It has been shown [1-3] that, if a plane electromagnetic wave is scattered on a finite dielectric object, structural resonances can be excited in the object (e.g., whispering gallery modes). They are associated with very high amplitudes of oscillating EM fields in the dielectric and its vicinity. Their maxima exceed values reached in resonant cavities of typical linear accelerators by several orders of magnitude. Therefore, one can think of applying these fields to accelerate charged particles [1-3]. Many other applications of the whispering gallery modes are described in [4-6].
As for the proposals given in [1-3], both light produced by lasers and microwaves are conceivable. However, it is difficult to achieve the required synchronization of wave particle in the optical frequency range. In the microwave frequency range, this mechanism would require excessive total excitation energy and so may not be practical.
In this paper we demonstrate that the last mentioned problem can be overcome by locating the dielectric object in a resonant cavity. This appeals to traditional accelerating structures used in SLAC, see
cavities. Unfortunately, since superconductivity of the walls disappears if the magnetic field on the wall exceeds a critical value, the maximal values of accelerating fields in these highly complicated cavities are not much higher than those reached in SLAC.
The presence of a dielectric in the central part of the resonance cavity shifts the magnetic field maximum from regions close to the metallic wall towards the dielectric surface. This considerably lowers skin effect losses in the wall. Even though additional losses due to dielectric heating are introduced, total losses would nevertheless be lower if one could apply ultra low loss dielectrics (with
) as described in [7,8]. In that case, a resonator quality reaching 
can be obtained, as compared to 
for SLAC.
either dielectric or vacuum layers. Each dielectric layer is assumed to be homogeneous. We introduce a spherical coordinate system 
with its origin at the cavity center. The layers are bounded by
, up to 
for the metallic boundary, see 
, and complex fields proportional to
):
is the angular frequency, and 
denotes complex dielectric permittivity
we obtain the most general form of the TE field in the uniform dielectric:
are vector spherical harmonics as defined by Eqation (9.119) in [
involves both the transverse and radial component:
are spherical harmonics, 
, 
is a positive integer related to the integer 
by 
involves both the transverse, and radial component:
, these losses will be neglected. Thus the wall is assumed to be perfectly conducting. This means that it carries no electric or magnetic field. Then continuity of the tangential components of the electric field and the normal components of the magnetic induction at each interface require vanishing of these components at the boundary of 
th layer at
. In view of (4)-(9), and (12)-(19) this will be the case if the following boundary conditions at 
are fulfilled:
we must choose
or 
at
.
, the boundary conditions (20) lead to
This defines the resonant frequency 
of either TE or TM modes, depending only on 
and
.
, 
must also depend on
etc. and therefore condition (22) cannot be fulfilled. In fact, for the same reason, we can assume that also the remaining functions in conditions (20) are non-vanishing. Therefore these conditions can be satisfied by choosing
and 
are normalization factors.
:
matrices are non-singular:
and 
are solutions of the Bessel equation
th interface between dielectrics, this relation defines the vector 
at the lower layer in terms of that at the upper one. Using this relation successively for
, we can express all 
vectors in terms of
, see (21). In view of this requirement, Equation (25) for 
can be written as
, and 
are components of the vector
. This vector is defined by (32) and (31) if
:
, 
is defined by (31), i.e., is given by the last factor in (34).
and 
will have non-zero solutions if and only if its determinant vanishes,
is given by either of Equation (33), which are equivalent. Like all remaining coefficients 
and
, also 
will be proportional to the normalization factor
, see (32) and (33).
, 
and 
in (33) and (35) are given by (31) and the resonant frequency 
defined by (35) can be found from
and
, we obtain the corresponding equations for the TM modes.
, along with the pertinent linear algebra for
. We did it for
, see the following section, and also for
, by using Mathematica.
in (19) is continuous at each dielectric interface, 
will have jumps, due to discontinuities in
. However, the radial component of the electric displacement 
will be continuous. This will also be true of the TE modes where the radial displacement is identically zero. These facts imply the vanishing of surface charges at each dielectric interface. And this in turn means that the multi-layer dielectric structure resembles (and can approximate) a smooth dielectric with some permittivity profile
, in spite of jumps in
.
is arbitrary.
and dielectric permittivity
, i.e., for
, and
.
and 
in (36) and (37), we obtain equations defining the resonant frequency 
and the amplitude coefficient
.
