Design Study of a Dielectric Laser Undulator

Dielectric laser acceleration (DLA) achieves remarkable gradients from the optical near fields of a grating structure. Tilting the dielectric grating with respect to the electron beam leads to deflection forces and the DLA structure can be utilized as a microchip undulator. We investigate the beam dynamics in such structures analytically and by numerical simulations. A crucial challenge is to keep the beam focused, especially in direction of the narrow channel. An alternating phase focusing scheme is optimized for this purpose and matched lattice functions are obtained. We distinguish synchronous operation with phase jumps in the grating and asynchronous operation with a strictly periodic grating and well-designed synchronicity mismatch. Especially the asynchronous DLA undulator is a promising approach, since a simple, commercially available grating suffices for the focusing lattice design. We pave the way towards experiments of radiation generation in these structures and provide estimates of the emitted radiation wavelength and power. The analytical models are validated by numerical simulations in the dedicated DLA simulation tool DLAtrack6D and Astra, where the underlying laser fields are computed by CST Studio.


I. INTRODUCTION
The working principle of modern accelerator light sources relies on undulators transforming the energy of an electron beam into short-wavelength photon radiation [1]. Large-scale facilities like the E-XFEL [2] and the Swiss-FEL [3] use periodically alternating magnets with approximately 1 T field strength to induce a transverse wiggling motion of the particle beam. With a typical period length of about one centimeter, magnetic undulators reach several tens to hundreds of meters total length. Active undulator designs use microwave [4], terahertz [5], or optical [6] electromagnetic fields to drive a wiggling motion of the electron beam. In order to miniaturize accelerator light sources, especially optical laser undulators are a promising concept since they potentially reach GeV/m acceleration and deflection gradients which allow to realize ultrashort undulator periods [7].
Dielectric laser acceleration (DLA) utilizes the periodic laser field distribution in a dielectric grating to accelerate electrons in excess of 1 GV/m field strength [8]. Tilting the grating with respect to the electron beam direction induces transverse deflection forces such that the structure effectively serves as a DLA undulator [9]. Thus, DLA technology has the potential to reduce the dimensions of an X-ray free electron laser to table-top size [10]. However, owing to the tiny dimensions and the laser induced defocusing forces beam transport through a DLA lattice remains challenging [11]. Magnetic quadrupole or solenoid lenses can neither mitigate the inherent beam divergence within the aperture nor compensate the defocusing induced by the nonlinear laser fields. Thus, * steffen.schmid@tu-darmstadt.de † niedermayer@temf.tu-darmstadt.de DLA undulators require an entirely laser-based focusing scheme as described in Refs. [12][13][14]. This article discusses how such a focusing scheme, i.e. alternating phase focusing (APF), can be adapted to DLA undulator structures, where the reference particle moves on an undulating trajectory. The ensemble of all other particles is then supposed to execute stable betatron oscillations around the reference particle. To achieve this, the APF for DLA formalism [13,14] is generalized to periodically curved reference trajectories. Moreover, borrowing from spatial harmonic focusing [12], mismatch in the synchronicity condition leads to a drifting synchronous phase which is modeled as an asynchronous APF scheme. This design study is intended to pave the way from the conceptual idea [7] towards a first experiment generating radiation from DLA undulators. Therefore, the study adopts machine parameters of the accelerator research experiment (ARES) at the R&D facility for short innovative bunches and accelerators at DESY (SIN-BAD) [15,16]. ARES generates a E = 107 MeV electron beam with Q = 0.5 pC bunch charge, σ t = 0.75 fs bunch length, ε x/y = 1 nm transverse geometric emittance, and σ E /E = 0.06 % energy spread. Furthermore, the silica grating design assumes λ 0 = 2 µm drive-laser wavelength as commonly used for DLA experiments in the accelerator on a chip (ACHIP) collaboration [17][18][19][20].
The structure of the paper is as follows. Section II recapitulates the theory of tilted DLA structures from Refs. [11,21] and provides an optimized grating design for a high-gradient fused silica DLA undulator. Section III discusses the single particle dynamics for both a synchronous and an asynchronous DLA undulator lattice. The investigation provides analytical formulas for the DLA undulator parameter K und . Section IV generalizes the DLA alternating phase focusing scheme [13] to DLA undulators which are shown to provide a FODO lat- Figure 1. CST Studio simulation [22] of the field between two opposing dielectric gratings for aẑ polarized drive-laser with amplitude E0 in a). In b), the field strength is compared to a cosine wave with e1(0, 0) complex amplitude, showing that the fundamental Fourier coefficient is dominant.
tice with the same periodicity as the undulator period. A subsequent beam-matching study reveals the specifications and limitations of the investigated DLA undulator designs. Finally, section V shows particle tracking simulations for the DLA design parameters that were identified as most suitable with respect to experimental prospects. The paper concludes with an outlook on the feasibility of a DLA undulator experiment at ARES and its potential implications on future applications of DLA undulators in large-scale facilities, such as the E-XFEL.

