2. Equations of Motion and Modes of Electromagnetic Field

Consider a bunch of electrons accelerated in wakefields generated by a laser pulse propagating from left to right along the 0z axis of a capillary waveguide filled with plasma with a constant electron concentration n. The entrance face of the capillary is at the point z = 0. The motion of electrons in cylindrically symmetric wakefields behind the laser pulse is determined by the relativistic equations of motion, which have the following form in Cartesian coordinates [Reference Andreev and Kuznetsov20]:

(1) $\begin{array}{}\frac{\text{d}{P}_z}{\text{d}\tau }={\partial }_{\xi }\varphi ,\end{array}$

(2) $\begin{array}{}\frac{\text{d}\xi }{\text{d}\tau }={\gamma }_e^{-1}{P}_z-1,\end{array}$

(3) $\begin{array}{c}\frac{\text{d}{P}_x}{\text{d}\tau }=\left(\frac{x}{\rho }\right){\partial }_{\rho }\varphi ,\frac{\text{d}{P}_y}{\text{d}\tau }=\left(\frac{y}{\rho }\right){\partial }_{\rho }\varphi ,\end{array}$

(4) $\begin{array}{c}\frac{\text{d}x}{\text{d}\tau }={\gamma }_e^{-1}{P}_x,\frac{\text{d}y}{\text{d}\tau }={\gamma }_e^{-1}{P}_y,\end{array}$

where $\varphi =\varphi \left(\xi ,\rho ,\tau \right)$ is the normalized on mc ²/e wakefield potential [Reference Andreev, Gorbunov, Kirsanov, Nakajima and Ogata21]; x = k_px, y = k_py, k_p = ω_p/c, and ${\omega }_p=\sqrt{4\pi n{e}^2/m}$ ; e, m, and c are electron charge, mass, and speed of light, respectively; P_α, α = x, y, z, are dimensionless on mc components of electrons momentum; ${\gamma }_e=\sqrt{1+{\left(P\right)}^2}$ is the electron gamma factor; $\xi =k_p\left(z-ct\right)$ , τ = ω_pt, ρ = k_pr, and $\tilde{\xi }=k_0\left(z-ct\right)$ , and $\tilde{\tau }={\omega }_0\tau$ (where k ₀ = ω₀/c, ${\omega }_0=2\pi c/{\lambda }_0$ is the laser pulse frequency, and λ ₀ is the laser wavelength) are dimensionless coordinates which will be used hereafter.

In the considered case of propagation of cylindrically symmetric laser pulses in capillary waveguide angular harmonics [Reference Veisman, Kuznetsov and Andreev22, Reference Veysman and Andreev23] of the wake potential, ϕ are absent and it can be determined using the following simple equation, written under the assumption of a constant electron density inside the capillary [Reference Joshi, Corde and Mori2, Reference Andreev, Gorbunov, Kirsanov, Nakajima and Ogata21, Reference Andreev, Cros, Gorbunov, Matthieussent, Mora and Ramazashvili24]:

(5) $\begin{array}{}\left(\frac{{\partial }^2}{\partial {\xi }^2}+1\right)\varphi =\frac{{\left(a_{\perp }\right)}^2}{4},\end{array}$

where $a_{\perp }=e{E}_{\perp }/\left(m{\omega }_{0}c\right)$ is the dimensionless transverse (with respect to direction of propagation 0z) complex amplitude of electromagnetic field strength, slowly varying in time scale ${\omega }_0^{-1}$ and at space length λ ₀. Solution of (5) is

(6) $\begin{array}{}\varphi \left(\xi ,\rho ,\tau \right)=\frac{1}{4}\underset{\infty }{\overset{\xi }{\int }}\sin \left(\xi -{\xi }^{\prime }\right){\left(a_{\perp }\right)}^2\left({\xi }^{\prime },\rho ,\tau \right)\text{d}{\xi }^{\prime }.\end{array}$

Assuming that the maximal power P ₀ of the laser pulse is much less than the critical power P_cr for the relativistic self-focusing, $P_0\ll P_{\text{cr}}=0.017{\left({\omega }_0/{\omega }_p\right)}^2$ TW, and disregarding all nonlinear process of laser field propagation inside a capillary waveguide, one can express ${\left(a_{\perp }\right)}^2$ in terms of the sum of its radial eigenmodes as follows [Reference Veisman, Kuznetsov and Andreev22, Reference Abrams25–Reference Lotov27]:

(7) $\begin{array}{}{\left(a_{\perp }\right)}^2=a_{\max }^2{\left(\sum _{n}4{\widetilde{ℭ}}_n\left(\xi ,\tau \right)D_n\left(\rho \right)\right)}^2,\end{array}$

where a _max is the maximum value of the module of dimensionless complex amplitude of electromagnetic field strength.

