BALANCED HOMODYNE DETECTION
5.5 Output mode cleaner cavities
5.5.3 OMC cavity filtering performance Incident mode spectrum
The filtering performance of the OMC is determined by the cavity finesse. However, the actual amount of junk light that makes it onto the OMC DCPDs is dependent on the amount incident as well.
For the aLIGO OMC, the filtering requirement is principally on the AS beam from the IFO. The recycling cavity lengths are chosen in such a way that the f2sideband is critically coupled to the dark port, while the f1 sideband mostly remains in the symmetric side (though the leakage to the dark port isn’t zero). Consequently, the
FM1 FM2
CM1 CM2
Incident Beam
OMC TRANS
ROC L1
Figure 5.7: Proposed OMC optical layout. To preserve symmetry, the mirrors labelled FM1 and FM2 are flat, and CM1 and CM2 have identical curvatures. The geometrical design parameters, namely the angle of incidence θi, the length L1, and the radius of curvature of the curved mirrors, ROC, are indicated in green. In addition to these, the OMCdesign has to determine the power transmissivities of the FMs,t2
F and CMs,t2
C.
following power law model was assumed for the power fraction in various carrier HOMs:
Pf0(n) [W/W]=
0 n= 0,
7×10−5 n= 1, 1.8×10−(3+
n 4.8)
n ≥ 2.
(5.10)
For the f2 sideband, the following model was assumed for transmission from the PRC to the SRC (the SRC cavity’s resonance structure is neglected):
Tf2(n)=
1 n=0,
1.7×10−1 n=1, 7×10−(1+
n
2) n≥ 2.
(5.11)
Finally, for the f1 sideband, it was assumed that Tf1 = Tf2/1000. This model neglects (i) mode-healing / mode-harming effects of the SRC, and (ii) the actual transmissivity of the f1 sideband with the SRC included. For the BHD setup, the OMC also needs to clean up the LO field, which is in fact picked off from the
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 TEM mode index
10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3
Power at dark port [W per 1W@BS]
carrier f1f2
Figure 5.8: Assumed model spectrum of power in HOMs. It must be emphasized that this model is extracted from a mode-scan done on the eLIGO interferometer’s output field. The 40m interferometer may have a different spectrum, in which case the transmission of the OMC for HOMs will have to be re-evaluated.
symmetric side of the interferometer. At first glance, this may suggest that the LO field will have strong f1content because the PRC is designed to be resonant for this field. However, assuming that the LO field is picked off from the transmission of a HRmirror (as is proposed), we estimate that the power of the f1content in the LO beam will be ≈ 1000× lower than the power of the f2 field in the antisymmetric port beam, and hence, we did not change the spectral model assumed. The model assumed is shown in Figure5.8.
Cavity round-trip length
The cavity round-trip length should be chosen such that the RF sideband TEM00 modes are not resonant in the OMC when the carrier is resonant. For the 40m, the RF sideband frequencies are (i) f1 = 11.066209 MHz and (ii) f2 = 5× f1.
To convert frequency offset from the carrier to phase offset, we use the relation φSBRT = 2πfSBLRT
c . (5.12)
The OMCpowertransmissivity is given by TOMC =
t2
F
1−r2
Cr2
Fe−iφ
2
, (5.13)
so we can just examine what this looks like for the TEM00mode of the RF sidebands as a function of the round-trip length of the cavity.
2.0 2.1 2.2 2.3 2.4 2.5 2.6 RoC of curved mirrors [m]
0.7 0.8 0.9 1.0 1.1 1.2
Round trip cavity length [m]
Proposed 40m aLIGO OMC
10 6 10 5 10 4
Transmitted power [W per 1W@BS]
40.324 20.161 13.441Sagitta [ m]10.081 8.065 6.720
(a)
2.0 2.1 2.2 2.3 2.4 2.5 2.6
RoC of curved mirror [m]
107 106 105 104
Transmitted power [W / 1W @ BS]
Sum of all modes Proposed operating point
0 5 10 15HoM index (m+n) 40.324 20.161 13.441 10.081Sagitta [ m] 8.065 6.720
(b)
0.6 0.7 0.8 0.9 1.0 1.1 1.2
Round-trip cavity length [m]
107 106 105 104
Transmitted power [W / 1W @ BS]
Sum of all modes Proposed operating point
0 5 10 15HoM index (m+n)
(c)
Figure 5.9: OMC design modeling results. Figure 5.9(a) Heatmap of the OMC transmission as a function of the parameters L1 and ROC. For the 40m RF side- band frequencies, the appropriate choice of L1and ROC are 0.875 m and 2.575 m respectively, with θi = 4.042◦. Figures 5.9(b) and 5.9(c) show one-dimensional slices of the heatmap in Figure5.9(a). The recommended design parameter choice is indicated by a vertical pink line. The various HOM resonances are shown by colored lines, with their sum shown in a thick grey line. The different linestyles are used to indicate different sideband frequencies.
Higher-order modes
Equation 5.12 can be generalized for higher order HG modes by realizing that the mn−th HOM experience an additionalphase shift of (m+ n) ×ζ relative to the TEM00 mode, with ζRT being the round-trip cavity Gouy phase which can be calculated from an ABCD matrix (not to be confused withζ, the homodyne phase, used elsewhere in this thesis). This number is always additive, whereas the phase offset from the carrier due to the field being an RF sideband is a signed number, depending on whether we are considering the upper or lower sideband. So the φ that enters the cavity transmission equation may be broken up into the following components:
φRTmn(f)= φ00RT(0)+ 2πfSB
RFLRT
c +(m+n)ζRT. (5.14) Plugging this into Eq. 5.13forTOMC, we get the power transmissivity of the OMC cavity.
The outcome of the modeling is shown in Figure5.9, with the suggested operating point indicated. We chose to preserve the curvature specification for the mirrors to be identical to the aLIGO OMC [76], 2.575 m, and varied the cavity length to account for the different sideband frequencies at the 40m. The cavity round-trip length is chosen to be 0.875 m. For comparison, the aLIGO OMC design parameters are indicated by dashed cyan lines. The one-dimensional slices along the dashed green lines are shown in Figure5.9(b)and Figure5.9(c), with the vertical pink lines indicating the suggested design parameters.