Optical Networks - _E_2 Nonlinear Effects on Pulse Propagation_156

Optical Networks - _E_2 Nonlinear Effects on Pulse Propagation_156

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E.2 Nonlinear Effects on Pulse Propagation 773 This can be rewritten in the form A(z, t) = A z e 1 + 2 t β 1 z Tz 2 e z (E.9) Comparing with (E.6), we see that A(z, t) is also the envelope of a chirped Gaussian pulse for all z > 0 , and the chirp factor κ remains unchanged. However, the width of this pulse increases as z increases if β 2 κ > 0 . This happens because the parameter governing the pulse width is now T 2 z = 1 + T 2 0 2 z( 1 + iκ) 1 = T 2 0 1 + β 2 T 2 0 2 + β 2 z T 2 0 2 , (E.10) which monotonically increases with increasing z if β 2 κ > 0 . A measure of the pulse broadening at distance z is the ratio T z /T 0 . The analytical expression (2.13) for this ratio follows from (E.10). E.2 Nonlinear Effects on Pulse Propagation So far, we have understood the origins of self-phase modulation (SPM) and cross- phase modulation (CPM) and the fact that these effects result in changing the phase of the pulse as a function of its intensity (and the intensity of other pulses at different wavelengths in the case of CPM). To understand the magnitude of this phase change or chirping and how it interacts with chromatic dispersion, we will need to go back and look at the differential equation governing the evolution of the pulse shape as it propagates in the fiber. We will also find that this relationship is important in understanding the fundamentals of solitons in Section 2.6. We will consider pulses for which the magnitude of the associated (real) electric field vector is given by (E.3), which is | E ( r , t) | = J(x, y) [ A(z, t)e i(ω 0 t β 0 z) ] .
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