(8) $\begin{array}{}{\widetilde{ℭ}}_n\left(\xi ,\tau \right)=C_n{F}_{\Vert }\left(\tilde{\xi }+{\Phi }_n\left(\tilde{\tau }\right)\right)\exp \left(-i{\Phi }_n\left(\tilde{\tau }\right)\right),\end{array}$

where $\tilde{\xi }={\gamma }_{\text{ph}}\xi ,\tilde{\tau }={\gamma }_{\text{ph}}\tau$ , ${\gamma }_{\text{ph}}={\omega }_0/{\omega }_p$ is the gamma factor of the plasma wave, $F_{\Vert }\left(\tilde{\xi }\right)$ is the longitudinal envelop of the laser pulse before its entrance into the guiding structure, and C_n are modes coefficients, determined by boundary conditions at the guiding structure entrance z = 0 [Reference Veysman, Cros, Andreev and Maynard7, Reference Andreev, Cros, Gorbunov, Matthieussent, Mora and Ramazashvili24]. The expressions for phase factors Φ_n and radial modes D_n(r) in the vicinity of the guiding structure axis 0z in the considered case of symmetric propagation of laser fields in capillary waveguides can be written as

(9) $\begin{array}{}{D}_n=J_0\left(\frac{u_{0,n}\rho }{\left(k_{p}R\right)}\right),\end{array}$

(10) $\begin{array}{c}{\Phi }_n=\left(\frac{\tilde{\tau }}{2}\right)\left(k_{\perp n}^2+1-{\epsilon }_c\right),k_{\perp n}=\frac{u_{0,n}}{ℛ}\left(1-i\frac{{\mu }_w}{ℛ}\right),\end{array}$

where J ₀ and u _0,n are zero-order Bessel function and its n-th root, respectively; $ℛ=k_{0}R$ ; R is the inner radius of a capillary; ${\epsilon }_c=1-{\gamma }_{\text{ph}}^{-2}$ is the permittivity of a plasma inside the capillary waveguide; and factor ${\mu }_w=\left({\epsilon }_w+1\right)/\left(2\sqrt{{\epsilon }_w-1}\right)$ is dependent on capillary wall properties, described by its permittivity ε_w.

Below we assume that the transverse size σ_r of the accelerated electron bunch is much smaller than the characteristic transverse scale of the wakefield determined by the driving laser spot size r ₀. For this case, expanding the wakefield potential near the axis 0z over power series of the value r/r ₀, one can omit all terms except quadratic one $\varphi \left(r⟶0\right)\sim {\left(r/r_0\right)}^2$ . This corresponds to a linear dependence of the radial focusing force $F_{\rho }=\partial \varphi /\partial \rho$ on distance ρ from capillary axis. In accordance with (3) and (4), the electron trajectory in the transverse plane xy is determined in this case by the following equations:

(11) $\begin{array}{}\frac{{\text{d}}^{2}x}{\text{d}{\tau }^2}+\frac{\text{d}\ln {\gamma }_e}{\text{d}\tau }\frac{\text{d}x}{\text{d}\tau }+{\Omega }^{2}x=0,\end{array}$

(12) $\begin{array}{c}\Omega \left(\xi ,\tau \right)=\sqrt{\frac{\alpha \left(\xi ,\tau \right)}{{\gamma }_e\left(\xi ,\tau \right)}},{\alpha \left(\xi ,\tau \right)=-{\rho }^{-1}{\partial }_{\rho }\varphi |}_{\rho =0}.\end{array}$

The equation for y coordinate coincides with equation (11) after replacement of x by y. Equation (11) describes the betatron oscillations of electrons in the transverse plane xy with the frequency Ω.

In (12), it is assumed that α > 0; that is, electrons are injected and accelerated in the focusing phase of the wakefields. As is known (see, for example, [Reference Veysman, Andreev, Maynard and Cros28]), near the entrance to the capillary, matched to the laser spot size, the laser and wakefields are not regular enough due to the beats of high eigenmodes. This leads here to the inevitable formation of defocusing regions. To exclude the influence of such regions on the propagation of electrons, it is necessary to inject electrons at a sufficient distance from the entrance to the capillary, at a certain length L _inj, sufficient to filter higher modes due to their damping stipulated by energy losses through the walls of the capillary. In what follows, we assume that such filtering has been done.

It can be shown [Reference Veisman and Andreev18] that, in the case of relativistic electrons moving along the 0z axis with a speed close to the speed of light, it is possible to separate their longitudinal and transverse motion that greatly simplifies the analysis of their betatron oscillations.

