Chapter 4: Optical Injection Phase-Lock Loop Theory
In Fig. 4.7 (a) and (b), the increase in the loop gain results in an improvement of the low frequency phase noise suppression, as the phase error spectrum values are smaller. This improvement should be extended to all frequencies if the loop parameters are such that the loop is operating inside a stable region. However, as the gain is increased towards its critical value, the OPLL becomes more susceptible to phase perturbations as the phase margin of the loop is reduced. This effect can be seen from the peaking appearing in the phase error spectrum curve for high frequencies. As the loop gain tends to the critical value, the peak tends to infinity and the loop becomes unstable. Comparing Fig. 4.7 (a) and (b), it is possible to notice that the second order loop presents even further reduction in the phase noise contents for low frequencies than the modified first order loop. In fact, the phase error spectrum tends to zero in the case of a second order type II loop and to a finite value in the case of the modified first order loop. This extra reduction is due to the inclusion of an integrator with high DC gain in the loop. Therefore, the static phase error of the second order type II loop tends to zero, offering improved low frequency phase noise suppression.
The phase error variance can be obtained from the phase error spectrum by:
^lpLL = r jo p u .( fW
(4.4.10)By substituting eq. (4.4.9) into (4.4.10), the phase error variance for an OPLL system can be written as:
= + (4.4.11)
where is the laser summed linewidth. The integrals Iq p l l and ^ „ .o p l l are given by:
^OPLL ~
J
df
(4.4.12)/
K-opu.= \ H m t d f
(4.4.13)Fig. 4.8 (a) and (b) illustrate the dependence of the phase error variance on delay time for modified first order and second order type II loops. In the modified first order loop model, the parameter for the plots is the time constant T. In the second order type II loop model, the parameter for the plots is the zero-delay damping ratio The
loop gain is set for each value of delay to be 10 dB below its critical value. The remaining parameters are the same as those used for Fig. 4.7. Numerical integration was used to evaluate eq. (4.4.12) and (4.4.13). It can be seen that the decrease of T,
reflecting the broadening of the loop filter bandwidth, can improve the phase noise suppression of the modified first order system. Basically, a sharp variation in the separation of the variance curves is observed from high T, tending to a limit for lower T,
suggesting that phase noise control is limited by the shot noise contribution. The curves for the phase error variance of second order type II loops are very closely spaced, indicating that the OPLL phase error variance is not very sensitive to different values of <^0- The system tends to present lower values of phase error variance for over damped second order OPLLs. The type II systems offer superior tracking and acquisition characteristics to the modified first order loops but they are more sensitive to the loop propagation delay as can be seen from the slopes of the curves in Fig. 4.8 (a) and (b) [4.17].
In Chapter 2, the assumption made to solve the OPLL equation was that the phase erro r
Q{t)
is small enough to allow the linearisation o f the sinusoidal characteristic of the phase detector (photodetector). However, if the real response is taken into account, the phase detector gain can decrease dramatically depending on the mean phase error and, as a result, the loop can loose lock. In fact, around 7t/2 rad, the phase detector output signal is negligible and the loop can not control the output phase of the slave laser to track the phase variations o f the master laser, resulting in loss of phase lock condition. If the phase error reaches 7t/2 rad, the phase detector response can present two distinct outcomes. The first one is that the loop reacquires lock as the phase error value drops back to the linear region of the same sector of the phase detector response. The second one is that the phase error jum ps one or more cycles of the sinusoidal response, acquiring lock again in the linear region of one of the next sectors of the phase detector transfer function.This outcome is known as cycle slipping [4.7,4.10,4.18] and depends on the amount of phase error present in the system. Therefore, for loops with good phase noise suppression, that is, with low values of phase error variance, the linear assumption for the phase detector response is valid and cycle slipping is rare. The average time between cycle slipping events can be obtained for the first order, modified first order and second order type II loops [4.7]. Eq. (4.4.14a) fits well for first order and modified first order loops and can be obtained directly from statistical analysis, assuming that the phase error is stationary inside an interval (-7C,7t) and has zero average. By means of the manipulation of a non-linear, stochastic partial differential equation and assuming that
Chapter 4; Optical Injection Phase-Lock Loop Theory
the variance of the system is low, the average time between cycle slips for first order and modified first order loops is given by [4.7,4.23]: