presence of the undulator field only is:

$$\begin{eqnarray}&&{z}_{j}(t)=c{\bar{\beta }}_{z}t-\displaystyle \frac{{\bar{a}}_{u}^{2}}{4{\gamma }_{j}^{2}{k}_{u}{\bar{\beta }}_{z}^{2}}\left(\displaystyle \frac{1-{u}_{e}^{2}}{1+{u}_{e}^{2}}\right)\mathrm{sin}(2{k}_{u}c{\bar{\beta }}_{z}t).\end{eqnarray} \tag{ 9 }$$

The oscillatory term in (9) describes the 'figure-of-eight' longitudinal jitter motion of the electron in a non-helical undulator associated with coupling to harmonics of the radiation field [12].

A co-propagating radiation field is similarly defined using the same non-unit vector basis ${\bf{f}}$ as the sum over harmonics of the fundamental resonant field, i.e. ${\bf{E}}={\sum }_{n}{{\bf{E}}}_{n}$, where:

$$\begin{eqnarray}&&{{\bf{E}}}_{n}(z,t)=\displaystyle \frac{{\rm{i}}}{\sqrt{2}}({\bf{f}}\ {{ \mathcal E }}_{n}(z,t){{\rm{e}}}^{{\rm{i}}n({k}_{r}z-{\omega }_{r}t)}-{\rm{c.c.}}).\end{eqnarray} \tag{ 10 }$$

The scaled energy evolution of the jth electron in the transverse plane-wave radiation field of (10) may then be written as:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\gamma }_{j}}{{\rm{d}}t}=-\displaystyle \frac{e}{{mc}}\displaystyle \sum _{n}{{\boldsymbol{\beta }}}_{\perp j}\cdot {{\bf{E}}}_{n},\end{eqnarray} \tag{ 11 }$$

using equations for the electron motion (8) and (9), the electric field (10) and the identity:

$$\begin{eqnarray}&&{{\rm{e}}}^{{\rm{i}}x\mathrm{sin}(\phi )}=\displaystyle \sum _{n=-\infty }^{\infty }{J}_{n}(x){{\rm{e}}}^{{\rm{i}}n\phi },\end{eqnarray} \tag{ 12 }$$

the equation for the electron energy (11) simplifies to:

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{\gamma }_{j}}{{\rm{d}}t} & = & -\displaystyle \frac{e}{4{mc}}{\left(\displaystyle \frac{2}{1+{u}_{e}^{2}}\right)}^{1/2}\displaystyle \frac{{\bar{a}}_{u}}{{\gamma }_{j}}\displaystyle \sum _{n}\left({{ \mathcal E }}_{n}{{\rm{e}}}^{{\rm{i}}n{\theta }_{j}}\left((1+{u}_{e}^{2})\displaystyle \sum _{m=-\infty }^{\infty }{J}_{m}(b){{\rm{e}}}^{-{\rm{i}}(n-1+2m){k}_{u}{\bar{\beta }}_{z}{ct}}\right.\right.\\ & & +\left.\left.(1-{u}_{e}^{2})\displaystyle \sum _{m=-\infty }^{\infty }{J}_{m}(b){{\rm{e}}}^{-{\rm{i}}(n+1+2m){k}_{u}{\bar{\beta }}_{z}{ct}}\right)+{\rm{c.c.}}\right),\end{eqnarray} \tag{ 13 }$$

where ${\theta }_{j}=({k}_{r}+{k}_{u}){\bar{\beta }}_{z}{ct}-{\omega }_{r}t$ is the ponderomotive phase. Resonant, non-oscillatory terms, which do not average to zero over an undulator period occur only for $n\pm 1+2m=0$, so that on averaging over an undulator period equation (13) simplifies further to:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\gamma }_{j}}{{\rm{d}}t}=-\displaystyle \frac{e}{4{mc}}{\left(\displaystyle \frac{2}{1+{u}_{e}^{2}}\right)}^{1/2}\displaystyle \frac{{\bar{a}}_{u}}{{\gamma }_{j}}\displaystyle \sum _{n}\;{{JJ}}_{n}({{ \mathcal E }}_{n}{{\rm{e}}}^{{\rm{i}}n{\theta }_{j}}\ +{\rm{c.c.}}),\end{eqnarray} \tag{ 14 }$$

