2.1 Interferometer as a weak force probe

In order to have a firm basis for understanding how quantum noise influences the sensitivity of a GW detector it would be illuminating to give a brief description of the interferometers as weak force/tiny displacement meters. It is by no means our intention to give a comprehensive survey of this ample field that is certainly worthy of a good book, which there are in abundance, but rather to provide the reader with the wherewithal for grasping the very principles of the GW interferometers operation as well as of other similar ultrasensitive optomechanical gauges. The reader interested in a more detailed description of the interferometric techniques being used in the field of GW detectors might enjoy reading this book [12Jump To The Next Citation Point] or the comprehensive Living Reviews on the subject by Freise and Strain [59Jump To The Next Citation Point] and by Pitkin et al. [123].

2.1.1 Light phase as indicator of a weak force

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Figure 1: Scheme of a simple weak force measurement: an external signal force G pulls the mirror from its equilibrium position x = 0, causing displacement δx. The signal displacement is measured by monitoring the phase shift of the light beam, reflected from the mirror.

Let us, for the time being, imagine that we are capable of measuring an electromagnetic (e.g., light) wave-phase shift δϕ with respect to some coherent reference of the same frequency. Having such a hypothetical tool, what would be the right way to use it, if one had a task to measure some tiny classical force? The simplest device one immediately conjures up is the one drawn in Figure 1View Image. It consists of a movable totally-reflective mirror with mass M and a coherent paraxial light beam, that impinges on the mirror and then gets reflected towards our hypothetical phase-sensitive device. The mirror acts as a probe for an external force G that one seeks to measure. The response of the mirror on the external force G depends upon the details of its dynamics. For definiteness, let the mirror be a harmonic oscillator with mechanical eigenfrequency Ω = 2πf m m. Then the mechanical equation of motion gives a connection between the mirror displacement x and the external force G in the very familiar form of the harmonic oscillator equation of motion:

∫ t M ¨x + M Ω2 x = G(t), =⇒ x(t) = x (t) + dt′χ (t − t′)G (t′), (1 ) m 0 0 xx
where x0(t) = x (0)cosΩmt + p(0)∕(M Ωm )sin Ωmt is the free motion of the mirror defined by its initial displacement x(0) and momentum p(0) at t = 0 and
′ χxx(t − t′) = sinΩm--(t-−-t) , t ≥ t′, (2 ) M Ωm
is the oscillator Green’s function. It is easy to see that the reflected light beam carries in its phase the information about the displacement δx(t) = x(t) − x(0) induced by the external force G. Indeed, there is a phase shift between the incident and reflected beams, that matches the additional distance the light must propagate to the new position of the mirror and back, i.e.,
2ω δx δx δϕ = ---0-- = 4π --, (3 ) c λ0
with ω0 = 2πc∕λ0 the incident light frequency, c the speed of light and λ0 the light wavelength. Here we implicitly assume mirror displacement to be much smaller than the light wavelength.

Apparently, the information about the signal force G(t) can be obtained from the measured phase shift by post-processing of the measurement data record δ&tidle;ϕ(t) ∝ δx(t) by substituting it into Eq. (1View Equation) instead of x. Thus, the estimate of the signal force &tidle;G reads:

[ ] &tidle; M-c- ¨&tidle; 2 &tidle; G = 2ω0 δϕ + Ωm δϕ . (4 )
This kind of post-processing pursues an evident goal of getting rid of any information about the eigenmotion of the test object while keeping only the signal-induced part of the total motion. The above time-domain expression can be further simplified by transforming it into a Fourier domain, since it does not depend anymore on the initial values of the mirror displacement x(0) and momentum p (0 ):
[ ] G&tidle;(Ω) = M--c Ω2m − Ω2 δϕ&tidle;(Ω ), (5 ) 2 ω0
∫ ∞ A (Ω ) = dtA(t)eiΩt (6 ) −∞
denotes a Fourier transform of an arbitrary time-domain function A (t). If the expected signal spectrum occupies a frequency range that is much higher than the mirror-oscillation frequency Ωm as is the case for ground based interferometric GW detectors, the oscillator behaves as a free mass and the term proportional to Ω2m in the equation of motion can be omitted yielding:
2 G&tidle;f.m.(Ω) = − M--cΩ-δ &tidle;ϕ . (7 ) 2ω0 Ω