II. TILTED DLA GRATING CELL
Laser-driven DLA undulators utilize Lorentz forces induced by optical near fields in tilted dielectric gratings to generate a wiggling beam trajectory [9]. Figure 1 a) visualizes the laser field in one tilted DLA grating cell simulated using CST Studio [22]. The working principle of DLA undulators relies either on synchronous or on asynchronous grating structures. In a synchronous structure the beam particles travel along the green line exactly one grating cell per laser period inẑ direction. Thus, the beam velocity v z = βc, the longitudinal grating constant λ z = 2π/k z , and the laser wavelength λ 0 = 2π/k 0 fulfill the synchronicity condition In an asynchronous structure the laser wavelength λ 0 does not match (1) which introduces a smooth phase drift of the optical fields at the particle positions. The following investigation provides an overview of the basic properties for both synchronous and asynchronous DLA undulators. The reciprocal grating vector k g = k xx + k zẑ in Fig. 1 a) is parallel to the y = 0 symmetry plane and tilted by an angle α with respect to the beam path alonĝ z. The periodic modulation of the laser field E num z alonĝ z in Fig. 1 b) mainly corresponds to the fundamental harmonic of the reciprocal grating vector k z . Thus, the Fourier coefficient with the phasorẼ z for the drive-laser field characterizes the effective interaction field E ana z of the DLA undulator. A general beam dynamics description based on the non-linear, time-dependent Lorentz force in such setups is rather cumbersome. However, in phase-synchronous structures the Panofsky-Wenzel theorem [23] allows to simplify the analysis. Referring to the investigation in Ref. [11], the effective electromagnetic interaction corresponds to a scalar potential the particle charge q, and an arbitrary phase ϕ. Unlike the definition of V used in Ref. [11], the force −∇V acting on a synchronous particle with z = s + β ct and y = 0 is constant in time and anti-/parallel to the grating vector k g . Hence, electrons passing a tilted diffraction grating experience both transverse deflection and longitudinal acceleration. The relative phase ϕ between the electron beam and the laser field determines the sign and strength of the force. Effective electromagnetic interaction between the drive-laser field and the electrons requires maximizing the structure coefficient |e 1 (α)| defined as (2) evaluated at x = y = 0 for the grating tilt angle α. Figure 2 a) shows the results of a numerical optimization study for the geometry in Fig. 2 b) using CST Studio [22]. The DLA cell consist of two opposing silica diffraction gratings with r = 2.0681 for a laser wavelength of λ 0 = 2 µm and a longitudinal grating wave-number k z according to (1) for β ≈ 1. The transverse grating wave number k x gets larger as the tilt angle α increases. The optima for the grating tooth width w t = 1.0 µm and for the tooth height h t = 1.5 µm are similar to typical specifications of commercially available NIR spectrometer gratings [24]. In Fig. 2 a) the coefficient |e 1 (α)| is maximum at zero tilt and decreases as α increases. Qualitatively, this degradation is a consequence of the laser field attenuation from the inner grating surface towards the beam path indicated as a green arrow. Combining (1), (4), and (5) yields the field attenuation constant k y = k 0 β −2 cos −2 α − 1 inŷ direction. The attenuation limits the maximum beam channel width to ∆y ∼ λ 0 for which the numerical geometry optimization yields a good compromise at ∆y = 1.2 µm.  [22] for the structure coefficient |e1 (α)| of a silica grating at different tilt angles α for â z polarized drive-laser with amplitude E0 in a). The grating design in b) uses a tooth width of wt = 0.6λg, a tooth height of ht = 0.75λ0, and a gap width of ∆y = 0.6λ0 .

III. DLA UNDULATOR LATTICE
In order to achieve a wiggling beam trajectory, the deflection needs to switch sign twice per undulator period length [7]. This can be implemented on a tilted grating structure in two ways: either, one can introduce phase jumps, as in an alternating phase focusing scheme for a synchronous lattice [13], or one can introduce a continuous phase drift by a mismatch of the laser wavelength and the grating period in (1) resulting in an asynchronous lattice. Figure 3 shows a lattice for the synchronous scheme with ∆z = λ z /2 drift spaces equivalent to phase shifts ∆ϕ = π in order to accomplish the alternating deflection. Hence, a full undulator period comprises two blocks of n ∈ N + DLA grating cells separated by λ z /2 which corresponds to the undulator wavelength λ und = (2n + 1) λ z . As it is the case for magnetic undulators, the DLA undulator requires λ und /4 sections at its entrance and exit to avoid imposing a transverse drift motion on the beam [25].
The blue line in Fig. 4 visualizes the transverse deflection x = dx/dz across one wavelength λ und = 554 µm for the reference particle located at (x 0 , y 0 = 0, s 0 ; ϕ 0 ) in a synchronous DLA undulator with n = 138 cells per segment. The trajectory corresponds to a periodic sequence of λ und /2 sections in which the particle experiences the acceleration dp x /dt = ±∂V /∂x. Since the longitudinal velocity is almost constant, z (t) ≈ s 0 + β ct, and the transverse deflection remains negligibly small, |x (t) − x 0 | 2π/k x , the motion resembles the dynamics of a uniformly accelerated particle. If applicable, direct integration of p x = ∂V /∂x dt with ∂V /∂x evaluated at (x 0 , 0, z (t) ; ϕ 0 ) yields an analytical expression for x (z) which is the piece-wise linear function in Fig. 4.
