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\begin{document}
{\large \bf Fast Estimate of the Locking Range of PhaseCOMB Self-Pulsations}

ede, 08.01.03

I consider the locking of mode-beating pulsations to harmonic external signals of frequency $f_\text{ext}$. To numerically determine the locking range by stepping through $f_\text{ext}$ is very time consuming. In case of the Adler model of locking (see my text from 17.11.02), this task can be solved much easier. In this case, the instantaneous frequency (= inverse temporal separation of two subsequent maxima) follows
\begin{eqnarray} \label{adlermodell}
\text{\bf Adler model: } ~~ f(t) - f_\text{ext}= \delta  + p \sin(\varphi ), 
\hspace{3em}
\varphi (t) = \int_{t0}^t \left[f(t')-f_\text{ext}\right] dt'+ \varphi _0. 
\end{eqnarray}
In this case, the locking range is just $2p$, and the borders of the locking cone are $\delta \pm p$.

How to determine the parameters $\delta , p$ numerically? It is easy: Choose any $f_\text{ext}$ outside of the locking range (but not too far of it), calculate f(t) over a sufficiently long time (that $\varphi $ changes by more than $2\pi $), than it holds
\begin{eqnarray}
\delta = \frac{\max(f)+\min(f)}{2}, ~~~
p = \frac{\max(f)-\min(f)}{2}.
\end{eqnarray}

I checked this idea with a particular point of operation (without LSHB yet). The $f(t)$ is easily obtained from the half-height slopes by the rlab-lines

\footnotesize
\begin{verbatim}
from file lock.r on sim-3r:~ede/ldsl_on_store/locking/030107/.

jittede = period(xoft[pulsnum1:pulsnum2]);
F=1e-9*jittede.frq.fr;                            // SPfrequenz in GHz
tt=[2:jittede.njit]; tt1=[1:jittede.njit-1];

// t in ns, f-fext in MHz :
outdat=[tt1'/F,-(jittede.slope[tt;1]-jittede.slope[tt1;1])*(F*1e9*F*1e3)];      
plot(outdat);
\end{verbatim}
\normalsize

\mb{0.5}{
For simplicity, I used current modulation. Without modulation $F=52.76$ GHz. I start with 500 MHz higher modulation. The resulting $f(t)-f_\text{ext}$ in MHz vs. $t$ in ns is drawn righthand. Just by inspecting, I find roughly $fmax=-310$ and $fmin=-710$, i.e,. $\delta \approx -510$ and $p \approx 200$, thus locking range estimate 200,  MHz each. 

To come closer to the locking range, we change $f_\text{ext}$ by nearly $fmax$ to 52.46 GHz. 
} \hfill \fig{0.45}{lockrange1}

\mb{0.5}{
The outcome is drawn right. Indeed, we are very close to the border of the cone ! A more precise estimate of the parameters is possible now from the Fig. In particular, the opposite border of the locking cone is expected at -400 MHz.
} \hfill \fig{0.45}{lockrange2}

\mb{0.5}{
Going to $f_\text{ext}= $ 52.07 GHz, we indeed arrive close outside the opposite side of the cone, as drawn in the righthand Figure. 
} \hfill \fig{0.45}{lockrange3}

\mb{0.5}{
What happens if we go very far from the locking cone, say to $f_\text{ext}= $ 51 GHz? The righthand Fig shows that this still gives resonable parameters, e.g. about 440 MHz for the locking range (modulation span of frequencies). 
} \hfill \fig{0.45}{lockrange4}



{\bf Conclusion}

PhaseCOMB (without LSHB) locks according to the Adler model. This allows to determine the borders of the locking cone from {\em only one} transient with an arbitrary $f_\text{ext}$ outside of the locking cone. In addition, the calculation time for the transient can be kept as short as e.g. 5 ns, because the external frequency can be taken sufficiently far away from the border of the cone. 
\end{document}
