Application of transfer matrix and transfer function analysis to grating-type dielectric laser accelerators: ponderomotive focusing of electrons

TL;DR

This paper evaluates the applicability of transfer matrix methods to dielectric laser accelerators and introduces a transfer function approach for better modeling of electron dynamics, including a phase-independent focusing structure.

Contribution

It demonstrates the limitations of transfer matrices for DLA and proposes a transfer function framework, along with a novel phase-independent focusing design.

Findings

Transfer matrices are insufficient for describing DLA electron dynamics.

Transfer functions provide a more general and effective modeling tool.

A new focusing structure works for all electron phases.

Abstract

The question of suitability of transfer matrix description of electrons traversing grating-type dielectric laser acceleration (DLA) structures is addressed. It is shown that although matrix considerations lead to interesting insights, the basic transfer properties of DLA cells cannot be described by a matrix. A more general notion of a transfer function is shown to be a simple and useful tool for formulating problems of particle dynamics in DLA. As an example, a focusing structure is proposed which works simultaneously for all electron phases.

Equations

$${\vec{X}}_2=\mathcal{R}{\vec{X}}_1,$$

$$\left(\begin{array}[]{c}x_{2}\\ x^{\prime}_{2}\end{array}\right)=\left(\begin{array}[]{cc}R_{11}&R_{12}\\ R_{21}&R_{22}\end{array}\right)\left(\begin{array}[]{c}x_{1}\\ x^{\prime}_{1}\end{array}\right).$$

$${\cal F}=\left(\begin{array}[]{cc}1&0\\ -1/f&1\end{array}\right)$$

$${\cal O}=\left(\begin{array}[]{cc}1&s\\ 0&1\end{array}\right).$$

$${\mathcal{R}}_{\mathrm{FODO}}$$

$$=$$

$$\frac{1}{f_{\mathrm{FODO}}}=\frac{R_{21}}{R_{11}}=\frac{(1/f{)}^{2}s}{1-s/f-(s/f{)}^2}$$

$$\frac{1}{f_{\mathrm{FODO}}}\approx {\left(\frac{1}{f}\right)}^{2}s,$$

$$p_0=\frac{m{\beta }_{0}c}{\sqrt{1-{\beta }_0^2}},$$

$$\delta =\frac{p-p_0}{p_0}.$$

$$\left(x^{\prime },y^{\prime }\right)=\left(\frac{dx}{dz},\frac{dy}{dz}\right).$$

$${\Phi }_1={\omega }_0{t}_1.$$

$$X_2=\mathcal{R}{X}_1,$$

$${\mathcal{R}}_{\text{tot}}={\mathcal{O}}_{m+1/2}{\mathcal{R}}^n{\mathcal{O}}_{m+1/2}{\mathcal{R}}^n,$$

$${\omega }^{\prime }=\frac{{\omega }_0}{(2n+2m+1)}$$

$$x_2$$

$$x_2^{\prime }$$

$$y_2$$

$$y_2^{\prime }$$

$${\Phi }_2$$

$${\delta }_2$$

$$C$$

$${\beta }_z$$

$$\Delta p_x$$

$$\Delta p_y$$

$$\Delta p_z$$

$$(x,y,z)=(x_1+x_1^{\prime }(z-z_1),y_1+y_1^{\prime }(z-z_1),z).$$

$${\vec{F}}_{\text{p}}\sim -\frac{1}{{\omega }^2}\nabla (|{\vec{F}}_0{|}^2).$$

$$F_0=⟨F_x⟩\sim \Delta p_x\sim \Delta x^{\prime }=x_2^{\prime }-x_1^{\prime },$$

$$(x_1,0,0,0,{\Phi }_1,0)\stackrel{\mathcal{R}}{\to }(x_2,x_2^{\prime },0,0,{\Phi }_2,{\delta }_2),$$

$$\nabla (F_0^2)\sim \frac{\partial }{\partial x_1}\left[{\left(x_2^{\prime }\right)}^2\right].$$

Full text

I Introduction

Several recent proof-of-principle experiments demonstrate the possibility of accelerating electrons in a laser-driven dielectric structure England et al. (2014); Wootton et al. (2016). One class of such dielectric laser accelerator (DLA) structures is the grating-type structure, in which a unit cell is iterated in one dimension, as in the recently developed single grating, dual-grating, and dual pillar structures Wootton et al. (2016). On basis of these successful experiments, compact laser driven accelerators are envisioned (see for example Fig. 4. in Ref. Breuer and Hommelhoff (2013)). A working device will require, in addition to acceleration, beam focusing, and possibly beam diagnostics sections and feedback beam steering. To design a complete DLA beamline, a mathematical description of electron trajectory throughout the whole device is necessary. For conventional radio-frequency (RF) accelerators, several mathematical tools were developed over the years to effectively describe the single particle and beam trajectories Brown and Servranckx (1984); Wille (2000); Wiedemann (2015); usp (2017). One such tool is the transfer matrix; it used to describe the particle transfer properties of the various building blocks of a beamline. For grating-type DLAs, the natural building block is the unit cell of the grating Joannopoulos et al. (2008). Here, interesting questions arise: what are the particle transfer properties of a DLA unit cell, and can they be described by a matrix? This problem has been partially addressed in Ken Soong’s PhD thesis Soong (2014), where the transfer matrix of a unit cell of a double-grating accelerator structure is calculated. In this pioneering work the adequacy of linear approximation is not discussed, and a 25-attosecond electron bunch is assumed, with length less than 1% of the grating period, evading the problem of distribution of phases. The purpose of the present work is to pursue further this interesting idea.

