The Optical Communications Reference

the TM mode is smaller than that fo r the TE mode with

respect to the same v. That means the TE mode is slightly

better confined in the core tha n the TM mode. The

power carried by the TM mode is obtained, from Eqs.

(2.2.6) and (2.2.30) by

P ¼

b

2

u3

0

ð

N

N

1

n

2





H

y





2

dx: (2.2.39)

2.2.2 Rectangular waveguides

2.2.2.1 Basic equations

In this section the analytical method, which was proposed

by Marcatili [5], to deal with the three-dimensional

optical waveguide, as shown in Fig. 2.2.10, is described.

The important assumption of this method is that the

electromagnetic field in the shaded area in Fig. 2.2.10

can be neglected since the electromagnetic field of the

well-guided mode decays quite rapidly in the cladding

region. Then we do not impose the boundary conditions

for the electromagnetic field in the shaded area.

We first consider the electromagnetic mode in which

E

x

and H

y

are predominant. According to Marcatili’s

treatment, we set H

x

¼ 0 in Eqs. (2.2.3) and (2.2.4).

Then the wave equation and electromagnetic field rep-

resentation are obtained as

v

2

H

y

vx

2

þ

v

2

H

y

vy

2

þðk

2

n

2



b

2

ÞH

y

¼ 0; (2.2.40)

H

x

¼ 0

E

x

¼

um

0

b

H

y

þ

1

u3

0

n

2

b

v

2

H

y

vx

2

E

y

¼

1

u3

0

n

2

b

v

2

H

y

vxvy

E

z

¼

j

u3

0

n

2

vH

y

vx

H

z

¼

j

b

vH

y

vy

:

8

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

:

(2.2.41)

On the other hand, we set H

y

¼ 0 in Eqs. (2.2.3)

and (2.2.4) to consider the electromagnetic field in

Fig. 2.2.9 Dispersion curves of TE and TM modes in the slab waveguide.

Fig. 2.2.10 Three-dimensional rectangular waveguide.

70

SECTION TWO Optical Waveguides

which E

y

and H

x

are predominant. The wave equation

and electromagnetic field representation are given by

v

2

H

x

vx

2

þ

v

2

H

x

vy

2

þðk

2

n

2



b

2

ÞH

x

¼ 0; (2.2.42)

H

y

¼ 0

E

x

¼

1

u3

0

n

2

b

v

2

H

x

vxvy

E

y

¼

um

0

b

H

x



1

u3

0

n

2

b

v

2

H

x

vy

2

E

z

¼

j

u3

0

n

2

vH

x

vy

H

z

¼

j

b

vH

x

vx

:

8

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

:

(2.2.43)

The modes in Eqs. (2.2.40) and (2.2.41) are described as

E

x

pq

( p and q are integers) since E

x

and H

y

are the

dominant electromagnetic fields. On the other hand, the

modes in Eqs. (2.2.42) and (2.2.43) are called E

y

pq

since

E

y

and H

x

are the dominant electromagnetic fields. In the

following section, the solution method of the dispersion

equation for the E

x

pq

mode is described in detail, and only

the results are shown for the E

y

pq

mode.

2.2.2.2 Dispersion equations for E

x

pq

and E

y

pq

modes

Since the rectangular waveguide shown in Fig. 2.2.10 is

symmetrical with respect to the x- and y-axes, we analyze

only regions j – k. We first express the solution fields,

which satisfy the wave equation (2.2.40),as

H

y

¼

8

<

:

A cosðk

x

x 

f

Þcosðk

y

y 

j

Þ region j

A cosðk

x

a 

f

Þe



g

x

ðxaÞ

cosðk

y

y 

j

Þ region k

A cosðk

x

x 

f

Þe



g

y

ðydÞ

cosðk

y

d 

j

Þ region l

(2.2.44)

where the transverse wavenumbers k

x

, k

y

, g

x

, and g

y

and

the optical phases f and j are given by

8

>

<

>

:

k

2

x

 k

2

y

þ k

2

n

2

1



b

2

¼ 0 region j

g

2

x

 k

2

y

þ k

2

n

2

0



b

2

¼ 0 region k

k

2

x

þ

g

2

y

þ k

2

n

2

0



b

2

¼ 0 region l

(2.2.45)

and

f

¼ðp  1Þ

p

2

ðp ¼ 1; 2; .Þ

j

¼ðq  1Þ

p

2

ðq ¼ 1; 2; .Þ:

8

<

:

(2.2.46)

We should note here that the integers p and q start from 1

because we follow the mode definition by Marcatili. To

the contrary the mode number m in Eq. (2.2.12) for the

slab waveguides starts from zero. By the conventional

mode definition, the lowest mode in the slab waveguide

is the TE

m¼0

mode (Fig. 2.2.7(a)) which has one electric

field peak. On the other hand, the lowest mode in the

rectangular waveguides is E

x

p¼1;q¼1

or E

y

p¼1;q¼1

mode

(Fig. 2.2.11) which has only one electric field peak along

both x- and y-axis directions. Therefore in the mode

definition by Marcatili, integers p and q represent the

number of local electric field peaks along the x- and

y-axis direction s.

When we apply the boundary conditions that the

electric field E

z

fð1=n

2

ÞvH

y

=vx should be continuous at

x ¼ a and the magnetic field H

z

fvH

y

=vx should be

continuous at y ¼ d, we obtain the following dispersion

equations:

k

x

a ¼ðp  1Þ

p

2

þ tan

1



n

2

1

g

x

n

2

0

k

x



; (2.2.47a)

k

y

d ¼ðq  1Þ

p

2

þ tan

1



g

y

k

y



: (2.2.47b)

Transversal wavenumbers k

x

, k

y

, g

x

, and g

v

are related, by

Eq. (2.2.45) as

g

2

x

¼ k

2

ðn

2

1

 n

2

0

Þk

2

x

; (2.2.48)

g

2

y

¼ k

2

ðn

2

1

 n

2

0

Þk

2

y

: (2.2.49)

Fig. 2.2.11 Mode definitions and electric field distributions in

Marcatili’s method.

71

Planar optical waveguides CHAPTER 2.2