Signal Extraction
3.2 The RSE Interferometer
3.2.1 Asymmetry
Figure 3.8 reduces to a single cavity for LIGO I, whose end mirror is the Michelson mirror. The carrier and RF sidebands are resonant in this cavity, and the carrier is held on a dark fringe in the Michelson. The Michelson phase q)_ for the RF sidebands,
however, is given by
cP- =
Dmod(ll- l2)=
Dmod6c c (3.29)
where 6 is a macroscopic asymmetry, also known as the "Schnupp asymmetry." Cou- pling of RF sideband power to the dark port is accomplished by choosing the asymme- try such that the transmittance of the Michelson is roughly equal to the transmittance of the power mirror, sin(¢-)2 :=:;j TPRM, since the RF sidebands are not resonant in the arms (rae r:::: 1). This makes the power cavity nearly optimally coupled for the RF sidebands. Because the RF sidebands are used at the dark port as the local oscillator (LO) for the gravitational wave signal, it's important that the amount of power present due to the RF sidebands is large compared to any other source of
"junk" light. Junk light generates shot noise without adding sensitivity to signal, so it's desired that the dominant source of shot noise will be due to the presence of the RF sidebands. Sources of junk light are contrast defect and higher order transverse spatial modes.
RSE is a bit more complicated, since the sidebands now need to transmit through a coupled cavity system. Consider first the broadband RSE case, where the carrier is resonant in the signal cavity. There are a couple of options. First, the asymmetry can be made large enough such that the reflectivity of the Michelson is zero. The coupled cavity is effectively reduced to a single cavity with the power and signal mirrors acting as its input and output mirrors. If the power and signal cavity mirrors have similar enough transmittances, the total transmitted power can be fairly good.
The asymmetry is set by the following equation.
c/J-
= Dmod6 = 1f12
c 6
=
Amod4
(3.30) (3.31)
The lengths of the power and signal cavity are chosen such that the round trip phase for the RF sidebands satisfies a resonant condition. Given the sign convention used in Figure 3.8, resonance is satisfied by a round trip phase equal to an odd integer 1r,
due to the sign from the power recycling mirror.
(
27r2(lprc
+
lsec)fmod1f) ( )
</Jcarrierprc
+
</Jcarrier.ec ± C+
22=
2n+
1 7r (3.32)l l 1Amod
pre
+
sec=
n -2- (3.33)
The round-trip carrier phases are some integer 21r in the broadband case. Of note is the final factor of 1r /2. This comes from the fact that the Michelson, although the magnitude of its transmission is unity, adds a phase of 1r /2 for each transmission, as evidenced by the factor of i in Eq. (3.28).
In general, though, the transmittance of the power and signal mirrors can be different enough that the transmittance of the coupled cavity on the bright fringe can become quite low. In this case the asymmetry must be properly matched in the coupled cavity system. One way to think about this is that, viewed from the power cavity, the signal cavity, formed by the Michelson and the signal mirror, looks like a mirror. The asymmetry is chosen such that the transmittance of the signal cavity compound mirror again is approximately equal to the transmittance of the power mirror, which then optimally couples the sideband transmission to the dark port. The general solution can be quite complex. As evidenced by the tile plot of Figure 2.3, this can involve off-resonant cavities, such that there are three variables to solve for: the asymmetry and the phases of the power and signal cavities. Simpler, analytical solutions can be found by setting the individual cavities on resonance or anti-resonance. There are two solutions for the reflectivity, rprm
=
±rsec, since a resonant cavity reflectivity can have either sign.The first solution assumes the power and signal cavity are both resonant for the RF sidebands. Solving this equality gives
r C
(A
r prm+
r sem )u
= - -
arccos bs 2 nmoo 1+
r prcr semAbs(3.34)
This usually is a fairly small number, on the order of a couple centimeters for typical mirrors and modulation frequencies in the 10's of MHz. The value of the Michelson
transmittance is then higher than either the power or signal mirrors. This allows a certain consistency, that being that both cavities, as viewed from the other, appear to be under-coupled. The sign on reflection is then positive, and the resonance of the cavities is preserved.
The second solution has a much larger asymmetry, such that the Michelson has a transmittance near unity. How the solution is found depends on which mirror, power
or signal, has the higher transmittance. First assume the power mirror has the higher transmittance. Since a resonant cavity has a higher transmission than either of the single mirrors, the signal cavity can be made to match its transmission to the power mirror. However, with a large asymmetry, the signal cavity is over-coupled, and the sign flip from the signal cavity requires that the power cavity be anti-resonant without this particular effect. This is in the same manner as the resonance condition for the carrier in the power cavity. In this case, the solution is
C ( r sem - r prm )
6
= - -
arccos Abs 2nmod 1-TprmTsemAbs
(3.35)
If the situation is reversed, that is the power mirror has a lower transmittance than the signal mirror, the solution is found to be
C ( r prm - r sem )
6
= - -
arccos Abs 2Dmod 1-TprmTsemAbs
(3.36)
As an example, the transmittance of the coupled cavity system is shown in Fig- ure 3.9 as a function of the asymmetry. This plot uses the power and signal mirrors for the optimized broadband interferometer, with the power mirror transmittance of
Tprm = 8.6% and a signal cavity mirror transmittance of Tsem = 5%. The frequency of the RF sidebands is at
!mod=
81 MHz, and the losses are assumed to be zero.Designing to the point of optimal coupling is, in fact, not a good practice. Optics invariably have characteristics (losses, reflectivities) which are not exactly what were ordered. Furthermore, losses can change in time. The desire to have strong transmis- sion to the dark port does constrain the asymmetry to a value close to the optimal
0.9 ....... , ........ ..... .
a.8 a.7
I ... : ..... ./. ..
I : I
·· ·· /
;:
I . I I I /
. ...... /.:.:/ .
/ /·
/ /
. I
... · ...... /. .. ···>······<·········<····
. I
. I
a.2 ...... -· --··r··
a.1 a a
/ / :
a.1
/ ·
/ .
a.2
I I
a.3 a.4 a.s a.6 a.7 Asymmetry,o (m)
· · _ <>.=a. <>.=a
__ <>.=1<12, <>.=a
a.8 a.9
Figure 3.9: Power/signal coupled cavity transmittance as a function of asymmetry.
The phases of the power and signal cavities are indicated as one-way phases in the legend. Tprm = 8.6%, Tsem = 5%, and the RF modulation frequency is 81 MHz.
asymmetry, so these formulas give a reasonable starting point.