II The transfer matrix formalism

In conventional RF accelerators particle motion is described relative to a reference trajectory Brown and Servranckx (1984); Hemsing et al. (2014). The reference trajectory defines a coordinate system which is in general curvilinear, with the distance along the trajectory described by coordinate $S$ (following the notation in Ref. Hemsing et al. (2014)), and with orthogonal coordinates $x,y$ describing the particle position in the transverse plane. The particle on the reference trajectory has reference energy ${\cal E}_{0}$ (corresponding to reference momentum $p_{0}$). The relative position of electrons on the reference trajectory with respect to the beam center is measured by $s$. The electron location in the six-dimensional phase space comoving with the electron beam is characterized by the vector $\vec{X}=(x,x^{\prime},y,y^{\prime},s,\eta)^{\mathrm{T}}$ Hemsing et al. (2014), where $x^{\prime}=dx/dS$ and $y^{\prime}=dy/dS$ are the small angles of deflection from the reference trajectory, and $\eta=\Delta{\cal E}/{\cal E}_{0}$ is the relative energy deviation (other authors Brown and Servranckx (1984); Wille (2000) use relative momentum deviation $\delta=\Delta p/p_{0}$ instead of $\eta$; in the ultrarelativistic limit $\eta=\delta$). Note that all coordinates of $\vec{X}$ are small and the reference particle is described by $\vec{X}=(0,0,0,0,0,0)^{\mathrm{T}}$.

In conventional RF accelerators, basic properties of a beamline section can be described by first-order beam transport optics Brown and Servranckx (1984), using linear approximation:

[TABLE]

where $\vec{X}_{1}$ describes a particle at the entrance of the section, $\vec{X}_{2}$—at the exit of the section, and $\cal R$ is a linear transfer function, which is represented by a $6\times 6$ transfer matrix. Phenomena not captured by this approximation can be described by second-order optics Brown and Servranckx (1984) or by detailed numerical particle tracing.

Often in the literature a reduced form of the $\cal R$ matrix is used Brown and Servranckx (1984); Wille (2000); Wiedemann (2015); usp (2017), where, as a starting point of the analysis, chromatic effects are neglected ($\delta=0$, $\eta=0$), and only $(x,x^{\prime})$ phase plane is considered:

[TABLE]

In the context of classical optics, such formulation is called ray transfer matrix analysis (or ABCD matrix analysis) and is used to describe the propagation of light rays and Gaussian beams in the paraxial approximation Brooker (2007). Beam transfer through a thin lens of focal length $f$ is described by the matrix

[TABLE]

($1/f$ is called optical power or focusing power). A free drift region of length $s$ with no optical elements is described by the transfer matrix

[TABLE]

One of the common building blocks used in design of RF accelerator beamlines is the FODO array (focusing–drift–defocusing–drift) Brown and Servranckx (1984); Wille (2000); Wiedemann (2015); usp (2017), which has an overall focusing effect, see Fig. 1.

The transfer function of the FODO array is the mathematical composition of the transfer functions of its four building blocks. Composition of linear functions is equivalent to matrix multiplication:

[TABLE]

The focusing power of the FODO structure is

[TABLE]

If the focal length is much larger than the length of the drift region, $f\gg s$, the expression simplifies to

[TABLE]

so in the thin and weak lens approximation, the focusing power of the FODO structure is proportional to the square of the constituent lens’ focusing power. This result will be recalled in Section V.

III Electron transfer analysis for grating-type dielectric laser accelerators

Let us try to develop a methodology, similar to the one outlined in Sect. II, to describe electron transfer through a grating-type DLA. In this context it is natural to use a Cartesian coordinate system, see Fig. 2.

The structure is driven by laser pulses from the direction perpendicular to the electron beam. It is assumed here that the structure exhibits no large-scale resonances such as guided-mode resonances Szczepkowicz (2016), so that the filling times are shorter than the laser pulse length. With this assumption stationary (time-harmonic) calculation of the electromagnetic field is appropriate, and one can obtain realistic time-dependent field by multiplying the stationary result by the laser pulse envelope. This scaling of the result is not carried out here, as it would not affect the conclusions. Let $(\tilde{E}_{x},\tilde{E}_{y},\tilde{E}_{z},\tilde{B}_{x},\tilde{B}_{y},\tilde{B}_{z})$ represent the stationary solution of the electromagnetic field in a given structure; $E_{x}(x,y,z,t)=\Re[\tilde{E}_{x}(x,y,z)e^{i\omega_{0}t}]$ etc. For a start, assume that electron velocity is perfectly aligned with $\hat{z}$. If the velocity $\beta_{0}c$ is tuned perfectly to the grating period $\lambda_{\text{p}}$ and laser wavelength $\lambda_{0}$, then $\beta_{0}=\lambda_{\text{p}}/\lambda_{0}=k_{0}/k_{\text{p}}$, assuming that the DLA is operated at first spatial harmonic Breuer et al. (2014). Let us call $\beta_{0}c$ the reference velocity, corresponding to the reference momentum

[TABLE]

