Nonlinear Fiber Optics - 4 ed. Agrawal

490 Chapter 12. Novel Nonlinear Phenomena
Figure 12.31: Supercontinuum spectra at the output of a highly birefringent microstructured
fiber for three different angles of linearly polarized 200-fs input pulses from the slow axis of the
fiber. Dashed lines indicate the ZDWL for the fiber mode polarized along the slow and fast axes.
(After Ref. [128]; c○2003 AIP.)
percontinuum at the fiber output exhibit different SOPs [131]. This nonlinear coupling
can also occur in ideal isotropic fibers, with a perfectly circular core and exhibiting no
birefringence, if the SOP of the input pulse is not linear so that both polarization modes
are excited [132]. Its mathematical description requires that we solve two coupled generalized
NLS equations of the form Eq. (12.4.1). Using the Jones vector and the Pauli
matrices, as defined in earlier chapters, they can be written in a vector form as
∂|A〉
∂z
+ 1 2
= i 2
(
∂
α + iα 1
∂t
(
Δβ + iΔβ 1
∂
∂t
)
|A〉 +
M
∑
m=2
)
σ 1 |A〉 + i
i m−1 β m ∂ m |A〉
m! ∂t m
(
∂
γ + iγ 1
∂t
)
|Q(z,t)〉, (12.4.6)
where |Q(z,t)〉 is related to the third-order nonlinear response of the fiber and has the
following form if we neglect the anisotropic part of the Raman response:
|Q(z,t)〉 = 2 3 (1 − f R) [ 〈A|A〉 ] |A(z,t)〉 + 1 3 (1 − f R) [ 〈A ∗ |A〉 ] |A ∗ (z,t)〉
+ f R |A(z,t)〉
∫ t
−∞
h R (t −t ′ )〈A(z,t ′ )|A(z,t ′ )〉dt ′ . (12.4.7)
The birefringence effects are included in Eq. (12.4.6) through the quantities Δβ and
Δβ 1 , where Δβ = β 0x − β 0y is given in Eq. (6.1.13) and Δβ 1 = β 1x − β 1y accounts for
the difference in the group velocities of two orthogonally polarized pulses.
Numerical simulations based on Eqs. (12.4.6) and (12.4.7) reveal that the SOP of
a specific spectral component in a supercontinuum is generally elliptical even when a
linearly polarized pulse is launched initially into the fiber. Moreover, it can vary widely
for different spectral components. In one study, the ellipticity, defined as
e p (ω)=〈Ã(L,ω)|σ 3 |Ã(L,ω)〉 / 〈Ã(L,ω)|Ã(L,ω)〉, (12.4.8)