Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics
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6.5 Rectangular Dielectric Waveguides 351
U
(y )
4
= A
4
cos(k
x
x + φ)e
jk
y4
y
e
−jβz
, (6.156)
U
(y )
5
= A
5
e
jk
x5
x
cos(k
y
y + ψ)e
−jβz
. (6.157)
The field components in the five regions are expressed as
E
xi
=
∂
2
U
(y )
i
∂x∂y
, (6.158)
E
yi
=
∂
2
U
(y )
i
∂y
2
+ k
2
U
(y )
i
=
¡
k
2
i
− K
2
yi
¢
U
(y )
i
=
¡
β
2
+ k
2
xi
¢
U
(y )
i
, (6.159)
E
zi
=
∂
2
U
(y )
i
∂z∂y
= −jβ
∂U
(y )
i
∂y
, (6.160)
H
xi
= −jω²
i
∂U
(y )
i
∂z
= −ω²
i
βU
(y )
i
, (6.161)
H
zi
= jω²
i
∂U
(y )
i
∂x
. (6.162)
On the boundaries x = ±a, the tangential components E
y
, E
z
, and H
z
must
be continuous. Considering the weakly guiding conditions, k
xi
¿ β and
E
z
¿ E
y
, we neglect E
z
and obtain
E
y 1
(a)=E
y 3
(a) −→ ²
1
A
1
cos(k
x
a+φ)=²
3
A
3
e
−jk
x3
a
,
H
z1
(a)=H
z3
(a) −→ ²
1
k
x
A
1
sin(k
x
a+φ)=j²
3
k
x3
A
3
e
−jk
x3
a
,
E
y 1
(−a)=E
y 5
(−a) −→ ²
1
A
1
cos(k
x
a−φ)=²
5
A
5
e
−jk
x5
a
,
H
z1
(−a)=H
z5
(−a) −→ ²
1
k
x
A
1
sin(k
x
a−φ)=j²
5
k
x5
A
5
e
−jk
x5
a
.
(6.163)
Similarly, on the boundaries y = ±b, the tangential components are E
x
, E
z
,
H
x
, and H
z
. Since k
xi
¿ β, E
x
¿ E
z
, and H
z
¿ H
x
, we have
E
z1
(b) = E
z2
(b) −→ k
y
A
1
sin(k
y
b + ψ) = jk
y 2
A
2
e
−jk
y2
b
,
H
x1
(b) = H
x2
(b) −→ ²
1
A
1
cos(k
y
b + ψ) = ²
2
A
2
e
−jk
y2
b
,
E
z1
(−b) = E
z4
(−b) −→ k
y
A
1
sin(k
y
b − ψ) = jk
y 4
A
4
e
−jk
y4
b
,
H
x1
(−b) = H
x4
(−b) −→ ²
1
A
1
cos(k
y
b − ψ) = ²
4
A
4
e
−jk
y4
b
.
(6.164)
These are two sets of fourth-order homogeneous simultaneous linear equa-
tions. They have nontrivial solutions only when both of the determinants of
the coefficients vanish, which gives the following two eigenvalue equations for
TM
(y )
mn
or E
(y )
mn
modes:
tan(2k
x
a − mπ) =
k
x
(τ
3
+ τ
5
)
k
2
x
− τ
3
τ
5
, (6.165)
tan(2k
y
b − nπ) =
²
1
k
y
(²
4
τ
2
+ ²
2
τ
4
)
²
2
²
4
k
2
y
− ²
2
1
τ
2
τ
4
. (6.166)
In addition, equations (6.152) give
τ
2
i
+ k
2
x
= ω
2
µ
0
(²
1
− ²
i
) i = 3, 5, (6.167)
352 6. Dielectric Waveguides and Resonators
Figure 6.15: Solution of rectangular dielectric waveguide by means of circular
harmonics.