Having solved the equations of electron motion in the laser wakefields, one can determine the normalized transverse emittance [Reference Mehrling, Grebenyuk, Tsung, Floettmann and Osterhoff16–Reference Veisman and Andreev18]:

(13) $\begin{array}{}\begin{array}{c}{\epsilon }_n=\sqrt{2\left({\epsilon }_{x,n}^2+{\epsilon }_{y,n}^2\right)},\\ {\epsilon }_{x,n}=\overline{{\gamma }_e}{\epsilon }_x,\\ {\epsilon }_x=\sqrt{\overline{x^2}\overline{x^{\prime 2}}-{\overline{x{x}^{\prime }}}^2},\end{array}\end{array}$

of the bunch of electrons, which determines the degree of their angular and spacial spread, important for applications. Here, $x^{\prime }=\text{d}x/\text{d}z=\dot{x}/\dot{z}=P_x/P_z$ for considered relativistic electrons moving along axis 0z; the bar above some value X means ensemble averaging; $\overline{X^2}=N_b^{-1}{\sum }_i{\left(X_i-\overline{X}\right)}^2,\overline{X}=N_b^{-1}{\sum }_i{X}_i$ , where index i refers to i-th particle; N_b is the full number of particles; and ${\epsilon }_n=2{\epsilon }_{x,n}=2{\epsilon }_{y,n}$ for considered cylindrically symmetric case.

3. Wakefields in Capillaries and Parametric Excitations of Betatron Oscillations

Taking in mind (6)–(10), one can write down the following expression for wakefield potential ϕ:

(14) $\begin{array}{c}\varphi \left(\xi ,\rho ,\tau \right)={\int }_{\infty }^{\xi }\sin \left(\xi -{\xi }^{\prime }\right)\frac{a_0^2\left({\xi }^{\prime },\rho ,\tau \right)}{4}{\stackrel{⌣}{\kappa }}_g\left({\xi }^{\prime },\rho ,\tau \right)\left(1+\sum _{n,k>n}{\stackrel{⌣}{\mu }}_{nk}\left({\xi }^{\prime },\rho ,\tau \right)\cos \left({\Omega }_{nk}\tau \right)\right)\text{d}{\xi }^{\prime },{\stackrel{⌣}{\kappa }}_g\left(\xi ,\rho ,\tau \right)=\frac{{\sum }_n{\tilde{C}}_n^2\left(\tau \right)F_{\left(n\right)}^2\left(\xi ,\tau \right)D_n^2\left(\rho \right)}{{\left({\sum }_n{C}_n{F}_{\left(n\right)}\left(\xi ,\tau \right)D_n\left(\rho \right)\right)}^2},\\ {\stackrel{⌣}{\mu }}_{nk}\left(\xi ,\rho ,\tau \right)=\frac{2{\tilde{C}}_n\left(\tau \right){\tilde{C}}_k\left(\tau \right)F_{\left(n\right)}\left(\xi ,\tau \right)F_{\left(k\right)}\left(\xi ,\tau \right)D_n\left(\rho \right)D_k\left(\rho \right)}{{\sum }_n{\tilde{C}}_n^2\left(\tau \right)F_{\left(n\right)}^2\left(\xi ,\tau \right)D_n^2\left(\rho \right)},\end{array}$

where a ₀ is the nonoscillating part of the module of complex amplitude $\left(a_{\perp }\right)$ ,

(15) $\begin{array}{c}{a}_0\left(\xi ,\rho ,\tau \right)=a_{\max }\sum _{n}2C_n{F}_{\left(n\right)}\left(\xi ,\tau \right)D_n\left(\rho \right);{\tilde{C}}_n=C_n{e}^{-k_n^{\prime }\tilde{\tau }},\\ k_n^{\prime }=\frac{1}{2}\left(\Re k_{\perp ,n}^2+{\gamma }_{\text{ph}}^{-2}\right),\\ k_n^{\prime }=\frac{1}{2}\left(\Im k_{\perp ,n}^2\right),\\ {\Omega }_{nk}={\gamma }_{\text{ph}}\left(k_k^{\prime }-k_n^{\prime }\right),\end{array}$

and k_⊥,n is given by (10):

(16) $\begin{array}{}{F}_{\left(n\right)}\left(\xi ,\tau \right)=\exp \left(\frac{-{\left(\delta \xi +\tau k_n^{\prime }\right)}^2}{L_{\xi }^2}\right),\end{array}$

where $\delta \xi =\xi -{\xi }_{\text{Lc}}$ , with ${\xi }_{\text{Lc}}$ as the coordinate on ξ of the laser pulse center; expression (16) for $F_{\Vert n}$ is written, for simplicity, with assumption of Gaussian longitudinal envelop of the laser pulse and taking in mind smallness of the parameter ${ℛ}^{-1}\ll 1$ ; and $L_{\xi }=k_{p}c{t}_{\text{FWHM}}/\sqrt{2\ln 2}$ , where t _FWHM is the full width at half of the maximum intensity of laser pulse.

