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CALCULATION OF FIELD COMPONENTS IN AN ANISOTROPIC
CYLINDRICAL DIELECTRIC RESONATOR
A.1 BASIC EQUATIONS
These notes deal with the basic time-harmonic electromagnetic field equations in a linear
anisotropic medium (Tobar and Mann 1991). Assuming the dielectric medium is anisotropic with
no off diagonal terms in the permittivity tensor, lossless and in a source free region, the electric and
magnetic field satisfy Maxwell's two curl equations;
  H = jD
(A.1),
  E = -jH
(A.2),
with
D =
r 0 0 
0  0 
0 0 z 
0E
(A.3),
.H = 0
(A.4).
together with Gauss's law in a charge free region;
.D = 0
; as well as
E, H and D are the electric field, magnetic field and electric flux density vector-phasors
respectively. The constant scalars r,  and z are the radial, azimuthal and axial relative
permittivities respectively, and the constant scalars 0 and  are the permittivity of free space and
the isotropic permeability of the medium respectively.
When solving these equations with respect to an anisotropic medium it is assumed that the
cylindrical crystal is orientated with its c-axis parallel to the z-axis of the cylinder. Hence the
relative permittivity parallel to the c-axis is given by z and perpendicular to the c-axis is given by
r where  = r as well.
The amount of difficulty can be greatly reduced by realising that only two field components directed
along a constant direction are needed to derive all other components (Auda and Kajfez, 1986). In
cylindrical co-ordinates the easiest direction to solve for is in the z direction.
Taking the curl of (A.1) it becomes;
(.H) - 2H = j   D
(A.5)
Utilising the fact that  = r the z component of   D is equal to the z component of r0  E
which from (A2) equals jz0Hz , and .H = 0. Substituting this into (A.5), the z component of
(A.5) can be written as;
2
2Hz + kr Hz = 0
where kr = (r0
)1/2
.
(A.6),
2
Taking the curl of (A.2) it becomes;
(.E) - 2E = -j   H
(A.7)
To calculate the z component of (A.7) is more complicated because the divergence of the electric
field is not constant for an anisotropic dielectric. In cylindrical co-ordinates the z component of
(.E) can be written as;
1  Er  2Er 1  2E   2Ez
 +
(.E)z =  r
+
+
 z2
  z  z r r  z 
(A.8)
However the divergence of the electric flux density is zero, hence utilising (A.3) the z component of
(.D) is equal to zero and can be written as;
 2Ez
1  Er  2Er 1  2E 
 + z 2 = 0
(.D)z = r r
+
+r
 z 
z
  z  z r
(A.9)
Eliminating the term in brackets from (A.8) and (A.9) the z component of (.E) becomes;
(.E)z =
 2Ez  z
1 - 
 z2  r 
(A.10)
Hence by substituting (A.1) and (A.10) into (A.7) the z component of (A.7) can be written as;
2Ez +
 2Ez z

2
 - 1 + kz Ez = 0
2
 z  r

(A.11),
where kz = (z0)1/2 .
The axial field components in the resonator, Ez and Hz , can thus be obtained by solving (A.6) and
(A.11). Once these components have been solved, the following relations derived from Maxwell's
equations are used to calculate the rest (Auda and Kajfez, 1986);
jr  Ez  2Hz
 2

+
 2 + kr2 Hr = r

 z r
 z

 Hz 1  2Ez
 2

2
+r
 2 + kr  E = j
r
 z 
 z

-j  Hz  2Ez
 2

+
 2 + kr2 Er = r

 z r
 z

2
 Ez 1  2Hz


+ r
 2 + kr2 H = -jr
r
 z 
 z

The solution of Ez and Hz is readily accomplished using the separation of variables method.
A.2 SOLUTION OF THE ANISOTROPIC WAVE EQUATION
(A.12)
(A.13)
(A.14)
(A.15)
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IN CYLINDRICAL CO-ORDINATES
In this section, the method of separation of variables is used to obtain solutions for the anisotropic
wave equations given by (A.6) and (A.11).
A.2.1 Solution of Hz
Equation (A.6) is of the same form for the isotropic case and is given by;
1    Hz 1  2Hz  2Hz
2


r  r r  r  + r2  2 +  z2 + kr Hz = 0
(A.16).
The z component of the magnetic field can not see the permittivity parallel to the c-axis, and hence
(A.16) is independent of z, with the effective dielectric constant equal to r.
The method of separation of variables seeks a solution to (A.16) of the form,
Hz = RH(r) FH() ZH(z)
(A.17)
Dividing (A.16) by Hz then substituting in (A.17), it becomes;
1 d  dRH
1 d2FH
1 d2ZH
2


rRH dr r dr  + r2FH d2 + ZH dz2 + kr
= 0
(A.18)
The third term is explicitly independent of r and . It is also necessarily independent of z, if (A.18)
is to sum to zero for all (r,,z). Thus;
1 d2ZH
2
(A.19),
ZH dz2 = -H
where H is a constant. Substituting (A.19) into (A.18), then multiplying through by r2 , it
becomes;
r d  dRH
1 d2FH
2
2