Referring to the particle motion in conventional magnetic undulators [25] with x = K und /γ sin (2πz/λ und ), the equivalent undulator parameter K und for the syn-  x ′ / mrad sync. operation async. operation chronous DLA lattice may be obtained from the fundamental Fourier coefficient of x (z) in Fig. 4 as with the electron charge q, the rest mass m, and the synchronous phase Note that in contrast to Ref. [13], where the synchronous phase is a lattice property, the synchronous phase is determined by the injection properties of the highly stiff beam. Furthermore, the particle trajectory contains contributions of higher harmonics (HH) with odd mode numbers m HH ∈ {3, 5, . . . } and amplitudes proportional to 1/m 2 HH . Similar to its magnetic equivalent, the DLA undulator parameter is proportional to the undulator wavelength λ und and the effective electromagnetic field strength |e 1 (α)|. In addition, K sync und varies with the grating tilt angle α and the synchronous phase ϕ s such that it vanishes completely if α = 0 or ϕ s ∈ {90 • , 270 • }. For the investigated silica structure it exists an optimum at α ≈ 27 • which maximizes the undulator parameter K und .
The orange dashed line in Fig. 4 for x (z) of an asynchronous DLA undulator is sinusoidal and thus has only the fundamental Fourier component. Because of the inherent phase drift, the grating structure does not require additional drift spaces to flip the deflection force. The synchronous phase increases every grating cell length λ z by ∆ϕ s = 2π λ z /λ und . Integration of the deflection force dp x /dt = ∂V /∂x ∝ cos [2π/λ und z (t)] results in an undulator parameter In contrast to the synchronous undulator, K async und represents a deflection which is approximately 20 % smaller than the maximum of K sync und and does not depend on the synchronous phase ϕ s .
Modeling the beam dynamics in asynchronous DLA structures requires additional caution since the aforementioned approximation of the electromagnetic interaction as a scalar potential (3) is not necessarily valid. A complete description of the relativistic electron dynamics in an asynchronous DLA laser field premises on the more general Hamiltonian [26] H (x, p x , y, p y , ct, γ; z) with the canonical momenta p in units of mc and the scaled vector potential a ≡ q/ (mc) A. Referring to Appx. A, the geometry of an asynchronous DLA undulator motivates a vector potential in the form of proportional to the laser field strength E 0 and a polarization vector ξ. The injection phase ϕ 0 is defined by the time at which the reference particle enters the undulator. As discussed in Appx. B, the dynamics of (9) yields a coupled system of six nonlinear differential equations which can be solved numerically as shown in Fig. 5. Typically, the energy variation across one laser wavelength qE 0 λ 0 is much smaller than the electron rest energy mc 2 . A laser field strength in range of the silica damage threshold E 0 ∼ 1−10 GV/m [8,27] at λ 0 = 2 µm in (11) yields a 0 ∼ 10 −3 . Following the idea of Ref. [28], the particle dynamics might be approximated treating the laser field as a perturbation in a 0 1. The approximation of (9) in Appx. B for a relativistic electron beam including terms O γ 0 −2 and O a 0 2 yields for the transverse coordinate of the reference particle with k und = 2π/λ und as the undulator wave number and the initial phase offset at z 0 = 0 defined as The approximation (12) evaluated for the design parameters in Fig. 5 shows a reasonable agreement with the numerically computed solution of (9). The analytical perturbation approach also reproduces fundamental aspects of the beam dynamics and provides basic insights on the parameter dependencies in asynchronous DLA undulators as follows: First, the undulator wavelength λ und in (13) corresponds to the beat wave due to the deviation from the synchronicity condition (1). Hence, variation of the drive-laser wavelength λ 0 in an experiment allows direct adjustment of λ und . The orange dashed line in Fig. 5 shows the effect of doubling the laser detuning to ∆k 0 /k 0 = 0.72% which reduces the effective undulator wavelength by a factor 1/2.
Second, the amplitude of the cosine terms in (12) represents the asynchronous DLA undulator parameter K async und . The first term ξ x a 0 with a 0 ∝ λ 0 represents the deflection induced by the transverse field component. Electromagnetic RF [4], MIR [6], and THz [5] driven undulators use this effect to wiggle the particle beam. Since DLA structures use NIR wavelengths λ 0 ∼ 2 µm and typically k x k und the influence of the first term is negligibly small compared to the second term ξ z a 0 k x /k und . Identifying k x = tan α k z and ξ z E 0 = |e 1 (α)| the second term reproduces the undulator parameter (8) in the relativistic limit β ≈ 1. Hence, for the investigated DLA structures the simplified modeling approach based on the scalar potential V provides reasonable results and, thus, can be a-posteriori justified. Further analysis of (9) in Ref. [29] indicates that our simulation code DLA-track6D [11] based on the scalar potential obtained by the Panofsky-Wenzel theorem [23] provides an adequate tool to model the beam dynamics not only in synchronous, but also in asynchronous DLA structures generally. Using DLAtrack6D, parameter studies for various DLA undulator designs can be performed in a numerically efficient way, giving rise to simple brute-force optimization.
Third, equation (12) shows that the initial phase offset ϕ creates an initial deflection angle, i.e. a transverse drift proportional to K und /γ sin ϕ. The green dotted line in Fig. 5 shows how an offset ∆x = 0.1 µm for α = 27 • tilt angle, equivalent to ∆ϕ = 10 • , influences the transverse drift angle. Furthermore, a second order effect, O a 0 −2 , results in the drift motion which is present in the analytical solution (blue line) as well as in the numerical solution (black line). Thus, for a finite bunch length, the asynchronous DLA undulator introduces a transverse beam divergence inx direction which fundamentally limits the length of the undulator.