In accordance with (6)–(10), beats of electromagnetic field modes inside a capillary waveguide due to phase factors Φ_n (8) and (10) give rise to terms $\sim \cos \left({\Omega }_{nk}\tau \right)$ in (14). These terms lead to oscillations with capillary length z of the maximum (on coordinate ξ) value of the wakefield potential at capillary axis.

Assuming that the center of the electron bunch is injected in the focusing phase of the wakefield in the vicinity of the maximum of the longitudinal acceleration force, we can obtain the following expression for the potential of the wakefield seen by these electrons at the moment τ:

(17) $\begin{array}{c}\varphi \left({\xi }_{\text{inj}},\rho ,\tau \right)=a_{\max }^{2}A{\kappa }_g\left({\xi }_{\text{inj}},\rho ,\tau \right)\left(1+\sum _{n,k>n}{\mu }_{nk}\left({\xi }_{\text{inj}},\rho ,\tau \right)\cos \left({\Omega }_{nk}\tau \right)\right),{\kappa }_g\left({\xi }_{\text{inj}},\rho ,\tau \right)={\sum }_{n}4C_n^2{\Gamma }_n\left(\tau \right)D_n^2\left(\rho \right)\sin \left({\xi }_{\text{inj}}+\frac{\tau }{2}\left(\frac{1}{{\gamma }_{\text{ph}}^2}-\frac{1}{{\gamma }_e^2}+\frac{u_{0n}^2}{{ℛ}^2}\right)\right),\\ {\mu }_{nk}\left({\xi }_{\text{inj}},\rho ,\tau \right)=\frac{8}{{\kappa }_g}{C}_n{C}_k{D}_n\left(\rho \right)D_k\left(\rho \right){\Gamma }_{nk}\left(\tau \right)\sin \left({\xi }_{\text{inj}}+\frac{\tau }{2}\left(\frac{1}{{\gamma }_{\text{ph}}^2}-\frac{1}{{\gamma }_e^2}+\frac{u_{0n}^2+u_{0k}^2}{2{ℛ}^2}\right)\right),\\ {\Gamma }_n\left(\tau \right)=\exp \left(-2\frac{u_{0,n}^2}{{ℛ}^3}{\mu }_w{\gamma }_{\text{ph}}\tau \right),\\ {\Gamma }_{nk}\left(\tau \right)=\exp \left(-\frac{u_{0,n}^2+u_{0,k}^2}{{ℛ}^3}{\mu }_w{\gamma }_{\text{ph}}\tau -\frac{1}{2}{\left(\frac{\tau {\Omega }_{nk}}{{\gamma }_{\text{ph}}{L}_{\xi }}\right)}^2\right),\end{array}$

where ξ _inj is the distance between the point of injection of the center of the electron bunch at τ = 0 and the point of maximum of the accelerating force (which is also the point of zero transverse force $F_{\perp }$ , i.e., the point of the boundary between the focusing and defocusing phases of the wakefield potential); γ_e is the gamma factor of an electron bunch, the transverse and longitudinal dimensions of which are assumed to be much smaller than the corresponding characteristic dimensions of the wakefield that makes it possible to disregard the energy spread of electrons in the bunch; and $A=\sqrt{\pi /2}{L}_{\xi }{e}^{-L_{\xi }^2/8}/4$ , its maximum value $\max A\approx 0.38$ at $L_{\xi }=2$ (resonance condition of plasma wake excitation).

From (17), one can obtain the following expression for the maximum in the ξ value of the wakefield potential on the capillary axis ${\varphi }_{m\xi }\left(\tau \right)={\max }_{\xi }\varphi \left(\xi ,\rho =0,\tau \right)$ , assuming that the first mode has the largest amplitude:

(18) $\begin{array}{}{\varphi }_{m\xi }\left(\tau \right)=4a_{\max }^{2}A\left(\sum _n{C}_n^2{\Gamma }_n\left(\tau \right)\cos \left(\frac{{\Omega }_{1n}}{{\gamma }_{\text{ph}}}\tau \right)+2\sum _{n,k>n}{C}_n{C}_k{\Gamma }_{nk}\left(\tau \right)\cos \left(\frac{{\Omega }_{1n}+{\Omega }_{1k}}{2{\gamma }_{\text{ph}}}\tau \right)\cos \left({\Omega }_{nk}\tau \right)\right),\end{array}$

where Ω₁₁ = 0 and ${\Omega }_{nk>n}={\gamma }_{\text{ph}}\left(u_{0,k}^2-u_{0,n}^2\right)/\left(2{ℛ}^2\right)$ in accordance with (10) and (15).

