Rosenzweig 1994 0226
Rosenzweig 1994 0226
Rosenzweig 1994 0226
Abstract
1. Introduction
where ε = ( ε min + ε max ) 2 and ∆ε = ε max − ε min . The geometry of this system, which is
assumed, for the purpose of discussion, infinite in the x dimension, is shown schema-
tically in Figure 1, with metallic mirrors chosen to allow the outer dielectric boundary
condition to be approximated as perfectly conducting. This is a powerful model to begin the
analysis, as its solutions can easily be generalized to include any periodically varying
permittivity.
AAAAAAAAAAA
y
Metallic mirror
b
Dielectric mask
d
a
electon beam
Symmetry plane
z
Figure 1. Schematic representation of the top half of one period of a side-injected laser
acceleration resonant structure using a modulated dielectric mask, with width in the x
dimension much greater than λ 0 , and a metallic mirror.
In this model, we can find the symmetric TM modes of the structure by assuming a
separable form of the longitudinal electric field Ez = E0Y ( y ) Z(z)eiωt , where we have
ignored the x dependence, assuming it occurs on a scale much larger than a free-space
wavelength. The familiar wave solution obtained for the vertical dependence of Ez is, thus
assuming a field null at the mirror-mask boundary;
[
Y ( y ) = sin ky ( y − (b + a)) , ] (2)
d2Z 2 ∆ε
2 + k0 ε + cos( k0 z ) − ky2 Z = 0, (3)
dz 2
which can be recognized as the Mathieu equation.
Since we are interested in maximizing the first higher spatial harmonic, the solution
to the Mathieu equation of most interest has two zeros in a period interval. This solution to
Eq. 3 can be written approximately as[12]
ky ≅ k0 ε − 1 + 485 ∆ε 2 . (5)
If the thickness of the dielectric mask is also chosen to yield ky b = nπ 2, n odd, then
∂ Ez ∂ y = 0 vanishes at the vacuum-dielectric interface. This allows for a smooth transition
to the kz = k0 spatial harmonic within the gap which, since it corresponds to ky = 0, is a
constant - i.e., the acceleration due to this resonant component is independent of transverse
position in the gap. The gap half-width a is constrained by the relation k0 a = mπ , m
integer, a condition arising from requiring mirror symmetry of the variation of the
fundamental ( ky = k0 , kz = 0 in the free-space gap region) spatial harmonic, which we have
minimized by our choice of mask thickness. Inside of the gap, therefore, the largest
component of the field is a kz = k0 standing wave whose forward component can
resonantly accelerate relativistic electrons with maximum force
4 πe 2QI
Fz = −eEz ≅
[ ]
, (6)
c 1 + ( ∆ε 2 8)
where I is the incident laser intensity, the average is over a period of the structure, and we
have assumed that the coupling of the laser to the cavity is unity.
For the purpose of example, a list of parameters is given in Table 1 which describe
the salient characteristics of a side-injected laser accelerator based on a modulated dielectric.
In addition, the accelerating field profiles as a function of z and y are shown in Figs. 2(a)
and (b), respectively. The shortness of the laser pulse (which at 2.5 psec is chosen to give
high breakdown threshold) compared to the electron transit time through the structure (67
psec) introduces an additional problem of requiring a correlation between the transverse and
longitudinal laser intensity profile to ensure that the accelerating field is present at the
correct time in the structure. This can be accomplished through use of a dielectric "stair-
case" to produce the necessary delay in the arrival time of the laser power at larger values of
z . Care must be taken in specification of this component, however, as with any component
which affects the phase of the incoming light. The maintanence of spatial coherence across
the laser wave-fronts is critical to this scheme.
2.50
2.00
1.50
1.00
Ez (GV/m)
0.50
0.00
-0.50
-1.00
-1.50
-2.00
-2.50
0.0 0.2 0.4 0.6 0.8 1.0
k0z/2π
3.00
2.00
1.00
Ez (GV/m)
0.00
-1.00
-2.00
Gap region
Dielectric mask
-3.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
y (µm)
2. (a) Longitudinal dependence at the symmetry plane ( y = 0) of Ez for parameters given
in Table 1. The offset from zero of the oscillation is due to the (kz = 0 ) fundamental space
harmonic. (b) Vertical dependence of Ez for parameters given in Table 1. The modulation
of Ez is due to the presence of the fundamental space harmonic.
It should be noted that this type of accelerator is conceptually similar to a drift-tube
linac[13], in that the component of the accelerating field which can resonantly interact with
a relativistic electron is created by periodically diminishing the zero-mode excitation.
Although only the forward wave component of this standing wave can impart net
acceleration to high energy electrons, the backward wave component can provide second
order transverse focusing for accelerating electrons, as discussed below.
