All the lengths described in other pages were so far simple optical distances. By optical length here I mean the equivalent distance between high-reflection-coated surfaces that takes into account the presence of the substrate and of their refractive indexes. Therefore these lengths can be quite different from the real distances between the payload. The payload distance can be defines as the distance between the suspension point (center of mass) of the mirrors, along the beam propagation direction.

Here I summarize all the computation needed to convert from optical lengths to payload distances.

Beam Splitter

Some care is needed when dealing with the beams around the beam splitter. Here I assume the HR face to be the one facing the power recycling mirror. I assume the beam to be centered on the HR face, which seems the most reasonable configuration.


The incoming beam has an angle of incidence of 45 degrees with respect to the BS normal direction. From Snell's law one gets that the refracted beam makes an angle of

sin t2 = 1/n sin t2 = 1/n * 1/sqrt(2)   ---> t2 = 29.21 degrees

assuming a refractive index equal to 1.45.

Referring to the left drawing of the previous picture, being d the beam splitter thickness, some distances are relevant:

d1 = d / cos t2 = 1.1457 d
d2 = d1 cos (45 - t2) = 1.102 d

In particular d1 is the distance traveled inside the substrate. The corresponding optical lenght is given by n*d1.

Some distances from the center of mass are useful. Referring to the right drawing:

d1' = d / (2 sqrt(2)) = 0.354 d
d2' = d2 - d1' = 0.748 d

Optical lengths

This plot shows the definition of payload distances D. The thickness of each mirror is denoted by d. The optical lengths L used in all other pages are:

• Distance PR-BS. Let's start from the distance between centers of mass. On the PR side, half thickness must be subtracted, since the HR face looks inward. On the BS side, only d1' must be subtracted, since the HR face looks toward the PR mirror:
  L_PR-BS = D_PR-BS - d1' - dpr/2
• Distance BS-SR. Again, on the SR side only half thickness must be removed. On the BS face the situation is a bit more complex. The AR face is d2' closer to SR mirror than the center of mass, but the beam travels for a distance d1 inside the BS:
  L_SR_BS = D_SR_BS - d2' + n d1 - dsr/2
• Distance BS-NI. On the NI side, the AR face is half thickness closer to BS than the center of mass, but the additional distance traveled inside the substrate must be considered. On the BS side, the same considerations made for BS-SR distance hold:
  L_BS_NI = D_BS_NI - d2' + n d1 - di/2 + n di
• Distance BS-WI. On the WI side the same consideration of the NI hold. On the BS side, the HR face is immediately met, therefore only a distance difference d1' is relevant:
  L_BS_WI = D_BS_WI - d1' - di/2 + n di

Since the thickness of all mirrors are of the order of centimeters, the differences coming from the above formulas are relevant and must be considered when studying the feasibility of a lengths/modulations configuration.

Allowed ranges

From A. Paoli talk the following "restricted" ranges of possible payload distances are possible without interference between the large flange on the tower bottom part and the concrete:

D_BS_PR = 4.90 - 6.70 m
D_BS_NI = 5.95 - 6.85 m
D_BS_WI = 5.15 - 6.05 m
D_BS_SR = 5.90 - 6.10 m

If we allow the large flange to be partially closed by the concrete floor and we only request the small flange to be free, the "extended" possible ranges are:

D_BS_PR = 4.30 - 7.30 m
D_BS_NI = 5.35 - 7.45 m
D_BS_WI = 4.55 - 6.65 m
D_BS_SR = 5.30 - 6.70 m


In the following plots the PR-BS distance is changed over all the allowed extended range. All four distances are then plotted:

• thick solid lines show points where the distance is within the restricted allowed range
• thin solid lines show points where the distance is within the extended allowed range
• dotted lines show point outside all allowed ranges
• circles give the current (Virgo) nominal distances
  

Small Schnupp - SB2 resonant in PRC and SRC option

Solution 1

The optical lengths needed for this configuration can be found here. In summary:

LPRC = 11.953 m
Schnupp = 0.040 m
LSRC = 11.033 / 12.872 m (for f2 = 81.5 MHz)


If we consider only the restricted ranges, the possible values for the distance PR-BS is quite small, and limited by the allowed ranges of input tower displacement. To obtain a signal recycling cavity with a length of 11.033 m (corresponding to the 81.5 MHz second modulation), the SR tower should be displaced quite a lot: indeed its distance from BS should be decreased by 1.2 meters, and this is not inside even the extended range for BS-SR.

If we allow also the NI to be outside the restricted range, there is one possible solution which implies a displacement of the SR tower of about 0.6 meters. In this case the needed displacements are:

BS-PR = 6.357 m (+0.357 m)
BS-SR = 5.360 m (-0.640 m)
BS-NI = 5.444 m (-0.956 m)
BS-WI = 5.480 m (-0.120 m)

Solution 2

The second optimal lengths for the 81.5 MHz modulation is at 12.872 m. The corresponding plot is:


As before the signal recycling tower should be displaced, but of a smaller amount in this case: about 0.7 meters. The corresponding solution is however still inside the extended range for SR:

BS-PR = 5.798 m (-0.202 m)
BS-SR = 6.641 m (+0.641 m)
BS-NI = 6.002 m (-0.398 m)
BS-WI = 6.038 m (+0.438 m)

Solution 3

This suggests that the longer optimal SRC length is the one which can be closer to the present value. Indeed a solution compatible with SR tower range is found for a modulation frequency of 244.5 MHz:


The corresponding displacements are all inside the restricted range:

BS-PR = 5.798 m (-0.202 m)
BS-SR = 6.028 m (+0.028 m)
BS-NI = 6.002 m (-0.398 m)
BS-WI = 6.038 m (+0.438 m)

Small Schnupp - SB2 resonant only in SRC option

The optical lengths needed for this configuration can be found here. In summary:

LPRC = 11.953 m
Schnupp = 0.040 m
LSRC = 11.947 m

There is only one choice of the optical signal recycling length, regardless of the chosen second modulation frequency.


Since PRC length and Schnupp asymmetry are the same as the previous configuration, the same allowed range for PR-BS and BS-NI, BS-WI are present. The SR tower should be displaced of about 0.3 meters to meet the optimal condition, and this is inside the extended region:

BS-PR = 5.798 m (-0.202 m)
BS-SR = 5.716 m (-0.284 m)
BS-NI = 6.002 m (-0.398 m)
BS-WI = 6.038 m (+0.438 m)

Otherwise the SR tower can be maintained fixed by allowing the NI to be outside the restricted range:

BS-PR = 6.078 m (+0.078 m)
BS-SR = 5.995 m (-0.005 m)
BS-NI = 5.723 m (-0.677 m)
BS-WI = 5.759 m (+0.159 m)

Large Schnupp option

Again the corresponding lengths and modulations can be found here. In summary:

LPRC = 11.954 m
Schnupp = 0.854 m
LSRC = 11.954 m

The corresponding plot is:



In this case there is a allowed range of distances (close to the present one) which meet the requirements on optical lengths. In particular it is possible to avoid any displacement of the SR tower:


BS-PR = 6.078 m (+0.078 m)
BS-SR = 6.002 m (+0.002 m)
BS-NI = 6.130 m (-0.270 m)
BS-WI = 5.352 m (-0.248 m)