The expression (18) with τ = k_pz, taking into account only the first two modes, well describes the main details of the oscillations of the wakefield potential due to mode beats, including amplitude, phase, and frequency (see Figure 1, where the case of a silicon capillary with a radius matched to the laser spot size (when r ₀/R ≈ 0.61) is considered). Particularly, for this case, one has from (18) with n, k = 1, 2:

(19) $\begin{array}{}\frac{{\varphi }_{m\xi }\left(z\right)}{{\varphi }_{m\xi }}\left(z=0\right)\approx 0.67e^{-0.2y}+0.023e^{-1.07y}\cos \left(1.2y\right)+0.25e^{-0.64y-0.18y^2}\cos \left(0.6y\right)\cos \left(96y\right),\hspace{1em}y\equiv \frac{z}{L_{\text{ph}}},\end{array}$

where $L_{\text{ph}}={\lambda }_0{\gamma }_{\text{ph}}^3$ is the dephasing length [Reference Andreev and Gorbunov29]. Nevertheless, the finer structure of oscillations of the wakefield amplitude, caused by higher modes, can be essential, as will be shown below, as long as it can leads to parametric excitation of betatron oscillations of electrons propagating in a capillary. This structure, due to the first 6 modes, can also be described with high accuracy by the expression (18) for the capillary length $z/L_{\text{ph}}>0.07$ , as seen from Figure 1. For $z/L_{\text{ph}}<0.07$ , the interference of higher modes leads to an irregular structure of the laser and wakefields, which determines the need for their filtering, as indicated above. In particular, for the case considered in Figure 1, we can write expressions for the damping factors of the first few modes in the form ${\Gamma }_n\left(z\right)=\exp \left(-{\Upsilon }_{n}z/L_{\text{ph}}\right)$ with ${\Upsilon }_1\approx 0.202,{\Upsilon }_2\approx 1.07,{\Upsilon }_3\approx 2.62,{\Upsilon }_4\approx 4.87,{\Upsilon }_5\approx 7.80,{\Upsilon }_6\approx 11.43,{\Upsilon }_7\approx 15.8,{\Upsilon }_8\approx 20.8,{\Upsilon }_9\approx 26.5,{\Upsilon }_{10}\approx 32.9,{\Upsilon }_{15}\approx 75.2$ . Therefore, for $z>0.1L_{\text{ph}}$ , one has ${\Gamma }_{10}<5\times {10}^{-2},{\Gamma }_{15}<{10}^{-3}$ that means effective filtering of high modes for such distances.

Figure 1: The maximum in the ξ value of the wakefield potential on the capillary axis, ${\varphi }_{m\xi }\left(z\right)$ , normalized to the value ${\varphi }_{m\xi }\left(z=0\right)$ , as a function of the propagation length z = ct of the laser pulse in the capillary, for the dimensionless maximum amplitude of the laser field a _max = 0.5, laser wavelength λ ₀ = 0.8 μm, plasma wave gamma factor ${\gamma }_{\text{ph}}=80$ , exponential width of the Gaussian transverse distribution of the laser field at the capillary entrance r ₀ = 50 μm, the full width of the laser pulse at half its maximum intensity (at z = −0), t _FWHM = 80 fs; and a silicon capillary waveguide (ε_w = 2.25) has a radius of R = 82 μm (the ratio r ₀/R ≈ 0.61 is close to the condition for the best laser energy input into a capillary [Reference Dorchies, Marquès and Cros26]). The length z along the capillary is normalized on the dephasing length $L_{\text{ph}}={\lambda }_0{\gamma }_{{\text{ph}}^3}\approx 41\text{cm}$ . The complete numerical solution (14) (with τ = k_pz) and the simplified analytical solution (18) with 2 and 6 modes are shown by different curves (see figure legend).

From (17), one can obtain an expression for the coefficient α (12) of the linear dependence of the focusing force near the capillary axis on the radius ρ and, therefore, an expression for the betatron frequency Ω (12) in equation (11) for betatron oscillations. This expression can be written as

(20) $\begin{array}{c}{\Omega }^2\left(\tau \right)={\Omega }_{\beta 0}^2\left(\tau \right)K_g\left(\tau \right)\left(1+\sum _{n,k>n}{\nu }_{nk}\left(\tau \right)\cos \left({\Omega }_{nk}\tau \right)\right),{\Omega }_{\beta 0}\left(\tau \right)=\frac{2a_{\max }}{k_{p}R}\sqrt{\frac{A}{{\gamma }_e\left(\tau \right)}},\\ K_g\left(\tau \right)=\sum _n{C}_n^2{u}_{0,n}^2{\Gamma }_n\left(\tau \right)\sin \left({\xi }_{\text{inj}}+\frac{\tau }{2}\left(\frac{1}{{\gamma }_{\text{ph}}^2}-\frac{1}{{\gamma }_e^2\left(\tau \right)}+\frac{u_{0n}^2}{{ℛ}^2}\right)\right),\\ {\nu }_{nk}\left(\tau \right)=K_g^{-1}\left(\tau \right)C_n{C}_k\left(u_{0,n}^2+u_{0,k}^2\right){\Gamma }_{nk}\left(\tau \right)\sin \left({\xi }_{\text{inj}}+\frac{\tau }{2}\left(\frac{1}{{\gamma }_{\text{ph}}^2}-\frac{1}{{\gamma }_e^2\left(\tau \right)}+\frac{u_{0n}^2+u_{0k}^2}{2{ℛ}^2}\right)\right).\end{array}$

The dynamic of gamma-factor ${\gamma }_e\left(\tau \right)$ of an electron bunch can be estimated, taking into account longitudinal equation of motion, as Veisman and Andreev [Reference Veisman and Andreev18]:

(21) $\begin{array}{}{\gamma }_e\left(\tau \right)={\gamma }_{\text{inj}}+2{\gamma }_{\text{ph}}^{2}A{a}_{\max }^2\left(\sin \left({\xi }_{\text{inj}}+\frac{\tau }{2}\left({\gamma }_{\text{ph}}^{-2}+\frac{u_{01}^2}{{ℛ}^2}\right)\right)-\sin \left({\xi }_{\text{inj}}\right)\right),\end{array}$

where γ _inj is the electrons gamma factor at the moment of injection.

Equation (20) shows that the squared frequency of betatron oscillations of electrons in a capillary waveguide

(22) $\begin{array}{}{\Omega }_{\text{cap}}^2\left(\tau \right)={\Omega }_{\beta 0}^2\left(\tau \right)K_g\left(\tau \right)\end{array}$

is modulated in time τ with frequencies Ω_nk and amplitudes ν_nk. In accordance with the common theory of parametric oscillator Landau and Lifschitz [Reference Landau and Lifschitz30], this leads to parametric excitation of electrons betatron oscillations (the frequency Ω (20) changes with time τ due to the change in electrons gamma factor stipulated by their acceleration, due to attenuation of modes and dispersion of their group velocities and due to the difference between electron bunch velocity, which is close to the speed of light and the plasma wave phase velocity, manifested in dependence on τ of the factors K_g and ν_nk in equation (20), but this change occurs adiabatically at betatron period [Reference Veisman and Andreev18], and therefore, we assume that the concept of parametric resonance is applicable for the case under consideration) provided Ω_cap (τ) is close (or becomes close) to resonance frequencies (higher order resonances with lower increments are also possible for frequencies which are multiples of frequencies (23) [Reference Landau and Lifschitz30]):

(23) $\begin{array}{}{\Omega }_{Rnk}=\frac{{\Omega }_{nk}}{2}.\end{array}$

Maximum increments S_nk of parametric resonances at frequencies (23) can be written, in accordance with Landau and Lifschitz [Reference Landau and Lifschitz30], as

(24) $\begin{array}{}{S}_{nk}=\frac{{\nu }_{nk}{\Omega }_{Rnk}}{4}.\end{array}$

The increments s_nk of the betatron oscillation amplitude are nonzero in a certain interval of betatron oscillation frequencies, in the vicinity of the resonance frequencies Ω_Rnk (23). This interval is specified by the condition that, in the following (25) expressions for s_nk root, arguments exceed zero:

(25) $\begin{array}{}{s}_{nk}=\sqrt{S_{nk}^2-{\left(\Omega -{\Omega }_{Rnk}\right)}^2},\end{array}$

where Ω_Rnk are given by (23) and S_nk are given by (24).

If $r_{\beta ,0}\left({\tau }_0\right)$ is the amplitude of the betatron oscillations of electrons injected at τ = τ ₀ at the point k_pL _inj = τ ₀ along the capillary length, then at a certain length $z=ct>L_{\text{inj}}$ inside the capillary, at the time τ = k_pz, this amplitude can be estimated as $r_{\beta }\left(\tau \right)=r_{\beta ,0}\left({\tau }_0\right)G_{nk}\left(\tau ,{\tau }_0\right)$ , where G_nk is the highest resonance growth factor for a given γ _inj:

(26) $\begin{array}{}{G}_{nk}\left(\tau ,{\tau }_0\right)=\exp \left({\int }_{{\tau }_0}^{\tau }{s}_{nk}\left({\tau }^{\prime }\right)\text{d}{\tau }^{\prime }\right).\end{array}$

The growth factors arising from the beatings of the first few modes in a matched capillary waveguide are shown in Figure 2.