τ
2
i
+ k
2
y
= ω
2
µ
0
(²
1
− ²
i
) i = 2, 4. (6.168)
From the above six equations, (6.165) to (6.168), we can solve for the trans-
verse phase and decaying coefficients k
x
, k
y
, τ
2
, τ
3
, τ
4
, τ
5
. Then the longitu-
dinal phase coefficient β can be obtained by means of (6.152) as
β
2
= ω
2
µ
0
²
1
− k
2
x
− k
2
y
. (6.169)
Similarly, the following eigenvalue equations for TM
(x)
mn
or E
(x)
mn
modes can
be obtained:
tan(2k
x
a − mπ) =
²
1
k
x
(²
5
τ
3
+ ²
3
τ
5
)
²
3
²
5
k
2
x
− ²
2
1
τ
3
τ
5
, (6.170)
tan(2k
y
b − nπ) =
k
y
(τ
2
+ τ
4
)
k
2
y
− τ
2
τ
4
. (6.171)
6.5.3 Solution of Rectangular Dielectric Waveguide
by Means of Circular Harmonics [33, 48]
There are different methods for the solution of rectangular dielectric waveg-
uide problems. The circular-harmonic computer analysis given by J.E.Goell
is a more accurate approach for this problem, and is an example for the
method to solve problems in rectangular coordinates by means of circular
harmonics. [33]
Goell’s analysis is based on expansion of electromagnetic field in terms
of a series of circular harmonics. The electric and magnetic fields inside
the core are matched to those outside the core at appropriate points on the
rectangular boundary of the waveguide to yield the eigenvalue equations. The
eigenvalue equations are then solved by a computer to give the propagation
characteristics and the field configuration of various modes. The geometry of
the Problem is shown in Fig. 6.15.
6.5 Rectangular Dielectric Waveguides 353
In region 1, the core, the scalar wave functions are standing waves in ρ
direction, expressed by series of the Bessel functions,
U
1
=
∞
X
n=0
A
n
J
n
(T ρ) sin(nφ + χ)e
j(ωt−βz)
, (6.172)
V
1
=
∞
X
n=0
B
n
J
n
(T ρ) sin(nφ + ψ)e
j(ωt−βz)
. (6.173)
In region 2, the cladding, the scalar wave functions are decaying fields in ρ
direction, expressed by series of the modified Bessel functions of the second
kind,
U
2
=
∞
X
n=0
C
n
K
n
(τρ) sin(nφ + χ)e
j(ωt−βz)
, (6.174)
V
2
=
∞
X
n=0
D
n
K
n
(τρ) sin(nφ + ψ)e
j(ωt−βz)
. (6.175)
Since the waveguide is symmetrical about both the x axis and the y axis. the
fields must be even function or odd function in φ direction, i.e., χ = 0 and
ψ = π/2 or χ = π/2 and ψ = 0.
The relations among the longitudinal phase coefficient, β, the transverse
phase coefficient in region 1, T , and the transverse decaying co efficient in
region 2, τ , are given by
β
2
+ T
2
= k
2
1
= ω
2
µ
1
²
1
, β
2
− τ
2
= k
2
2
= ω
2
µ
2
²
2
. (6.176)
The field components can be found by substituting (6.172), (6.173) for
the core and (6.172), (6.173) for the cladding into the Borgni’s formulas for
circular cylindrical coordinates (4.196) to (4.201).
Because of the symmetrical property of the waveguide, the field matching
need only be performed in one quadrant, 0 < φ < π/2. The boundary
x = a corresponds to 0 < φ < φ
c
and the boundary y = b corresponds to
φ
c
< φ < π/2, where φ
c
is the angle which a radial line to the corner of the
rectangular in the first quadrant makes with the φ = 0, i.e., the x axis. Refer
to Fig. 6.15.