where:

$$\begin{eqnarray}&&{{JJ}}_{n}={(-1)}^{\tfrac{n-1}{2}}\left((1+{u}_{e}^{2}){J}_{\tfrac{n-1}{2}}(\xi )-(1-{u}_{e}^{2}){J}_{\tfrac{n+1}{2}}(\xi )\right),\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\xi =\displaystyle \frac{n{\bar{a}}_{u}^{2}}{2(1+{\bar{a}}_{u}^{2})}\displaystyle \frac{1-{u}_{e}^{2}}{1+{u}_{e}^{2}}.\end{eqnarray} \tag{ 16 }$$

The 1D wave equation is used to model the plane wave radiation field evolution and is given by:

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\partial }^{2}}{\partial {z}^{2}}-\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{\partial }^{2}}{\partial {t}^{2}}\right){\bf{E}}=\displaystyle \frac{{\mu }_{0}}{\sigma }\displaystyle \frac{\partial {{\bf{J}}}_{\perp }}{\partial t},\end{eqnarray} \tag{ 17 }$$

where σ is the transverse area of the co-propagating planar radiation field and electron beam with transverse current density of ${{\bf{J}}}_{\perp }=-{ec}{\sum }_{j=1}^{N}{{\boldsymbol{\beta }}}_{\perp }\delta ({\bf{r}}-{{\bf{r}}}_{j}(t))$. The transverse components of the electric field and transverse current density are defined by ${E}_{\perp }=\sqrt{2}\ {\bf{E}}\cdot {{\bf{f}}}^{* }$ and ${J}_{\perp }=\sqrt{2}\ {\bf{J}}\cdot {{\bf{f}}}^{* }$ respectively. In the 1D limit, the wave equation (17) simplifies to:

$$\begin{eqnarray}&&\left(\displaystyle \frac{{\partial }^{2}}{\partial {z}^{2}}-\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{\partial }^{2}}{\partial {t}^{2}}\right){E}_{\perp }=\displaystyle \frac{{\mu }_{0}}{\sigma }\displaystyle \frac{\partial {J}_{\perp }}{\partial t}.\end{eqnarray} \tag{ 18 }$$

By using the transverse velocity (8), the harmonic fields (10) and by neglecting the backward wave as detailed in [13] then, using the Bessel identity (12), the wave equation (18) reduces to a wave equation for each harmonic envelope ${{ \mathcal E }}_{n}$:

$$\begin{eqnarray}\left(\displaystyle \frac{\partial }{\partial z}+\displaystyle \frac{1}{c}\displaystyle \frac{\partial }{\partial t}\right){{ \mathcal E }}_{n} & = & \displaystyle \frac{e{\mu }_{0}{c}^{2}{\bar{a}}_{u}}{\sqrt{2}\sigma {(1+{u}_{e}^{2})}^{3/2}}\displaystyle \sum _{j=1}^{N}\displaystyle \frac{{{\rm{e}}}^{-{\rm{i}}n{\theta }_{j}}}{{\gamma }_{j}}\left((1+{u}_{e}^{2})\displaystyle \sum _{m=-\infty }^{\infty }{J}_{m}(b){{\rm{e}}}^{{\rm{i}}(n-1+2m){k}_{u}{\bar{\beta }}_{z}{ct}}\right.\\ & & +\left.(1-{u}_{e}^{2})\displaystyle \sum _{m=-\infty }^{\infty }{J}_{m}(b){{\rm{e}}}^{{\rm{i}}(n+1+2m){k}_{u}{\bar{\beta }}_{z}{ct}}\right)\ \delta (z-{z}_{j}(t)),\end{eqnarray} \tag{ 19 }$$

