The Optical Communications Reference

6 is commonly expressed as a percentage. The numer-

ical aperture NA is related to the relative refractive-

index difference 6 by

NA ¼

q

max

y n

1

ffiffiffiffiffiffi

2

D

p

: (2.1.4)

The maximum angle for the propagating light within the

core is given by

f

max

y

q

max

=n

1

y

ffiffiffiffiffiffi

2

D

p

. For typical optical

waveguides, NA ¼ 0.21 and q

max

¼ 12



(f

max

¼ 8.1



)

when n

1

¼ 1.47, 6 ¼ 1% (for n

0

¼ 1.455).

2.1.2 Formation of guided modes

We have accounted for the mechanism of mode con-

finement and have indicated that the angle f must not

exceed the critical angle. Even though the angle f is

smaller than the critical angle, light rays with arbitrary

angles are not able to propagate in the waveguide. Each

mode is associated with light rays at a discrete angle of

propagation, as given by electromagnetic wave analysis.

Here we describe the formation of modes with the ray

picture in the slab waveguide [1], as shown in Fig. 2.1.2.

Let us consider a plane wave propagating along the

z-direction with inclination angle f. The phase fronts of

the plane waves are perpendicular to the light rays. The

wavelength and the wavenumber of light in the core are

l/n

1

and kn

l

(k ¼ 2p/l) respectively, where l is the

wavelength of light in vacuum. The propagation con-

stants along z and x (lateral direction) are expressed by

b

¼ kn

1

cos

f

; (2.1.5)

k

¼ kn

1

sin

f

: (2.1.6)

Before describing the formation of modes in detail, we

must explain the phase shift of a light ray that suffers

total reflection. The reflection coefficient of the totally

reflected light, which is polarized perpendicular to the

incident plane (plane formed by the incident and

reflected rays), as shown in Fig. 2.1.3, is given by [2]

r ¼

A

r

A

i

¼

n

1

sin

f

þ j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n

2

1

cos

2

f

 n

2

0

q

n

1

sin

f

 j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n

2

1

cos

2

f

 n

2

0

q

: (2.1.7)

When we express the complex reflection coefficient r as

r ¼ exp(jF), the amount of phase shift F is obtained as

F

¼2tan

1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n

2

1

cos

2

f

 n

2

0

q

n

1

sin

f

¼2tan

1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

D

sin

2

f

 1:

s

(2.1.8)

where Eq. (2.1.3) has been used. The foregoing phase

shift for the totally reflected light is called the Goos–

Ha

¨

nchen shift [1, 3].

Let us consider the phase difference between the two

light rays belonging to the same plane wave in Fig. 2.1.2.

Light ray PQ, which propagates from point P to Q, does

not suffer the influence of reflection. On the other hand,

light ray RS, propagating from point R to S, is reflected

two times (at the upper and lower core-cladding in-

terfaces). Since points P and R or points Q and S are on

the same phase front, optical paths PQ and RS (including

the Goos–Ha

¨

nchen shifts caused by the two total re-

flections) should be equal, or their difference should be

an in tegral multiple of 2p. Since the distance between

points Q and R is 2a/tan f  2a tan f the distance be-

tween points P and Q is expressed by

‘

1

¼



2a

tan

f

 2a tan

f



cos

f

¼ 2a



1

sin

f

 2 sin

f



:

(2.1.9)

Also, the distance between points R and S is given by

‘

2

¼

2a

sin

f

: (2.1.10)

The phase-matching condition for the optical paths PQ

and RS then becomes

ðkn

1

‘

2

þ 2

F

Þkn

1

‘

1

¼ 2m

p

; (2.1.11)

where m is an integer. Substituting Eqs. (2.1.8)–(2.1.10)

into Eq. (2.1.11) we obtain the condition for the prop-

agation angle f as

Fig. 2.1.2 Light rays and their phase fronts in the waveguide.

Fig. 2.1.3 Total reflection of a plane wave at a dielectric interface.

58

SECTION TWO Optical Waveguides