Let $\delta$ denote electron’s relative deviation from the reference momentum:

[TABLE]

Electron position in the transverse plane is described by $(x,y)$, and the slope of the trajectory is described by

[TABLE]

In a radio-frequency accelerator, particle bunch duration $\tau$ is $\sim 3$ orders of magnitude smaller than the period of the driving electromagnetic wave: $\tau\ll T_{0}\approx 10^{-10}$ s. In contrast to this, in DLA, the inequality is reversed: $\tau\gg T_{0}\approx 10^{-14}$ s, due to limitations of the present day electron sources (see eg. Hoffrogge et al. (2014)); another limiting factor is the space charge force Breuer et al. (2014). As a result, in DLA electrons in a bunch populate all phases. In the context of grating-type DLAs, phase appears more important than longitudinal position of the electron along the grating, so it will be convenient to use a parameter $\Phi$ (radians) instead of $S$ (meters) to describe electron’s longitudinal degree of freedom. Let us define $\Phi_{1}$ of an electron as the phase of the electromagnetic field at the moment $t_{1}$ when the electron enters the unit cell of the grating:

[TABLE]

For an electron with $x^{\prime}=0,y^{\prime}=0$ and reference momentum $p=p_{0}$, traversing the unit cell from $z=z_{1}$ to $z=z_{1}+\lambda_{p}=z_{2}$, the phase increases from $\Phi_{1}$ to $\Phi_{1}+2\pi=\Phi_{2}$. Note that in contrast to $x$, $x^{\prime}$, $y$, $y^{\prime}$ and $\delta$, the parameter $\Phi$ is not small; it is analogous to the parameter $S$ defined in Sect. II, not the small parameter $s$. A parameter analogous to $s$ would be $\phi=\Phi_{2}-\Phi_{1}-2\pi$.

The set of parameters $(x_{1},x^{\prime}_{1},y_{1},y^{\prime}_{1},\Phi_{1},\delta_{1})$ fully describes the classical motion state of a particle at the entrance of the unit cell. Therefore there exists a transfer function $\cal R$, such that

[TABLE]

where $X_{1}=(x_{1},x^{\prime}_{1},y_{1},y^{\prime}_{1},\Phi_{1},\delta_{1})^{\mathrm{T}}$ are the parameters of the electron at the entrance of the unit cell, and $X_{2}=(x_{2},x^{\prime}_{2},y_{2},y^{\prime}_{2},\Phi_{2},\delta_{2})^{\mathrm{T}}$ are the parameters of the electron at the exit of the unit cell, see Fig. 2. A matrix-like notation is used here, where one-column matrix $X_{2}$ is the result of operator ${\cal R}$ acting on one-column matrix $X_{1}$.

Using (23), the properties of $\cal R$ can be studied numerically (particle tracing) even without explicit formulas for $\cal R$, by specifying sets of example parameters $\{X_{1}\}$ and calculating corresponding sets of $\{X_{2}\}$. Explicit formulas for $\cal R$ are given in Appendix A; these formulas were used in subsequent analysis.

IV Example transfer function analysis: a double column structure

Let us now apply the concepts of Sect. III to a specific example of a grating-type DLA: the double-column structure described in Ref. Leedle et al. (2015). Figure 3 shows the unit cell.

The columns are long enough so that the system can be described in two dimensions $(z,x)$, assuming infinite column extension in the $\hat{y}$ direction Leedle et al. (2015). The $y$ coordinate is not significant and will be set to 0. Let us study some of the properties of the transfer function of the unit cell. First, the electromagnetic field is calculated using finite element method. Then the transfer function is applied to sample input parameters using equations given in Appendix A. Suppose the incoming electrons are parallel to the $\hat{z}$ direction: $x^{\prime}_{1}=0,y^{\prime}_{1}=0$, and have reference momentum: $p=p_{0}$, so that $\delta_{1}=0$. For a start let’s choose an initial phase $\Phi_{1}=0$ and a set of initial electron positions: $\{x_{1,i}\}=\{-50~{}\mathrm{nm},-25~{}\mathrm{nm},0~{}\mathrm{nm},+25~{}\mathrm{nm},+50~{}\mathrm{nm}\}$. The result of applying $\cal R$ to $X_{1,i}=(x_{1,i},0,0,0,\Phi_{1},0)^{\mathrm{T}}$ is $X_{2,i}=(x_{2,i},x^{\prime}_{2,i},0,0,\Phi_{2,i},\delta_{2,i})^{\mathrm{T}}$. With this set of calculated parameters various plots are possible. An example is shown in Fig. 4, where in (a) pairs $(x_{1},x^{\prime}_{1})$ are plotted, while (b) shows $(x_{2},x^{\prime}_{2})$ pairs (black curve).

Subsequently, another initial phase $\Phi_{1}$ is selected and the procedure is repeated, with results plotted in different color in the same Figure.

