5.3 Losses Between Fiber Joints
231
Fig. 5.12 Optical power loss when fiber ends are separated longitudinally by a gap s
identical step-index fibers is found to be
L F = −10 log
a
a + x
2
= −10 log
a
a + s tanθ A
2
= −10 log
1 +
s
a
sin
−1
N A
n
−2
(5.33)
where a is the fiber radius, NA is the numerical aperture of the fiber, and n is the
refractive index of the material between the fiber ends (usually either air or an index
matching gel).
Example 5.9 Two identical step-index fibers each have a 25-μm core radius and
an acceptance angle of 14°. Assume the two fibers are perfectly aligned axially and
angularly. What is the insertion loss for a longitudinal separation of 0.025 mm?
Solution The insertion loss due to a fiber gap between can be found by using
Eq. (5.33). For a 0.025-mm = 25-μm gap
L F = −10 log
25
25 + 25 tan14
◦
2
= 1.93 dB
When the axes of two joined fibers are angularly misaligned at the joint, the
optical power that leaves the emitting fiber outside of the solid acceptance angle
of the receiving fiber will be lost. For two step-index fibers that have an angular
misalignment θ, the optical power loss at the joint has been shown to be [21]
L F = −10 log
cosθ
1
2
−
1
π
p
1 − p
2
1/2 −
1
π
arcsin p − q
1
π
y
1 − y
2
1/2 +
1
π
arcsiny +
1
2
(5.34)
where