Proceedings of the 14th Annual Meeting of Particle Accelerator Society of Japan
August 1-3, 2017, Sapporo, Japan

PASJ2017 TUP115

DIELECTRIC LASER ACCELERATION OF ELECTRONS WITHDOUBLE GRATING STRUCTURE∗
Zhaofu Chen† , The University of Tokyo, Tokyo, Japan
Kazuyoshi Koyama, Mitsuru Uesaka,The University of Tokyo, Tokai, Japan
Mitsuhiro Yoshida, High Energy Accelerator Research Organization, Tsukuba, Japan
Abstract
The dielectric laser accelerator (DLAs) has attracted increasing interest in the recent years due to its potential tominiaturize the accelerator. We propose a double grating
structure powered by obliquely incident laser for subrelativistic electrons. The structure enables a symmetric accelerating
mode which results in lower energy spread.
Figure 1: Cross section of the grating discussed in this paper
and the longitudinal electric field of the accelerating mode.

INTRODUCTION
Dielectric laser accelerator (DLA) with dielectric structure has great potential for the miniaturization of accelerator
with its high accelerating gradient enabled by the high damage thresholds of dielectric materials in optical region [1, 2].
Recently, by leveraging the well-developed industrial fabrication capabilities and the commercially available highpower lasers, accelerating gradient of 690 MV/m has beenachieved with fused silica dual-grating DLA for relativisticelectrons [3, 4], and accelerating gradient of 370 MV/m has
been obtained with silicon dual-pillar DLA for nonrelativistic electrons [5, 6].
Here, we present a double grating structure for the acceleration of subrelativistic electrons. The structure is a modification from the double structure experimentally demonstrated
by Leedle et al. [6–11], but with a oblique-angle incident
laser. The oblique angle enables easier fabrication of thegrating and the double grating structure result in a rather
uniform accelerating field across the channel.

SINGLE GRATING
To understand the accelerating mode in double grating
which is the superposition of modes from each single grating, we start from the evanescent modes at single grating.
Figure 1 shows the single grating and the longitudinal electric field of the accelerating mode. The laser is incident fromabove in the x direction, the electron is propagating alongz direction. For simplicity, we constraint our discussion on
the case of TM polarization in which the electric field isperpendicular to the groove direction, which is required forthe acceleration.
At a single grating with period Λ, the incident plane wave
with wavelength λ excites a series of spatial harmonics in

its vicinity with field profiles given by:Hy,n (x, z) =Ex,n (x, z) =Ez,n (x, z) =H0 e−jk x, n x−jkz, n zkzHyω−k xHy
ω

(1)
(2)
(3)

with θ the incident angle, n = 0, ±1, ±2, . . . the order number, k 0 = 2π/λ the wave number in free space, the z component of the wave vector k z,n = k 0 sin(θ) + nk p withk p = 2π/Λ.
In the longitudinal direction, the phase velocity of theseharmonics could be written as v ph,n = ω/k z,n . The synchronicity condition between the electron and spatial harmonic with order number s results in
k z,s = k0 /βs
k x,s = ik0 /(βs γs )

(4)
(5)

where βs is the electron velocity vs normalized to the speedof light c. We can describe the synchronicity condition withthe ratio of grating period Λ to laser wavelength λ:Λβn=λ1 − β sin θ(6)∗ Work supported by KAKENHI, Grant-in-Aid for Scientific Research (B)

15H03595† chen.zhaofu@nuclear.jp

Figure 2: Required grating period Λ vs incident angle θ.

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Proceedings of the 14th Annual Meeting of Particle Accelerator Society of Japan
August 1-3, 2017, Sapporo, JapanPASJ2017 TUP115

The accelerating mode profiles can be given bypx
1
HdHy,s (x) = 2 Hu Hd sinh+ ln( )Γs 2Hu√r1Hd2 Hu Hd µ0xsinh+ ln( )Ex,s (x) =βs0Γs 2Hu√ri2 Hu Hd µ01HdxEz,s (x) =cosh+ ln( )βs γ s0Γs 2
HuFigure 3: Simulated longitudinal electric field in optimized
SiO2 single grating structure (a) and accelerating gradientaround it (b).