The main question that motivated the described investigations was: is transfer matrix description suitable for grating-type DLA structures? The answer follows easily from Fig. 4. The transfer function does not in general transform $(0,0,0,0,0,0)^{\mathrm{T}}$ into $(0,0,0,0,0,0)^{\mathrm{T}}$, so it is not a linear function and it cannot be described by a matrix. Even if $\Phi$ is excluded from the set of transformed parameters and one looks for a reduced $\cal R^{\prime}$ operating in the $(x,x^{\prime})$ space, Fig. 4 shows that in general ${\cal R^{\prime}}(0,0)^{\mathrm{T}}\neq(0,0)^{\mathrm{T}}$, so matrix description is not possible. For example, an electron entering the unit cell with phase $\Phi_{1}=-\frac{2}{3}\pi$ and zero slope leaves the cell with nonzero slope $x^{\prime}_{2}\approx 0.0004$. What is more, neither $\cal R$ nor $\cal R^{\prime}$ belong to the wider class of affine transforms (linearity with an offset), because the plots in Fig. 4(b) are not rectilinear. Here and in subsequent considerations chromatic effects are neglected: $\delta_{1}=0$ is assumed.

Let us compare the calculated transfer properties with optical transfer properties of glass solids, Fig. 5.

As can be seen form comparison of Figs. 4 and 5, the accelerator unit cell, depending on the incoming electron’s phase $\Phi_{1}$, acts as a converging lens for $\Phi_{1}\in(-\frac{1}{3}\pi,0)$, a diverging lens for the opposite phase $\Phi_{1}\in(+\frac{2}{3}\pi,+\pi)$, an upward-deflecting nonlinear prism (larger deflection for larger $|x_{1}|$) for $\Phi_{1}\approx-\frac{2}{3}\pi$, and a downward-deflecting nonlinear prism for $\Phi_{1}\approx+\frac{1}{3}\pi$.

V Ponderomotive focusing in grating-type dielectric laser accelerators

In conventional accelerators the primary method of focusing is alternating gradient focusing (also called strong focusing), where lensing quadrupole magnets generate field gradient $\partial B/\partial x,\partial B/\partial y$, and are arranged along the beam direction $z$ with alternating polarity. This is an implementation of the FODO focusing principle described in Section II. Alternating gradient focusing will be used in planned hybrid accelerator experiments, where a RF beamline will be matched to grating-type DLAs Ody et al. (2017); Prat et al. (2017). Of course, the ultimate goal is to develop compact accelerators employing optical-frequency focusing. At present, laser focusing is in early development stage, with conceptual and simulation work under way Plettner et al. (2009); Soong et al. (2012); Wootton et al. (2017), and a first proof-of-principle experiment with parabolic grating McNeur et al. (2016). One major problem with focusing in DLA is the same as with acceleration: as yet the phase of electrons in not controlled experimentally, and a shift of phase by $\pi$ reverses the force of the electromagnetic field on the particle and turns focusing into defocusing, so only a fraction of electrons is focused. Is it possible to focus electrons with different phases $\Phi$ at the same time?

An interesting property of a FODO structure is that it keeps its focusing properties if the forces are reversed: both $\cal ODOF$ and $\cal OFOD$ are focusing transformations. Suppose an electron enters a DLA structure shown in Fig. 6, and the unit cell has similar transfer properties as in Fig. 4.

The transfer function of the whole structure is

[TABLE]

where again matrix-like notation is used, with multiplication representing mathematical composition of functions, ${\cal R}^{n}$ denoting the composition of $n$ single cell transfer functions $\cal R$, and ${\cal O}_{m+1/2}$ denoting the linear drift operator (4) for $s=(m+\frac{1}{2})\lambda_{\text{p}}$. If, for an electron with phase $\Phi$, $\cal R$ has focusing properties, then ${\cal R}^{n}$ is also focusing (for $n$ small enough so that dephasing McNeur et al. (2016) is not significant). The drift section ${\cal O}_{m+1/2}$ advances the electron phase by $2\pi m+\pi$, so in the second ${\cal R}^{n}$ section is defocusing—just like in a FODO structure. If another electron enters the same structure with phase $\Phi+\pi$, the structure acts on it as DOFO. For both electrons the structure acts as a converging lens. Consider now an electron with such phase $\Phi^{\prime}$ that the unit cell acts as a nonlinear upward-deflecting prism. Now the whole structure cannot be classified as FODO. After traversing the first ${\cal R}^{n}$ section, the electron is deflected upwards, ${\cal O}_{m+1/2}$ reverses the phase, and in the second ${\cal R}^{n}$ section the electron is deflected downwards. However, because the “prism” ${\cal R}^{n}$ is nonlinear, its action is stronger away from the $x=0$ line and the overall effect of ${\cal R}_{\text{tot}}$ is again a converging lens. A similar argument applies to an electron entering the structure with $\Phi^{\prime}+\pi$ phase. This reasoning, based on $(x,x^{\prime})$ plots, is purely geometric, but a chromatic effect ($\delta\neq 0$) also plays a role in focusing, as shown in Appendix C.

The phase-independent focusing effect of ${\cal R}_{\text{tot}}$ is shown in Fig. 7.