The longitudinal dynamics of electrons in this type of structure are more interesting
than in high gradient microwave accelerators, because the phase slippage is significant even
for fairly relativistic electrons, due to the shortness of the wavelength. This slippage can be
counteracted at low energies, as one can taper the periodicity length of the dielectric mask in
z to satisfy kz (z) = 2 π d(z) = ω v(z), matching the phase velocity of the accelerating
component of the field to the velocity of the electrons. If the field amplitude is also ramped
positively in the transrelativistic injection region, then an initially nearly monochromatic,
continuous beam of low energy electrons may be adiabatically captured into small regions
of phase, allowing a for very short micro-bunches with minimal energy spread. This type
of capture and acceleration scheme is in fact typical of proton linacs (such as an RFQ[13],
or a drift-tube linac), where the transrelativistic region also has many spatial periodicity
lengths.
Table 1. Example parameters for a high gradient, resonant side-injected laser accelerator.
γ ′ γ ′′ 1 γ ′ 2
y ′′ + y ′ + + y = 0. (7)
γ γ 2 γ
Here the prime indicates the derivative with respect to distance along z , and we have
defined for ultra-relativistic electrons the normalized energy gain averaged over a period of
the structure, γ ′(z) ≡ −eEz (z) me c 2 . The term proportional to γ ′ is responsible for
adiabatic damping of the motion, and the terms proportional to γ ′′ and ( γ ′ ) and due to the
2
focusing
0.60
no focusing
0.50
σy (µm)
0.40
0.30
0.20
0.10
0.00
0.0 1.0 2.0 3.0 4.0 5.0
z (mm)
3. Evolution of rms envelope (solid line) in a linearly rising field region (Lr = 5 mm) with
all other parameters as given in Table 1. The dashed line gives envelope in the absence of
the focusing and accelerating electromagnetic field.
As can be seen from this example, it is the electromagnetic focusing which allows
the propagation of the beam through the gap of the structure. We must emphasize that the
physical basis of this effect is beneficial only symmetric structures. In asymmetric systems
like the inverse Smith-Purcell accelerator (which have been proposed due to ease of
coupling the laser to the structure), transverse electromagnetic forces will produce
uncompensated deflections due to ponderomotive effects, not focusing. It is also interesting
to note that the transverse emittance required for this scheme is very small, many orders of
magnitude below those injected into rf electron linacs. This is a general feature of
acceleration at optical wavelengths, as the beam size must scale with the wavelength to pass
through any acceleration structure. Likewise, the longitudinal emittance of an electron
bunch captured and accelerated in an optical wave will be much smaller than what is
achieved in microwave devices. One can view this scaling both as a technical challenge,
and as a desirable result of laser acceleration, since beams of such small phase space
volume would undoubtedly lead to many advances in electron beam-based sciences.
In addition to independent particle dynamics, one must be concerned with collective
effects, most importantly the space-charge fields of the beam which are deleterious at low
energy, and also the beam-excited transverse modes in an accelerator, which can lead to an
instability known commonly as beam break-up[18] (BBU). The geometry of the beam,
which is much larger in the x than the y -dimension, by itself will mitigate the space-charge
effects in comparison to a cylindrically symmetric beam[[14]. Also, it is well known that
short wavelength structures electromagnetically couple more strongly to the beam current
due to the proximity of the walls of the structure to the beam axis. Since the structure under
consideration is open horizontally, and will only confine well the mode being externally
pumped[8], any beam excited transverse modes should radiate away, and not feed back on
the beam dynamics. Thus this type of structure should have a high threshold current for
BBU, which is important for allowing high levels of beam-loading, and thus high power
efficiency.
5. Conclusions
The promise of the system we have described in this paper is to obtain a compact,
laser excited, high gradient, inexpensive electron accelerator with attractive injection and
transport properties. Most of the demands on the laser system, which in the example uses
only 200 µJ per pulse, can be met with today's commercial systems, with some issues
requiring additional attention, such as temporal flattening of the laser pulse to provide a
uniform accelerating field over most of the laser irraditation time. In addition, the example
we have provided here uses 1 µ m light (typical of that produced by a Nd:YLF laser), while
it is clear that initial proof-of-principle experiments will be easier if performed with longer
wavelengths, easing the structure manufacture and alignment, as well as the injected
electron source requirements, which are beyond the state of the art in our example.
However, there is much reason to believe that the dielectric structures we have described
are not overly challenging to build; nanofabrication techniques developed in recent years,
for advanced applications such as gradient index optics[19], and microcavity lasers[20],
can also meet most of the requirements for construction of the dielectric masks.
The experimental development of this type of accelerator should be within reach of
the advanced accelerator community, as the concepts underlying it are nearly identical to
known linear accelerators. The innovation here lies entirely in the desire to use inexpensive
optical radiation, which requires a rethinking of the accelerator structure, and coupling
system design to meet the demands of beam acceleration and transport as they are scaled to
shorter wavelengths.
Acknowledgements
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