On the first quadrant of the rectangular cylindrical boundary, the longi-
tudinal tangential components of the electric and magnetic fields are E
z
, H
z
,
and the transverse tangential components are given by
E
t
= E
ρ
sin φ + E
φ
cos φ for 0 < φ < φ
c
, (6.177)
E
t
= −E
ρ
cos φ + E
φ
sin φ for φ
c
< φ < π/2, (6.178)
H
t
= H
ρ
sin φ + H
φ
cos φ for 0 < φ < φ
c
, (6.179)
H
t
= −H
ρ
cos φ + H
φ
sin φ for φ
c
< φ < π/2. (6.180)
354 6. Dielectric Waveguides and Resonators
The four boundary equations are given by
E
z1
|
S
= E
z2
|
S
, (6.181)
H
z1
|
S
= H
z2
|
S
, (6.182)
E
ρ1
sin φ + E
φ1
cos φ = E
ρ2
sin φ + E
φ2
cos φ for 0 < φ < φ
c
,
−E
ρ1
cos φ + E
φ1
sin φ = −E
ρ2
cos φ + E
φ2
sin φ for φ
c
< φ < π/2,
¾
(6.183)
H
ρ1
sin φ + H
φ1
cos φ = H
ρ2
sin φ + H
φ2
cos φ for 0 < φ < φ
c
,
−H
ρ1
cos φ + H
φ1
sin φ = −H
ρ2
cos φ + H
φ2
sin φ for φ
c
< φ < π/2.
¾
(6.184)
Selecting discrete points on the boundary, substituting field components
derived from functions U and V of (6.172) to (6.175), yields a set of linear
equations,
E
LA
A − E
LC
C = 0, (6.185)
H
LB
B + H
LD
D = 0, (6.186)
E
TA
A + E
TB
B − E
TC
C − E
TD
D = 0, (6.187)
H
TA
A + H
TB
B − H
TC
C − H
TD
D = 0. (6.188)
where A, B, C and D are N element column matrices of the field coefficients
a
n
, b
n
, c
n
and d
n
, respectively, N is the number of space harmonics to be
considered depends upon the accuracy of the computation. E
LA
, E
LC
, H
LB
,
H
LD
, E
TA
, E
TB
, E
TC
, E
TD
, H
TA
, H
TB
, H
TC
and H
TD
are M ×N matrices,
M is the number of discrete points on the boundary to be selected depends
upon the accuracy of the computation. The elements of the above matrices
are given by
e
LA
mn
= J
n
(T ρ
m
) sin(nφ
m
+ χ)
e
LC
mn
= K
n
(τρ
m
) sin(nφ
m
+ χ)
h
LB
mn
= J
n
(T ρ
m
) cos(nφ
m
+ χ)
h
LD
mn
= K
n
(τρ
m
) cos(nφ
m
+ χ)
e
TA
mn
= −β[J
0
n
(T ρ
m
) sin(nφ
m
+ χ)R + J
n
(T ρ
m
) cos(nφ
m
+ χ)P ]
e
TB
mn
= ωµ
0
[J
n
(T ρ
m
) sin(nφ
m
+ χ)R + J
0
n
(T ρ
m
) cos(nφ
m
+ χ)P ]
e
TC
mn
= β[K
0
n
(τρ
m
) sin(nφ
m
+ χ)R + K
n
(τρ
m
) cos(nφ
m
+ χ)P ]
e
TD
mn
= −ωµ
0
[K
n
(τρ
m
) sin(nφ
m
+ χ)R + K
0
n
(τρ
m
) cos(nφ
m
+ χ)P ]
h
TA
mn
= ω²[J
n
(T ρ
m
) cos(nφ
m
+ χ)R − J
0
n
(T ρ
m
) sin(nφ
m
+ χ)P ]
h
TB
mn
= −β[J
0
n
(T ρ
m
) cos(nφ
m
+ χ)R − J
n
(T ρ
m
) sin(nφ
m
+ χ)P ]
h
TC
mn
= −ω²
0
[K
n
(τρ
m
) cos(nφ
m
+ χ)R − K
0
n
(τρ
m
) sin(nφ
m
+ χ)P ]
h
TD
mn
= β[K
0
n
(τρ
m
) cos(nφ
m
+ χ)R − K
n
(τρ
m
) sin(nφ
m
+ χ)P ]
where
χ = 0 or χ = π/2,
6.5 Rectangular Dielectric Waveguides 355
Figure 6.16: Normalized dispersion curves for a rectangular dielectric waveg-
uide.
