Optimization of Cascaded Regenerative Links based on Phase

Note that the regenerative function depends only on the phase of the input signal. The parameter m represents the amplitude ratio between the signal and its idler and has a critical role in defining the phase noise suppression properties of the PSA. Our recent analytical study (171) (see section 5.1.1) has identified m = 1/(M−1) as the optimum value, which provides a flat phase response at the alphabet points of the signal, en- abling maximum phase noise suppression and regenerative performance along the link. The amplitude and phase response of an 8-level PSA with an optimized m-parameter are depicted in Figs. 5.1(a)–5.1(b). Its phase response has a periodic staircase shape with plateau regions around the signal alphabet points. This periodicity suggests that only constellation diagrams with phase symmetry can be handled by the specific PSA scheme, although it can become the building block of more complex configurations that will allow dealing with rectangular M-QAM signals (68).

5.2.1

Optimization of Cascaded Regenerative Links based on Phase

Sensitive Amplifiers

Here we develop an analytical method for optimizing phase sensitive amplifiers for regeneration in multilevel phase encoded transmission systems. The model accu- rately predicts the optimum transfer function characteristics and identifies operating tolerances for different signal constellations and transmission scenarios. The results demonstrate the scalability of the scheme and show the significance of having simul- taneous optimization of the transfer function and the signal alphabet. The model is general and can be applied to any regenerative system.

In the above equations m corresponds to the amplitude ratio of the interfering signal-idler pair of waves. It is an optimization parameter that defines the slope of

5.2 Phase regeneration 0.8 0.9 1.0 1.1 1.2 (b) 2 0 O u t p u t p h a s e , o u t Input phase, in Regenerative function Linear function: out = in (a) 0 2 2 N o r m a l i z e d a m p l i t u d e r e s p o n s e , r o u t / r in Input phase, in 0

Figure 5.1: PSA response - (a) Normalized phase response and (b) normalized amplitude response of an 8-level phase sensitive amplifier as a function of the phase of the input signal. 0 O u t p u t a n g l e , o u t 0 0 2 2 P a r a m e t e r , m 0 0.4 In p u t a n g l e , in

Figure 5.2: Periodicity and slope characteristics of the PSA phase transfer function - with varying input phaseφinand regenerative parameter m with M = 8 phase levels.

5.2 Phase regeneration 4 8 12 16 0.0 0.2 0.4 0.6 0.8 1.0 b) m=m opt m=m cr o u t = F ( i n ) input angle, in a) P h a se T F , m OPT m CR P a r a m e t e r , m a n d o p e r a t i o n a l d i f f u si o n , constellation size, M

Figure 5.3: The PSA transfer function- a) PSA phase transfer function for 8-PSK for- mat. The optimal value of the parameter mopt = 1/7 (shown by green solid line) demon-

strates a plateau centered at the alphabet points, whereas a critical choice mcr = 0.33

(shown by red dashed line) degrades performance. b) Dependence of the optimal and critical values of m and the attraction region’s maximum half-widthδmaxon the order M

5.2 Phase regeneration 4 8 12 16 0.0 0.2 0.4 0.6 0.8 1.0 b) m=m opt m=m cr o u t = F ( i n ) input angle, in a) P h a se T F , m OPT m CR P a r a m e t e r , m a n d o p e r a t i o n a l d i f f u si o n , constellation size, M

Figure 5.4: a) PSA phase transfer function for 8-PSK format. The optimal value of the parameter mopt= 1/7 (shown by green solid line) demonstrates a plateau centered at

the alphabet points, whereas a critical choice mcr= 0.33 of the parameter (shown by red

dashed line) is expected to lead to poorer regenerative performance. b) Dependence of the optimal and critical values of m and the attraction region’s maximum half-width∆maxon

the order M of the PSK modulation format.

the phase transfer function of the PSA near the alphabet points and characterizes its regenerative properties (see Fig. 5.2-5.4). More specifically, the phase transfer func- tion creates a periodic attractive potential around a set of special stationary points that are unaffected by the nonlinear transformation. Both, the stationary points and the at- tractive potential, contribute to the phase squeezing performance of the PSA. Clearly, the optimum set of values for parameter m has to be identified to maximize the re- generation efficiency. Since the alphabet is discrete, it is also necessary to adapt the characteristics of the nonlinear element to the employed signal format and vice versa. Firstly, the nonlinear transfer function of the regenerator is adjusted to the signal al- phabet. For this, the corresponding attraction regions should be centred at the corre- sponding alphabet pointsφ∗, the stationary points of the transformation, as defined by the conditions: F′′(φ∗) = 0 and F(φ∗) =φ∗. Thus, starting from Eq.5.2, we derive the optimum constellation given by the set φ∗= lπ/M, l ∈ Z. The next step is to ensure stability of the nonlinear transformation at the alphabet points for effective suppression of the phase noise. This condition leads to the inequality|F′(φ∗)| < 1, from which the

5.2 Phase regeneration

variation limits of the regenerative parameter m are derived as:

|m| < mcr, mcr= 2

M− 2 (5.3)

here m > 0 if l = 2k and m < 0 if l = 2k + 1, with k∈ Z. The points are superstable, i.e. F′(φ∗) = 0, when

|mopt| = 1

M− 1 (5.4)

The superstable case creates operational plateaus around the alphabet points of the transfer function that enable maximum phase noise suppression. For m̸= moptplateaus cannot be defined, however, provided that |m| < mcr, partial suppression of the phase noise is still feasible within the limits of each attraction region, since |F′(φ∗)| < 1. An analytical expression can be derived for the attraction region’s half-widthδ in the limit of δ ≪ 1 by performing perturbation analysis on the equation F′(φ∗+δ)| = 1, resulting in: δ ≃ √ ( 1−−1 − m + mM 1 + m  )/ (_mM3₍₁− m) 2(1 + m)3 ) (5.5) here for simplicity we consider the absolute value m =|m|. The parameter δ has the meaning of the maximum phase noise distortion that can be suppressed by the regener- ator. It acquires the maximal valueδmaxfor the plateau condition m = mopt, whereas for sub-optimal m values it narrows and tends to zero when m approaches mcr. Fig. 5.4a) depicts the phase transfer functions of regenerative PSAs with eight discrete phase states. The green solid line corresponds to the moptselection of the regenerative param- eter. The condition,|F′(φ∗)| < 1results in simultaneous suppression of phase-to-phase and phase-to-amplitude noise conversion mechanisms. The red dashed line has been taken for the critical value mcr, above which the PSA elements amplify phase noise and degrade the system performance. Fig. 5.4b) shows the dependence of the optimal and critical values of the regenerative parameter m on the constellation size M. One can see that for high-order modulation formats the gap between mopt and mcr is nar- rowed, making the PSA optimization more critical. The variation of the corresponding half-widthδmax is also depicted in the same figure.

5.2 Phase regeneration