1.1 Analysis and design of advanced gravitational-wave in- terferometers: beating the Standard Quantum Limit
1.1.5 Sagnac interferometers as speed meters
by the squeeze factor (as in Caves’ original proposal for conventional interferometers [11]; see also KLMTV [15]), thereby enhancing the sensitivity by the same factor (in power) for frequencies below the sloshing frequency. We also show that, using two detuned FP cavities as optical filters in the output can enhance high-frequency performance greatly. As a by-product of our research, in Sec. 6.7, we work out the most general frequency-dependent rotation angle in quadrature fields achievable by detuned (high finesse) FP cavities, and give a prescription for solving for the corresponding filter parameters needed — an issue left untackled by KLMTV [15].
Finally, Purdue and I study the influence of optical losses in speed-meter interferometers. The mirror quality thought achievable by the next decade (10 ppm loss per bounce) dictates that the sloshing cavity, as well as the (optional) output filters have lengths of kilometers, in order to achieve a sensitivity a factor 5 (in amplitude) below the Standard Quantum Limit. This reduces the practical- ity of adding these cavities. On the other hand, these speed meters were able to achieve a broadband QND performance with one such additional cavity (one less than the KLMTV interferometers); and were found to be significantly less susceptible to losses than the KLMTV interferometers, due to the shape of their transfer functions. The full noise spectra of speed meter designs, with optical losses included are summarized and compared with KLMTV interferometers (QND position meters) in Sec. 6.5.3. (See Fig. 1.4).
aRN bRNbLN aLN
aRE
bRE aLE bLE BS
SRM PRM
z Laser
p q PD
ITM ETM
ITM ETM
aRN bRN
bLN
aLN
aRE
bRE aLE
bLE
BS
SRM PRM
ETM ETM
z Laser
p q PD
Figure 1.5: Signal recycled Sagnac interferometers with optical delay lines (upper panel) or ring- shaped Fabry-Perot cavities (lower panel) in the arms.
in the Sagnac noise curve that cannot be mimicked by signal recycled Michelson topologies [37]. As a consequence, despite the attractiveness of all-reflective optics, little effort has been made to shift away from the much more mature Michelson topology and build Sagnac interferometers in major third-generation interferometers.
As has long been known, Sagnac interferometers only sense thetime-dependentpart of test-mass motion. Surprisingly, until the work described in Chapter 7 of this thesis, nobody seems to have seriously realized that this implies that Sagnac interferometers are speed meters automatically — without the need of any additional kilometer-scale cavities. This fact follows naturally as we explain how a Sagnac gravitational-wave interferometer works.
In a Sagnac interferometer (see Fig. 1.5), the input light beam is split in two by the beamsplitter;
the two beams can be denoted R (“right propagation”) and L (“left propagation”). The R beam is sent into the North (N) arm first, and then fed into the East (E) arm; while the L beam enters the two arms in the opposite order, E first and N second. When the two beams recombine at the beamsplitter, the phase gained by each of them separately can be written as
δφR ∼ xN(t) +xE(t+τarm), (1.18) δφL ∼ xE(t) +xN(t+τarm), (1.19) where τarm is the (average) time each photon stays in the arm, xN(t) and xE(t) are the (tiny) differences of the North and East arm lengths to their reference values (which resonates with the carrier laser). The output signal will then be proportional to
δφR−δφL ∼[xN(t)−xN(t+τarm)]−[xE(t)−xE(t+τarm)], (1.20) which is sensitive to thechange of arm-length difference during the light’s travel. As we infer from Eq. (1.20), for motions with frequencies much lower than 1/τarm, a speed measurement is obtained;
and at higher frequencies the signal contains a combination of speed and higher time derivatives of position.
Nobody before has taken seriously this speed-meter-like response function of the Sagnac inter- ferometer and asked for its quantum-mechanical implications [QND performance]. In Chapter 7, it takes only a trivial calculation to confirm the “quantum speed meter” performance of ideal Sagnac in- terferometers, i.e., a performance similar to that of the Michelson speed meter in Chapter 6, namely, a uniform beating of the Standard Quantum Limit in a broad frequency band, with ordinary ho- modyne detection. [See Sec. 7.3, for performances of ideal Sagnac interferometers; example noise spectra are also shown in Fig. 1.6.] In particular, signal recycled Sagnac interferometers with ring cavities (lower panel of Fig. 1.5) in the arms can be shown to have the same input-output relation as
10 20 50 100 200 500 1000 f (Hz)
0.1 1 10
1 [S h(f)/S SQL(100Hz)]1/2
10 20 50 100 200 500 1000
f (Hz) 0.1
1 10
1 [S h(f)/S SQL(100Hz)]1/2
Figure 1.6: Quantum noise of ideal Sagnac interferometers with optical delay lines (DL for short, upper panel) or ring-shaped Fabry-Perot cavities (FP for short, lower panel) in the arms. In both configurations, a total circulating power of 820 kW and input (power) squeeze factor of e2R = 10 is assumed; solid curves stand for signal recycled Sagnac interferometers, dashed curves stand for Sagnac interferometers without signal recycling, solid straight lines stand for the Standard Quantum Limit, and gray curves stand for a fiducial Michelson speed meter for comparison. For the DL scheme, the light bounces for B= 60 times in each arm, and the signal recycling amplitude reflectivity is ρ= 0.12; in the FP scheme, the power transmissivity of the input test-mass mirror isT = 0.0564, and the signal recycling amplitude reflectivity isρ= 0.268. In this case, the Michelson Speed Meter curve coincides with the signal recycled Sagnac curve.
the Michelson speed meters (lower panel of Fig. 1.6). In this way, Sagnac interferometers might well be the easiest-to-build QND interferometers. This, combined with the promise of all-reflective op- tics, can make the Sagnac interferometer a strong candidate for third-generation gravitational-wave interferometers. [Technical issues, including the influence of optical losses, will have to be analyzed thoroughly, although it is plausible that, like other speed meters, the Sagnac interferometer is also less susceptible to optical losses than QND position meters. ]
Interestingly, a careful comparison between Sagnac interferometers and the first gedanken ex- periment of Braginsky and Khalili in Ref. [27] will reveal an enlightening resemblence between the two schemes. In fact, the Sagnac interferometer can in some sense be regarded as its practical im- plementation in optics. Khalili, in an independent but subsequent work [28], also realized this link, and deduced the quantum speed meter performance of Sagnac interferometers.