Integrated Optics - Applied Optics Fundamentals and Device Applications - page 214

the circular dielectric waveguide modes by a sum of modified Bessel func-
tions inside and outside the waveguide core. Matching the fields along the
perimeter of the core provided computer solutions. Schlosser's analysis was
for the case of a dielectric rod in air and applied to index of refraction differ-
ences more applicable to microwaves. Bartling solved the problem in terms
of a Green's dyadic but did not calculate specific cases.
After some simplifying assumptions, Marcatili solved Maxwell's equations
in closed form for rectangular cross section dielectric waveguides of various
aspect ratios. Proper selection of the index of refraction results in a wave-
guide supporting only the fundamental modes in a family of two types of
hybrid transverse electric magnetic (TEM) modes polarized at right angles.
Marcatili's analysis provides analytical results in a relatively simple form
and is preferred since the calculations required are simpler. Goell suggests
that the analysis may not provide the accuracy desired for the lowest order
mode, however [3], as shown in Figure 5.4. Figure 5.4 provides solutions from
Marcatili's transcendental and closed form equations and Goell's circular
harmonic solutions for an aspect ratio (channel width to height ratio) of 2,
with an index difference in the channel greater than 1.05 times that of the
surrounding dielectric. The two methods indicate values of a normalized
propagation constant (in terms of the transverse and waveguide propagation
constants)

k

2

−
−

k

2

⎛
⎜

k
k

⎞
⎟

⎛
⎜

n
n

⎞
⎟

z 4 z 1

p

2

= ≅ − − (5.7)

1 1

2 2

k k

1 4 4 4

1.2

n 4

y

x

E 11΄ E 11

n 4 n 4

n 1

b

y

x

E 21΄ E 21

1.0

y

n 1

x

E 21΄ E 21

< n 4 < n 1

1.03

0.8

a = 2 b

0.6

2α
= λ

n 1

k 1

y

x

E 12΄ E 12 4

4

0.4

y

x

E 22΄ E 22

0.2

0

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

2 b

B = b

λ ( n 1 - n 4 ) ½

=

a

FIGURE 5.4
Propagation constant for several modes of rectangular dielectric waveguide. Transcendental
solution …… ; closed form solutions ____ ; Goell's computer solution.