This structure is a converging lens that exhibits both geometric and “phase” aberrations. The focal lengths for the structure ${\cal O}_{1/2}{\cal R}{\cal O}_{1/2}{\cal R}$, as shown in Fig. 7(b), lie in the range 30 mm–35 mm, so the focusing effect is very weak. The focal lengths for the structure ${\cal O}_{5+1/2}{\cal R}^{8}{\cal O}_{5+1/2}{\cal R}^{8}$, as shown in Fig. 7(c), lie in the range 48 $\mu$m–70 $\mu$m, so the focusing effect is three orders of magnitude stronger. This shows that grouping of the unit cells is critical (see also Ref. Naranjo et al. (2012)). The effect of grouping is even stronger than for a thin lens FODO structure described by Eq. (18) (see also Appendix D). However, grouping increases the chance of electron collision with the dielectric. It is likely that the geometry of the unit cell (Fig. 3) could be optimized for better transfer and focusing performance, but this is left for future work. Also, in the presented approach boundary field effects were neglected. This is justified for large structures like ${\cal O}_{5+1/2}{\cal R}^{8}{\cal O}_{5+1/2}{\cal R}^{8}$, but the calculation of ${\cal O}_{1/2}{\cal R}{\cal O}_{1/2}{\cal R}$ may be inaccurate. Boundary field effects can be handled with the transfer function approach by introducing intermediate boundary cells ${\cal B}_{\pm}$, as shown in Fig. 8.

In this approach, the transfer function ${\cal O}_{5+1/2}{\cal R}^{8}{\cal O}_{5+1/2}{\cal R}^{8}$ is an approximation of the more accurate ${\cal O}_{3+1/2}{\cal B}_{+}{\cal R}^{8}{\cal B}_{-}{\cal O}_{3+1/2}{\cal B}_{+}{\cal R}^{8}{\cal B}_{-}$.

The structure shown in Fig. 6, with its converging property, cannot in general (for arbitrary $\Phi$) be classified as FODO (see Fig. 9), but along with FODO it belongs to a wider class of focusing setups based on ponderomotive force Mulser (1990); Macchi (2013) (Mulser (1990) gives historical references). Quantitative similarities and differences between the classical ponderomotive force and focusing force of the ${\cal O}_{m+1/2}{\cal R}^{n}{\cal O}_{m+1/2}{\cal R}^{n}$ structure are discussed in Appendix D.

As noted by Hartman and Rosenzweig Hartman and Rosenzweig (1993), other alternating focusing schemes used in radio frequency accelerators, like radio-frequency quadrupole (RFQ) focusing Wangler (2008), or alternating phase focusing Wangler (2008); Swenson (1976), are also based on ponderomotive force. In the context of DLA, a ponderomotive focusing scheme has already been studied for photonic band-gap accelerators Naranjo et al. (2012). For grating-type DLA, the idea was considered in Ref. Breuer et al. (2014) (citing Swenson (1976); Naranjo et al. (2012)), but specific implementation was not proposed.

Ponderomotive focusing of electrons in the transverse plane is analogous to the redistribution of sand on a Chladni plate Chladni (1787). A grain of sand on a vibrating plate is subject to alternating force whose amplitude is a function of position on the plate, and diffuses towards regions of smaller amplitude, finally settling in the nodal regions. Similarly, an electron traversing a ${\cal O}_{m+1/2}{\cal R}^{n}{\cal O}_{m+1/2}{\cal R}^{n}{\cal O}_{m+1/2}{\cal R}^{n}\ldots$ structure with reference velocity $\beta_{0}c$ is subject to an alternating force of frequency

[TABLE]

(because the spatial period of the structure is $(2n+2m+1)\lambda_{\text{p}}$), and is attracted in the transverse plane towards regions of smaller force amplitude—smaller electromagnetic field. The field is stronger close to the dielectric surfaces, and for double grating-like structures the minimum of the transverse force lies in the electron channel between the two surfaces.

VI Conclusion and outlook

Transfer matrices are known to be useful for the description of particle motion through the segments of conventional RF accelerators. A similar description is proposed here for grating-type DLAs: linear transfer functions, represented by matrices, are replaced by nonlinear transfer functions; matrix multiplication is replaced by numerical function composition; these differences are hardly noticeable with a compact, matrix-like notation. The approach facilitates quantitative description of electron transfer through a DLA structure, where, in the first approximation, the transfer properties of larger units are easily determined from the transfer properties of the DLA unit cell. Hopefully this approach will make easier the conceptual and simulation work on new designs, and help in clear presentation and discussion of the properties of new DLA structures. One example of presentation of transfer properties are the $(x,x^{\prime})$ plots, sometimes called “trace space plots” in the RF accelerator literature; such plots are already entering the DLA literature Ody et al. (2017), and can naturally be produced with the transfer function approach described here.

In Sect. V the transfer function approach led naturally to the idea of building a FODO-like DLA structure, which focuses electrons irrespective of the phase. The converging force in the proposed setup is yet another example of ponderomotive force. Further work is required to optimize the geometry. One approach would be to drive the structure symmetrically from two sides by employing distributed Bragg reflectors Prat et al. (2017).

In this paper the transfer function is applied only to lensing properties of DLA structures. Of course the primary function of DLAs is to accelerate: to increase $\delta$. Here it was assumed that $\delta_{1}=0$ and $\delta_{2}$ was not analyzed. Hopefully the described formalism with its six parameters $(x,x^{\prime},y,y^{\prime},\Phi,\delta)$ will also be useful to describe acceleration schemes. Here a major challenge is the phase distribution of electrons, which results in only a fraction of electrons being accelerated. To address this issue, methods to compress the particle bunch are investigated Prat et al. (2017) to obtain single-phase particles. More generally, a method is needed to redistribute the electron phases to populate several narrow $\Phi$ subsets separated by $2\pi$. Alternatively, perhaps an accelerating scheme working for all incoming $\Phi$ could be invented. Formulation of these challenges using $(x,x^{\prime},y,y^{\prime},\Phi,\delta)$ may accelerate progress in this field.

Acknowledgements.

I am grateful to Martin Kozák, Joshua McNeur and Peter Hommelhoff for inspiring discussions during my visit in Erlangen in May 2016. I am grateful to Wrocław Networking and Supercomputing Center for granting access to the PLATON computing infrastructure.

Appendix A Transfer function equations

The transfer function defined by Eq. (23) can put into the following explicit form (derived in Appendix B):

[TABLE]

In Eqations (26), the following auxiliary quantities were used: $C$ is the trajectory deflection cosine = $\hat{z}\cdot\hat{v}_{1}$, $\beta_{z}$ is the relative longitudinal velocity, $\Delta p_{x},\Delta p_{y},\Delta p_{z}$ is the momentum change of the electron. The formulas for these auxiliary quantities are:

[TABLE]

The complex-valued functions $(\tilde{E}_{x},\tilde{E}_{y},\tilde{E}_{z},\tilde{B}_{x},\tilde{B}_{y},\tilde{B}_{z})$ represent the stationary solution of the electromagnetic field in a given structure; $E_{x}(x,y,z,t)=\Re[\tilde{E}_{x}(x,y,z)e^{i\omega_{0}t}]$ etc. The components of the electromagnetic field under the integrals are taken at the electron position parameterized by $z$:

[TABLE]

It is assumed here that the motion of the electrons is piecewise linear, with straight line trajectory within one unit cell, from $z_{1}$ to $z_{2}=z_{1}+\lambda_{p}$; although the electron accumulates momentum during its flight through the cell, in calculation the accumulated momentum is added only at the exit of the cell; this is equivalent to the Euler method of solving differential equations (a first-order Runge-Kutta method). This method is numerically less efficient than the conventional fourth-order Runge-Kutta algorithm, but the formulas are simpler, easier to derive, analyze, expand in series, and this facilitates elementary physical insight.

The validity of the Euler approximation was checked for the calculations of Sect. IV and V by subdividing the unit cell into 4 sub-cells, and calculating the unit cell transfer function as a composition ${\cal R}={\cal R}_{4}{\cal R}_{3}{\cal R}_{2}{\cal R}_{1}$, where ${\cal R}_{1}$ is the transfer function from $z$ to $z+\frac{1}{4}\lambda_{\text{p}}$, etc. Such refinement did not influence the $(x_{2},x^{\prime}_{2})$ plots in Figs. 4 and 7(c). On the other hand, the refinement did quantitatively influence the calculation shown in Fig. 7(b), where the accelerator segment consisted of only two elementary cells. In this case the calculation converged for $n\approx 50$ subdivision segments, and this large number of segments was used to produce Fig. 7(b).

Equations (26) contain small dimensionless parameters $x^{\prime}$, $y^{\prime}$, $\delta$, $\Delta p_{i}/p_{0}$. In textbooks on conventional accelerators such equations are usually expanded in Taylor series and higher order terms are dropped Wille (2000). For the purposes of this paper Taylor expansion of Eq. (26) would not be productive. Note that linearization of the transfer function is not possible, as discussed in Sect. IV.

Appendix B Derivation of the transfer function equations

Assuming the electron trajectory is linear within the unit cell (or its subset, see previous section), as the electron travels from $z_{1}$ to $z_{2}$, its transverse position $x$ increases from $x_{1}$ to $x_{2}$, with $x_{2}=x_{1}+\Delta x=x_{1}+\frac{\Delta x}{\Delta z}\Delta z=x_{1}+x^{\prime}_{1}(z_{2}-z_{1})$. Similarly, $y_{2}=y_{1}+y^{\prime}_{1}(z_{2}-z_{1})$.

$C$ is the cosine of the deflection of electron trajectory from the $\hat{z}$ direction, $C=\hat{z}\cdot\hat{v}_{1}=(0,0,1)\cdot\frac{(v_{1x},v_{1y},v_{1z})}{v_{1}}$, where $\vec{v}_{1}$ is the velocity of the electron at the entrance of the cell. It follows that $C=\frac{v_{1z}}{v_{1}}=\frac{v_{1z}dt}{v_{1}dt}=\frac{dz}{\sqrt{dx^{2}+dy^{2}+dz^{2}}}=\frac{1}{\sqrt{(dx/dz)^{2}+(dy/dz)^{2}+(dz/dz)^{2}}}=\frac{1}{\sqrt{{x^{\prime}_{1}}^{2}+{y^{\prime}_{1}}^{2}+1}}$.

Momentum and velocity at the entrance of the cell are related by $p_{1}=\frac{1}{\sqrt{1-{\beta_{1}}^{2}}}m\beta_{1}c$, or $\beta_{1}=\frac{p_{1}}{\sqrt{{p_{1}}^{2}+m^{2}c^{2}}}$, or, using the definition of $\delta$ (Eq. 20), $\beta_{1}=\frac{p_{0}(1+\delta_{1})}{\sqrt{{p_{0}}^{2}(1+\delta_{1})^{2}+m^{2}c^{2}}}$. The $z$ component of the relative velocity is $\beta_{1z}=\frac{\beta_{1z}}{\beta_{1}}\beta_{1}=C\frac{p_{0}(1+\delta_{1})}{\sqrt{{p_{0}}^{2}(1+\delta_{1})^{2}+m^{2}c^{2}}}$.

