The hold-in range of the PLL must be greater than +/- 4MHz in order to bring the arm cavity to its resonance. 

(Hold-in range is the range of frequencies over which the PLL can track the input signal.)

However as I mentioned in the past elog (see this entry), the PLL showed a small hold-in range of about +/- 1MHz which is insufficient.

In this entry I explain what is the limitation factor for the hold-in range and how to enlarge the range.

(Requirement for hold-in range )

 We have to track the frequency of the green beat signal and finally bring it to a certain frequency by controlling the cavity length of the arm.

For this purpose we must be able to track the beat signal at least over the frequency range of 2*FSR ~ +/- 4MHz.

Then we will be able to have more than two resonances, in which both the end green and the PSL green are able to resonate  to the arm at the same time. 

And if we have just two resonances in the range, either one of two resonances gives a resonance for both IR and green. At this phase we just bring it to that frequency while tracking it.

  Theoretically this requirement can be cleared by using our VCO because the VCO can drive the frequency up to approximately +/- 5MHz (see this entry)

 The figure below is an example of resonant condition of green and IR. The VCO range should contain at least one resonance for IR.

(In the plot L=38.4m is assumed)

(an issue) 

However the measured hold-in range was about +/- 1MHz or less. This is obviously not large enough.

According to a textbook[1], this fact is easily understandable.

The hold-in range is actually limited by gains of all the components such as a phase detector's, a control filter's and a VCO's gain.

Finally it is going to be expressed by,

                         [hold-in range] = G_pd * G_filter * G_vco

 At the PD (Phase Detector which is a mixer in our case) the signal does not exceed G_pd [V] because it appears as G_pd * sin(phi).

When the input signal is at the edge of the hold-in range, the PD gives its maximum voltage of G_pd to maintain the lock.

Consequently the voltage G_pd [V] goes through to G_filter [V/V] and G_vco [Hz/V].

This chain results the maximum pushable frequency, that is, hold-in range given above equation.

In our case, the estimated hold-in range was 

                      [hold-in range] ~ 0.4 [V] * 3 [V/V] * 1 [MHz/V]

                          = 1.2 [MHz]

This number reasonably explains what I saw.

In order to enlarge the hold-in range, increase the gain by more than factor of 5. That's it.

* reference [1]  "Phase-Locked Loops 6th edition" Rolan E. Best