# Dependence on Frequency of the Electromagnetic Field Distribution inside a Cylindrical Cavity

Authors: Fabrizio Consoli; Luigi Celona; Giovanni Ciavola; Santo Gammino; Fabio Maimone; Rocco Salvatore Catalano; Sebastiano Barbarino; David Mascali; Livia Tumino
Journal: Active and Passive Electronic Components
 (2008)

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## Abstract

To explain the relevant changes in the electron cyclotron resonance ion source behaviour for small variations of the exciting radiation frequency, we determine the spatial distribution of the field within the cavity for every resonant mode.


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## Body

## 1. Introduction

In we want to study
the electromagnetic field distribution in a perfectly conductive cylindrical
cavity, for every resonant mode excited by a rectangular waveguide operating in
the microwave range, through an aperture placed off axis on the top circular
base (see Figure1).Scheme of the top base of the cylindrical cavity with the feeding waveguide,a =  6.5 cm, L (cavity length) = 45 cm, ag = 2bg = 15.8 mm, xB = −  40.4 mm, yB
= −  28.95 mm. (b) Mode
distribution near to 14 GHz, usual source excitation frequency.
(a)(b)This situation is actually met
in the modern electron cyclotron resonance (ECR) ion sources, where plasma is
magnetically confined and excited by microwave fields. It was experimentally
observed [1] that the plasma formation, the consequent amount of particles
extracted from the source, and the related beam shape strongly depend on the
frequency of the electromagnetic wave feeding the cavity. Indeed, by
considering an ideal cavity, the frequency variation of the incoming radiation
causes the excitation of a discrete number of modes (see Figure 1(b)), each coupled
differently with the off-axis waveguide. Then, the electromagnetic field
distribution inside the cavity will be different for each resonant frequency. Therefore,
the particle motion will be affected by the selected mode, that is, by the
excitation frequency. On the purpose to give a quantitative explanation of the problem we propose, in the preliminary phase of this work, to give the description of the electromagnetic field within the cavity following Van Bladel’s approach [2]. We
consider as a reference for this study the experimental setup represented by
the SERSE ion source operating at INFN-LNS in Catania since 1998 [3, 4], but the
analysis described in the following is applicable to any similar apparatus.


## 2. Description of the Experimental Setup

The
particles inside the ion source cavity are subjected to a nonuniform confining
magnetostatic fieldB→. It is possible to consider it as generated by the
superimposition of a hexapole and two solenoids:(1)Bx=x(−B1z+2Sexy),By=−B1yz+Sex(x2−y2),Bz=B0+B1z2, where Sex
is a constant related
to the hexapole field, B0 and  B1  to the solenoids ones. For the sake
of simplicity, we consider here the presence of only one electron inside the
cavity. The equation describing its motion in a magnetostatic field is(2)dν→dt=qm0γ(ν)[ν→×B→], where ν→,  q,  and m0 are the velocity, the charge,
and the rest mass of  the electron, γ(ν)=[1−(ν/c)2]−0.5 is the
relativistic factor, c is the light
velocity in vacuum, and B→=Bzx^+Byy^+Bzz^ is given by the formulas (1). The particle trajectory achieved by the numerical solution of this differential equation with given initial
conditions has a projection in the xy plane for z = L/2, like that star shape represented in all the pictures
of Figure 2. By considering the different patterns achieved by varying these
initial conditions,  it was possible to observe that the motion shape and the
directions of the trajectories tips are quite similar one to the others because
they are determined by the magnetostatic field only.Modulus of the electric field without the term{1/[ 1 − ((ω/ωnνr)2]} in the plane z = L/2 for modes close to the
14 GHz frequency for 1 W of the incoming wave power. For each mode the maximum
field value is estimated after an equal time interval, supposed large enough to
neglect the second-order terms. The white normal axes indicate the mode
rotation caused by the coupling with the rectangular waveguide.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 3. Distribution of the Electromagnetic Field Within the Cavity

