The Optical Communications Reference

tan



kn

1

a sin

f



m

p

2



¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

D

sin

2

f

 1:

s

(2.1.12)

Equation (2.1.12) shows that the propagation angle of

a light ray is discrete and is determined by the wave-

guide structure (core radius a, refractive index n

1

re-

fractive-index difference 6 ) and the wavelength l of

the light source (wavenumber is k ¼ 2p/l) [4]. The

optical field distribution that satisfies the phase-

matching condition of Eq. (2.1.12) is called the mode.

The allowed value of propagation constant b [Eq.

(2.1.5)] is also discrete and is denoted as an eigenvalue.

The mode that has the minimum ang le f in Eq.

(2.1.12) (m ¼ 0) is the fundamental mode; the other

modes, having larger angles, are higher-order modes

(m  1).

Figure 2.1.4 schematically shows the formation of

modes (standing waves) for (a) the fundamental mode

and (b) a higher-order mode, respectively, through the

interference of light waves. In the fig ure the solid line

represents a positive phase front and a dotted line rep-

resents a negative phase front, respectively. The electric

field amplitude becomes the maximum (minimum) at

the point where two positive (negative) phase fronts in-

terfere. In contrast, the electric field amplitude becomes

almost zero near the core–cladding interface, since

positive and negative phase fronts cancel out each othe r.

Therefore the field distribution along the x-(transverse)

direction becomes a standing wave and varies periodically

along the z direction with the period

l

p

¼ð

l

=n

1

Þ=

cos

f

¼ 2

p

=

b

.

Since n

1

sin f ¼ sin q 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n

2

1

 n

2

0

q

from Fig. 2.1.1,

Eqs. (2.1.1) and (2.1.3) give the propagation angle as sin

sin

f



ffiffiffiffiffiffi

2

D

p

. When we introduce the parameter

x

¼

sin

f

ffiffiffiffiffiffi

2

D

p

; (2.1.13)

which is normalized to 1, the phase-matching Eq.

(2.1.12) can be rewritten as

kn

1

a

ffiffiffiffiffiffi

2

D

p

¼

cos

1

x

þ m

p

=2

x

: (2.1.14)

The term on the left-hand side of Eq. (2.1.14) is known

as the normalized frequency, and it is expressed by

v ¼ kn

1

a

ffiffiffiffiffiffi

2

D

p

: (2.1.15)

When we use the normalized frequency v, the propaga-

tion characteristics of the waveguides can be treated

generally (independent of each waveguide structure).

The relationship between normalized frequency v and x

(propagation constant b), Eq. (2.1.14), is called the dis-

persion equation. Figure 2.1.5 shows the dispersion

curves of a slab waveguide. The crossing point between

h

¼ðcos

1

x

þm

p

=2Þ=

x

and h ¼v gives x

m

for each mode

number m, and the propagation constant b

m

is obtained

from Eqs. (2.1.5) and (2.1.13).

It is known from Fig. 2.1.5 that only the fundamental

mode with m ¼ 0 can exist when v < v

c

¼ p/2. v

c

determines the single-mode condition of the slab wave-

guidedin other words, the condition in which higher-

order modes are cut off. Therefore it is called the cutoff

v-value. When we rewrite the cutoff condition in terms

of the wavelength we obtain

l

c

¼

2

p

v

c

an

1

ffiffiffiffiffiffi

2

D

p

: (2.1.16)

l

c

is called the cutoff (free-spac e) wavelength. The

waveguide operates in a single mode for wavelengths

longer than l

c

. For example, l

c

¼ 0.8 mm when the core

width 2a ¼ 3.54 mm for the slab waveguide of n

1

¼ 1.46,

D ¼ 0.3%(n

0

¼ 1.455).

2.1.3 Maxwell’s equations

Maxwell’s equations in a homogeneous and lossless di-

electric medium are written in terms of the electric field

e and magnetic field h as [5]

Fig. 2.1.4 Formation of modes: (a) Fundamental mode, (b) higher-

order mode.

59

Wave theory of optical waveguides CHAPTER 2.1