Nonlinear Fiber Optics - 4 ed. Agrawal

256 Chapter 7. Cross-Phase Modulation
where P 0 and P 20 represent the peak powers of the pump and probe pulses at z = 0.
The Pauli spin vector σ is defined using the unit vectors ê j in the Stokes space as
σ = σ 1 ê 1 +σ 2 ê 2 +σ 3 ê 3 , where the three Pauli matrices are given in Eq. (6.6.8). Using
Eqs. (7.6.7) through (7.6.9), the two Stokes vectors are found to satisfy [107]
∂ p
∂ξ + μ ∂ p
∂τ = 2 3 p 3 × p, (7.6.10)
∂ s
∂ξ = −4ω 2
3ω 1
(p − p 3 ) × s, (7.6.11)
where ξ = z/L NL is the distance normalized to the nonlinear length L NL =(γ 1 P 0 ) −1 ,
μ = L NL /L W , and p 3 =(p·ê 3 )ê 3 is the third component of the Stokes vector p. Whenever
p 3 = 0, p lies entirely in the equatorial plane of the Poincaré sphere, and the pump
field is linearly polarized. In deriving Eqs. (7.6.10) and (7.6.11), we made use of the
following identities [109]
|A〉〈A| = 1 2 [I + 〈A|σ|A〉·σ], (7.6.12)
|A ∗ 〉〈A ∗ | = |A〉〈A|−〈A|σ 3 |A〉σ 3 , (7.6.13)
σ(a · σ) =aI + ia × σ, (7.6.14)
where I is an identity matrix and a is an arbitrary Stokes vector.
The pump equation (7.6.10) is relatively easy to solve and has the solution
p(ξ ,τ)=exp[(2ξ /3)p 3 (0,τ − μξ)×]p(0,τ − μξ), (7.6.15)
where exp(a×) is an operator interpreted in terms of a series expansion [109]. Physically
speaking, the Stokes vector p rotates on the Poincaré sphere along the vertical
axis at a rate 2p 3 /3. As discussed in Section 6.3, this rotation is due to XPM-induced
nonlinear birefringence and is known as the nonlinear polarization rotation. If the pump
is linearly or circularly polarized initially, its SOP does not change along the fiber. For
an elliptically polarized pump, the SOP changes as the pump pulse propagates through
the fiber. Moreover, as the rotation rate depends on the optical power, different parts
of the pump pulse acquire different SOPs. Such intrapulse polarization effects have a
profound effect on the probe evolution.
The probe equation (7.6.11) shows that the pump rotates the probe’s Stokes vector
around p − p 3 , a vector that lies in the equatorial plane of the Poincaré sphere. As a
result, if the pump is circularly polarized initially, XPM effect becomes polarization
independent since p − p 3 = 0. On the other hand, if the pump is linearly polarized,
p 3 = 0, and p remains fixed in the Stokes space. However, even though the pump SOP
does not change in this case, the probe SOP can still change through XPM. Moreover,
XPM produces different SOPs for different parts of the probe pulse depending on the
local pump power, resulting in nonuniform polarization along the probe pulse profile.
The XPM-induced polarization effects become quite complicated when the pump is
elliptically polarized because the pump SOP itself changes along the fiber.
As an example, consider the case in which the pump pulse is elliptically polarized
but the probe pulse is linearly polarized at the input end. Assuming a Gaussian shape