Nonlinear Fiber Optics - 4 ed. Agrawal

316 Chapter 8. Stimulated Raman Scattering
such that ∫ ∞
0 h j(t) =1 for j = a and b. Physically, h a and h b represent the isotropic
and anisotropic parts of the nuclear response with the relative strengths f a and f b ,
respectively.
The tensor nature of χ (3) plays an important role in Eq. (8.5.1). For silica fibers,
the third-order susceptibility has the form given in Eq. (6.1.2). Using it in Eq. (8.5.1)
together with Eq. (8.5.2), the product χ (3)
ijklR(t) can be written as [5]:
[
χ (3)
1 − fR
ijklR(t) =χ(3) xxxx (δ ij δ kl + δ ik δ jl + δ il δ jk )
3
]
+ f a h a (t)δ ij δ kl + 1 2 f bh b (t)(δ ik δ jl + δ il δ jk ) , (8.5.3)
where the three components of χ (3)
ijkl
appearing in Eq. (6.1.2) were taken to be equal in
magnitude.
In the case of SRS, we should consider the pump and Stokes fields separately. We
follow the treatment of Section 7.6.1 and write the total electric field E and induced
polarization P NL in the form
E(t) =Re[E p exp(−iω p t)+E s exp(−iω s t)], (8.5.4)
P NL (t) =Re[P p exp(−iω p t)+P s exp(−iω s t)]. (8.5.5)
The dependence of P p and P s on the pump and Stokes fields is found by substituting
Eq. (8.5.4) in Eq. (8.5.1). The result is given by
P j = 3ε [
0
4 χ(3) xxxx c 0 (E j · E j )E ∗ j + c 1 (E ∗ j · E j )E j
+ c 2 (E ∗ m · E m )E j + c 3 (E m · E j )E ∗ m + c 4 (E ∗ m · E j )E m
], (8.5.6)
where j ≠ m and c 0 = 1 3 (1 − f R)+ 1 2 f b. The remaining coefficients are defined as
c 1 = 2 3 (1 − f R)+ f a + 1 2 f b, c 2 = 2 3 (1 − f R)+ f a + 1 2 f b ˜h b (Ω), (8.5.7)
c 3 = 2 3 (1 − f R)+ 1 2 f b + f a ˜h a (Ω), c 4 = 2 3 (1 − f R)+ 1 2 f b + 1 2 f b ˜h b (Ω), (8.5.8)
with Ω = ω j − ω k . In arriving at Eq. (8.5.6), we made use of the relation ˜h a (0) =
˜h b (0)=1.
We can now introduce the Jones vectors |A p 〉 and |A s 〉 as in Section 7.6.1. To keep
the following analysis relatively simple, we assume that both the pump and Stokes are
in the form of CW or quasi-CW waves and neglect the dispersive effects. With this
simplification, we obtain the following set of two coupled vector equations [189]:
d|A p 〉
dz
d|A s 〉
dz
+ α p
2 |A p〉 + i (
2 ω pb l · σ|A p 〉 = iγ p c 1 〈A p |A p 〉 + c 0 |A ∗ p〉〈A ∗ p|
)
+ c 2 〈A s |A s 〉 + c 3 |A s 〉〈A s | + c 4 |A ∗ s 〉〈A ∗ s | |A p 〉, (8.5.9)
+ α s
2 |A s〉 + i (
2 ω sb l · σ|A s 〉 = iγ s c 1 〈A s |A s 〉 + c 0 |A ∗ s 〉〈A ∗ s |
)
+ c 2 〈A p |A p 〉 + c 3 |A p 〉〈A p | + c 4 |A ∗ p〉〈A ∗ p| |A s 〉, (8.5.10)