2.1.2 Michelson interferometer

Above, we assumed a direct light phase measurement with a hypothetical device in order to detect a weak external force, possibly created by a GW. However, in reality, direct phase measurement are not so easy to realize at optical frequencies. At the same time, physicists know well how to measure light intensity (amplitude) with very high precision using different kinds of photodetectors ranging from ancient–yet–die-hard reliable photographic plates to superconductive photodetectors capable of registering individual photons [67]. How can one transform the signal, residing in the outgoing light phase, into amplitude or intensity variation? This question is rhetorical for physicists, for interference of light as well as the multitude of interferometers of various design and purpose have become common knowledge since a couple of centuries ago. Indeed, the amplitude of the superposition of two coherent waves depends on the relative phase of these two waves, thus transforming phase variation into the variation of the light amplitude.

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Figure 2: Scheme of a Michelson interferometer. When the end mirrors of the interferometer arms Mn,e are at rest the length of the arms L is such that the light from the laser gets reflected back entirely (bright port), while at the dark port the reflected waves suffer destructive interference keeping it really dark. If, due to some reason, e.g., because of GWs, the lengths of the arms changed in such a way that their difference was δL, the photodetector at the dark port should measure light intensity I δL Idark(δL) = -02 (1 − cos4π λ-).

For the detection of GWs, the most popular design is the Michelson interferometer [15Jump To The Next Citation Point, 12, 59Jump To The Next Citation Point], which schematic view is presented in Figure 2View Image. Let us briefly discuss how it works. Here, the light wave from a laser source gets split by a semi-transparent mirror, called a beamsplitter, into two waves with equal amplitudes, travelling towards two highly-reflective mirrors Mn,e1 to get reflected off them, and then recombine at the beamsplitter. The readout is performed by a photodetector, placed in the signal port. The interferometer is usually tuned in such a way as to operate at a dark fringe, which means that by default the lengths of the arms are taken so that the optical paths for light, propagating back and forth in both arms, are equal to each other, and when they recombine at the signal port, they interfere destructively, leaving the photodetector unilluminated. On the opposite, the two waves coming back towards the laser, interfere constructively. The situation changes if the end mirrors get displaced by some external force in a differential manner, i.e., such that the difference of the arms lengths is non-zero: δL = Le − Ln ⁄= 0. Let a laser send to the interferometer a monochromatic wave that, at the beamsplitter, can be written as

Elaser(t) = E0 cos(ω0t).
Hence, the waves reflected off the interferometer arms at the beamsplitter (before interacting with it for the second time) are2:
out -E0- En,e(t) = − √--cos(ω0t − 2ω0Ln,e∕c), 2
and after the beamsplitter:
Eout(t) − Eout(t) ω0δL Edark(t) = --n----√---e-----= E0 sin ----- sin (ω0t − ω0 [Ln + Le]∕c), 2 c Eount(t) +-Eoeut(t) ω0δL- Ebright(t) = √2-- = − E0 cos c cos(ω0t − ω0[Ln + Le]∕c).
And the intensity of the outgoing light in both ports can be found using a relation --- ℐ ∝ E2 with overline meaning time-average over many oscillation periods:
ℐ ( δL ) ℐ ( δL) ℐdark(δL ∕λ0) = --0 1 − cos4π --- , and ℐbright(δL ∕λ0) = -0- 1 + cos4π --- . (8 ) 2 λ0 2 λ0
Apparently, for small differential displacements δL ≪ λ0, the Michelson interferometer tuned to operate at the dark fringe has a sensitivity to 2 ∼ (δL∕ λ0) that yields extremely weak light power on the photodetector and therefore very high levels of dark current noise. In practice, the interferometer, in the majority of cases, is slightly detuned from the dark fringe condition that can be viewed as an introduction of some constant small bias δL0 between the arms lengths. By this simple trick experimentalists get linear response to the signal nonstationary displacement δx (t):
ℐ ( δL + δx) ℐdark(δx∕ λ0) = -0- 1 − cos 4π---0----- ≃ ( ) 2 λ(0 ) 2 δL0δx δx2 δL20 δx δx2 δL20 8π ℐ0 -λ2---+ 𝒪 λ2-,-λ2- = const. × 4π λ-+ 𝒪 -λ2-,-λ2- . (9 ) 0 0 0 0 0 0
Comparison of this formula with Eq. (3View Equation) should immediately conjure up the striking similarity between the response of the Michelson interferometer and the single moving mirror. The nonstationary phase difference of light beams in two interferometer arms δϕ (t) = 4π δx(t)∕λ 0 is absolutely the same as in the case of a single moving mirror (cf. Eq. (3View Equation)). It is no coincidence, though, but a manifestation of the internal symmetry that all Michelson-type interferometers possess with respect to coupling between mechanical displacements of their arm mirrors and the optical modes of the outgoing fields. In the next Section 2.1.3, we show how this symmetry displays itself in GW interferometers.