ρ
m
=
a
2 cos φ
m
, R = sin φ
m
, P = cos φ
m
for φ < φ
c
ρ
m
=
b
2 sin φ
m
, R = −cos φ
m
, P = sin φ
m
for φ > φ
c
In order to have nontrivial solutions to linear equations (6.185) to (6.188),
the determinant of the coefficients must be zero,
¯
¯
¯
¯
¯
¯
¯
¯
E
LA
0 −E
LC
0
0 H
LB
0 −H
LD
E
TA
E
TB
−E
TC
−E
TD
H
TA
H
TB
−H
TC
−H
TD
¯
¯
¯
¯
¯
¯
¯
¯
= 0. (6.189)
This is the eigenvalue equation of the problem. The longitudinal phase coef-
ficient was found by substituting test values in the equation, then Newton’s
method was used to find the roots to the desired accuracy. So it is recognized
to be the exact solution.
The normalized dispersion curves of some modes for a rectangular dielec-
tric waveguide are shown in Fig. 6.16. We find that the differences between
the two solutions are obvious for the near-cutoff region, and become negligible
when the operating frequency is high enough respect to its cutoff value. Be-
sides, for a waveguide with large aspect ratio, i.e., a wide and thin waveguide,
the approximate solution is acceptable.
356 6. Dielectric Waveguides and Resonators
Figure 6.17: Circular dielectric waveguide and optical fiber.
6.6 Circular Dielectric Waveguides and
Optical Fibers
Circular dielectric waveguides for millimeter waves can simply be made as a
uniform dielectric rod or dielectric wire with circular cross section, as shown in
Fig. 6.17(a). The most promising optical waveguides for long-distance trans-
mission of signals are the circular optical fibers including step-index fibers
and graded-index fibers. The theory of graded-index fiber relates to the wave
propagation in nonuniform medium and will not be included in this book.
The step-index fiber consists of a core of dielectric material with refractive
index n
1
and a cladding of different dielectric material whose refractive index
n
2
is slightly less than n
1
, as shown in Fig. 6.17(b). With this configuration,
for the guided-wave state, the total reflection condition is satisfied and the
fields are mainly confined in the core.
For guided modes, the fields in the core are radial standing waves with
Bessel function dependence, whereas the fields in the cladding are radial
decaying fields with modified Bessel function dependence. The thickness of
the cladding is usually large enough so that the fields on the outer surface
of the cladding are small enough. We assume that the cladding extends to
infinity. The mathematical–physical model of the step-index optical fiber as
well as the uniform circular dielectric waveguide is shown in Fig. 6.17(c).
6.6.1 General Solutions of Circular Dielectric
Waveguides
In uniform circular dielectric waveguides, only when the fields are uniform in
the transverse tangential direction of the boundary, i.e., ∂/∂φ = 0, can the
TE or TM mode alone satisfy the boundary conditions or exist independently,
which corresponds to the axially symmetric modes or meridional rays. When
6.6 Circular Dielectric Waveguides and Optical Fibers 357
the fields are nonuniform in the φ direction, i.e., ∂/∂φ 6= 0, the eigenmodes
must be hybrid, or so-called HEM modes, which corresponds to the axially
asymmetric modes or skew rays.
(1) Field Components and Eigenvalue Equations
We begin with the analysis of the circular dielectric waveguide shown in
Fig. 6.17(c). The core is denoted by region 1 with radius a; the cladding is
denoted by region 2 and extents to infinity in the ρ direction. In general, the
constitutive parameters of the core are ²
1
, µ
1
and those of the cladding are
²
2
, µ
2
. In uniform circular dielectric waveguides or step-index optical fibers,
the eigenmodes in most cases are hybrid modes with U 6= 0 and V 6= 0.