In Fig. 2 we show the required grating period as a functionof incident angle θ and normalized electron velocity βs . It isclear that the grating period becomes larger with an oblique
incident angle, thus the difficulty in fabrication could berelaxed, which is desirable for electron with low β [12].
With Γs = 1/ik x,s , we can rewritten the field profiles asHy,s (x, z) =Ex,s (x, z) =Ez,s (x, z) =H0 e−x/Γs e−j(kz, s z)r
1 µ0
H0 e−x/Γs e−j(kz, s z)
β s 0r−iµ0
H0 e−x/Γs e−j(kz, s z)βs γs 0(7)(8)(9)

The imaginary value of k x as shown in Eq. 5 indicates thatthe accelerating field is evanescent and fall off exponentiallywith an increasing distance from the grating surface in x direction, as shown in Fig. 3. The skew profile is not desirable
for accelerator since it results in large energy spread [13].

DOUBLE GRATING
To produce a symmetric mode, we use the double grating
as shown in Fig. 4. The superposition of evanescent modes
from each grating forms the accelerating mode inside thechannel, thus a more uniform profile could be obtained.

Figure 4: Left: Schematic of the double grating discussed
in this paper and the longitudinal electric field of the accelerating mode. Right: Accelerating gradient G a and deflecting
gradient G d across the channel.

(10)
(11)
(12)

where Hu and Hd are the amplitudes of modes at the channel
center from above and below, respectively. Eq. 10, Eq. 11and Eq. 12 indicate that the axis of the accelerating mode
may not be located at the channel center, depending on theamplitude of modes from single grating.
The Lorentz force experienced by the synchronous electron in the channel can be given byhiq√ 2 H H q µ0

Hdx1
sinh+ln()ei(ωt−kz, s z−φ0 ) Fx   βsuγ2 d0Γs2Hus  
 Fy  =    √h 0iq
 Fz   i2 Hu Hd q µ0Hx1i(ωt−k z, s z−φ0 ) dcosh+ln()e   β s γsΓ2Hsu0(13)where φ0 is the start phase of electron in the optical cycle.
For the case where Hd = Hu , the accelerating electricfield in the channel is symmetric, as shown in Fig. 4. Withqµ0uq
the accelerating gradient at channel center G a,0 = 2Hβ s γs
0 ,the Lorentz force experienced by the synchronous electron inthe channel can be rewritten as Eq. 14. It can be seen that theaccelerating mode has a rather uniform electric field profilearound the axis, which is desirable for the acceleration ofelectrons.

Fx  −G a,0 sinh(x/Γs ) cos(ωt − k z,s z − φ0 )/γs   
 Fy  = 
 (14)0  
 Fz   G a,0 cosh(x/Γs ) sin(ωt − k z,s z − φ0 )   ACCELERATION OF ELECTRONSHere, we focus on the case where the first order spatial
harmonic is used for the acceleration. As the electron gains
energy, the relative phase of the electron with respect tothe accelerating mode changes as the electron gains energy,
known as the dephasing process, which may lead to decreasing of acceleration gradient, or even deceleration. In thiscase, the energy gain depends not only the accelerating ratio between the maximum gradient and the electric field in
the grating material G a /E0 , but also the changing of therelative phase of electron with respect to the optical cycle.
To estimate the energy gain, we have to study the effect ofdephasing on the acceleration, i.e., how the gradient changes
with the distance upon dephasing.
Electrons with kinetic energy of 50 keV are injected into
the channel with width d = 150 nm. The grating has a lengthof 30 µm, including 71 periods. In the simulation, identical accelerating efficiency G a /E0 and deflecting efficiency
G d /E0 in Fig. 4 are used, with G a,0 /E0 = 0.015 at channel center. The Gaussian laser pulse with surface-normal- 678 -

Proceedings of the 14th Annual Meeting of Particle Accelerator Society of Japan
August 1-3, 2017, Sapporo, JapanPASJ2017 TUP115

Figure 5: The energy gain ∆E (left), lateral displacement ∆d (mid) and phase slippage ∆φ (right) of electron as a functionof start phase φ0 and initial distance d0 from the grating below. The white area represents those electrons which are deflected
and crash into the dielectric grating.
incidence has an average electric fieldt − τp /2 2z − w1 /2 2E0 (z, t) = E p exp −() − 2 ln(2)()
w1
τp(15)
where peak field E p = 6.7 GV/m, pulse duration τp =100 fs and waist radius w1 = 10 µm. In the synchronous
frame the average electric field experienced by electrons canbe written asz 2E0 (z) = E p exp −()(16)
wintwith
2 ln(2)
1+wint =22(βw1s cτp )! −1/2(17)

