Page 554 - Fiber Optic Communications Fund

Appendix B 535

where n is the linear refractive index of the fiber. For a single-mode fiber we have
2 2
2
2
n (r,) 2
+ + = (), (B.17)
x 2 y 2 c 2
where () is the propagation constant. Substituting Eqs. (B.16) and (B.17) in Eq. (B.13), we obtain
2
̃q(z, Ω) 2 2 3 (3) 3 ∗
2i +[ ()− ]̃q =− { (x, y)[̃q(z, Ω) ⊗̃q (z, −Ω) ⊗̃q(z, Ω)]}. (B.18)
0 0 2
z 4c
To remove the dependence of transverse field distributions, we multiply Eq. (B.18) by (x, y) and integrate
from −∞ to ∞ in the x–y plane to obtain
2
2
2
̃q [ ()− ]̃q 3 (3) ∗
0
i + =− [̃q(z, Ω) ⊗̃q (z, −Ω) ⊗̃q(z, Ω)], (B.19)
z 2 0 8c A
2
eff 0
where ∞ ∞
∫ ∫ (x, y)dxdy
2
A = −∞ −∞ . (B.20)
eff ∞ ∞
∫ ∫ (x, y)dxdy
4
−∞ −∞
The second term on the left-hand side of Eq. (B.19) can be approximated as
2
2
[ ()− ]̃q [()+ ][()− ]̃q
0 0 0
= ≅[()− ]̃q. (B.21)
0
2 2
0 0
The above approximation is valid if the difference between () and is quite small. If the spectral width
0
of the optical signal is comparable with or larger than , the above approximation could be incorrect. When
0
the spectral width Δ≪ , we can approximate () as a Taylor series around and retain the first three
0 0
terms,
2 2 3 3
()= + ( − )+ ( − ) + ( − ) , (B.22)
0 1 0 0 0
2 6
where n
d |
= | (B.23)
n n |
d |= 0
is known as the nth-order dispersion coefficient (see Chapter 2). Substituting Eqs. (B.21) and (B.22) in
Eq. (B.19), we obtain
( ) 2 (3)
̃q 2 2 3 3 3( +Ω) ∗
0
i + Ω+ Ω + Ω ̃ q =− [̃q(z, Ω) ⊗̃q (z, −Ω) ⊗̃q(z, Ω)]. (B.24)
1
2
z 2 6 8c A
eff 0
If we include the fiber losses by treating the refractive index n as complex with its imaginary part being
frequency independent, Eq. (B.24) is modified as
( ) 2 (3)
̃q 2 2 3 3 ̃q 3( +Ω) ∗
0
i + Ω+ Ω + Ω ̃ q =−i − [̃q(z, Ω) ⊗̃q (z, −Ω) ⊗̃q(z, Ω)], (B.25)
1
2
z 2 6 2 8c A
eff 0
where is the fiber loss coefficient related to the imaginary part of the refractive index through Eq. (10.37). We
2
2
have assumed to be independent of frequency. Since Ω ≪ , ( +Ω) ≃ + 2 Ω. Now, performing
0 0 0 0
the inverse Fourier transform, we obtain
( ) 2 3 2
q q q 2 i q i2 (|q| q) i
3
i + 1 − 2 + i|q| q = − + q, (B.26)
z t t 2 6 t 3 0 t 2