Region 1 (core): In the core region or guided-wave region, 0 ≤ ρ ≤ a,
the wave function dependence on ρ is uniquely determined as the Bessel
function of the first kind, J
n
(T ρ), and the coefficient of N
n
(T ρ) must be zero
because the axis ρ = 0 is included in the region. The angular dependence of
the wave functions must be e
jnφ
, where n is a positive or negative integer,
since the whole circumference is included in the region. The longitudinal
dependence is supposed to be e
−jβz
, for the traveling waves along +z. Then
the functions U
1
and V
1
are
U
1
= AJ
n
(T ρ)e
jnφ
e
−jβz
, V
1
= BJ
n
(T ρ)e
jnφ
e
−jβz
. (6.190)
The six field components become
E
ρ1
=
·
−jβT AJ
0
n
(T ρ) +
ωµ
1
n
ρ
BJ
n
(T ρ)
¸
e
jnφ
e
−jβz
, (6.191)
E
φ1
=
·
βn
ρ
AJ
n
(T ρ) + jωµ
1
T BJ
0
n
(T ρ)
¸
e
jnφ
e
−jβz
, (6.192)
E
z1
= T
2
AJ
n
(T ρ)e
jnφ
e
−jβz
, (6.193)
H
ρ1
=
·
−
ω²
1
n
ρ
AJ
n
(T ρ) − jβT BJ
0
n
(T ρ)
¸
e
jnφ
e
−jβz
, (6.194)
H
φ1
=
·
−jω²
1
T AJ
0
n
(T ρ) +
βn
ρ
BJ
n
(T ρ)
¸
e
jnφ
e
−jβz
, (6.195)
H
z1
= T
2
BJ
n
(T ρ)e
jnφ
e
−jβz
. (6.196)
Region 2 (cladding): In the cladding, a ≤ ρ ≤ ∞, the wave function
dependence on ρ is uniquely determined as the modified Bessel function of the
second kind, K
n
(τρ), and the coefficient of I
n
(τρ) must be zero because ρ →
∞ is included in the region. The angular dependence and the longitudinal
dependence of the wave functions must be the same as those in region 1
satisfying the boundary conditions. Then the functions U
2
and V
2
are
U
2
= CK
n
(τρ)e
jnφ
e
−jβz
, V
2
= DK
n
(τρ)e
jnφ
e
−jβz
. (6.197)
358 6. Dielectric Waveguides and Resonators
The six field components become
E
ρ2
=
·
−jβτCK
0
n
(τρ) +
ωµ
2
n
ρ
DK
n
(τρ)
¸
e
jnφ
e
−jβz
, (6.198)
E
φ2
=
·
βn
ρ
CK
n
(τρ) + jωµ
2
τDK
0
n
(τρ)
¸
e
jnφ
e
−jβz
, (6.199)
E
z2
= −τ
2
CK
n
(τρ)e
jnφ
e
−jβz
, (6.200)
H
ρ2
=
·
−
ω²
2
n
ρ
CK
n
(τρ) − jβτDK
0
n
(τρ)
¸
e
jnφ
e
−jβz
, (6.201)
H
φ2
=
·
−jω²
2
τCK
0
n
(τρ) +
βn
ρ
DK
n
(τρ)
¸
e
jnφ
e
−jβz
, (6.202)
H
z2
= −τ
2
DK
n
(τρ)e
jnφ
e
−jβz
. (6.203)
To satisfy Helmholtz’s equations, the relations among β, the longitudinal
phase coefficient, T , the transverse phase coefficient in region 1, τ, the trans-
verse decaying coefficient in region 2, and k