where $\theta =({k}_{r}+{k}_{u})z-{\omega }_{r}t$ is the ponderomotive phase of the fundamental wavelength. Resonant terms are only seen to occur for $n\pm 1+2m=0$ and, as m is integer, the harmonic numbers n are therefore odd. Applying this resonant condition yields:

$$\begin{eqnarray}&&\left(\displaystyle \frac{\partial }{\partial z}+\displaystyle \frac{1}{c}\displaystyle \frac{\partial }{\partial t}\right){{ \mathcal E }}_{n}=\displaystyle \frac{e{\mu }_{0}{c}^{2}{\bar{a}}_{u}}{\sqrt{2}\sigma {(1+{u}_{e}^{2})}^{3/2}}\displaystyle \sum _{j=1}^{N}\displaystyle \frac{{{\rm{e}}}^{-{\rm{i}}n\theta }}{{\gamma }_{j}}{{JJ}}_{n}\ \delta (z-{z}_{j}(t)).\end{eqnarray} \tag{ 20 }$$

The scaling of [15, 16] is now applied using the FEL parameter $\rho ={\gamma }_{r}^{-1}{({\bar{a}}_{u}{\omega }_{p}/4{{ck}}_{u})}^{2/3}$, where ${\omega }_{p}$ is the peak non-relativistic plasma frequency of the electron beam. The wave equation for field (20) is also averaged over a radiation wavelength by assuming the field envelope does not change in this interval. The independent variables are the scaled distance through the FEL $\bar{z}=z/{l}_{g}$, and scaled position in the electron beam rest-frame ${\bar{z}}_{1}=(z-c{\bar{\beta }}_{z}t)/{\bar{\beta }}_{z}{l}_{c}=2\rho {\theta }_{j}$, where ${l}_{g}={\lambda }_{u}/4\pi \rho$ and ${l}_{c}={\lambda }_{r}/4\pi \rho$ are respectively the gain length and cooperation length of the FEL interaction at the fundamental (n = 1) in an helical undulator (u_e = 1) [16]. Clearly, and as shown from the scaling below, these lengths are different for interactions at harmonics and in an elliptical undulator.

Introducing the scaled harmonic radiation envelopes:

$$\begin{eqnarray}&&{A}_{n}=\sqrt{\displaystyle \frac{1+{u}_{e}^{2}}{2}}\displaystyle \frac{{\bar{a}}_{u}e{{ \mathcal E }}_{n}}{4\;{{mc}}^{2}{k}_{u}{(\rho {\gamma }_{r})}^{2}},\end{eqnarray} \tag{ 21 }$$

the scaled electron energy ${p}_{j}=({\gamma }_{j}-{\gamma }_{r})/\rho {\gamma }_{r}$ and using the definition of the ponderomotive phase θ, the scaled equations for the 1D FEL interaction in an elliptically polarised undulator including harmonic radiation fields are given by:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{\theta }_{j}}{{\rm{d}}\bar{z}}={p}_{j},\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{p}_{j}}{{\rm{d}}\bar{z}}=-\displaystyle \sum _{n,\mathrm{odd}}{\alpha }_{n}({A}_{n}{{\rm{e}}}^{{\rm{i}}{\theta }_{j}}+{\rm{c.c.}}),\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{\partial }{\partial \bar{z}}+\displaystyle \frac{\partial }{\partial {\bar{z}}_{1}}\right){A}_{n}={\alpha }_{n}\ \chi ({\bar{z}}_{1})\langle {{\rm{e}}}^{-{\rm{i}}{\theta }_{j}}\rangle ,\end{eqnarray} \tag{ 24 }$$

where ${\alpha }_{n}$ are ellipticity dependent coupling parameters given by:

$$\begin{eqnarray}&&{\alpha }_{n}=\displaystyle \frac{{{JJ}}_{n}}{1+{u}_{e}^{2}},\end{eqnarray} \tag{ 25 }$$

and $\chi ({\bar{z}}_{1})=I({\bar{z}}_{1})/{I}_{{pk}}$ is the beam current scaled with respect to its peak value [13]. There is one wave equation of type (24) for each harmonic considered. Notice from (21), that the harmonic field envelopes ${{ \mathcal E }}_{n}$ are scaled so that the $| {A}_{n}{| }^{2}$ are proportional to the power of the elliptically polarised harmonic radiation fields over the full range of u_e, from planar to helical polarisation.