, the average energies associated with the electric and the magnetic field in the cavity are equal:
was chosen so as to satisfy:
as a function of 
is presented for three spherical cavities with dielectric spheres. Note that 
is 
independent (due to degeneracy). This fact is one of the reasons why the spherical resonator shown in 
of a relativistic electron crossing the spherical cavity shown in 
and electron velocity
:
was chosen at the center of the dielectric sphere, and the electron moving along the 
axis was passing just above the dielectric sphere as shown in 
, is equal to the real part of 
is the electron trajectory segment within the cavity, 
is the 
component of the electric field 
given by (4) and 
is given by (43). Thus
if 
is chosen so that the accelerating phase
. With this choice, the relativistic electron is never decelerated within the spherical cavity, see 
obtained with our normalization (42) are shown in Figures 6 and 7.
given by (43), due to interaction with this field. A nice feature of our model is that the field in question is described analytically by (4)-(9) so that the pertinent equations of the transversal motion can easily be integrated numerically.
and 
acting on each electron will be superpositions of the external fields and the fields due to the electron charge and current. However, in the lowest approximation (and particularly for not too large beam densities) the latter fields can be neglected. Furthermore, if as in our case, the transversal deflections are small, the deflecting fields can be calculated on the unperturbed trajectory given by (43). It will also be assumed that the electron mass
, is time independent within the spherical cavity. With these approximations, and within the Cartesian coordinate system 
with its center at
, the 
axis along the unperturbed trajectory, and the 
axis along
, the electron’s transversal motion will be described by
, the field components 
, etc. are given by the real parts of Equations (4), (8) and (9), taken at
, and
. Integrating these equations with zero initial conditions we end up with 
, and
.
and 
of the deflecting force. They correspond to 
cm, 
cm, and
, for which the the accelerating field will be our reference value
, both 
and 
are odd functions. Therefore in this case of maximal acceleration, there will be no transversal velocity increments over the cavity, 
. At the same time the velocity components 
and 
in (51) will be even functions of 
tending to zero as
. Hence the transversal deflections 
and 
will be increasing functions tending to constants as
, see 
for which 
and 
will be increasing functions soon reaching their limiting values 
and 
for
. The corresponding transversal motions 
and 
will soon become uniform for
, leading to much larger deflections at
, see 
is given by (52), 
is the assumed value of the effective accelerating field, and
. For large values of
, small 
requires 
to be sufficiently large. Assuming that 
, see 
of the spacing between the dielectric sphere and the electron trajectory, 
cm, the required minimal electron energy is given by
MV/m, the transversal displacements will be smaller than 
of the spacing in question, if the electron energy 
GeV, i.e., for typical output energies from SLAC. Whether the real dielectric can withstand this value of 
is another question beyond the scope of this paper. More comments will be given later on.
of its resonant cavities:
is the resonant angular frequency of the ideal cavity
, 
is the corresponding reso-
is the time-averaged energy stored in the cavity 
is time-averaged cavity power loss.
is given by
is the skin depth, 
is conductivity of the wall, and the magnetic field intensity 
refers to the ideal cavity, i.e., its normal component is vanishing
.
:
and height
. In that case, the cylindrically symmetric 
TM mode used for acceleration is given by
, 
is a Bessel function, and 
and 
are cylindrical coordinates (cylindrical axis along
).
on an ideally conducting cylindrical wall requires that 
(the smallest zero of
) which defines the angular resonant frequency in terms of
. Equations (62) and (60) lead to the well known formula for the quality of the cylindrical pill box cavity
and 
for copper wall in room-temperature), this formula leads to
. The corresponding values for spherical cavities with ideal dielectric spheres and the same values of 
and 
reach much larger values, see 
. The non-vanishing value of 
leads to
, for 
defined by (35). In view of Equation (2) this implies a complex value of
.
for the energy 
being dissipated rather than generated. Using this result we find for the power losses in the dielectric:
. It is approximately 
independent.
and
. Dielectrics with such ultra small losses were investigated in [
is due to the power loss in the metallic wall and that in the dielectric sphere:
and using (55) we obtain
is the total Q-factor of the spherical cavity. Values of 
versus 
for three spherical cavities with dielectric spheres and 
cm are shown in 
.
but should have little effect on the total quality of the spherical cavity
. The latter is defined by losses in the dielectric, see (66) where
. The resonant frequency should not be drastically changed either, as the EM fields at the iris
are very small fractions of their maxima, see 
, the time-averaged energy stored in the cavity, see (56), is the same in both situations. Therefore the normalization factor 
in (61) and (62) will first be chosen so that
erg).
for the cylindrical resonator shown in 
is given by the right hand side of (45) in which 
is defined by (61) with
, and
. The result is
is the time at which the electron passes the center of the cavity.
and 
the maximal effective accelerating fields (equal to
) for the spherical cavity with a dielectric sphere and the traditional cylindrical cavity, for any values of the average energies in the cavities, 
and
. In view of the fact that 
in (61) is proportional to 
we can write, using the definition (55) of
,
, the “gain factor”, is given by
and 
are the resonant frequencies of the spherical and the cylindrical cavities 
and 
and 
are the corresponding power losses. They are equal to the powers that must be supplied from external sources to sustain the oscillations. They should be as large as possible to avoid breakdown in the dielectric or at the metallic wall. Further research is necessary to give an estimate of the ratio
. We can only hope that it is not smaller than unity.
we obtain 
and
kV/m to be used in (70). The resulting values of the gain factor are shown in 
.
and
. The optimal parameters found were: 
cm, 
cm, and
.
Kuliński for useful discussions.