Cross-phase modulation

When two or more optical waves propagate inside optical fibers, they can interact thanks to the nonlinear response of induced polarization. We saw in section 2.3.1 that FWM results in the generation of additional frequencies when the phase matching condition is satisfied. Moreover new waves can be generated through stimulated Raman or Brillouin scattering which will be described in the following chapters. Processes such as FWM, Raman or Brillouin scattering, or harmonic generation generate new waves at the expense of the

energy of the input waves. On the other hand cross phase modulation (XPM) can couple two waves without any energy transfer between them. Moreover XPM can occur not only between two optical waves with different wavelengths, but also between two polarization states of single frequency. The first experimental observation of XPM was made using two CW lasers and a 15 km silica fiber [22]. A few years later spectrum changes of ps pulses were observed due to the XPM [23]. In order to explain the effect of cross phase modulation let us consider two linearly polarized optical fields at different wavelength propagating along the fiber. Then we can express the total electric field as:

1 1 2 2

( , ) exp( ) exp( ) . .

E z t =E i t

ω

+E i t

ω

+c c (2.37)

Now by inserting this expression into the equation for induced polarization (2.34) we can see that induced nonlinear polarizations oscillating at initial frequencies

ω

₁ and

ω

₂will be as follows:

( )

₍₃₎ 2 2

( )

3 1 0 (3 1 6 2 ) 1 exp 1 . . NL P ω = ε χ E + E E ⎡_⎣ i tω +c c⎤_{⎦ (2.38)}

( )

₍₃₎ 2 2

( )

3 2 0 (6 1 3 2 ) 2 exp 2 . . NL P ω = ε χ E + E E ⎡_⎣ iωt +c c⎤_{⎦ (2.39)}

From these expressions we can see that nonlinear susceptibility is dependent on the intensities of both waves and effective susceptibility can be expressed as (j=1,2):

( )₁ ( )₁ ( )₃

(

2 2

)

3

6 3

eff NL E−j Ej

χ = χ + χ = χ + χ + (2.40)

It’s clear that refractive index of the wave gets modified both by the intensity of the wave itself, and also by the intensity of the co-propagating wave at another optical frequency [2, 5]:

(

2 2

)

' 2 3 / 2 2 j NL j j j n n n E E₋ Δ ≈ χ ≈ + (2.41)

where n is a nonlinear parameter of the fiber. Change in the refractive index of the wave at '₂

1

ω

due to its own intensity, E₁2, is a result of SPM, and it is represented in the expression (2.41) by the first term. On the other hand, second term responsible for XPM changes refractive index twice more efficiently with the intensity of co-propagating wave, E₂ 2. As a result phase of the wave is modulated by XPM twice more effective than due to the SPM. As in the case of SPM, where there is change in spectrum of the pulse due to its own intensity, XPM results also in a modification of the spectrum of the copropagating pulses. Moreover, cross phase interaction between two pulses happens only when two pulses physically overlap in time with each other. Similar to SPM effect, spectrum of the pulses develops multipeak structure and broadens. However, spectral shape is modified this time by both SPM and XPM. In order to clearly see the effect of XPM on the pulse spectrum it is convenient to consider two pulses with sufficiently large difference in peak powers. This situation can be considered as interaction between strong pump and weak probe pulses. In this case XPM contribution to

the pump spectrum can be neglected, as well as SPM contribution to the probe pulse spectrum. By looking at the probe spectrum we can see how XPM induced by the pump pulse modifies the probe spectrum. By solving numerically the equations describing pulse propagation in the presence of nonlinear effects as well as experimental demonstrations it was shown that probe pulse spectrum becomes chirped. The sign of the chirp as well as its magnitude strongly depends on which part of the pump pulse overlaps with the probe pulse during interaction and on the pump pulse peak power [2]. Thus, induced probe pulse chirp due to the XPM depends on initial delay of the two pulses and on the group velocity mismatch between two pulses. If the probe pulse mainly interacts with the trailing edge of the pump pulse it gets positive chirp due to the XPM. As a result probe pulse has only blue-shifted components in its spectrum. On the contrary, when the leading edge of the pump pulse interacts with the probe pulse its spectrum develops red shifted components and negative chirp. In the case when the pump pulse fully passes through the probe pulse, induced chirp is zero at the pulse center and its magnitude is small across the entire probe pulse. Pulse spectrum is broadened symmetrically, but only edges of the pulse contribute to the new frequencies generation. As tails carry relatively small amount of energy pulse spectrum preserves its shape at the carrier frequency.

Described pulse spectrum modifications are considered only by the XPM alone, but in many situations fiber nonlinearities and dispersion properties of the fiber act together on the shape and spectrum of the propagating pulse. More complex transformations in spectrum and temporal shape of the pulse are expected when effects from XPM, SPM and GVD are combined. Since only the phase of the pulse is modified as a result of XPM and SPM effects, the pulse shape remains unchanged, while the spectrum is modified. However, in the presence of GVD different spectral components of the pulse, generated through XPM and SPM, propagate at different velocities. As a result the shape is severely distorted in an asymmetric manner [24]. It was shown that actually GVD reduced the extent of asymmetry in pulse spectrum compared to the situation when XPM acts alone.

In document Generation and application of dynamic gratings in optical fibers using stimulated Brillouin scattering (Page 35-37)