IV. ALTERNATING-PHASE FOCUSING
Due to the small aperture in the y-coordinate, external beam optics elements cannot compensate the laser induced defocusing forces [13]. Consequently, an entirely laser-based focusing scheme such as APF [13,14] is required as otherwise the electron beam widens and most particles will be lost in the grating material. The following discussion generalizes the APF scheme for a sinusoidal wiggling reference trajectory.
A paraxial approximation of the Hamiltonian in comoving coordinates [13] yields (15) with δr T = (δx, δy, δs) representing the distance to a reference particle located at r 0 T = (x 0 , 0, s 0 + βct 0 ). The synchronous DLA potential implies by definition a fixed point of motion at δr = 0 where −∇V s = 0. In the vicinity of r 0 including terms O δr 2 the synchronous DLA potential simplifies to which results in a coupled set of linear equations of motion In accordance with Ref. [13], at zero tilt angle α = 0 equation (18) decouples to Hill's equations for a quadrupole lens [30] aligned in y/s direction with focusing constant K α=0 . For a tilted grating with α = 0, diagonalization ofM α provides analytical solutions for the beam dynamics. The tilted grating acts as a thick defocusing quadrupole lens in y-direction, which yields the general solution where tilt angle dependent focusing constant is given by and c i are constants to be determined from the initial conditions. Figure 6 shows a plot of the focusing constant of a synchronous DLA undulator, inserting the results of Fig. 1 a) into (20). The maximum at α ≈ 27 • tilt angle originates from a compromise between increasing tan α and the thereby decreasing |e 1 (α)|. The electron trajectory in x-/s-direction is the superposition of a uniform drift motion and the particle dynamics in a focusing quadrupole lens. The drift direction δx/δs = − cot α coincides with the translation symmetry axis of the grating which is perpendicular to k g . Equivalent to a translation motion along the xdirection in a non-tilted grating, an electron drifting perpendicularly to k g in a tilted grating acquires zero phase advance δϕ = k z δs + k x δx = 0 and its relative momentum p remains constant. Regarding the quadrupole-like terms, a longitudinal motion δs translates directly to a transverse motion δx = γ 2 tan α δs. The synchronous DLA undulator lattice shown in Fig. 3 forms a sequence of focusing and defocusing sections effectively acting like a FODO channel. Each drift  section of length l O = λ z /2 shifts the synchronous phase by ∆ϕ s = π which alternates sign of the focusing constant ±K (α). Hence, the undulator wavelength λ und determines the length of the de-/focusing sections as An eigenvector analysisT η = η for the FODO transport matricesT provides the Courant-Snyder parameters η T = β ,α,γ [31] of the electron beam matched into the DLA undulator lattice. Optimizing l D/F with respect to the beam size σ = εβ as discussed in Ref. [13] yields the undulator wavelength λ und for which the maximum of the betafunctionβ max = max β (z) is at its minimum value. Figure 7 shows the minimumβ max of a matched electron beam for E 0 = 10 GV/m laser field strength. The smallest achievable valueβ max = 826 µm appears for the maximum focusing constant at α = 27 • and ϕ s = π/2. However, according to (6) the undulator parameter K sync und vanishes at ϕ s = π/2. Conversely, the maximum undulator parameter K sync und at ϕ s ∈ {0, π} implies zero focusing K (α) = 0 which is whyβ max diverges. Consequently, efficient operation of the synchronous DLA undulator requires a compromise between focusing and deflection strength, which is set via ϕ s by the injection properties (cf. (7)). Forβ max ≈ 1 mm indicated with the red dashed line in Fig. 7, the optimal FODO lattice cell length yields λ und ∼ 0.6 mm.
An asynchronous DLA undulator effectively functions like a spatial harmonic focusing device as discussed in Ref. [12]. The phase drift ϕ s = k und z creates a continuously varying focusing function K (α, z) = K 0 cos (k und z) with K 0 ≡ K (α)| ϕs=π/2 . This results basically in the equation of motion for ponderomotive focusing in Ref. [29]. The efficiency of spatial harmonic focusing in DLA structures typically suffers form sharing the available laser power between the phase synchronous acceleration field and the phase asynchronous focusing field [12]. However, for the application in an undulator this drawback vanishes since the asynchronous field provides both beam wiggling and focusing. The total transport matrixT of an asynchronous DLA undulator  is basically a matrix product of multiple single DLA cell transport matrices with ∆ϕ s = 2πλ z /λ und phase shifts in between. Thus, a numerical evaluation ofT facilitates the same eigenvector analysis for η as used for the synchronous design. Figure 8 shows the minimum ofβ max for the asynchronous DLA undulator. The absolute minimum of β max is 952 µm and appears for α = 27 • at λ und ≈ 0.55 mm. In contrast to the synchronous design, the asynchronous undulator parameter K async und ≈ 0.3 is nonzero for all injection phases. As in Ref. [13], the betafunction diverges when λ und → 0 and it also diverges for an undulator wavelength λ und 0.8 mm when the FODO lattice reaches its over-focusing stability limit. Hence, the practical working range of the asynchronous DLA undulator indicated by the red dashed line restricts the achievable undulator wavelength to 0.45 mm ≤ λ und ≤ 0.64 mm.