Figure 2: The growth factors G_nk of betatron oscillations amplitudes as function of gamma-factors of injected electrons γ _inj, calculated by formulas (20)–(25). G_nk are estimated for a capillary waveguide with the same parameters as in Figure 1, for the duration of electron acceleration $\Delta \tau =\tau -{\tau }_0=0.2k_p{L}_{\text{ph}}$ (where τ ₀ = k_pL _inj indicates the start of acceleration; τ ₀ = 3800 ≈ 0.093k_pL _ph in calculations). The growth factors are due to the beating of the modes with the numbers n, k indicated on the legend.

From Figure 2, it is clear that beating of higher-order modes ( $n,k>n,k\ge 5$ ) leads to very high increments of parametric instability linear theory predicting high growth factors of parametric instability and becomes inapplicable when the amplitudes of betatron oscillations increase so strongly that they become comparable with the capillary radius R for electrons with relatively low injection energies γ _inj < 150. Instability regions are within the intervals γ _inj < 700, 10³ < γ _inj < 210³, 910³ < γ _inj < 2.510⁴.

Figure 3 shows the dynamics of changes in the normalized emittance ε_n (13), the root-mean-square radius of the accelerated electron bunch ${\left(r^2\right)}^{1/2}$ , and the average increase in the electron energy during acceleration $\left(E\right)-E_{\text{inj}}$ (where E _inj is the energy of the electron bunch at the moment of injection). Curves are shown for electrons with different injection energies propagating in a matched capillary waveguide. Electron dynamics in the wakefields given by the expression (14) was determined by means of numerical solution of the system of equations (1)–(4) for each electron of the bunch.

Figure 3: Dependencies on the acceleration length z of (a) normalized emittance ε_n, (b) root-mean-square radius of the accelerated electron bunch ${\left(r^2\right)}^{1/2}$ , and (c) average energy gain of electrons during acceleration $\left(E\right)-E_{\text{inj}}=m{c}^2\left({\gamma }_e-{\gamma }_{\text{inj}}\right)$ for matched capillary waveguide. Curves are shown for different energies of injection of electron bunches $E_{\text{inj}}=m{c}^2{\gamma }_{\text{inj}}$ (see legend). Electrons in accelerated bunches have Gaussian distribution in both longitudinal and transverse directions with dimensionless characteristic sizes k_pσ_ξ = 0.14 and k_pσ_r = 0.21, respectively. Electron bunches are injected at the time ${\tau }_0=k_p{L}_{\text{inj}}$ at the point $k_p{L}_{\text{inj}}=0.093k_p{L}_{\text{ph}}=3800$ at the length of the capillary, with zero initial emittance. The center of the electron bunch is injected into the focusing phase of the wakefield at a distance of ξ _inj = 0.2 from the point of maximum of the accelerating force. Other parameters are the same as for Figure 1.

To avoid defocusing due to high (with numbers over 10) modes near the capillary entrance, the electron injection was shifted along capillary length from the capillary entrance by the value $L_{\text{inj}}\approx 0.093L_{\text{ph}}$ . For the selected value of L _inj, modes that violate the regular structure of the wakefield near the capillary entrance are effectively filtered, while the lower modes still survive, and their beating leads to parametric excitation of betatron oscillations of electrons in the corresponding resonance regions. In particular, electrons with γ _inj = 500 and γ _inj = 1500 undergo parametric excitation of their betatron oscillations due to the beats of 1.4 and 1.3 modes, respectively, as follows from the results shown above in Figure 2. This excitement manifests itself in a sharp increase in the rms radius ${\left(r^2\right)}^{1/2}$ and normalized emittance ε _n (see thick and thin solid curves on Figures 3(a) and 3(b)).

On the contrary, electrons with γ _inj = 850 and γ _inj = 7000 are not subjected to parametric excitation of their betatron oscillations since they are outside the parametric instability regions shown in Figure 2. For these electrons, both the root-mean-square radius ${\left(r^2\right)}^{1/2}$ and the normalized emittance ε_n are bounded, respectively, by the values $k_p{\left(r^2\right)}^{1/2}\left(\tau =0\right)=0.21$ and ε_n < 1 mm·mrad.

4. Estimation of Synchrotron Radiation

Regular calculation of the emission spectrum of accelerated electrons requires integration over exact trajectories of electrons [Reference Esarey, Shadwick, Catravas and Leemans31], but estimates of the intensities of the corresponding spectral lines, characteristic frequencies, and spectral widths can be made on the basis of the assumption of harmonic oscillations of electrons with constant energies and amplitudes.