1
and k
2
, the phase coefficients in
unbounded media of regions 1 and 2, are given as
β
2
+ T
2
= k
2
1
= ω
2
µ
1
²
1
, β
2
− τ
2
= k
2
2
= ω
2
µ
2
²
2
. (6.204)
The boundary conditions at the boundary between regions 1 and 2,
E
z1
(a)=E
z2
(a), H
z1
(a)=H
z2
(a), E
φ1
(a)=E
φ2
(a), H
φ1
(a)=H
φ2
(a),
give the following four boundary equations:
T
2
J
n
(T a)A + τ
2
K
n
(τa)C = 0, (6.205)
T
2
J
n
(T a)B + τ
2
K
n
(τa)D = 0, (6.206)
βn
a
J
n
(T a)A + jωµ
1
T J
0
n
(T a)B −
βn
a
K
n
(τa)C − jωµ
2
τK
0
n
(τa)D = 0, (6.207)
−jω²
1
T J
0
n
(T a)A +
βn
a
J
n
(T a)B + jω²
2
τK
0
n
(τa)C −
βn
a
K
n
(τa)D = 0. (6.208)
These are four simultaneous homogeneous linear equations, which have non-
trivial solutions when the determinant of the coefficients vanishes, that is
¯
¯
¯
¯
¯
¯
¯
¯
¯
T
2
J
n
(T a) 0 τ
2
K
n
(τa) 0
0 T
2
J
n
(T a) 0 τ
2
K
n
(τa)
βn
a
J
n
(T a) jωµ
1
T J
0
n
(T a) −
βn
a
K
n
(τa) −jωµ
2
τK
0
n
(τa)
−jω²
1
T J
0
n
(T a)
βn
a
J
n
(T a) jω²
2
τK
0
n
(τa) −
βn
a
K
n
(τa)
¯
¯
¯
¯
¯
¯
¯
¯
¯
= 0.
(6.209)
This is the general eigenvalue equation of the uniform circular dielectric
waveguide or step-index optical fiber, which can be rewritten as the following
6.6 Circular Dielectric Waveguides and Optical Fibers 359
transcendental equation:
·
²
1
J
0
n
(T a)
T aJ
n
(T a)
+
²
2
K
0
n
(τa)
τaK
n
(τa)
¸·
µ
1
J
0
n
(T a)
T aJ
n
(T a)
+
µ
2
K
0
n
(τa)
τaK
n
(τa)
¸
−
n
2
β
2
ω
2
·
1
(T a)
2
+
1
(τa)
2
¸
2
=0. (6.210)
From equation (6.204), we have
β
2
·
1
(T a)
2
+
1
(τa)
2
¸
=
k
2
1
(T a)
2
+
k
2
2
(τa)
2
= ω
2
·
²
1
µ
1
(T a)
2
+
²
2
µ
2
(τa)
2
¸
. (6.211)
Combining (6.211) and (6.210), β can be elimination, and the other form of
the eigenvalue equation is given by:
·
²
1
J
0
n
(T a)
T aJ
n
(T a)
+
²
2
K
0
n
(τa)
τaK
n
(τa)
¸·
µ
1
J
0
n
(T a)
T aJ
n
(T a)
+
µ
2
K
0
n
(τa)
τaK
n
(τa)
¸
− n
2
·
²
1
µ
1
(T a)
2
+
²
2
µ
2
(τa)
2
¸·
1
(T a)
2
+
1
(τa)
2
¸
= 0. (6.212)
From (6.204), we have the relation between T and τ
T
2
+ τ
2
= k
2
1
− k
2
2
= ω
2
(µ
1
²
1
− µ
2
²
2
), or (T a)
2
+ (τa)
2
= V
2
, (6.213)
where V = ωa
√
µ
1
²
1
− µ
2
²
2
is the normalized frequency for circular dielectric
waveguides. Combining equations (6.212) and (6.213), we can have the roots
of T and τ.
