Some important features of the nonlinear

By dropping the loss and the cubic dispersion terms from Eq. (5.98) and using appropriate transformations the following dimensionless nonlinear Schrodinger equation can be derived:

-P"A

IP"I

. du 1

where s are the normalized spatial and (local) time variables defined by9

9 The physical meaning of the coefficient “a” in these given transformations can be found from Eq. (5.104b). It is actually the pulse duration parameter and, therefore, in principle can be selected according to the pulse shapes. Nevertheless, it is often defined by assuming a sech2 pulse shape, i. e. a = 1.76/Atj (see Sec. 2.1).

5 = a2|p"|z = (Ati/1.76)-2|P"|z (5.104a)

and

s = a(t - P'z) = (Ati/1.76)-1 (t - p'z) (5.104b)

(5.104c) Eq. (5.103) reflects the combined effect of the second-order group velocity dispersion and the relatively weak nonlinearity on the propagating pulses. As summarized in Table 5.5, according to the different signs that the dispersion parameter may have and the essential feature of the initial pulses, distinct solutions can be derived from this equation (Blow and Doran 1987, Tomlinson 1989).

Table 5.5 Combined effects of group velocity dispersion and nonlinearity*

Pulse

types Ordinary pulses Inverse pulses t

Dispersion (u —> 0 for t» Ati) (u —> 1 for t» Ati)

p" <0 Solitons Modulational instabilities

p" >0 Enhanced dispersion Dark solitons

* After Blow and Doran (1987).

t This category of pulses, having a sudden drop of intensity on a continuous radiation background, may also be termed as dark pulses.

We may concentrate on the situation where the incident pulses are the ordinary pulses. [Investigations of dark solitons which only exist for the inverse pulses can be found in the papers by Weiner et al (1988), and Tomlinson et al (1989)].

For P" < 0, Eq. (5.103) can be rewritten as

. du ld2u , .2

= (5-105)

One of the most important features of the above nonlinear Schrodinger equation is that it has analytical solutions for the initial condition (Satsuma and Yajima 1974):

u(0,s) = Nsech(s) (5.106)

where N is an integral. For example, with N = 1,

ui(£,s) = e_I^2sech(s) (5.107)

and for N = 2,

I2(U = 4e-iV2 _„

ch(4s) + 4ch(2s) + 3cos(4£) (5.108)

The wave relating to the solution for N - tv (tv > 1) is often called the TVth-order soliton. Solitons are stable with respect to small perturbations in the initial input condition, such as in Eq. (5.106) N is changed to N + £, (Zakarov and Shabat 1972).

The fundamental (N = 1) soliton remains unchanged in its shape, while for higher order solitons (N > 2) the pulse shape changes periodically during propagation. In the latter case, although for different N values the pattern of the changes of pulse shape varies dramatically a common spatial period (known as soliton period) exists, at which the pulses are restored to their original shapes. The soliton period is given by (Satsuma and Yajima 1974)

(5.109) In real space, using Eq. (5.104a), this expression leads to

,2 Zo

or (Mollenauer and Stolen 1982)

_ ZEr Ati 1 1 “ 2 L 1.76J |p"| (5.110a) Zo = TtAti Xl.76 _c_ _ nT?? (ftAtj)2c _ 0.322(TtAtjc)2 |D| " X2|D| " X|Y| (5.110b) n2

Solitons in the second half period make a mirror image of those in the first half and so for the purpose of calculation only the traces in £ < tc/4 region need to be considered. For the higher-order solitons, at the position of the exact half period (N - 1) peaks exist, all of them sitting on the top of a broader pedestal background (Tan 1987). Such a feature, obviously, may be utilized to make a judgement of the order of the solitons observed in experiments.

For producing a N = 1 soliton, from the transformation given by Eqs. (5.104), the associated actual incident optical pulse should be

!<&(0,t)l = 1.76

the associated peak amplitude of which is given by

A2 = I <&(0,0) 12 =

The above expression is identical to

'■-sS <=••«

where XVac is the vacuum wavelength for the carrier (Mollenauer and Stolen 1982). Similarly, for the N = 2 soliton, through using Eqs. (5.108) and (5.104), it can be readily derived that

I2 = 22Ii (5.113)

In general, for the Tvth-order soliton, the peak power for the initial optical pulses should satisfy

lft = u2I, (5.114)

A significant feature of the higher-order solitions is that they all experience a self­ compression process immediately after entering the medium. The higher the soliton order, the larger the compression will be. For a N = 2 soliton, the shortest peak occurs at the half soliton period. As N increases such a peak appears earlier and earlier. After reaching the shortest peak position, the temporal profile of the solitons collapse and is involved into a complicated pulse splitting process. While N = 1 soliton is of considerable potential use in optical digital communication (Hasegawa and Tappert 1973, Mollenauer et al

1986), a direct application of higher order solitons is in the pulse compression scheme (Mollenauer et al 1983, Gouveia-Neto et al 1988)10.

In the cases where p" > 0, Eq. (5.103) becomes

. du 1 02U , , ,2 zc 1 1 ex

^=‘2a?+|u|u (5ll5)

Because for ordinary pulses 32u/3s2 < 0, there is no possible cancellation between the two terms on the right-hand side of the above equation, and so no self-compression occurs for propagating pulses. (In some publications, to differ from the cases where only SPM is present, the coexistence of both positive GVD and SPM is also termed as dispersed self-

^However, a noticeable disadvantage of the pulse compression using higher-order solitons is that the compressed pulses are always accompanied by relatively large pedestals.

phase modulation). The frequency chirp caused by SPM and that by GVD under these situation will enhance each other, and the eventual result from the interplay between SPM and the positive GVD is that a rectangular-shaped pulse having largely stretched, linear frequency chirp is formed (Natatsaka et al 1981, Nelson et al 1983). Such a feature has been widely utilised in the fibre-grating pair pulse compression scheme (Grischowsky and Balant 1982; also see Appendix D).

Part III