234
5 Optical Power Coupling
5.3.3 Single-Mode Fiber Losses
As is the case in multimode fibers, in single-mode fibers the lateral (axial) offset
misalignment presents the most serious loss. This loss depends on the shape of the
propagating mode. For Gaussian-shaped beams the loss between identical fibers is
[22]
L SM,lat = −10 log
ex p
−
d
w
2
(5.38)
where the spot size w is the mode-field radius defined in Eq. (2.34), and d is the lateral
displacement shown in Fig. 5.9. Because the spot size is only a few micrometers in
single-mode fibers, low-loss coupling requires a very high degree of mechanical
precision in the axial dimension.
Example 5.12 A single-mode fiber has a normalized frequency V = 2.20, a core
refractive index n 1 = 1.47, a cladding refractive index n 2 = 1.465, and a core diameter
2a = 9 μm. What is the insertion loss of a fiber joint having a lateral offset of d =
1 μm? For the mode-field diameter use the expression
w = a(0.65 + 1.619V
−3/2
+ 2.879V
−6
).
Solution First, using the above expression for the mode-field diameter,
w = 4.5[0.65 + 1.619(2.20)
−3/2
+ 2.879(2.20)
−6
] = 5.27 μm
w.
Then, from Eq. (5.38),
L SM,lat = −10 log
exp
−(l/5.27)
2
= 0.156 dB
For angular misalignment in single-mode fibers, the loss at a wavelength λ is [22]
L SM,ang = −10 log
exp
−
π n 2 wθ
λ
2
(5.39)
where n 2 is the refractive index of the cladding, θ is the angular misalignment in
radians shown in Fig. 5.9, and w is the mode-field radius.
For a gap s with a material of index n 3 , and letting G = s/kw
2 , the gap loss for
identical single-mode fiber splices is
L SM,gap = −10 log
64n
2
1 n
2
3
(n 1 + n 3 )
4
G 2 + 4
(5.40)