2.1.3 Gravitational waves’ interaction with interferometer

Let us see how a Michelson interferometer interacts with the GW. For this purpose we need to understand, on a very basic level, what a GW is. Following the poetic, yet precise, definition by Kip Thorne, ‘gravitational waves are ripples in the curvature of spacetime that are emitted by violent astrophysical events, and that propagate out from their source with the speed of light’ [13Jump To The Next Citation Point, 110]. A weak GW far away from its birthplace can be most easily understood from analyzing its action on the probe bodies motion in some region of spacetime. Usually, the deformation of a circular ring of free test particles is considered (see Chapter 26: Section 26.3.2 of [13] for more rigorous treatment) when a GW impinges it along the z-direction, perpendicular to the plane where the test particles are located. Each particle, having plane coordinates (x,y) with respect to the center of the ring, undergoes displacement δr ≡ (δx,δy) from its position at rest, induced by GWs:

1- 1- δx = 2 h+x, δy = − 2h+y, (10 ) 1 1 δx = --h×y, δy = --h×x. (11 ) 2 2
Here, h+ ≡ h+(t − z∕c) and h × ≡ h× (t − z∕c) stand for two independent polarizations of a GW that creates an acceleration field resulting in the above deformations. The above expressions comprise a solution to the equation of motion for free particles in the tidal acceleration field created by a GW:
1 [ ] δ¨r = -- (¨h+x + ¨h×y)ex + (− ¨h+y + ¨h×x)ey , 2
with ex = {1,0}T and ey = {0,1}T the unit vectors pointing in the x and y direction, respectively.
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Figure 3: Action of the GW on a Michelson interferometer: (a) h+-polarized GW periodically stretch and squeeze the interferometer arms in the x- and y-directions, (b) h ×-polarized GW though have no impact on the interferometer, yet produce stretching and squeezing of the imaginary test particle ring, but along the directions, rotated by ∘ 45 with respect to the x and y directions of the frame. The lower pictures feature field lines of the corresponding tidal acceleration fields ∝ ¨h+,×.

For our Michelson interferometer, one can consider the end mirrors to be those test particles that lie on a circular ring with beamsplitter located in its center. One can choose arms directions to coincide with the frame x and y axes, then the mirrors will have coordinates (0,Ln ) and (Le, 0), correspondingly. For this case, the action of the GW field on the mirrors is featured in Figure 3View Image. It is evident from this picture and from the above formulas that an h×-polarized component of the GW does not change the relative lengths of the Michelson interferometer arms and thus does not contribute to its output signal; at the same time, h+-polarized GWs act on the end masses of the interferometer as a pair of tidal forces of the same value but opposite in direction:

1- ¨ 1- ¨ Gn = − 2Mn h+Ln, Ge = 2Me h+Le.

Assuming Ge = − Gn = G, Mn = Me = M, and Le = Ln = L, one can write down the equations of motion for the interferometer end mirrors that are now considered free (Ω ≪ Ω m GW) as:

M ¨x = G, M ¨y = − G,
and for the differential displacement of the mirrors δL = Le − Ln = x − y, which, we have shown above, the Michelson interferometer is sensitive to, one gets the following equation of motion:
¨ ¨ M δL = 2G (t) = M h+ (t)L (12 )
that is absolutely analogous to Eq. (1View Equation) for a single free mirror with mass M. Therefore, we have proven that a Michelson interferometer has the same dynamical behavior with respect to the tidal force ¨ G (t) = M h+ (t)L ∕2 created by GWs, as the single movable mirror with mass M to some external generic force G (t).

The foregoing conclusion can be understood in the following way: for GWs are inherently quadruple and, when the detector’s plane is orthogonal to the wave propagation direction, can only excite a differential mechanical motion of its mirrors, one can reduce a complicated dynamics of the interferometer probe masses to the dynamics of a single effective particle that is the differential motion of the mirrors in the arms. This useful observation appears to be invaluably helpful for calculation of the real complicated interferometer responses to GWs and also for estimation of its optical quantum noise, that comprises the rest of this review.

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