In order to compare the two lattice designs Fig. 9 shows the optimumβ max for α = 27 • tilt angle at the corre- sponding radiation wavelength [25] λ rad = λ und and the maximum peak power [32] P rad ≈ qπ ε 0 for the average current I of a Q = 0.5 pC electron bunch with σ t = 0.75 fs pulse duration [16]. Varying λ und for the asynchronous DLA undulator directly affects the FODO lengths l D/F . However, the amplitude of the focusing function K = K 0 remains fixed. Hence, the optimumβ max (λ rad ) approaches a local minimum at λ rad ≈ 6.5 nm and starts diverging outside the interval 2 nm λ rad 10 nm. Varying λ und for the synchronous DLA undulator requires adjustment of the synchronous phase ϕ s in a way that √ K ∝ √ sin ϕ s ∝ 1/λ und yields a constant betatron phase advance θ ≡ √ K l D/F , which facilitates to keep the betafunctionβ max ∝ 1 √ K at its minimum, until K reaches its upper limit max [K (α)]. Thus, a smaller radiation wavelength λ rad yields smallerβ max . For equal peak radiation power, the optimumβ max of the asynchronous design is slightly smaller than for the synchronous undulator. However, the synchronous design allows to trade off peak radiation power for a smaller value ofβ max . Table I shows the specifications and performance parameters of both optimized DLA undulator designs for the ARES accelerator injection parameters. The synchronous DLA undulator yields the same peak power as the asynchronous design if both undulator parameters (6) and (8) are equal which is the case for ϕ s ≈ 38 • synchronous phase. The damage threshold fluence 1.85 J/cm 2 for a silica grating [33] limits the maximum field strength to E 0 ∼ 10 GV/m and necessitates to utilize ultra-short laser pulses with τ ∼ 100 fs pulse duration. Reducing the peak field to E 0 = 1 GV/m relaxes the short pulse requirement to τ ∼ 10 ps at the cost of weaker focusing and smaller K und . Even though the damage threshold restricts the drive-laser power, pulse front tilted laser beams [34] offer an elegant solution to maximize the interaction length without increasing the pulse duration τ . In exactly the same way tilted pulse front lasers enhance the energy gain in accelerating grating structures [35] the technique can be applied for DLA undulators as well.
For both designs the performance critically depends on the maximum achievable focusing constant K (α). The ARES electron beam specified with ε y = 1 nm geometric emittance reaches for E 0 ∼ 10 GV/m a beam size of σ y ≈ 1 µm inside the DLA structure. Thus, nonnegligible particles losses in the dielectric material of the beam channel with ∆y = 1.2 µm gap width will occur. At E 0 ∼ 1 GV/m the expected beam size is about twice as large. Consequently, maximizing the laser field strength E 0 is a crucial aspect for the undulator's efficacy. The achievable photon wavelength covers a spectral range from XUV to soft X-ray radiation. For E 0 ∼ 10 GV/m the DLA undulator light source reaches up toP rad ∼ 100 W peak power, converting to E rad ∼ 0.1 pJ photon pulse energy per 0.5 pC, 0.75 fs electron bunch.

V. PARTICLE TRACKING SIMULATIONS
Limited by the laser damage threshold of silica, we assume 10 GV/m field strength for a sufficiently short, pulse front tilted drive-laser illumination. With a total length of z = 1.25 cm, the dimensions of the simulated DLA undulator lattice are similar to the typical size of commercially available transmission spectrometer gratings [24]. In order to determine the Courant-Snyder parameters η of the injected electron distribution, the numerical studies initially consider a Gaussian phase space distribution with an idealized, hypothetical emittance of ε y = 25 pm and ε x = ε z = 0 represented by N p = 10 5 macroparticles. Based on the matched parameters η, a subsequent, fully numeric simulation study provides the beam dynamics of the experimental setup for the ARES  electron beam with ε x = ε y = 1 nm transverse emittance and σ t = 0.75 fs bunch length.
First, we employ the semi-analytical particle tracking code DLAtrack6D [11] which provides a fast opportunity to investigate and optimize the non-linear beam dynamics inside the DLA undulator lattice. Figure 10 a) compares the reference trajectory for three different setups of a synchronous DLA undulator lattice as shown in Fig. 3. According to (6) and (18), the transverse deflection K und and the focusing strength K α of the undulator lattice depend on the synchronous phase ϕ s and the undulator period λ und . Figure. 10 b) shows the corresponding beam size σ y for the reference electron distribution with ε y = 25 pm emittance. The particle trajectory for λ und = 491 µm and ϕ s = 90 • (orange) corresponds to a purely focusing setup without transverse deflection. As visible in the strictly periodic σ y -envelope function, perfectly matched injection was attained.
The simulation results for λ und = 546 µm and λ und = 274 µm demonstrate that for the synchronous phase ϕ s = 38.2 • (cf. Tab. I) the DLA lattice provides both deflection in Fig. 10 a) and periodic focusing. According to (7), the transverse deflection in x-direction affects ϕ s which, if not compensated, results in a drifting synchronous phase. The investigation of the dynamics in tilted gratings in Ref. [11] shows that a deflection induced phase drift converts to a coherent oscillation in the x-direction as well. However, the longitudinal period λ u in Ref. [11] is larger than the undulator period λ und . Appropriate adjustment of the drift spaces allows to mitigate the phase drift which the transverse deflection along the x-axis induces in (7). A numerical optimization of the DLA lattice design yields a consecutive alternation of the drift space lengths by δλ z ≈ +9%/ − 7% for λ und = 546 µm and δλ z ≈ +2%/ − 2% for λ und = 274 µm. Nevertheless, a constant drift motion in x-direction, as visible for the larger undulator period λ und = 546 µm, remains. Moreover, slight mismatch caused by the non-linear field distribution of the DLA structure overlay the strictly periodic envelope oscillation of the beam size σ y with periodicity λ und in Fig. 10 b). Since the FODO segment length corresponding to λ und = 546 µm was obtained from an optimization of the Courant-Snyder parameters for the given DLA cell in Fig. 2 with α = 27 • , deviating from that segment length to λ und = 274 µm results in a suboptimal (larger) beam size.