The slope at the exit of the cell is $x^{\prime}_{2}=\frac{dx}{dz}=\frac{p_{2x}}{p_{2z}}=\frac{p_{1x}+\Delta p_{x}}{p_{1z}+\Delta p_{z}}=\frac{{p_{1x}}/{p_{1z}}+{\Delta p_{x}}/{p_{1z}}}{{p_{1z}}/{p_{1z}}+{\Delta p_{z}}/{p_{1z}}}=\frac{x^{\prime}_{1}+{\Delta p_{x}}/{p_{1z}}}{1+{\Delta p_{z}}/{p_{1z}}}=\frac{x^{\prime}_{1}+{\Delta p_{x}}/{Cp_{1}}}{1+{\Delta p_{z}}/{Cp_{1}}}=\frac{x^{\prime}_{1}+{\Delta p_{x}}/{Cp_{0}(1+\delta_{1})}}{1+{\Delta p_{z}}/{Cp_{0}(1+\delta_{1})}}$. The expression for $y^{\prime}_{2}$ is analogous.

The momentum at the exit of the cell is

$p_{2}=\sqrt{(p_{1x}+\Delta p_{x})^{2}+(p_{1y}+\Delta p_{y})^{2}+(p_{1z}+\Delta p_{z})^{2}}$

${}=p_{1z}\sqrt{\left(\frac{p_{1x}}{p_{1z}}+\frac{\Delta p_{x}}{p_{1z}}\right)^{2}+\left(\frac{p_{1y}}{p_{1z}}+\frac{\Delta p_{y}}{p_{1z}}\right)^{2}+\left(\frac{p_{1z}}{p_{1z}}+\frac{\Delta p_{z}}{p_{1z}}\right)^{2}}$

${}=Cp_{1}\sqrt{\left(x^{\prime}_{1}+\frac{\Delta p_{x}}{Cp_{1}}\right)^{2}+\left(y^{\prime}_{1}+\frac{\Delta p_{y}}{Cp_{1}}\right)^{2}+\left(1+\frac{\Delta p_{z}}{Cp_{1}}\right)^{2}}$

${}=Cp_{0}(1+\delta_{1})\sqrt{\left(x^{\prime}_{1}+\frac{\Delta p_{x}}{Cp_{0}(1+\delta_{1})}\right)^{2}+\left(y^{\prime}_{1}+\frac{\Delta p_{y}}{Cp_{0}(1+\delta_{1})}\right)^{2}+\left(1+\frac{\Delta p_{z}}{Cp_{0}(1+\delta_{1})}\right)^{2}}$,

so the relative momentum deviation is by definition (20) $\delta_{2}=p_{2}/p_{0}-1$

${}=C(1+\delta_{1})\sqrt{\left(x^{\prime}_{1}+\frac{\Delta p_{x}}{Cp_{0}(1+\delta_{1})}\right)^{2}+\left(y^{\prime}_{1}+\frac{\Delta p_{y}}{Cp_{0}(1+\delta_{1})}\right)^{2}+\left(1+\frac{\Delta p_{z}}{Cp_{0}(1+\delta_{1})}\right)^{2}}-1$

The phase increases from $\Phi_{1}=\omega_{0}t_{1}$ to $\Phi_{2}=\omega_{0}t_{2}$, and $\Phi_{2}=\Phi_{1}+\omega_{0}\Delta t=\Phi_{1}+\omega_{0}\frac{\Delta z}{\beta_{z}c}=\Phi_{1}+\frac{k_{0}(z_{2}-z_{1})}{\beta_{z}}$.

During its flight through the cell the electron receives momentum $(\Delta p_{x},\Delta p_{y},\Delta p_{z})$ from the electromagnetic field, where $\Delta p_{x}=\int F_{x}dt=\int\frac{F_{x}}{dz/dt}dz=\int\frac{F_{x}}{c\beta_{z}}dz=\frac{(-e)}{c}\int\frac{1}{\beta_{z}}(E_{x}+\beta_{y}cB_{z}-\beta_{z}cB_{y})dz=\frac{(-e)}{c}\int(\frac{1}{\beta_{z}}E_{x}+y^{\prime}_{1}cB_{z}-cB_{y})dz$. The electromagnetic field components under the integral are taken at the electron location, parameterized by $z$: $E_{x}=E_{x}(x,y,z,t)=\Re\{\tilde{E}_{x}(x,y,z)\exp[i\omega_{0}t]\}=\Re\{\tilde{E}_{x}(x(z),y(z),z)\exp[i\omega_{0}t(z)]\}=\Re\{\tilde{E}_{x}(x_{1}+x^{\prime}_{1}(z-z_{1}),y_{1}+y^{\prime}_{1}(z-z_{1}),z)\exp[i(\Phi_{1}+\frac{k_{0}(z-z_{1})}{\beta_{x}})]\}$, and similarly for $E_{y}$ and $E_{z}$. The real-part operator $\Re$ is additive and in the expression for $\Delta p_{x}$ can act as the final operation: $\Delta p_{x}=\Re\{\frac{(-e)}{c}\int(\frac{1}{\beta_{z}}\tilde{E}_{x}+y^{\prime}_{1}c\tilde{B}_{z}-c\tilde{B}_{y})\exp[i(\Phi_{1}+\frac{k_{0}(z-z_{1})}{\beta_{x}})]dz\}$. The derivation of expressions for $\Delta p_{y}$ and $\Delta p_{z}$ is similar.

