Biophotonics_Concepts_to_Applications

D¼

n^21 n^22

2n^21



n 1 n 2

n 1

ð 3 : 11 Þ

The approximation on the right-hand side reduces this expression forΔto that of
the step-indexfiber. Thus, the same symbol is used in both cases. Forα=∞, inside
the core Eq. (3.10) reduces to the step-index profile n(r) = nl.

3.2.2 Graded-Index Numerical Aperture

Whereas for a step-indexfiber the NA is constant across the core end face, in
graded-indexfibers the NA is a function of position across the core end face. Thus
determining the NA is more complex for graded-indexfibers. Light incident on the
fiber core end face at a position r will be a guided mode only if it is within thelocal
numerical apertureNA(r) at that point. This parameter is defined as [ 6 ]

NA(rÞ¼n^2 ðrÞn^22

(^1) = 2
NAð 0 Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ðÞr=aa
q
for ra
¼0 for r[a
ð 3 : 12 Þ
where theaxial numerical apertureis defined as
NAð 0 Þ¼n^2 ð 0 Þn^22
(^1) = 2
¼ n^21 n^22

 1 = 2

n 1

ffiffiffiffiffiffi

2 D

p

ð 3 : 13 Þ

Thus, the numerical aperture of a graded-indexfiber decreases from NA(0) to
zero for values of r ranging from thefiber axis to the core edge. In a graded-index
fiber the number of bound modes Mgis [ 1 ]

Mg¼

a

aþ 2

2 pa

k

 2

n^21 D

a

aþ 2

V^2

2

ð 3 : 14 Þ

Typically a parabolic refractive index profile given byα= 2.0 is selected by
fiber manufacturers. In this case, Mg=V^2 /4, which is half the number of modes
supported by a step-indexfiber (for whichα=∞) that has the same V value.

Example 3.8Consider a 50-μm diameter graded-index fiber that has a

parabolic refractive index profile (α= 2). If thefiber has a numerical aperture

NA = 0.22, how many guided modes are there in thefiber at a wavelength of

1310 nm?

3.2 Graded-Index Optical Fibers 65