A Fourier analysis of the reference trajectory yields the spectral components of the transverse particle deflection x (z). Figure 11 shows the numerical results for the effective undulator parameter K und of the contributing oscillation wavelengths λ und . In very good agreement with (6), the fundamental harmonic oscillation in the DLA lattice design for λ und = 546 µm corresponds to an effective undulator parameter of K und ≈ 0.29. Furthermore, at λ und ∈ {182 nm, 109 nm, . . . } the spectrum shows contributions of uneven higher harmonics which are typical for the synchronous undulator setup. According to (6), the undulator parameter scales linearly with the undulator wavelength such that λ und = 274 µm results in K und ≈ 0.14. Figure 12 shows that the phase space ellipses of the three different setups can be obtained from Poincaré cross-sections (after each FODO period λ und ) of the phase space coordinates (y, y ) for a particle with ε y = 25 pm single-particle emittance. The solid lines are obtained from fitting the points to an ellipse, which shows that the single particle emittance as a Courant-Snyder invariant is preserved. Moreover, matching at injection is at least approximately confirmed. As expected by design, the purely focusing DLA lattice with λ und = 491 µm  (18). Figure 13 shows DLAtrack6D simulations of the asynchronous DLA undulator lattice. According to (12), electrons injected into the asynchronous DLA undulator at the laser phase of maximum focusing, ϕ s = 90 • , acquire a transverse momentum offset in x-direction. However, simulations show that a linear ramp-up of the laser field amplitude across one undulator period λ und as shown in Fig. 13 c) allows to mitigate the resulting drift motion. The intensity distribution of the drive-laser pulse in a practical DLA experiment automatically introduces a reduction of field strength at the entrance and exit of the DLA undulator. In the numerical model, we simplified the ramp-up as linear. However, the model can easily be extended to a more realistic, e.g. Gaussian, shape. Regarding the deflection and focusing properties, the simulation results are to be interpreted in the same way as the aforementioned synchronous DLA setup. Evaluation of the numerical simulation data for λ und = 546 µm yields K und ≈ 0.28 andβ max ≈ 1 mm and, thereby, reproduces the deflection and focusing properties listed in Tab. I.
In order to validate the results of the semi-analytic DLAtrack6D model for the asynchronous setup, Fig. 13 additionally shows simulation results of the fully numeric particle tracking code Astra [36]. The Astra simulation uses a time-dependent representation of the laser field distribution inside the beam channel of the DLA lattice (see Appx. C describing the concatenation of single cell fields for the entire structure). Since modeling the ramp-up of the field strength would extend the computational domain to the memory limit of Astra, the tracking simulation starts in the fully periodic region of the DLA lattice for z > λ und . The initial particle distribution is obtained from a DLAtrack6D simulation of the ramp-up section. The detuning required for the asynchronous undulator was chosen according to (13) as    Fig. 13 shows that the undulator wavelength, the oscillation amplitude, and the beam width agree very well. Hence, we consider designing both synchronous and asynchronous lattices with DLAtrack6D as legitimate, while a cross-check with a much slower full field simulation tool should be reserved for a finalized design.
With the current ARES beam parameters, losses in the beam transport through the investigated DLA undulator lattice are unavoidable. Limited by the focusing strength of the DLA lattice in y direction, roughly 50% of the initial particle bunch are lost during injection. Because the spatial extend of the beam is not negligibly small compared to the characteristic dimensions 2π/k x and 2π/k z of the DLA cells, it is impracticable to match the entire electron bunch to the optics functions of the DLA FODO lattice and there is a dynamical aperture that is smaller than the physical aperture. Figure 14 shows the transverse momentum x vs. ϕ (cf. (14)) distribution in the center and the end of the DLA undulator. Since the dynamical evolution of each particle described by (12) depends crucially on the initial parameters x 0 and ct 0 , the oscillations of a beam that is not microbunched are incoherent and the generated photons subsequently are also incoherent. In order to obtain coherence, microbunching prior to injection is required.
The electron distribution at the rear end of the asynchronous DLA undulator in Fig. 14 b) provides approximately 17% transmission with the ARES beam injection. Additionally to the injection itself, most of the particle losses occur within the first couple undulator wavelengths z 2λ und . Doubling the total length in the DLAtrack6D simulation to L ≈ 2.5 cm still provides 17% and for L ≈ 6 cm approximately 16.5% of the injected electrons remain. Hence, scalability of the confinement, within its limitations of the dynamic aperture, is confirmed. Figures 15 and 16 compare a) DLAtrack6D and b) Astra simulation results for the transverse phase spaces x − x and y − y at z ≈ 9.3 mm. The laser field interaction imprints the transverse grating periodicity with λ x = 2 µm/ tan 27 • ≈ 3.9 µm on the electron beam's x − x phase space distribution in Fig. 15. For the transverse y −y coordinates in Fig. 16, the focusing properties of the DLA lattice maintain a spatially confined beam.
While the results with regard to phase space shape are very similar, the Astra simulation reaches only 8.3% transmission rate as compared to 17% in DLAtrack6D. This discrepancy is related to edge field effects at the transverse boundaries of the beam channel |y| → 0.6 µm which are not considered in the DLAtrack6D model (the DLAtrack6D model assumes a cosh-shaped transverse profile, which is inaccurate close to the channel boundary). Furthermore, the Astra simulation in Fig. 15   predicts a larger beam divergence in x-direction than DLAtrack6D in Fig. 15 a). This indicates that the investigated DLA lattice design cannot completely compensate the transverse particle drift motion (12) which the laser field imposes on the particles during injection. Since a mismatched beam injection can induce both particle losses as well as transverse drift motion, a fully numeric tracking optimization of the lattice design and injection properties might mitigate the these discrepancies. In an experiment, the injection parameters are usually tweaked manually for best performance.

VI. CONCLUSION
In summary, this study considers two different concepts of laser driven undulator designs based on either synchronous or asynchronous tilted-grating APF-DLA lattices. The theoretical investigation of the beam dynamics in such structures contributes to a collection of basic design guidelines for experimental setups, which can be implemented for example at the DESY-ARES accelerator. First, a wave-optical simulation of the silica grating allows to optimize the DLA cell geometry for maximum interaction between the electron beam and the drive-laser field. Second, the analytical analysis of the beam dynamics yields approximate formulas for the undulator parameter and the focusing strength. Application of these formulas shows that the DLA undulator can be operated as an APF focusing channel. Third, the focusing properties of the DLA undulator impose direct restrictions on the achievable undulator wavelength. For the parameters considered in this study, the optimization of the beam focusing yields an undulator wavelength of approximately λ und = 0.6 mm which will generate XUV and soft X-ray photons. Fourth, semi-analytical DLA-track6D simulations and fully numeric particle tracking e.g. with Astra facilitate to match the Courant-Snyder parameters of the injected electron bunch to the beam optics functions of the DLA lattice. Beam dynamics simulations are a crucial step in the design process, since mismatched Courant-Snyder parameters would result in excessive particle losses. Last but not least, a comparison of both synchronous and asynchronous undulator concepts reveals the advantages and drawbacks of the two approaches. The asynchronous DLA undulator is very promising since its experimental setup is comparably simple and can be implemented by bonding two commercially available diffraction gratings together. Synchronous DLA undulators require a dedicated grating structure design and fabrication, but offer more design options to optimize the beam dynamics. In order to obtain coherent radiation output, the beam has to be bunched on the optical scale, such that it undulates coherently. Moreover, its transverse size (or better transverse emittance) needs to be sufficiently small to fit in one of the "transverse buckets". Such a matching is possible at ARES, when the accelerator reaches its full performance [37]. With the larger bunch as considered here, the electrons will not oscillate with a proper phase relation and therefore incoherent radiation is obtained. Detecting this radiation and characterizing its spectral properties is, however, already an experimental challenge worthy of addressing.
Although the predicted power output is miniscule, the simple design of an asynchronous DLA undulator lattice qualifies it as a promising concept for a demonstration experiment at ARES. A setup based on two commercially available silica diffraction gratings like in Ref. [24] allows to conduct a comparably simple radiation generation experiment. However, the size limitation of the beam channel and the laser induced beam divergence will result in significant electron losses inside and behind the DLA structure. Utilizing a λ 0 = 5 µm MIR drive-laser currently developed at DESY-ARES rather than the current 2 µm one offers one possibility to lift design restrictions. Due to the scale in-variance of Maxwell's equations, the relative electron beam size in the gap of the DLA undulator decreases like σ y /∆y ∝ λ 0 −3/4 which reduces the particle losses in the structure. Likewise the radiation wavelength increases like λ rad ∝ λ 0 1/2 . However, such setups require different grating materials since silica looses its transparency for MIR wavelengths. Another approach for DLA experiments considers the injection of preconditioned electron beams in form of microbunch trains [37]. Utilizing tailored electron beams for DLA undulators is a promising concept, because both the particle transmission rate as well as the spectral coherence of the X-ray emission potentially profit. Considering the impact on large scale X-ray light sources, it is rather unlikely that tilted DLA gratings will become a replacement for conventional magnetic undulators. For instance, the E-XFEL utilizes a Q = 1 nC bunch charge, ε n x = 1.4 mm mrad normalized emittance electron beam in order to generateP rad ∼ 100 GW peak radiation power with ultra large brilliance [2]. Hence, the particle beam properties deviate significantly from the working range of the investigated DLA structures. However, small scale DLA setups open new research prospects by pushing the photon pulse duration from σ t ∼ 100 fs [2] into the subfemtosecond regime. Especially the combination of DLA undulators with DLA electron accelerators is a promising concept, since a complete light source could be integrated on a single chip. Such a device offers the potential to enable typical large scale pump-probe experiments in The following section contains supplementary information regarding an ansatz for the scaled vector potential in an asynchronous DLA undulator. The geometry and symmetry of the DLA grating structure motivate the analytical form of the vector potential ansatz (10) in Sec. III.