Appendix C Are variations in $\delta$ significant for focusing?

In Section V the forces on an electron traversing a ${\cal O}_{m+1/2}{\cal R}^{n}{\cal O}_{m+1/2}{\cal R}^{n}$ structure are discussed, and it is shown that the overall effect is focusing. The argument, based on $(x,x^{\prime})$ plots for a single cell, is purely geometric, assuming $\delta=0$ and thus neglecting the ,,chromatic effects”. However, the calculations leading to Fig. 7 are exact in the sense that full transfer function is used (equations (26)), so in the calculation $\delta$ is nonzero (except the entrance of the cell). Is focusing modified by chromatic effects ($\delta\neq 0$)? To answer this question, let us “spoil” the transformation (26) by assuming $\delta_{2}=\delta_{1}$ instead of Eq. (26f). This means that now $\delta$ is forced to remain constant, equal to the initial zero value, and that the phase $\Phi$ advances in each elementary cell by exactly $2\pi$. The result is shown in Fig. 10.

The structure still has focusing properties, but the result is significantly different than for the correct transformation, and the average focusing power decreases by a factor of $\sim 2$. So the ,,geometric argument”, while essentially correct, does not capture all focusing factors, and chromatic effects are also important.

Appendix D Ponderomotive focusing and ponderomotive force – quantitative analysis

Suppose a particle is subject to an oscillating force $\vec{F}=\vec{F}_{0}\cos\omega t$, whose amplitude $\vec{F}_{0}$ varies spatially on length scales larger than the amplitude of the $\omega$–oscillation of the particle. Under these circumstances an effective, average force on the particle arises, called the ponderomotive force (see e.g. Mulser (1990); Macchi (2013)):

[TABLE]

For a high-energy particle traversing a FODO-like DLA structure described in Sect. V, the transverse defecting force is a function of transverse position $(x,y)$ and oscillates with frequency $\omega^{\prime}$ given by Eq. (25), causing small-amplitude electron oscillation in the $(x,y)$ plane, so the basic requirements for ponderomotive force are satisfied. The distinction between a single ,,FODO” cell and repeated ,,FODOFODO…” structure does not affect the physical focusing mechanism and should not affect the terminology. There is however one significant difference between the classical ponderomotive force and the present situation: the oscillation of the focusing force is not harmonic, as shown in Fig. 11.

This sheds doubt on the applicability of Eq. (29) to the present situation. Te derivation of this equation Mulser (1990); Macchi (2013) should be reconsidered, allowing for non-harmonic force oscillations, which is beyond the scope of this paper. Nevertheless, let us numerically check three features of ponderomotive focusing occuring in the ${\cal O}_{5+1/2}{\cal R}^{8}{\cal O}_{5+1/2}{\cal R}^{8}$ structure, and compare them with Eq. (29).

(1) Let us reduce the amplitude of force oscillation $F_{0}$ by half by reducing driving laser amplitude $E_{0}$ (see Fig. 3) by half. The calculation yields the result that the average focusing power of the structure decreases by a factor of 4.2, signifying the decrease of the ponderomotive focusing force by the same factor. This result is close to the value of 4 expected from Eq. (29).

(2) Let us shorten the structure approximately by half: ${\cal O}_{3+1/2}{\cal R}^{4}{\cal O}_{3+1/2}{\cal R}^{4}$. This increases the oscillation frequency by a factor of 2. The result is that the average focusing power decreases by a factor of 7.5. This is actually closer to $2^{3}$ than to the value $2^{2}$ expected from Eq. (29) and questions the applicability of this equation to non-harmonic oscillating forces.

(3) Let us, for the structure ${\cal O}_{5+1/2}{\cal R}^{8}{\cal O}_{5+1/2}{\cal R}^{8}$, calculate the gradient $\nabla(F_{0}^{2})=\frac{\partial}{\partial x}(F_{0}^{2})$. The force amplitude $F_{0}$ is equal to the average force exerted on the electron traversing an elementary DLA cell

[TABLE]

where an approximate form $x^{\prime}_{2}=x^{\prime}_{1}+\Delta p_{x}/p_{0}$ of Eq. (26b) was used. For simplification, a transfer function for $x^{\prime}_{1}=0$ is considered here:

[TABLE]

so the force amplitude $F_{0}$ is simply proportional to $x^{\prime}_{2}$ for an elementary cell, where $x^{\prime}_{2}$ is a function of six parameters: $x^{\prime}_{2}=x^{\prime}_{2}(x_{1},0,0,0,\Phi_{1},0)$. The final equation for the gradient, neglecting multiplicative constants, is

[TABLE]

This functional dependence is plotted in Fig. 12.

If Eq. (29) was strictly valid, the plots in Figs 7(c) and 12 should be the same up to a multiplicative constant. While both plots indicate focusing, there are quantitative differences, so Eq. (29) is not strictly valid for DLA ponderomotive focusing force.

There could be one more reason for the inaccuracy of Eq. (29) in the present situation. Perhaps the transverse oscillation amplitude of the electron is too large. This hypothesis mav be verified in future work.