Nonlinear Fiber Optics - 4 ed. Agrawal

178 Chapter 6. Polarization Effects
gree of modal birefringence, B m = |n x − n y |, and the orientation of x and y axes change
randomly over a length scale ∼10 m unless special precautions are taken.
In polarization-maintaining fibers, the built-in birefringence is made much larger
than random changes occurring due to stress and core-shape variations. As a result,
such fibers exhibit nearly constant birefringence along their entire length. This kind
of birefringence is called linear birefringence. When the nonlinear effects in optical
fibers become important, a sufficiently intense optical field can induce nonlinear
birefringence whose magnitude is intensity dependent. Such self-induced polarization
effects were observed as early as 1964 in bulk nonlinear media [1] and have been studied
extensively since then [2]–[10]. In this section, we discuss the origin of nonlinear
birefringence and develop mathematical tools that are needed for studying the polarization
effects in optical fibers assuming a constant modal birefringence. Fibers in which
linear birefringence changes randomly over their length are considered in Section 6.6.
6.1.1 Origin of Nonlinear Birefringence
A fiber with constant modal birefringence has two principal axes along which the fiber
is capable of maintaining the state of linear polarization of the incident light. These
axes are called slow and fast axes based on the speed at which light polarized along
them travels inside the fiber. Assuming n x > n y , n x and n y are the mode indices along
the slow and fast axes, respectively. When low-power, continuous-wave (CW) light is
launched with its polarization direction oriented at an angle with respect to the slow
(or fast) axis, the polarization state of the CW light changes along the fiber from linear
to elliptic, elliptic to circular, and then back to linear in a periodic manner (see Figure
1.9) over a distance known as the beat length and defined as L B = λ/B m . The beat
length can be as small as 1 cm in high-birefringence fibers with B m ∼ 10 −4 .Inlowbirefringence
fibers, typically B m ∼ 10 −6 , and the beat length is ∼1m.
If we assume that the longitudinal (or axial) component E z of the electromagnetic
field remains small enough that it can be ignored in comparison with the transverse
components, the electric field associated with an arbitrarily polarized optical wave can
be written as
E(r,t)= 1 2 ( ˆxE x + ŷE y )exp(−iω 0 t)+c.c., (6.1.1)
where E x and E y are the complex amplitudes of the polarization components of the
field oscillating at the carrier frequency ω 0 .
The nonlinear part of the induced polarization 1 P NL is obtained by substituting Eq.
(6.1.1) in Eq. (2.3.6). In general, the third-order susceptibility is a fourth-rank tensor
with 81 elements. In an isotropic medium, such as silica glass, only three elements are
independent of one another, and the third-order susceptibility can be written in terms
of them as [10]
χ (3)
ijkl = χ(3) xxyyδ ij δ kl + χ xyxyδ (3)
ik δ jl + χ xyyxδ (3)
il δ jk , (6.1.2)
1 Polarization induced inside a dielectric medium by an electromagnetic field should not be confused with
the state of polarization of that field. The terminology is certainly confusing but is accepted for historical
reasons.