Nonlinear Fiber Optics - 4 ed. Agrawal

184 Chapter 6. Polarization Effects
Figure 6.1: Schematic illustration of a Kerr shutter. Pump and probe beams are linearly polarized
at 45 ◦ to each other at the input end. Polarizer blocks probe transmission in the absence of pump.
respect to the direction of pump polarization) become slightly different because of
pump-induced birefringence. The phase difference between the two components at
the fiber output manifests as a change in the probe polarization, and a portion of the
probe intensity is transmitted through the polarizer. The probe transmissivity depends
on the pump intensity and can be controlled simply by changing it. In particular, a
pulse at the pump wavelength opens the Kerr shutter only during its passage through
the fiber. As the probe output at one wavelength can be modulated through a pump
at a different wavelength, this device is also referred to as the Kerr modulator. It has
potential applications in fiber-optical networks requiring all-optical switching.
Equation (6.2.6) cannot be used to calculate the phase difference between the x and
y components of the probe because the pump and probe beams have different wavelengths
in Kerr shutters. We follow a slightly different approach and neglect fiber
losses for the moment; they can be included later by replacing L with L eff . The relative
phase difference for the probe at the output of a fiber of length L can always be written
as
Δφ =(2π/λ)(ñ x − ñ y )L, (6.2.8)
where λ is the probe wavelength and
ñ x = n x + Δn x , ñ y = n y + Δn y . (6.2.9)
As discussed earlier, the linear parts n x and n y of the refractive indices are different
because of modal birefringence. The nonlinear parts Δn x and Δn y are different because
of pump-induced birefringence.
Consider the case of a pump polarized linearly along the x axis. The x component
of the probe is polarized parallel to the pump but its wavelength is different. For this
reason, the corresponding index change Δn x must be obtained by using the theory of
Section 7.1. If the SPM contribution is neglected,
Δn x = 2n 2 |E p | 2 , (6.2.10)
where |E p | 2 is the pump intensity. When the pump and probe are orthogonally polarized,
only the first term in Eq. (6.1.4) contributes to Δn y because of different wavelengths
of the pump and probe beams [9]. Again neglecting the SPM term, Δn y becomes
Δn y = 2n 2 b|E p | 2 , b = χ xxyy/χ (3)
xxxx. (3)
(6.2.11)
If the origin of χ (3) is purely electronic, b = 1 3
. Combining Eqs. (6.2.8)–(6.2.11), the
phase difference becomes
Δφ ≡ Δφ L + Δφ NL =(2πL/λ)(Δn L + n 2B |E p | 2 ), (6.2.12)