RH dr r dr  + FH d2 + kr - H
(
) r2
= 0
(A.20)
The second term is a function of  only, and by the same argument;
1 d2FH
2
FH d2 = -m
(A.21) ,
where m is constant. Substituting (A.21) into (A.20) and multiplying throughout by RH , (A.20)
becomes;
d  dRH
r dr r dr  + ((kH r)2 - m2) RH = 0


(A.22),
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where,
2
2
2
kH = kr - H
(A.23)
The wave equation is thus separated into three equations, each of which determines one of the
functions RH ,FH or ZH . The first two, (A.19) and (A.21), are harmonic equations, whose solutions
are harmonic functions. The last equation (A.22) is a Bessel equation of the mth order, whose
solutions are Bessel functions.
A.2.2 Solution of Ez
Equation (A.11) is given by;
1    Ez 1  2Ez z  2Ez
2


r  r r  r  + r2  2 + r  z2 + kz Ez = 0
(A.24).
The method of separation of variables seeks a solution to (A.24) of the form,
Ez = RE(r) FE() ZE(z)
(A.25)
Dividing (A.24) by Ez then substituting in (A.25), it becomes;
1 d  dRE
1 d2FE z 1 d2ZE
2
r
 + 2
rRE dr  dr  r FE d2 + r ZE dz2 + kz
= 0
(A.26)
The third term is explicitly independent of r and . It is also necessarily independent of z, if (A.26)
is to sum to zero for all (r,,z). Thus;
1 d 2Z E
2
= -E
2
Z E dz
(A.27),
where E is a constant. Substituting (A.27) into (A.26), then multiplying through by r2 , it becomes;
r d  dRE
1 d2FE
2
2
r
 +
RE dr  dr  FE d2 + kz - E
(
) r2
= 0
(A.28)
The second term is a function of  only, and by the same argument;
1 d2FE
FE d2
= -m2
(A.29) ,
where m is constant. Substituting (A.29) into (A.28) and multiplying throughout by RE , (A.28)
becomes;
d  dRE
r dr r dr  + ((kE r)2 - m2) RE


= 0
(A.30),
5
 2
2
2
kE = kz - z E
r
where,
(A.31)
The wave equation is thus separated into three equations, each of which determines one of the
functions RE ,FE or ZE . The first two, (A.27) and (A.29), are harmonic equations, whose solutions
are harmonic functions. The last equation (A.30) is a Bessel equation of the mth order, whose
solutions are Bessel functions.
A.3 FIELD COMPONENTS
Consider the cylindrical dielectric resonator shown in figure A1.
z
3
2
2
1

r
h
3
d
Figure A1. The open dielectric crystal is analyzed in cylindrical coordinates {r,,z}. Resonant frequencies are solved
by matching tangential fields between regions 1 and 2, and regions 1 and 3.
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The field components Ez and Hz are periodic inside the dielectric and evanescent outside, also they
must satisfy the separation of variables solutions derived in the previous section. These
requirements are met by choosing Ez and Hz in the form;
Cos(m ) CosE z 
Sin(m  ) Sin E z
Cos(m  ) CosE z
E z2  C Km (k E r)
Sin(m  ) Sin Ez 
Cos(m)
Ez3 = E Jm(kEr)
exp(-z)
Sin(m)
Sin(m) Sin(H z)
Hz1 = B Jm(kHr)
Cos(m) Cos(H z)
E z1  AJ m (kE r)
Hz2 = D Km(koutr)
Hz3 = F Jm(kHr)
Sin(m) Sin(H z)
Cos(m) Cos(H z)
Sin(m)
exp(-z)
Cos(m)
(A.32a)
(A.32b)
(A.32c)
(A.32d)
(A.32e)
(A.32f)
To simplify the solutions of the resonant frequencies it has been assumed that H = E = , (which
also means kE2=(z/r)kH2 ).
To calculate the other field components equations (A.12) to (A.15) are used. To calculate the
resonant frequencies one follows the methods used in (Kajfez and Guillon,1986), (Garault and
Guillon, 1976) and (Tobar and Mann, 1991). Tangential field components are matched at the
boundary of regions 1 and 2, and regions 1 and 3, to obtain two coupled transcendental eigen-value
equations, to solve for the resonant frequencies.
References
Auda H. and Kajfez D., "Dielectric rod waveguides," in Dielectric Resonators, D. Kajfez and P.
Guillon, Ed., Artech House, Inc., pp 65-111, 1986.
Garault Y. and Guillon P., "Higher accuracy for the resonance frequencies of dielectric resonators,"
Electron. Lett., vol. 12, no. 18, pp. 475-476, Sept. 1976
Kajfez D. and Guillon P., Editors, Dielectric Resonators, Artech House, Inc., 1986.
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Tobar M.E. and Mann A.G., "Resonant frequencies of higher order modes in cylindrical anisotropic
dielectric resonators," IEEE. Trans. MTT., vol. 39, no. 12, Dec. 1991.