From equation.(2.2) with the second and third order dispersion parameters set to zero, the NLSE with normalised amplitude becomes
^ = — exp(-CK)|«7pf/ (2.25)
*
w here a is the fibre loss and the nonlinear length (inversely proportional to the input pow er), and has the solution
U (z, T ) =
U(0, r)exp[i^^,(z, n ]
(2.26)
w here U(0,T) is the amplitude at input, z = 0, and the nonlinear phase term is given by
(l>,,{z,T) = \ U i O , T f { L ^ J L , ,)
(2.27)
w ith L ^ th e effective length given in equation (2.8).
The solution to pulse evolution shows that the intensity profile o f the pulse, \U(z,T)\^
remains unaltered by the nonlinearity but the phase o f the pulse is dependent upon the
initial pow er through param eter and increases with distance, z, scaled by loss via the
^max - ^ e jf I ^N L (2.28)
i.e. w hen the nonlinear length equals the effective length the maximum phase change is unity.
The instantaneous frequency across the pulse is given by the rate of change of phase
(2.29)
The pulse has been chirped: the instantaneous frequency changes across the pulse dependent on both the distance travelled and on the rate of change o f intensity. Larger frequency deviations, or frequency broadening, will occur where the intensity is changing m ost rapidly, e.g. at the leading and trailing edges of pulses. If w e substitute the expression for super-Gaussian pulses (2.13) into equation (2.29) the chirp-induced upon the pulse is given by
8(û[T) = 2m 4# ~T~ 2 m -\ exp 2 m -\ 4 ^ N L J o . \^o JT (2.30)
For higher order Gaussian pulses (higher values o f m) the m ore square the pulses become and the greater they are chirped. Figure 2.4 displays the phase and chirp profile o f a
G aussian (m=7) and super-Gaussian pulse {m=3) at The nonlinear phase
follow s the shape of the input pulse intensity but the chirp has a more complicated behaviour. The leading edge of the pulses experience a reduction in frequency (red-shift) and the trailing edges experience a frequency increase (blue-shift), therefore broadening the spectrum o f the pulse. W ith higher order pulses, chirping is evident only at the pulse edges and not across the pulse centre (where the intensity is constant). SPM thus causes frequency broadening of the pulse spectrum, a rough estimate o f w hich can be found by obtaining the maximum frequency deviation caused by the nonlinear chirp from equation (2.30) giving
(2.31)
(2.32)
i.e. the amount of spectral broadening is proportional to the maximum nonlinear phase shift (this does not hold for super-Gaussian pulses as their spectra are not G aussian and
Ao) 1/To but it allows an approximate calculation for the amount of spectral broadening pulses will experience).
The nonlinear effect acts to distort the phase across optical pulses. These phase variations in time manifest themselves as instantaneous frequency deviations across the duration of the pulse, i.e. the pulse has becom e chirped. The extent of the chirping is dependent upon the distance propagated and the intensity profile o f the pulse; faster rises or falls in intensity give more pronounced chirping and correspondingly larger spectral broadening.
In an ideal, intensity modulated direct detection (IM/DD) transm ission system , with a receiver bandwidth wide enough to pass the complete spectrum of the propagated signal, all the information contained within the spectrum w ould then be available with which to reconstruct the pulses. Intensity detection is insensitive to the purely phase modulating effect from the nonlinearity. But in practical systems the receiver will have a certain bandw idth which is not necessarily large enough to pass all the frequencies o f spectrum of the nonlinearly chirped channel. The receiver filter bandwidth will be designed large enough to pass the majority of the signal but keep the amount o f noise to a minimum and in a multi-channel system, small enough to reject neighbouring channels. The finite bandw idth of the receiver filter will cut off frequencies outwith its passband and, unless it has a flat frequency response, distort those which lie within. This loss and corruption of inform ation will lead to distortions of the intensity profiles o f the pulses and affect system perform ance. The preservation of phase information is vital for coherent transm ission formats such as phase-shift keying: any variation from the ideal constant intensity level
will directly affect the bit-pattem, placing more strict tolerance limits upon CW fluctuations
super-Gaussian pulse shapes ■o m = l Time +ve ■ - in=3 in=
-ve -- frequency chirp
J- across pulses
F i g u r e 2 .4 G a u s s ia n (/n = l ) an d s u p e r - G a u s s i a n ( m= 3 ) p u ls e s h a p e s a n d d y n a m i c f r e q u e n c y c h a n g e (o r c h i rp ) in d u c e d by the S P M effec t.
2 . 4 . 1 . 1 SPM In The Presence o f D ispersion
If the dispersion parameter is re-introduced and the dispersion length is com parable to the nonlinear length, the evolution of pulses in this nonlinear and dispersive medium is qualitatively different than a purely dispersive or purely nonlinear fibre. The com bination of the tw o effects is now exam ined.
Dispersion causes different optical frequencies to travel at different velocities through the fibre. As shown in the previous section the nonlinearity im poses chirp and broadens the spectrum of pulses. The increase or decrease in frequency will be met with an increase or decrease in velocity of those spectral com ponents depending w hether the dispersion is anom alous or normal.
In the case of G aussian pulses it was dem onstrated that due to the action o f SPM the leading edge is red-shifted in frequency which travel faster in normal dispersion than the blue-shifted frequencies at the trailing edge which travel more slow ly. A pulse therefore experiences greater broadening in nonlinear normal dispersion com pared to its propagation within a region of linear dispersion.
In anom alous dispersion the lower frequencies travel at slow er velocities than the higher frequencies. For a G aussian pulse, the red-shifted leading edge is slow ed as the blue- shifted trailing edge travels faster. Instead of the pulse broadening, as in a dispersive m edium , the pulse is narrowed. The SPM -induced chirp described by equation (2.30), is positive in value and has the opposite sign to the linear dispersion-induced chirp of equation (2.31) when P2 < 0, i.e. anomalous dispersion. If the magnitudes o f the two
chirping processes can be made equal then they may cancel each other allowing the pulse to travel through the fibre undistorted. This balancing act is obtained exactly in ideal lossless fibre for a particular pulse shape described by a hyperbolic secant and referred to as a soliton pulse. It was first suggested for use in telecommunications over tw o decades ago [7].