Figure 1 plots the elliptical coupling parameters ${\alpha }_{n}$ as a function of the ellipticity parameter u_e for the resonant odd harmonics $n=1\ldots 7$ and for a range of rms undulator parameters ${\bar{a}}_{u}$. The coupling parameters agree with previous results in the helical and planar limits. It is worth noting that for the harmonic fields $n\gt 1$, and for larger undulator parameters ${\bar{a}}_{u}$, that the coupling is stronger for elliptically polarised undulators rather than the planar case of u_e = 0. This result is perhaps somewhat unexpected.

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If the equations for the elliptical model (22)–(24) are written in the absence of any harmonic interactions, i.e. for n = 1 only, then the elliptical coupling parameter ${\alpha }_{1}$ could be incorporated into the scaling to give a system of universally scaled equations with no free parameters [15]. In this case the FEL scaling parameter would now depend upon the elliptical coupling parameter for the fundamental as $\rho \propto {\alpha }_{1}^{2/3}$, so that the gain length of the interaction, and so also the saturation length z_sat, would scale as ${l}_{g},{z}_{\mathrm{sat}}\propto {\alpha }_{1}^{-2/3}$. The scaled saturation power would scale as $| A{| }_{\mathrm{sat}}^{2}\propto {\alpha }_{1}^{2/3}$.

In the simulations which follow, an electron pulse of charge 70 pC is assumed with a uniform current, $\chi ({\bar{z}}_{1})=1$, over scaled pulse length of ${\bar{l}}_{e}={l}_{e}/{l}_{c}=129$. A mean beam energy ${\gamma }_{r}=1500$ with zero energy spread and an FEL parameter of $\rho =2\times {10}^{-3}$ is used. Unless otherwise stated, the undulator has fixed rms undulator parameter of ${\bar{a}}_{u}=1.0$ independent of the undulator ellipticity, to give a fixed resonant radiation wavelength of ${\lambda }_{r}=16$ nm. A seed laser of scaled amplitude of ${A}_{0}={10}^{-4}$ was used to initiate the FEL interaction. This eliminates shot-to-shot variation of the radiation pulse saturation energy and saturation length which occurs when the interaction starts from noise, simplifying comparison with analysis and the results obtained from the solutions of the different numerical codes. The total scaled energy of an harmonic of the radiation pulse is defined by:

$$\begin{eqnarray}&&{E}_{n}(\bar{z})={\int }_{-\infty }^{+\infty }| {A}_{n}(\bar{z},{\bar{z}}_{1}){| }^{2}\;{\rm{d}}{\bar{z}}_{1}.\end{eqnarray} \tag{ 26 }$$

with the total given by the sum over the odd harmonics $E={\sum }_{n,{odd}}{E}_{n}$. As the electron pulse is many cooperation lengths long (${\bar{l}}_{e}=129$) and the interaction is seeded, the interaction will approximate a steady-state interaction where pulse effects are small. In this case, the scaled pulse energy at saturation, either for a particular harmonic component n or for the total, will be ${E}_{\mathrm{sat}}\approx \bar{{l}_{e}}| A{| }_{\mathrm{sat}}^{2}$. For an helical undulator in the steady-state, the scaled saturation power of the fundamental (n = 1) is $| A{| }_{\mathrm{sat}}^{2}\approx 1.37$. For the case considered here this gives a scaled pulse energy at saturation of ${E}_{\mathrm{sat}}\approx 177$.

In order to test the above scaling for the scaled saturation energy and saturation length, the equations (22)–(24) were solved numerically for the above parameters in the absence of any harmonic interaction for a range of undulator ellipticities. Figure 2 demonstrates that the numerical solutions are in very good agreement with the predicted scaling.