Rewrite the eigenvalue equation (6.212), and introducing a parameter χ,
χ =
²
r1
J
0
n
(T a)
T aJ
n
(T a)
+
²
r2
K
0
n
(τa)
τaK
n
(τa)
n
·
µ
r1
²
r1
(T a)
2
+
µ
r2
²
r2
(τa)
2
¸
=
n
·
1
(T a)
2
+
1
(τa)
2
¸
µ
r1
J
0
n
(T a)
T aJ
n
(T a)
+
µ
r2
K
0
n
(τa)
τaK
n
(τa)
. (6.214)
Substituting n from (6.210) into the above formula for χ, yields
χ =
ω
√
µ
0
²
0
β
r
²
r1
J
0
n
(T a)
T aJ
n
(T a)
+
²
r2
K
0
n
(τa)
τaK
n
(τa)
r
µ
r1
J
0
n
(T a)
T aJ
n
(T a)
+
µ
r2
K
0
n
(τa)
τaK
n
(τa)
. (6.215)
From the simultaneous boundary equations (6.205)–(6.208), we obtain the
relations among the field coefficients
C
A
=
D
B
= −
T
2
J
n
(T a)
τ
2
K
n
(τa)
, (6.216)
360 6. Dielectric Waveguides and Resonators
H
z
E
z
=
B
A
=
D
C
=
jβχ
ωµ
0
= j
r
²
0
µ
0
r
²
r1
J
0
n
(T a)
T aJ
n
(T a)
+
²
r2
K
0
n
(τa)
τaK
n
(τa)
r
µ
r1
J
0
n
(T a)
T aJ
n
(T a)
+
µ
r2
K
0
n
(τa)
τaK
n
(τa)
. (6.217)
Substituting them into (6.191)–(6.196) and (6.198)–(6.203), we have all the
field components with only one unknown coefficient A, which is determined
by the strength of the wave in the waveguide.
By applying the ab ove relations (6.216), (6.217), and the recurrence and
differential formulas of Bessel functions, (C.13) and (C.16),
J
n
(x) =
x
2n
[J
n−1
(x) + J
n+1
(x)], J
0
n
(x)=
1
2
[J
n−1
(x) − J
n+1
(x)],
we can express the field components in the core, (6.191)–(6.196), by
E
ρ1
= jβT A
·
1 + µ
r1
χ
2
J
n+1
(T ρ) −
1 − µ
r1
χ
2
J
n−1
(T ρ)
¸
e
jnφ
e
−jβz
, (6.218)
E
φ1
= βT A
·
1 + µ
r1
χ
2
J
n+1
(T ρ) +
1 − µ
r1
χ
2
J
n−1
(T ρ)
¸
e
jnφ
e
−jβz
, (6.219)
E
z1
= T
2
AJ
n
(T ρ)e
jnφ
e
−jβz
, (6.220)
H
ρ1
= −
β
2
T A
ωµ
0
χ+
k
2
β
2
²
r1
2
J
n+1
(T ρ)−
χ−
k
2
β
2
²
r1
2
J
n−1
(T ρ)
e
jnφ
e
−jβz
, (6.221)
H
φ1
= j
β
2
T A
ωµ
0
χ+
k
2
β
2
²
r1
2
J
n+1
(T ρ)+
χ−
k
2
β
2
²
r1
2
J
n−1
(T ρ)
e
jnφ
e
−jβz
, (6.222)
H
z1
= j
T
2
βχ
ωµ
0
AJ
n
(T ρ)e
jnφ
e
−jβz
, (6.223)
where k
2
= ω
2
µ
0
²
0
. By applying (C.15) and (C.18),
K
n
(x) = −
x
2n
[K
n−1
(x) − K
n+1
(x)], K
0
n
(x)=−
1
2
[K
n−1
(x) + K
n+1
(x)],
we can express the field components in the cladding, (6.198)–(6.203), by
E
ρ2
= jβτ C
·
1 + µ
r2
χ
2
K
n+1
(τρ) +
1 − µ
r2
χ
2
K
n−1
(τρ)
¸
e
jnφ
e
−jβz
, (6.224)
E
φ2
= βτ C
·
1 + µ
r2
χ
2
K
n+1
(τρ) −
1 − µ
r2
χ
2
K
n−1
(τρ)
¸
e
jnφ
e
−jβz
, (6.225)
E
z2
= τ
2
CK
n
(τρ)e
jnφ
e
−jβz
, (6.226)
H
ρ2
= −
β
2
τ
ωµ
0
C
χ+
k
2
β
2
²
r2
2
K
n+1
(τρ)+
χ−
k
2
β
2
²
r2
2
K
n−1
(τρ)
e
jnφ
e
−jβz
, (6.227)