In Figure1(a), the scheme of the
cylindrical cavity is shown with its feeding WR62 rectangular waveguide placed far
from the cavity axis. The fields in stationary conditions, inside
a lossless cavity in vacuum can be written as [2] (3)E→=−∑m[cωmωm2−ω2∫S(z^×E→TE10)⋅h→mdS∫V|e→m|2dV]e→m,H→=−1iωμ∑m[∫S(z^×E→TE10)⋅g→mdS∫V|g→m|2dV]g→m+∑m[1iμωωm2−ω2∫S(z^×E→TE10)⋅h→mdS∫V|h→m|2dV]h→m, where
the index m stands for a triple set
of indices, μ=4π10−7 H/m, c = 299792458 m/s, S is the excitation aperture surface, V is the cavity volume, ω  and ωm are the excitation and the mode
characteristic angular frequencies, h→m and e→m  are the
solenoidal magnetic and electric eigenvectors (with h→m=(c/ωm)∇→×e→m),    g→m is the
irrotational magnetic eigenvector, and (4)E→TE10=−Aieiωtωμagπsin⁡(πxag)y^
is the field of the dominant mode in
the rectangular waveguide. We indicate with ag the waveguide width (see Figure 1), and A is a constant related to the waveguide power. If the frequency of
the incoming wave does not coincide with one of the ωm in (3), the field in the ideal lossless cavity is the
sum of different finite terms, contributing to the total energy in the cavity.
If, otherwise, for a given m¯ the frequency ω coincides exactly with
ωm¯, the term
in (3) relative to this mode will diverge, that is, it will become much larger
than the others. In the time domain, it means that the energy in the cavity
constantly increases with time and with a rate depending on the coupling
between the incoming wave and the mode m¯ [5]. An accurate general representation of the fields at the resonance
can be therefore obtained by considering only the coefficient of the diverging
term. The general expression of the electric field  can be written as(5)E→TM=ieiωtAc2μagωL2πa×∑n,ν,r11−(ω/ωnνr)2εnωnνr2xnνJn+12(xnν){Re⁡{InνTM}Im⁡{InνTM}}×[rxnνaJn′(xnνρa)sin⁡(rπzL){sinnϕcosnϕ}e^ρ+rnρJn(xnνρa)sin⁡(rπzL){cosnϕ−sinnϕ}e^ϕ−Lxnν2πa2Jn(xnνρa)cos⁡(rπzL){sinnϕcosnϕ}e^z], for the  TM modes, with εn the Neumann’s factor (equal to 1 for n = 0, equal to 2 if n=1,2,…),  xnν the ν-root of the Bessel function of n order, and(6)InνTM=−∫Ssin⁡[π(x−xA)ag][e−i(n+1)ϕJn+1(xnνρa)+e−i(n−1)ϕJn−1(xnνρa)]dS, and for the TE modes(7)E→TE=ieiωtAc2μagωL2πa×∑n,ν,r11−(ω/ωnνr)2εnrxnν′ωnνr2(xnν′2−n2)Jn2(xnν′)×{Im⁡{InνTE}Re⁡{InνTE}}sin⁡(rπzL)⋅[nρJn(xnν′ρa){cosnϕ−sinnϕ}e^ρ−xnν′aJn′(xnν′ρa){sinnϕcosnϕ}e^ϕ], with xnν′ the  ν-root of the first derivative of the
Bessel function of n order and(8)InνTE=∫Ssin⁡[π(x−xA)ag][−ei(n+1)ϕJn+1(xnν′ρa)+ei(n−1)ϕJn−1(xnν′ρa)]dS. The integrals (6) and (8) represent the coupling for the TM and TE cases, giving a different amount of energy
to be transferred from the incoming electromagnetic wave to each mode. In Figure 2, the modulus of the electric
field without the term 1/(1−(ω/ωnνr)2) is  shown in the xy plane for z = L/2, for the modes with resonance between 13991.15 MHz and
14005.44 MHz. For each of them, the geometric degeneration is uniquely resolved
by the exciting waveguide, and the related rotation is indicated by two normal
white axes. The maximum modulus has been calculated for 1 W of the waveguide
power, and after a time interval large enough to neglect the second-order terms.
It can be an estimation of the actual energy coupled to each mode.


## 4. Conclusion

It has been shown that the field distribution
inside the cavity of an electron cyclotron resonance ion source can be changed
significantly by means of small frequency variations that have a huge effect both
on the spatial distribution of minima and maxima and on the amount of energy
coupled from the feeding waveguide to the cavity. The modes have a completely
different pattern. Preliminary calculations, not shown here, indicated that
some electromagnetic field configurations are more effective to accelerate the
confined electrons, because of their proper spatial intensity distribution
respect to the star shaped trajectory, representative of the particle motion in
presence of the magnetostatic field only. It has a direct effect on the rate of
ionisation and therefore on the plasma formation.



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*Source: 859052-2008-05-08.xml*