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Numerical solutions to the averaged elliptical model of equations (22)–(24) are now compared with the those generated by the unaveraged code Puffin [14], which is able to model an FEL interaction in an elliptically polarised undulator across a broad bandwidth radiation field that includes harmonic content. The unaveraged electron motion of the Puffin model includes any 'jitter' motion of equation (9) due to an elliptically polarised undulator.

As Puffin is an unaveraged FEL simulator, the effects of self amplified coherent spontaneous emission can be significant when modelling a 'flat-top' electron bunch which has discontinuities in the electron beam current. As these effects cannot be modelled in an averaged model, the electron bunch used in the Puffin simulations here is modified to have smooth ramp down in current over several radiation wavelengths at the electron bunch edges. This smooth ramping of the current significantly reduces the generation of any coherent spontaneous emission, enabling a better comparison between the two models.

In what follows, only the fundamental and third harmonics (n = 1, 3) are modelled using the above parameters. In the averaged model, the harmonic radiation content is obtained directly from the individual harmonic components, A_n. In the unaveraged model, however, access to the content of each harmonic is obtained by fourier filtering the broadband radiation field about a narrow bandwidth of the particular harmonic of interest (in this case for n = 3.)

Figure 3 plots the scaled pulse energy of the fundamental E₁, from the averaged and Puffin simulations as a function of scaled propagation distance through the interaction $\bar{z}$, for three different undulator ellipticities, ${u}_{e}=0,0.5,1.0$. Excellent agreement between the simulations is seen for all u_e, well into the saturated, nonlinear regime.

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The scaled radiation pulse energies E_n of the fundamental and third harmonic for both averaged and unaveraged simulations for the planar undulator (u_e = 0) are shown in figure 4. As previously seen in figure 3, the fundamental pulse energies E₁ of the averaged and unaveraged simulations are in excellent agreement. The third harmonic shows reasonable agreement in the decoupled linear regime until $\bar{z}\approx 11$. At this point in the averaged model, the electron bunching at the fundamental also begins to drive the third harmonic field with a growth rate ∼3 times that of the fundamental [17]. While there is evidence of similar enhanced harmonic growth in the unaveraged simulation, the effect is seen to be significantly less pronounced. As the interaction proceeds into the nonlinear, saturation regime for $\bar{z}\gt 13$, both simulations are seen to resume a similar evolution.

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It was noted from figure 1 that for larger undulator parameters ${\bar{a}}_{u}$, the coupling parameters ${\alpha }_{n}$ for harmonics maximise for an elliptical undulator configuration, ${u}_{e}\gt 0$. This increased coupling can be expected to decrease the gain length and increase the saturation pulse energies of harmonics for these elliptical polarisations. In particular, the gain length for the third harmonic in an undulator with parameter ${\bar{a}}_{u}=5.0$, should be minimised for an elliptical undulator with ${u}_{e}\approx 0.34$. From the above scaling (and writing ${l}_{g}({u}_{e})$, etc) the ratio of the two gain lengths ${l}_{g}(0.34)/{l}_{g}(0)=0.934$.

Both the averaged and unaveraged numerical models were also used to simulate both undulator ellipticities ${u}_{e}=(0,0.34)$ for the same value of ${\bar{a}}_{u}=5.0$. The results are shown in figure 5. The simulations are seen to agree well with each other in the linear regime with the elliptical undulator measured as having the shorter gain length ${l}_{g}(0.34)/{l}_{g}(0)\approx 0.931$, in good agreement with the value calculated from scaling.

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A similar scaling argument for the electron pulse energies at saturation gives ${E}_{3}(0.34)/{E}_{3}(0)=1.071$ which is more difficult to compare with the simulations of figure 5 due to the problem in defining the points of saturation. Note again, the difference in the simulation results between the averaged and unaveraged models as saturation is approached and the fundamental interaction drives that of the harmonic. The divergence between the two models is probably more pronounced in this case where ${\bar{a}}_{u}=5.0$, than that of figure 4 where ${\bar{a}}_{u}=1.0$.