Keiko, Kiwamu
I have run Kiwamu's length tolerance code (in CVS iscmodeling, ArmTolerance.m & analyseArmTorelance.m ) for the vertex ifo.
In his previous post, he montecarloed the arm lengths and saw the histogram of the sensing matrix and the demodulation phase between POP55 MICH and POP55 SRCL. From these plots, he roughly estimated that the tolerance is about 1 cm (sigma of the rondom gaussian) and in that case POP55 MICH and SRCL is separated by the demodulation phase 60150 degrees.
This time I put the length displacements of random gaussian on PRC, SRC, MICH lengths at the same time (Fig.1).
Fig. 1. History of random walk in PRC, SRC, MICH lengths parameter space. Same as Kiwamu's previous post, The position of the three degrees are randomly chosen with a Gaussian distribution function in every simulaton. This example was generated when \sigma = 1 cm for all the three lengths, where \sigma is the standard deviation of the Gaussian function. The number of simulation is 1000 times.
When the sigma is 1 cm, we found that the sensing matrix is quite bad if you look at Fig. 2. In Fig.2 row POP55, although the desired degrees of freedoms are MICH and SRCL, they have quite a bit of variety. Their separation in the demodulation phase is plotted in Fig.3. The separation in the demodulation phase varies from 40 degrees to 140 degrees, and around 270 degrees. It is not good as ideally we want it to be 90.
Fig. 2 Histgram of the sensing signal power in the matrix when 1 cm sigma rondom gaussian is applied on PRC, SRC, MICH lengths. x axis it the signal power in log10.
Fig.3 POP55 MICH and POP55 SRCL separation with the displacement sigma 1 cm.
Kiwamu suspected that PRC length as more strict tolerance than other two (SRC, MICH) for POP55, as 55MHz is fast and can be sensitive to the arm length change. So I ran the same montecarlo with SRC, MICH displacements but no PRC displacements when sigma is the same, 1cm. The results were almost same as above, nothing obvious difference.
With 2mm sigma, the signal power matrix and the POP55 MICH and POP55 SRCL separation in the demodulation phase look good (Fig. 4 and Fig. 5).
Fig.4 Signal power matrix when PRC, SRC, MICH lengths fractuate with random gaussian distribution with 2mm sigma. The signal powers are shown in log10 in x axis, and they do not vary very much in this case.
Fig.5 POP55 MICH and POP55 SRCL separation with the displacement sigma 2 mm. The separation of the two signal is 6090 degrees, much better than when sigma is 1 cm. We may need to check 60 degree separation is really ok or not.
PRC SRC MICH lengths tolerances of 2 mm in the real world will be very difficult !
Next I will check what happens on 3f signals.
Quote: 
Required arm length = 37.7974 +/ 0.02 [m]
This is a preliminary result of the estimation of the Arm length tolerance.
This number was obtained from a simulation based on Optickle.
Note that the simulation was done by considering misplacements in only the arm lengths while keeping PRCL, SRCL and MICH at the ideal lengths.
Therefore the tolerance will be somewhat tighter if misplacements in the central part are taken into account.
Next : check 3f signals, and include misplacements in PRCL, SRCL and MICH.
Figure.2 A sensing matrix of the 40m DRFPMI while changing the position of ETMX/Y by \sigma = 2 cm.
For convenience, only REFL11, AS55, POP11 and POP55 are shown. They are the designed signal ports that
mentioned in the aLIGO LSC document ( T1000298). In all the histograms, xaxis represents the optical gain in log scale in units of [W/m].
The yaxis is the number of events. The diagonal ports are surrounded by red rectangular window.
(Results2 : demodulation phase of MICH and SRCL on POP55)
Now a special attention should be payed on the MICH and SRCL signals on POP55.
Since MICH and SRCL are designed to be taken from POP55, they must be nicely separated in their demodulation phases.
Therefore the demodulation phase of MICH and SRCL has to be carefully examined.
The plot in Figure.3 is the resultant phase difference between MICH and SRCL on POP55 when \sigma_x = \sigma_y = 2 cm.
As shown in the plot the phase are always within a range of 60  120 deg, which satisfies my requirement (2) mentioned in the last section.
Figure.3 Difference in the demodulation phase of MICH and SRCL on POP55.
xaxis is the difference in the demodulation phase of MICH and SRCL, and yaxis the number of events.