For clarity, we use subscript s to represent the electrons
which are synchronous with the first order spatial harmonic,
and subscript e to represent instantaneous parameters for theelectrons traversing the grating. Here, we assume ∆βe /βe 1, which is true when the electron’s energy gain over onewavelength of the incident field is well below the electron’srest energy m0 c2 , i.e., G a  m0 c2 /λ ≈ 496 GV/m. In a
DLA driven by an infrared laser, the general acceleration
gradient G a < 10 GV/m, so that the assumption is valid.
Since the electrons’ transverse velocity is much smaller
than the longitudinal velocity, so we can neglect the contribution of the transverse velocity to the total kinetic energy
in the longitudinal motion equation. From the theorem ofkinetic energy, we obtain the longitudinal motion equation,where φ = ωt − k z z − φ0 represents the phase of electron in
the optical cycledγe m0 c2= Re(Fz )dz
dφ1γe= k0−pdz
βsγe2 − 1asdγe m0 vx
1=Re(Fx )dz
βe cvxdx=dz
βe c(22)(23)

With Eq. 18, Eq. 19, Eq. 22 and Eq. 23, we could estimate
the electron parameters at the end of grating, neglecting thewake field and space charge effect. For one 50 keV electron
traveling through the channel, the energy gain, the lateraldisplacement and the phase slippage vs. the start phase φand x position are shown in Fig. 5. The cosh accelerating
field profile leads to a rather uniform energy gain aroundchannel center, which is useful for many applications. It
can be also seen that due to the transverse deflection, someelectrons will be deflected into the grating. The electronwave interaction dependence on start phase as shown in Fig. 5
determines that operation with the longitudinal focusing
leads to transverse defocusing, which is in agreement withEarnshaw’s theorem [14]. To stably accelerate electron,
external focusing component is needed.
In Fig. 6 we show the instantaneous parameters of one
electron launched at the channel center at an initial phaseof 3π/2 as a function of the longitudinal posizion z, including the kinetic energy Ek , the acceleration gradient G a , the(18)!
(19)

The transverse motion equation can be given bydγe m0 vx= Re(Fx )
dt
dx
= vx
dtd
By using dtd = βe c dz
, equation (20) and (21) can rewritten

(20)
(21)

Figure 6: Instantaneous electron parameters as a functionof z position. We show the instantaneous kinetic energy Ek(left: red curve), the acceleration gradient G a (left: bluecurve), the average laser electric field experienced by electron E0 (right: red curve) and the phase of electron in theoptical cycle φ (right: blue curve). The 50 keV electrons
are launched at start phase φ = 3π/2 at the channel centerx = 0 nm .

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Proceedings of the 14th Annual Meeting of Particle Accelerator Society of Japan
August 1-3, 2017, Sapporo, JapanPASJ2017 TUP115

Figure 7: Left: The normalized electron charge density after interaction. Mid: The normalized electron charge density
dependence on the positionx before (red) and after (blue) interaction. Right: The normalized electron charge density
dependence on the energy E. The electron beam density is normalized to 1.
laser electric field amplitude experienced by the electron E0
and the phase of electron in the optical cycle φ. The lateraldisplacement of the electron is zero due to the vanishingdeflection at channel center.It is shown that the dephasing
effect could be so severe that the electron is decelerated after
initial acceleration, i.e. the electron moves from the acceleration phase to the deceleration phase. By adjusting thegrating period according to electron energy variations, thedephasing problem could be alleviated and a higher energy
gain could be obtained.
To study the density distribution after interaction, a simplified 2D model is used in the estimation, with the laserand electron beam assumed to be uniform along y direction.
For one electron beam traversing the channel, the charge
density after interaction are shown in Fig. 7. The temporalenvelope and the spatial envelope are taken into account,which are identical with Eq. 15. A Gaussian electron bunchof transverse waist radius we = 30nm is launched at thechannel center. For simplicity, the density in electron bunchis set to be temporally constant and the pulse duration is one
optical cycle. The electron beam energy spread and beam
divergence are neglected. It can be seen that the proportionof electron around the channel center becomes higher afterinteraction, which is the result of sinh deflecting field. There
are two distinct charge peaks at channel center, with energy
48.6 keV and 51.4 keV, respectively.

CONCLUSION
We present the study of a double grating structure for theacceleration of 50 keV electrons. By using the obliquely
incident laser, the difficulty in fabrication of the grating
structure could be alleviated. The uniform accelerating fieldaround the channel center, which is formed by superpositionof the evanescent modes from the single grating at each side,is desirable for a variety of applications, such as table-top
cancer therapy and university-scale light sources.

ACKNOWLEDGEMENT
We thank R. Zhang, T. Shibuya and T. Natsui for helpful
discussions.

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