In accordance with [Reference Esarey, Shadwick, Catravas and Leemans31], the energy radiated in the direction 0z (angle θ = 0) in a unit solid angle $\text{d}\Theta$ in a unit frequency range dω (or in a unit energy range $\hslash \text{d}\omega$ ) during $N_{\beta }=z/{\lambda }_{\beta }$ betatron periods (where ${\lambda }_{\beta }=2\pi c/\Omega$ and Ω are the wavelength and frequency of betatron oscillations, respectively), due to synchrotron radiation of an electron, oscillating with constant amplitude r_β can be written as

(27) $\begin{array}{c}\frac{{\text{d}}^2{E}_{\omega }}{\hslash \text{d}\omega \text{d}\Theta }=\sum _n{Q}_n,Q_n=\frac{1}{137}\frac{4\omega }{{\omega }_n}\frac{{\gamma }_e^2{N}_{\beta }^2{R}_n\left(\omega \right)F_n\left({\alpha }_n\right)}{1+a_{\beta }^2},\\ a_{\beta }=\frac{{\gamma }_e\Omega r_{\beta }}{c},\end{array}$

where

(28) $\begin{array}{}{R}_n\left(\omega \right)={\left(\frac{\sin \left(\pi N_{\beta }\left(\omega /{\omega }_1-n\right)\right)}{\pi N_{\beta }\left(\omega /{\omega }_1-n\right)}\right)}^2,\end{array}$

is the resonance function,

(29) $\begin{array}{c}{\omega }_1=\frac{{\omega }_{\text{ef}}}{1+a_{\beta }^2},{\omega }_{\text{ef}}=2{\gamma }_e^2\Omega ,\end{array}$

and a_β is given by (27); the function

(30) $\begin{array}{c}{F}_n\left({\alpha }_n\right)=n{\alpha }_n{\left(J_{\left(n+1\right)/2}\left({\alpha }_n\right)-J_{\left(n+1\right)/2}\left({\alpha }_n\right)\right)}^2,{\alpha }_n=\frac{\omega }{{\omega }_1}\frac{a_{\beta }^2/4}{1+a_{\beta }^2/2},\end{array}$

and $J_{\nu }$ are Bessel functions of the order ν.

Figure 4 shows the computation of ${\text{d}}^2{E}_{\omega }/\left(\hslash \text{d}\omega \text{d}\Theta \right)$ for N_β = 4, with constant γ_e = 2100 and for different $k_p{r}_{\beta }=0.21$ and $k_p{r}_{\beta }=0.63,1.05$ , which estimates, respectively, the unexcited amplitude of electron oscillations and the amplitude of oscillations excited (with growth factors G = 3 and 5) due to parametric resonances, for the parameters shown in Figure 3.

Figure 4: Energy ${\text{d}}^2{E}_{\omega }/\left(\hslash \text{d}\omega \text{d}\Theta \right)$ , radiated in the direction 0z in a unit solid angle $\text{d}\Theta$ in a unit frequency interval dω, during N_β = 4 betatron periods, for different $k_p{r}_{\beta }=0.21,0.63,1.05$ (from left to right subfigures), ${\gamma }_e=2100,\text{and}\Omega k_p{L}_{\text{ph}}=60$ . Other parameters are the same as in Figure 3. The characteristic quantum energy $\hslash {\omega }_{\text{ef}}=260\text{eV}$ and the parameter $a_{\beta }=0.66,1.99,3.32$ for given calculations with $k_p{r}_{\beta }=0.21,0.63,1.05$ , respectively.

The value of betatron frequency Ω in a capillary waveguide, calculated by expressions (20) and (22), was 1.49 × 10⁻³ ( $\Omega k_p{L}_{\text{ph}}=60.0$ ), which corresponds to the frequency of betatron oscillations in the capillary at $\tau =0.28k_p{L}_{\text{ph}}$ of an electron injected with γ _inj = 1500 at ${\tau }_0=0.093k_p{L}_{\text{ph}}$ and for all other parameters that are the same as in Figure 3. The factor G = 5 roughly corresponds to the cases of parametric excitation of betatron oscillations, shown in Figure 3 by thick and thin solid curves. The value γ_e = 2100 corresponds to the moment of acceleration $\tau \approx 0.28k_p{L}_{\text{ph}}$ for electrons injected with γ _inj = 1500 at ${\tau }_0=0.093k_p{L}_{\text{ph}}$ (thin solid curve at Figure 3).

A significant increase in the amplitudes of the first few harmonics of the emitted synchrotron radiation with an increase in k_pr_β from 0.21 till 0.63 and 1.05 and an increase in the width of the emission spectrum are clearly seen. In the considered example of calculations with k_pr_β = 1.05 (corresponding to betatron oscillations growth factor G = 5 under conditions of Figure 3), photons are emitted in the X-ray spectral range with photon energies $\hslash \omega \in \left(90;650\right)\text{eV}$ for the level of the spectral energy flux ${\text{d}}^2{E}_{\omega }/\hslash \text{d}\omega \text{d}\Theta \sim {10}^5$ .

Such modifications of the synchrotron radiation spectrum with a change in the amplitude of the root-mean-square radius of an electron bunch accelerated in a capillary waveguide can serve as a diagnosis of betatron oscillations of the bunch.