Considering the special case of a non-tilted grating with α = 0, the wave vector of the electromagnetic field distribution in the DLA beam channel is perpendicular to the x-direction k ⊥x. Since the DLA grating geometry remains invariant for a mirror reflection in the y − zplane [12], the electromagnetic field decomposes into TE (E y , E z , B x ) and TM (B y , B z , E x ) polarized modes [38, pp. 37-39] which each fulfill Maxwell's equations. Hence, a reasonable ansatz for a fundamental spatial harmonic of the vector potential A α (x, y, z, ct) in a non-tilted DLA structure with grating wave-number k g and α = 0 is with the corresponding mode amplitudes A TE and A TM . Note that the ansatz represents a y-symmetric field distribution which is periodic in longitudinal z-direction and decays exponentially towards the center of the beam channel at y = 0.
A rotationRŷ (α) around the y-axis transforms the vector potential A 0 (x, y, z, ct) into the coordinate system (x,ỹ,z, ct) of an electron beam propagating at a tilt angle α with respect to the z-axis of the non-tilted grating. For the corresponding mode amplitudes in the new coordinates the rotationRŷ (α) yields and, thereby, defines the polarization vector ξ in (10). Note that the components of ξ are not fully independent since A α is a superposition of the TE and TM modes of the dielectric grating. Applying the identity kz = k g cos α, the transformation of the spatial coordinates reproduces the relation between the transverse and longitudinal grating wave-numbers kx = kz tan α from Ref. [11]. Finally, conversion of the laser field strength E 0 to the corresponding vector potential amplitude E0 k0c and scaling A α by q mc yields the scaled vector potential ansatz (10). This section addresses a perturbation theory model for the beam dynamics in an asynchronous DLA undulator. In order to provide a reasonable approximation, the analytical model requires a relativistic electron beam and a reasonably small vector potential amplitude a 0 → 0. In that case, the perturbation approach results in the approximation for the particle trajectories discussed in Sec. III.
Using the ansatz (10) in (12) for the canonical coordinates x (z), y (z), ct (z), p x (z), p y (z), and −γ (z) as functions of the independent variable z yield a system of six coupled nonlinear first order ordinary differential equations (ODEs). Considering a relativistic electron beam with γ 1, a series approximation for the right hand side of (B1) including terms O 1/γ 2 allows to simplify the ODEs to For the DLA undulator discussed in Sec. III the amplitude (11) of the scaled vector potential is small a 0 1. Hence, the solutions of (B2) can be approximated considering the laser field interaction as a perturbation of the electron trajectory. If the laser is switched off, a 0 = 0, the unperturbed electron follows the trajectory of a free particle x 0 (z). In order to approximate the solution for a 0 > 0, the straightforward perturbation method [39, pp. 355-359] substitutes a 0 → (B3) x (z) → x 0 (z) + x 1 (z) + 2 x 2 (z) (B4) y (z) → y 0 (z) + y 1 (z) + 2 y 2 (z) (B5) . . .
in (B2) and sorts the resulting terms by increasing powers in . Incremental computation of the perturbation terms x 1 (z), x 2 (z), etc. by reinserting the results of each previous calculation step in (B2) provides the beam dynamics corrections for laser field interaction. The procedure has been implemented in Mathematica [40] which yields the approximation (12) if all terms up to O 2 are taken into account. In most accelerator designs the relative energy gain of the particles qE0 mc 2 per accelerating field wavelength 2π/k 0 is comparably small. Thus, the fundamental assumption of the perturbation approach a 0 1 does not specifically apply to DLA undulators, but is also valid in the more general context of electromagnetic undulators like e.g. radio frequency wigglers as discussed in Ref. [28]. However, the approximation (12) fails describing the beam dynamics of a synchronous DLA undulator. If the synchronicity condition (1) is fulfilled, the undulator wave number (13) vanishes and, therefore, the oscillation amplitude in (12) diverges. The perturbation approach assuming small corrections x 1 (z), etc. is not valid anymore. However, the semi-analytical tracking approach for synchronous DLA structures in DLAtrack6D [11] provides a suitable workaround to compensate the weakness of the perturbation theory model for k und → 0.
Appendix C: Construction of 3D Field Maps for the Astra Simulations The fully numeric particle tracking simulation Astra [36] uses 3D field maps of the drive-laser distribution in the beam channel of the DLA undulator lattice. An electromagnetic field simulation in CST Studio [22] provides the complex field amplitudes for the time-harmonic laser field on a hexahedral mesh in the computational domain. The numerical model applies a magnetic boundary condition B x = B z = 0 in the symmetry plane of the DLA cell at y = 0 in order to reduce the computational effort and ensure a mirror-symmetric field distribution.
A post-processing step in Matlab [41] transforms the output of CST Studio to the input file format for 3D cavity field maps in Astra. First, mirroring the numerical simulation results at the y = 0 boundary extends field map to the upper half-space and provides the unit cell of the DLA grating. Figure 17 shows the electromagnetic field in one unit cell of a tilted silica grating. Second, a periodic repetition in x-direction extends the DLA unit cell to several transverse periods of the DLA lattice. This step is cumbersome but necessary, because Astra does not provide a periodic repetition of cavity field maps in transverse direction. However, periodic repetition in longitudinal z-direction is possible, such that Astra can model the complete asynchronous DLA undulator with upwards of 6000 silica grating periods in a single tracking simulation. The laser distribution of the DLA undulator lattice enters the Astra tracking simulation in form of time-dependent accelerator cavity field maps.